GS_Truss_(v0.5.5) -- Sage

%auto

def map2 (f, l2) :
    return map (lambda y : map (f, y), l2)

def map_dict (f, d) :
    return dict (zip (d.keys(), map (f, d.values())))
    
def map2_dict (f, d) :
    return dict (zip (d.keys(), map2 (f, d.values())))

def filter_dict (f, d) : 
    fd = {}
    for k in d.keys():
        if f (d[k]) : 
            fd[k] = d[k]
    return fd
    

def ND (sig_digits_10) : 
    return lambda x : N (x, digits = sig_digits_10)
    
def chop (tolerance = None) :
    def f_ (x) :
        t = sqrt ((parent(x)(1)).ulp()) if tolerance == None else tolerance
        return x if (abs(x) >= t) else parent(x)(0)
    return f_
 
def NDC (sig_digits_10, tolerance = None) : 
    return lambda x : N (chop(tolerance) (x), digits = sig_digits_10)
   
def sqrt_ulp (field = RDF) :
    return sqrt (field(1).ulp())
    

def dict_joint_bars (free_nodes, elems) : 
    nds_els = {}
    for fni in free_nodes :
        nds_els[fni] = []
    for (ei, ec) in elems.iteritems() : 
        (i0, i1) = ec
        if i0 in free_nodes :
            nds_els[i0].append (ei)
        if i1 in free_nodes :
            nds_els[i1].append (ei)
    return nds_els


def dict_equation_index (free_nodes, dimen = 3) : 
    return dict (zip (sorted (free_nodes), xrange (0, dimen * len (free_nodes), dimen)))  
      
def dict_index_equation (free_nodes, dimen = 3) : 
    return dict (zip (xrange (0, dimen * len (free_nodes), dimen), sorted (free_nodes)))  
      
def dict_unknown_index (elems) : 
    return dict (zip (sorted (elems.keys()), xrange (len (elems))))   

def dict_index_unknown (elems) : 
    return dict (zip (xrange (len (elems)), sorted (elems.keys())))   

def dict_label_index (lbls) : 
    return dict (zip (sorted (lbls), xrange (len (lbls))))   

def dict_index_label (lbls) : 
    return dict (zip (xrange (len (lbls)), sorted (lbls)))   

        
def others (l1, l2) :
    return [c for c in l2 if c not in l1]    

def list_indexed (inds, l) :
    return [l[i] for i in inds]

def all_indices (l) :
    return xrange (len (l))
    

def maxwells_rule (joints, supports, bars) : 
    dim = len (joints.values()[0]) 
    free_joints = others (supports, joints)
    mxw = dim*len (free_joints) - len (bars)
    print dim, '*', len (free_joints), '-', len (bars), ' == ', mxw, 
    if mxw != 0 :
        print ' !=  0'
    else :
        print 
    return 


def base_vectors (dim, field = RDF) :
    bv = zero_vector (field, dim)
    bvs = [copy (bv) for i in xrange (dim)]
    for i in xrange (dim) :
        bvs[i][i] = field (1)
    return bvs


def bar_length (xxi, xxj) : 
    return sqrt (sum (map (lambda xi, xj : (xj - xi)^2, xxi, xxj)))


def equilibrium_matrix (nodes, elems, free_nodes, field = RDF) : 
    nfj = len (free_nodes)
    nb = len (elems)
    ne_inc = dict_joint_bars (free_nodes, elems)
    ukws = dict_unknown_index (elems)
    dim = len (nodes.values()[0]) 
    eqs = dict_equation_index (free_nodes, dim)
    A = zero_matrix (field, dim*nfj, nb)
    
    for ni in free_nodes : 
        ei = eqs[ni]
        bni = ne_inc[ni]
        for bi in bni :
            if ni == elems[bi][0] :
                nj = elems[bi][1]
            else :
                nj = elems[bi][0]
            xxi = map (field, nodes[ni]) 
            xxj = map (field, nodes[nj]) 
            # if field == SR :
            #     if ni < nj :
            #         l = var ('l_%d%d' % (ni, nj))
            #     else :
            #         l = var ('l_%d%d' % (nj, ni))
            # else :
            l = sqrt (sum (map (lambda xi, xj : (xj - xi)^2, xxi, xxj)))
            cc = map (lambda xi, xj : (xj - xi) / l, xxi, xxj)
            ui = ukws[bi]
            A[ei : (ei+dim), ui] += matrix (dim, 1, cc)
            
    return A


def load_vector (loads, free_nodes, field = RDF) :
    dim = len (loads.values()[0]) 
    nfj = len (free_nodes)
    eqs = dict_equation_index (free_nodes, dim)
    b = zero_vector (field, dim*nfj)
    for li, ff in loads.iteritems() :
        ei = eqs[li]
        b[ei : (ei+dim)] = map (field, ff)
    return b


def augmented_equil_matrix (nodes, elems, loads, free_nodes, field = RDF) : 
    nfj = len (free_nodes)
    nb = len (elems)
    ne_inc = dict_joint_bars (free_nodes, elems)
    ukws = dict_unknown_index (elems)
    dim = len (nodes.values()[0]) 
    eqs = dict_equation_index (free_nodes, dim)
    A = zero_matrix (field, dim*nfj, nb + 1)
    
    for ni in free_nodes : 
        ei = eqs[ni]
        bni = ne_inc[ni]
        for bi in bni :
            if ni == elems[bi][0] :
                nj = elems[bi][1]
            else :
                nj = elems[bi][0]
            xxi = map (field, nodes[ni]) 
            xxj = map (field, nodes[nj]) 
            l = sqrt (sum (map (lambda xi, xj : (xj - xi)^2, xxi, xxj)))
            cc = map (lambda xi, xj : (xj - xi) / l, xxi, xxj)
            ui = ukws[bi]
            A[ei : (ei+dim), ui] += matrix (dim, 1, cc)
    
    for li, ff in loads.iteritems() :
        ei = eqs[li]
        ej = nb
        A[ei : (ei+dim), ej] = matrix (dim, 1, map (lambda x : -field(x), ff))
        
    A.subdivide ([], [nb])
    return A


def dict_bar_force (elems, forces) :
    ukws = dict_label_index (elems.keys()) 
    return dict ([(b, forces[i]) for b, i in ukws.iteritems()]) 


def truss_skd (nodes, elems, supports, loads, field = RDF) : 
    free_nodes = others (supports, nodes)
    A = equilibrium_matrix (nodes, elems, free_nodes, field)
    b = load_vector (loads, free_nodes, field)
    ff = A.solve_right (-b)
    return dict_bar_force (elems, ff)
    

def is_zz (x) :
    return x == 0

def is_zz_tol (tolerance = None) :
    def f_ (x) :
        t = sqrt ((parent(x)(1)).ulp()) if tolerance == None else tolerance
        return abs (x) < t
    return f_

 
def ref_wpp (m, is_zero_f = is_zz_tol(), in_place = False) :
    pivot = 0
    if in_place == True : 
        r = m
    else :
        r = copy (m)
    pivotlist = []
    for i in xrange (r.ncols()) :
        if pivot == r.nrows() :
            break
        maxrow = pivot
        for k in xrange (pivot+1, r.nrows()):
            if abs (r[k][i]) > abs (r[maxrow][i]) : 
                maxrow = k
        if pivot != maxrow :
            r.swap_rows (pivot, maxrow)
        if not is_zero_f (r[pivot, i]) :
            scale = r[pivot, i]
            r.rescale_row (pivot, 1/scale, i)
            for j in xrange (pivot+1, r.nrows()) :
                scale = r[j, i]
                r.add_multiple_of_row (j, pivot, -scale, i)
            pivotlist.append (i)
            pivot += 1
    return r, pivotlist

def ref_solution (Ar, piv) : 
    r = Ar.column (-1)
    nu = Ar.ncols() - 1
    if nu < Ar.nrows() :
        r = r[0:nu]
    np = len (piv)
    for j in xrange (np-1, -1, -1) :
        for jj in xrange (np-1, j, -1) :
            r[j] -= Ar[j, piv[jj]] * r[jj]
    return r


def rref_wpp (m, is_zero_f = is_zz_tol(), in_place = False) :
    pivot = 0
    if in_place == True :
        r = m
    else :
        r = copy (m)
    pivotlist = []
    for i in xrange (r.ncols()) :
        if pivot == r.nrows() :
            break
        maxrow = pivot
        for k in xrange (pivot+1, r.nrows()):
            if abs (r[k][i]) > abs (r[maxrow][i]) : 
                maxrow = k
        if pivot != maxrow :
            r.swap_rows (pivot, maxrow)
        if not is_zero_f (r[pivot, i]) :
            scale = r[pivot, i]
            r.rescale_row (pivot, 1/scale, i)
            for j in xrange (r.nrows()) :
                if j == pivot :
                    continue
                scale = r[j, i]
                r.add_multiple_of_row (j, pivot, -scale, i)
            pivotlist.append (i)
            pivot += 1
    return r, pivotlist

def rref_solution (Arr) :
    r = Arr.column (-1)
    nu = Arr.ncols() - 1
    if nu < Arr.nrows() :
        r = r[0:nu]
    return r
    
 
def truss (nodes, elems, supports, loads, is_zero_f = is_zz_tol(), field = RDF) : 
    free_nodes = others (supports, nodes)
    Ab = augmented_equil_matrix (nodes, elems, loads, free_nodes, field)
    rref_wpp (Ab, is_zero_f = is_zero_f, in_place = True)
    return dict_bar_force (elems, Ab.column (-1))

    
def pivot_columns (mr) : 
    nr = mr.nrows()
    pc = []
    for i in xrange (mr.nrows()) : 
        for j in xrange (i, mr.ncols()) :
            if mr[i, j] != 0 :
                pc.append (j)
                break
    return pc


def ref_ext_wpp (m, b = None, is_zero_f = is_zz_tol()) :
    pivot = 0
    if b == None :
        r = block_matrix (1, 2, [m, identity_matrix (m.base_ring(), m.nrows())])
    else :
        r = block_matrix (1, 3, [m, identity_matrix (m.base_ring(), m.nrows()), b])
    mncols = m.ncols()
    pivotlist = []
    for i in xrange (mncols) :
        if pivot == r.nrows() :
            break
        maxrow = pivot
        for k in xrange (pivot+1, r.nrows()):
            if abs (r[k][i]) > abs (r[maxrow][i]) : 
                maxrow = k
        if pivot != maxrow :
            r.swap_rows (pivot, maxrow)
        if not is_zero_f (r[pivot, i]) :
            scale = r[pivot, i]
            r.rescale_row (pivot, 1/scale, i)
            for j in xrange (pivot+1, r.nrows()) :
                scale = r[j, i]
                r.add_multiple_of_row (j, pivot, -scale, i)
            pivotlist.append (i)
            pivot += 1
    return r, pivotlist


def rref_ext_wpp (m, b = None, is_zero_f = is_zz_tol()) :
    pivot = 0
    if b == None :
        r = block_matrix (1, 2, [m, identity_matrix (m.base_ring(), m.nrows())])
    else :
        r = block_matrix (1, 3, [m, identity_matrix (m.base_ring(), m.nrows()), b])
    mncols = m.ncols()
    pivotlist = []
    for i in xrange (mncols) :
        if pivot == r.nrows() :
            break
        maxrow = pivot
        for k in xrange (pivot+1, r.nrows()):
            if abs (r[k][i]) > abs (r[maxrow][i]) : 
                maxrow = k
        if pivot != maxrow :
            r.swap_rows (pivot, maxrow)
        if not is_zero_f (r[pivot, i]) :
            scale = r[pivot, i]
            r.rescale_row (pivot, 1/scale, i)
            for j in xrange (r.nrows()) :
                if j == pivot :
                    continue
                scale = r[j, i]
                r.add_multiple_of_row (j, pivot, -scale, i)
            pivotlist.append (i)
            pivot += 1
    return r, pivotlist


def column_space (m, piv, rref = False, is_zero_f = is_zz_tol()) :
    cs = []
    for p in piv :
        cs.append (m.column (p))
    if rref == False : 
        return cs
    mcs = matrix (cs)
    mcsr, pp = rref_wpp (mcs, is_zero_f = is_zero_f, in_place = True)
    return mcsr.rows()
    
def row_space (mr, piv) :
    rs = []
    m = mr 
    for i in xrange (len (piv)) :
        rs.append (m.row (i))
    return rs

def left_kernel (mre, piv) :
    lk = []
    ii = mre 
    nr = ii.nrows()
    r0 = len (piv)
    for i in xrange (r0, nr) :
        lk.append (ii.row (i))
    return lk

def subs_r_ (m, nc, piv, np, i) : 
    v = vector (m.base_ring(), nc)
    v[i] = 1
    for j in xrange (np) :
        v[piv[j]] = -m[j,i]
    return v
    
def backsubs_ (m, nc, piv, np, i) : 
    v = vector (m.base_ring(), nc)
    v[i] = 1
    for j in xrange (np-1, -1, -1) :
        smj = 0 
        for jj in xrange (np-1, j, -1) :
            smj += m[j, piv[jj]] * v[piv[jj]]
        v[piv[j]] = -m[j,i] - smj
    return v

def kernel (mr, piv, backsubs = backsubs_) :
    k = []
    m = mr 
    nc = m.ncols()
    np = len (piv)
    piv_s = set (piv)
    for i in xrange (nc) :
        if not i in piv_s : 
            k.append (backsubs (m, nc, piv, np, i))
    return k

def subspace_bases (m, is_zero_f = is_zz_tol()) :
    mre, piv = ref_ext_wpp (m, is_zero_f = is_zero_f)
    rs = row_space (mre.subdivision (0, 0), piv)
    cs = column_space (m, piv, is_zero_f = is_zero_f)
    rk = kernel (mre.subdivision (0, 0), piv)
    lk = left_kernel (mre.subdivision (0, 1), piv)
    return rs, cs, rk, lk

def subspace_bases_r (m, is_zero_f = is_zz_tol()) :
    mre, piv = rref_ext_wpp (m, is_zero_f = is_zero_f)
    rs = row_space (mre.subdivision (0, 0), piv)
    cs = column_space (m, piv, is_zero_f = is_zero_f)
    rk = kernel (mre.subdivision (0, 0), piv)
    lk = left_kernel (mre.subdivision (0, 1), piv)
    return rs, cs, rk, lk

def subspaces (m, is_zero_f = is_zz_tol(), return_pivots = False) :
    mre, piv = ref_ext_wpp (m, is_zero_f = is_zero_f)
    nj3 = m.nrows()
    nb = m.ncols()
    rm = len (piv)
    rsb = row_space (mre.subdivision (0, 0), piv)
    rsn = nb
    rsd = rm
    rkb = kernel (mre.subdivision (0, 0), piv)
    rkn = nb
    rkd = nb - rm
    csb = column_space (m, piv)
    csn = nj3
    csd = rm
    lkb = left_kernel (mre.subdivision (0, 1), piv)
    lkn = nj3
    lkd = nj3 - rm
    rs = {'name' : 'Row space', 'degree' : rsn, 'dimension' : rsd, 'basis' : rsb} 
    cs = {'name' : 'Column space', 'degree' : csn, 'dimension' : csd, 'basis' : csb} 
    rk = {'name' : 'Kernel', 'degree' : rkn, 'dimension' : rkd, 'basis' : rkb} 
    lk = {'name' : 'Left kernel', 'degree' : lkn, 'dimension' : lkd, 'basis' : lkb} 
    if return_pivots == True :
        r = rs, cs, rk, lk, piv
    else :
        r = rs, cs, rk, lk
    return r

def subspaces_r (m, is_zero_f = is_zz_tol(), return_pivots = False) :
    mre, piv = rref_ext_wpp (m, is_zero_f = is_zero_f)
    nj3 = m.nrows()
    nb = m.ncols()
    rm = len (piv)
    rsb = row_space (mre.subdivision (0, 0), piv)
    rsn = nb
    rsd = rm
    rkb = kernel (mre.subdivision (0, 0), piv)
    rkn = nb
    rkd = nb - rm
    csb = column_space (m, piv, rref = True, is_zero_f = is_zero_f)
    csn = nj3
    csd = rm
    lkb = left_kernel (mre.subdivision (0, 1), piv)
    lkn = nj3
    lkd = nj3 - rm
    rs = {'name' : 'Row space', 'degree' : rsn, 'dimension' : rsd, 'basis' : rsb} 
    cs = {'name' : 'Column space', 'degree' : csn, 'dimension' : csd, 'basis' : csb} 
    rk = {'name' : 'Kernel', 'degree' : rkn, 'dimension' : rkd, 'basis' : rkb} 
    lk = {'name' : 'Left kernel', 'degree' : lkn, 'dimension' : lkd, 'basis' : lkb} 
    if return_pivots == True :
        r = rs, cs, rk, lk, piv
    else :
        r = rs, cs, rk, lk
    return r

def print_subspace (ss) :
    print '{0}{1}'.format (ss['name'], ':')
    print '   Vector space of degree', ss['degree'], 'and dimension', ss['dimension']
    if ss['dimension'] > 0 and ss['dimension'] <= 18 and ss['degree'] > 0 and ss['degree'] <= 18:
        print '   Basis:'
        print_list (ss['basis'], '     ')
        
def are_linearly_dependent (vl, is_zero_f = is_zz_tol()) :
    m = matrix (vl) 
    mre, piv = rref_wpp (m, is_zero_f = is_zero_f)
    return len (vl) != len (piv)

def is_in_subspace (v, sp, is_zero_f = is_zz_tol()) : 
    vl = [v]
    vl.extend (sp)
    return are_linearly_dependent (vl, is_zero_f = is_zero_f)

def has_component_in_subspace (v, sp, is_zero_f = is_zz_tol()) :
    r = False
    for vv in sp :
        s = v * vv
        if not is_zero_f (s) :
            r = True
            break
    return r


def all_bar_forces (primary_bars, redundant_bars, primary_forces) : 
    bfs = vector (primary_forces.base_ring(), len (primary_bars) + len (redundant_bars))
    for i in xrange (len (primary_bars)) :
        bfs[primary_bars[i]] = primary_forces[i]
    return bfs    


def diag_flex (elems, nodes, E, F, field = RDF) :
    ldelta = []
    for el in elems :
        ni, nj = elems[el]
        xi = nodes[ni] 
        xj = nodes[nj] 
        ldelta.append (bar_length (xi, xj) / (E * F))
    return diagonal_matrix (RDF, ldelta)


def diag_stiff (elems, nodes, E, F, field = RDF) :
    lk = []
    for el in elems :
        ni, nj = elems[el]
        xi = nodes[ni] 
        xj = nodes[nj] 
        lk.append (E * F / bar_length (xi, xj))
    return diagonal_matrix (RDF, lk)


def make_bars__ (nb, b0j, steps) :
    return [map (lambda x, y : x + y, b0j, map (lambda x : i * x, steps)) for i in xrange (nb)]


def schwedler_1 (r0, hh, n) :
    if not is_even (n) :
        print 'n must be even'
        return

    jsl = []
    bsl = []
    ssl = []
    rsl = [r0]
    
    h0 = hh[-1]
    rr = r0^2/(2*h0) + h0/2
    aa = rr - h0
    for h in hh[:-1] :
        rsl.append (sqrt (rr^2 - (aa+h)^2))
    hh.insert (0, 0.0)
    alpha = RDF (2*pi/n)
    for i in xrange (len (rsl)) :
        for j in xrange (n) :
            x = rsl[i] * cos (j*alpha)
            y = rsl[i] * sin (j*alpha)
            z = hh[i]
            jsl.append ((x, y, z))
    jsd = dict (zip (xrange (len (jsl)), jsl))
    
    for i in xrange (0, len (hh) - 2) :
        n0 = i*n
        for j in xrange (n) :
            bsl.append ((n0+j, n0+j+n))
        for j in xrange (n) :
            bsl.append ((n0+n+j, n*(i+1)+(n0+j+1)%n))
        for j in xrange (0, n, 2) :
            bsl.append ((n0+j, (n*(i+1)+(n0+j+1)%n)))
        for j in xrange (0, n, 2) :
            bsl.append ((n0+j, (n*(i+1)+(n0+j-1)%n)))
    bsd = dict (zip (xrange (len (bsl)), bsl))
           
    ssl = range (n)
    return jsd, bsd, ssl


def schwedler_2 (r0, hh, n, ccw = True) :        
    jsl = []
    bsl = []
    ssl = []
    rsl = [r0]
    
    h0 = hh[-1]
    rr = r0^2/(2*h0) + h0/2
    aa = rr - h0
    for h in hh[:-1] :
        rsl.append (sqrt (rr^2 - (aa+h)^2))
    hh.insert (0, 0.0)
    alpha = RDF (2*pi/n) 
    for i in xrange (len (rsl)) :
        for j in xrange (n) : 
            x = rsl[i] * cos (j*alpha)
            y = rsl[i] * sin (j*alpha)
            z = hh[i]
            jsl.append ((x, y, z))
    jsd = dict (zip (xrange (len (jsl)), jsl))
    
    for i in xrange (0, len (hh) - 2) : 
        n0 = i*n 
        for j in xrange (n) : 
            bsl.append ((n0+j, n0+j+n)) 
        for j in xrange (n) : 
            bsl.append ((n0+n+j, n*(i+1)+(n0+j+1)%n))
        if ccw == True :
            for j in xrange (0, n) : 
                bsl.append ((n0+j, (n*(i+1)+(n0+j+1)%n)))
        else :
            for j in xrange (0, n) : 
                bsl.append ((n0+j, (n*(i+1)+(n0+j-1)%n)))
    bsd = dict (zip (xrange (len (bsl)), bsl))
           
    ssl = range (n)
    return jsd, bsd, ssl


def schwedler_kind (r0, hh, n) :        
    jsl = []
    bsl = []
    ssl = []
    rsl = [r0]
    
    h0 = hh[-1]
    rr = r0^2/(2*h0) + h0/2
    aa = rr - h0
    for h in hh[:-1] :
        rsl.append (sqrt (rr^2 - (aa+h)^2))
    hh.insert (0, 0.0)
    alpha = RDF (2*pi/n) 
    for i in xrange (len (rsl)) :
        for j in xrange (n) : 
            x = rsl[i] * cos (j*alpha)
            y = rsl[i] * sin (j*alpha)
            z = hh[i]
            jsl.append ((x, y, z))
    jsd = dict (zip (xrange (len (jsl)), jsl))
    
    for i in xrange (0, len (hh) - 2) : 
        n0 = i*n 
        for j in xrange (n) : 
            bsl.append ((n0+j, n0+j+n)) 
        for j in xrange (n) : 
            bsl.append ((n0+n+j, n*(i+1)+(n0+j+1)%n))
    bsd = dict (zip (xrange (len (bsl)), bsl))
           
    ssl = range (n)
    return jsd, bsd, ssl


def schwedler_sind (r0, hh, n) :        
    jsl = []
    bsl = []
    ssl = []
    rsl = [r0]
    
    h0 = hh[-1]
    rr = r0^2/(2*h0) + h0/2
    aa = rr - h0
    for h in hh[:-1] :
        rsl.append (sqrt (rr^2 - (aa+h)^2))
    hh.insert (0, 0.0)
    alpha = RDF (2*pi/n) 
    for i in xrange (len (rsl)) :
        for j in xrange (n) : 
            x = rsl[i] * cos (j*alpha)
            y = rsl[i] * sin (j*alpha)
            z = hh[i]
            jsl.append ((x, y, z))
    jsd = dict (zip (xrange (len (jsl)), jsl))
    
    for i in xrange (0, len (hh) - 2) : 
        n0 = i*n 
        for j in xrange (n) : 
            bsl.append ((n0+j, n0+j+n)) 
        for j in xrange (n) : 
            bsl.append ((n0+n+j, n*(i+1)+(n0+j+1)%n))
        for j in xrange (0, n) : 
            bsl.append ((n0+j, (n*(i+1)+(n0+j+1)%n)))
        for j in xrange (0, n) : 
            bsl.append ((n0+j, (n*(i+1)+(n0+j-1)%n)))
    bsd = dict (zip (xrange (len (bsl)), bsl))
           
    ssl = range (n)
    return jsd, bsd, ssl


def hypar_xy (x, y, z_scale) :
    return x * y * z_scale 
    
def hypar_x2y2 (x, y, z_scale) :
    return (x^2 - y^2) * z_scale 
    
def surface_net (f, lx, ly, m, n) : 
    jsl = []
    bsl = []
    ssl = []

    m2 = m + 2
    n2 = n + 2
    xmin = -lx/2
    ymin = -ly/2
    hx = lx/(m2-1)
    hy = ly/(n2-1) 
    
    for j in xrange (n2) : 
        for i in xrange (m2) : 
            xx = xmin + i*hx
            yy = ymin + j*hy
            jsl.append ((xx, yy, f (xx, yy)))
    jsd = dict (zip (xrange (len (jsl)), jsl))

    for j in xrange (1, n2-1) : 
        n0 = j*m2 
        for i in xrange (m2-1) : 
            bsl.append ((n0+i, n0+i+1)) 
    for i in xrange (1, m2-1) : 
        for j in xrange (0, (n2-1)*m2, m2) : 
            bsl.append ((i+j, i+j+m2)) 
    bsd = dict (zip (xrange (len (bsl)), bsl))
        
    ssl = range (m2)
    for i in xrange (1, n2-1) :
        ssl.append (i*m2)
        ssl.append (i*m2 + m2 - 1)
    ssl.extend (range ((n2-1)*m2, n2*m2))

    return jsd, bsd, ssl


def make_loads__ (nl, n0, step, ld) :
    return zip (range (n0, n0 + nl*step, step), [ld for i in xrange (nl)])
    
def make_loads (generator) :
    ll = []
    for gg in generator :
        if len (gg) == 3 : 
            nl, n0, ld = gg
            step = 1
        else :
            nl, n0, step, ld = gg
        ll.extend (make_loads__ (nl, n0, step, ld))
    return dict (ll)
    
    
def tension_compression (bar_forces, tolerance = 1.5e-8) :
    tt = [] 
    cc = []
    zz = []
    for (bi, f) in bar_forces.iteritems() : 
        if abs (f) > tolerance : 
            if f > 0 :
                tt.append ((bi, f))
            else :
                cc.append ((bi, f))
        else :
            zz.append (bi)
    tt = sorted (tt, key = lambda x : x[1])
    cc = sorted (cc, key = lambda x : x[1])
    return (tt, cc, zz)

def unique_lp (l, tolerance = 1e-8) : 
    li = []
    if l != []: 
        (ai, av) = l[0]
        ll = [ai]
        for (bi, bv) in l[1:] :
            if abs (av - bv) < tolerance :
                ll.append (bi)
            else : 
                li.append (copy (ll))
                ll = [bi] 
                av = bv
        li.append (copy (ll))
    return li   


def plot_free_joints2d (pts, rr) :
    return sum ([circle (pt, rr, edgecolor = 'black', facecolor = 'white',
                 fill = True, aspect_ratio = 1) for pt in pts])

def plot_supports2d (pts, rr) :
    return sum ([circle (pt, rr, edgecolor = 'black', facecolor = 'black',
                 fill = True, aspect_ratio = 1) for pt in pts])

def plot_bars2d (bars, joints, color = 'black', thickness = 2) :
    return sum ([line2d ((joints[b[0]], joints[b[1]]), thickness = thickness,
                 color = color, aspect_ratio = 1) for b in bars])

def plot_loads2d (loads, joints, load_scale) :
    pls = []
    for li in loads :
        tail = joints[li]
        ff = loads[li]
        head = [tail[0] + load_scale*ff[0], tail[1] + load_scale*ff[1]]
        pls.append (arrow2d (tail, head, width = 3, color = 'crimson', aspect_ratio = 1))
    return sum (pls)
                                                 
def plot_truss2d (joints, bars, supports, loads = [], load_scale = 0.01) :
    prr = point2d (joints.values(), aspect_ratio = 1)
    rrx = (prr.xmax() - prr.xmin())
    rry = (prr.ymax() - prr.ymin())
    rr = rrx if rrx > rry else rry*3/2
    pfjs = plot_free_joints2d (list_indexed (others (supports, joints), joints), rr/120)
    psjs = plot_supports2d (list_indexed (supports, joints), rr/150)
    pbs = plot_bars2d (bars.values(), joints)
    pt = pfjs + psjs + pbs
    if loads != [] :
        pls = plot_loads2d (loads, joints, load_scale)
        pt += pls
    return pt
                 
def plot_truss_with_forces2d (joints, bars, supports, forces) :
    prr = point2d (joints.values(), aspect_ratio = 1)
    rrx = (prr.xmax() - prr.xmin())
    rry = (prr.ymax() - prr.ymin())
    rr = rrx if rrx > rry else rry*3/2
    pfjs = plot_free_joints2d (list_indexed (others (supports, joints), joints), rr/120)
    psjs = plot_supports2d (list_indexed (supports, joints), rr/150)
    pt = pfjs + psjs
    tt, cc, zi = tension_compression (forces)
    ti = unique_lp (tt)
    ti.reverse()
    ci = unique_lp (cc)
    if zi != [] :
        cl = Color ('orange')
        cl = cl.lighter (0.425)
        pbz = plot_bars2d (list_indexed (zi, bars), joints, color = cl, thickness = 3)
        pt += pbz
    if ti != [] :
        step = 1./len (ti)
        cl = Color ('darkblue')
        pbt = plot_bars2d (list_indexed (ti[0], bars), joints, color = cl, thickness = 3)
        for i in xrange (1, len (ti)) :
            cl = cl.lighter (step)
            pbt += plot_bars2d (list_indexed (ti[i], bars), joints, color = cl, thickness = 3)
        pt += pbt
    if ci != [] :
        step = 1./len (ci)
        cl = Color ('red')
        pbt = plot_bars2d (list_indexed (ci[0], bars), joints, color = cl, thickness = 3)
        for i in xrange (1, len (ci)) :
            cl = cl.lighter (step)
            pbt += plot_bars2d (list_indexed (ci[i], bars), joints, color = cl, thickness = 3)
        pt += pbt
    return pt


def plot_free_joints3d (pts, color = 'white', size = 27) : 
    return point3d (pts, color = color, size = size, aspect_ratio = 1, frame = False)
  
def plot_supports3d (pts, color = 'black', size = 23) : 
    return point3d (pts, color = color, size = size, aspect_ratio = 1, frame = False)
  
def plot_bars3d (bars, joints, color = 'black', thickness = 7) : 
    return sum ([line3d ((joints[b[0]], joints[b[1]]), thickness = thickness,
                 color = color, aspect_ratio = 1, frame = False) for b in bars]) 


def plot_arrow3d (ff, tail = (0, 0, 0), scale = 1, color = 'blue', thickness = 1) :
    fs = scale * ff
    nf = norm (fs)
    if nf != 0 :
        fw = (3.425 * thickness * nf)^(2/5)
        rd = fw / 50
        hl = 6.425 * rd
        hr = 1.625 * rd
        vtl = vector (tail)
        ar = arrow3d (vtl, vtl + fs, radius = rd, head_len = hl, head_radius = hr, color = color, aspect_ratio = 1, frame = False)
    else :
        ar = Sphere (0.001, color = 'white').translate (tail)
    return ar

def plot_loads3d (loads, joints, load_scale, pull = True, thickness = 1, color = 'crimson') : 
    pls = []
    for li in loads :
        ff = vector (loads[li]) 
        if pull == True :
            tail = joints[li]
        else : 
            head = joints[li]
            tail = [head[0] - load_scale*ff[0], head[1] - load_scale*ff[1], head[2] - load_scale*ff[2]]
        pls.append (plot_arrow3d (ff, tail, color = color, thickness = thickness, scale = load_scale))
    return sum (pls)


from sage.plot.plot3d.plot3d import axes
    
def plot_truss3d (joints, bars, supports, loads = [], load_scale = 0.01, with_axes = False, pull = True) : 
    pfjs = plot_free_joints3d (list_indexed (others (supports, joints), joints))
    psjs = plot_supports3d (list_indexed (supports, joints))
    pbs = plot_bars3d (bars.values(), joints)
    pt = pfjs + psjs + pbs
    if loads != [] :
        pls = plot_loads3d (loads, joints, load_scale, pull)
        pt += pls
    if with_axes == True :
        pt += axes (radius = 0.325)
    return pt

   
def plot_truss_with_forces3d (joints, bars, supports, forces) : 
    pfjs = plot_free_joints3d (list_indexed (others (supports, joints), joints))
    psjs = plot_supports3d (list_indexed (supports, joints))
    pt = pfjs + psjs 
    tt, cc, zi = tension_compression (forces)
    ti = unique_lp (tt)
    ti.reverse() 
    ci = unique_lp (cc)
    if zi != [] :
        cl = Color ('orange')
        cl = cl.lighter (0.625)
        pbz = plot_bars3d (list_indexed (zi, bars), joints, color = cl, thickness = 7) 
        pt += pbz
    if ti != [] :
        step = 1./len (ti) 
        cl = Color ('indigo')
        pbt = plot_bars3d (list_indexed (ti[0], bars), joints, color = cl, thickness = 7) 
        for i in xrange (1, len (ti)) :
            cl = cl.lighter (step)
            pbt += plot_bars3d (list_indexed (ti[i], bars), joints, color = cl, thickness = 7) 
        pt += pbt
    if ci != [] :
        step = 1./len (ci) 
        cl = Color ('red')
        pbt = plot_bars3d (list_indexed (ci[0], bars), joints, color = cl, thickness = 7) 
        for i in xrange (1, len (ci)) :
            cl = cl.lighter (step)
            pbt += plot_bars3d (list_indexed (ci[i], bars), joints, color = cl, thickness = 7) 
        pt += pbt
    return pt


def plot_disp_lines3d (joints, djoints, color = 'orange', thickness = 3) : 
    return sum ([line3d ((joints[i], djoints[i]), arrow_head = True, thickness = thickness,
                 color = color, aspect_ratio = 1, frame = False) for i in xrange (len (joints))]) 
                         
def plot_truss_with_displacements3d (joints, bars, supports, displs, 
                                     displ_scale = 1, plot_dlines = True) : 
    fjis = others (supports, joints)
    fjs = list_indexed (fjis, joints)
    pfjs = plot_free_joints3d (fjs, color = 'whitesmoke', size = 18)
    psjs = plot_supports3d (list_indexed (supports, joints))
    pbs = plot_bars3d (bars.values(), joints, color = 'whitesmoke', thickness = 4)
    ds = displ_scale
    dfjs = copy (fjs)
    dispc = map (chop(), displs)
    for i in xrange (len (dfjs)) :
        jnt = dfjs[i]
        dfjs[i] = (jnt[0] + ds*dispc[i*3], jnt[1] + ds*dispc[i*3 + 1], jnt[2] + ds*dispc[i*3 + 2])
    djs = copy (joints)
    for i in xrange (len (fjis)) :
        djs[fjis[i]] = dfjs[i] 
    pdfjs = plot_free_joints3d (dfjs)
    pdbs = plot_bars3d (bars.values(), djs)
    pt = pfjs + psjs + pbs + pdfjs + pdbs
    if plot_dlines == True :
        pdl = plot_disp_lines3d (fjs, dfjs)
        pt += pdl
    return pt


def plot_vectors3d (vs, color = 'green', thickness = 0.1, scale = 1.) : 
    pvs = []
    coord_axes = []
    coord_axes.append (plot_arrow3d (vector ((1, 0, 0)), (0, 0, 0), thickness = 0.01))
    coord_axes.append (plot_arrow3d (vector ((0, 1, 0)), (0, 0, 0), thickness = 0.01))
    coord_axes.append (plot_arrow3d (vector ((0, 0, 1)), (0, 0, 0), thickness = 0.01))
    for v in vs :
        pvs.append (plot_arrow3d (scale * v, (0, 0, 0), color = color, thickness = thickness))
    return sum (coord_axes) + sum (pvs)
    
    
def plot_vectors2d (vs, color = 'green', thickness = 1) :
    pvs = [] 
    for v in vs :
        pvs.append (v.plot (color = color, thickness = thickness, aspect_ratio = 1))
    return sum (pvs)


def print_indexed (l, i0 = 0, idx_len = 4, indent = '') :
    fstr = '{0}{1:<' + str (idx_len) + '} {2}'
    for i, e in enumerate (l) :
        print fstr.format (indent, str(i+i0) + ':', e)

def print_list (l, indent = '', braces = False) : 
    if l == [] :
        print '{0}{1}'.format (indent, '[]')
    else :
        if braces == True :
            print '{0}{1}'.format (indent, '[')
        for x in l :
            print '{0}{1}'.format (indent, x)
        if braces == True :
            print '{0}{1}'.format (indent, ']')
        
def print_dict (d, key_len = 4, sort_keys = True, indent = '') : 
    fstr = '{0}{1:<' + str (key_len) + '} {2}'
    for k in sorted (d.keys()) if sort_keys == True else d.keys():
        print fstr.format (indent, str(k) + ':', d[k])

joints = { 0: (0.0, 0.0, 0.0), 1: (2.0, 0.0, 0.0), 2: (0.0, 2.0, 0.0), 3: (0.0, 0.0, 2.0) }
print 'joints:'
print_dict (joints, indent = '   ')
print

bars = { 0: (0, 3), 1: (1, 3), 2: (2, 3) }
print 'bars:' 
print_dict (bars, indent = '   ')
print

supports = [0, 1, 2]
print 'supports:', supports
print

loads = { 3: (100.0, 100.0, 0.0) }
print 'loads:' 
print_dict (loads, indent = '   ')

joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 100.000000000000, 0.000000000000000)

joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 100.000000000000, 0.000000000000000)

plot_truss3d (joints, bars, supports, loads)

Click on 3-D image to make it live. Right-click on live image for a control menu.

plot_truss3d (joints, bars, supports, loads, with_axes = True)

bar_forces = truss (joints, bars, supports, loads)
bar_forces

{0: 200.0, 1: -141.4213562373095, 2: -141.4213562373095}

{0: 200.0, 1: -141.4213562373095, 2: -141.4213562373095}

print_dict (bar_forces)

0:   200.0
1:   -141.4213562373095
2:   -141.4213562373095

0:   200.0
1:   -141.4213562373095
2:   -141.4213562373095

plot_truss_with_forces3d (joints, bars, supports, bar_forces)

free_joints = others (supports, joints)
free_joints

[3]

[3]

maxwells_rule (joints, supports, bars)

3 * 1 - 3  ==  0

3 * 1 - 3  ==  0

A = equilibrium_matrix (joints, bars, free_joints)
show (A)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 0.0 & 0.7071067811865475 & 0.0 \\ 0.0 & 0.0 & 0.7071067811865475 \\ -1.0 & -0.7071067811865475 & -0.7071067811865475 \end{array}\right)$

b = load_vector (loads, free_joints)
show (b)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(100.0,\,100.0,\,0.0\right)$

x = -A \ b
show (x)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(200.0,\,-141.4213562373095,\,-141.4213562373095\right)$

bfs = dict_bar_force (bars, x)
print_dict (bfs)

0:   200.0
1:   -141.4213562373095
2:   -141.4213562373095

0:   200.0
1:   -141.4213562373095
2:   -141.4213562373095

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
show (Ab)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} 0.0 & 0.7071067811865475 & 0.0 & -100.0 \\ 0.0 & 0.0 & 0.7071067811865475 & -100.0 \\ -1.0 & -0.7071067811865475 & -0.7071067811865475 & -0.0 \end{array}\right)$

Abr, pivots = ref_wpp (Ab)
show (Abr)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} 1.0 & 0.7071067811865475 & 0.7071067811865475 & 0.0 \\ 0.0 & 1.0 & 0.0 & -141.4213562373095 \\ 0.0 & 0.0 & 1.0 & -141.4213562373095 \end{array}\right)$

pivots

[0, 1, 2]

[0, 1, 2]

xx = ref_solution (Abr, pivots)
xx

(200.0, -141.4213562373095, -141.4213562373095)

(200.0, -141.4213562373095, -141.4213562373095)

print_dict (dict_bar_force (bars, xx))

0:   200.0
1:   -141.4213562373095
2:   -141.4213562373095

0:   200.0
1:   -141.4213562373095
2:   -141.4213562373095

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 3 and dimension 3
   Basis:
     (0.0, 0.0, -1.0)
     (0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.7071067811865475, -0.7071067811865475)

Column space:
   Vector space of degree 3 and dimension 3
   Basis:
     (0.0, 0.0, -1.0)
     (0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.7071067811865475, -0.7071067811865475)

show (A)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 0.0 & 0.7071067811865475 & 0.0 \\ 0.0 & 0.0 & 0.7071067811865475 \\ -1.0 & -0.7071067811865475 & -0.7071067811865475 \end{array}\right)$

print_subspace (rs)

Row space:
   Vector space of degree 3 and dimension 3
   Basis:
     (1.0, 0.7071067811865475, 0.7071067811865475)
     (0.0, 1.0, 0.0)
     (0.0, 0.0, 1.0)

Row space:
   Vector space of degree 3 and dimension 3
   Basis:
     (1.0, 0.7071067811865475, 0.7071067811865475)
     (0.0, 1.0, 0.0)
     (0.0, 0.0, 1.0)

Abr.subdivision (0, 0)

[               1.0 0.7071067811865475 0.7071067811865475]
[               0.0                1.0                0.0]
[               0.0                0.0                1.0]

[               1.0 0.7071067811865475 0.7071067811865475]
[               0.0                1.0                0.0]
[               0.0                0.0                1.0]

print_subspace (rk)

Kernel:
   Vector space of degree 3 and dimension 0

Kernel:
   Vector space of degree 3 and dimension 0

print_subspace (lk)

Left kernel:
   Vector space of degree 3 and dimension 0

Left kernel:
   Vector space of degree 3 and dimension 0

joints = { 0: (-2.0, -1.0, 0.0), 1: (0.0, 0.0, 0.0), 2: (2.0, 1.0, 0.0), 3: (0.0, 0.0, 2.0) }
print 'joints:'
print_dict (joints, indent = '   ')
print

bars = { 0: (0, 3), 1: (1, 3), 2: (2, 3) }
print 'bars:' 
print_dict (bars, indent = '   ')
print

supports = [0, 1, 2]
print 'supports:', supports
print

loads = { 3: (100.0, 50.0, 0.0) }
print 'loads:' 
print_dict (loads, indent = '   ')

joints:
   0:   (-2.00000000000000, -1.00000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 1.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 50.0000000000000, 0.000000000000000)

joints:
   0:   (-2.00000000000000, -1.00000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 1.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 50.0000000000000, 0.000000000000000)

plot_truss3d (joints, bars, supports, loads, with_axes = True)

free_joints = others (supports, joints)
free_joints

[3]

[3]

maxwells_rule (joints, supports, bars)

3 * 1 - 3  ==  0

3 * 1 - 3  ==  0

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
show (Ab)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} -0.6666666666666666 & 0.0 & 0.6666666666666666 & -100.0 \\ -0.3333333333333333 & 0.0 & 0.3333333333333333 & -50.0 \\ -0.6666666666666666 & -1.0 & -0.6666666666666666 & -0.0 \end{array}\right)$

Abr, pivots = ref_wpp (Ab)
show (Abr)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} 1.0 & -0.0 & -1.0 & 150.0 \\ 0.0 & 1.0 & 1.3333333333333333 & -100.0 \\ 0.0 & 0.0 & 0.0 & 0.0 \end{array}\right)$

pivots

[0, 1]

[0, 1]

primary_system_bars = pivots
primary_system_bars

[0, 1]

[0, 1]

redundant_bars = others (pivots, bars)
redundant_bars

[2]

[2]

bar_forces = dict_bar_force (bars, ref_solution (Abr, pivots))
print_dict (bar_forces)

0:   150.0
1:   -100.0
2:   0.0

0:   150.0
1:   -100.0
2:   0.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces)

A = equilibrium_matrix (joints, bars, free_joints)
show (A)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} -0.6666666666666666 & 0.0 & 0.6666666666666666 \\ -0.3333333333333333 & 0.0 & 0.3333333333333333 \\ -0.6666666666666666 & -1.0 & -0.6666666666666666 \end{array}\right)$

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (-0.6666666666666666, -0.3333333333333333, -0.6666666666666666)
     (0.0, 0.0, -1.0)

Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (-0.6666666666666666, -0.3333333333333333, -0.6666666666666666)
     (0.0, 0.0, -1.0)

print_subspace (rs)

Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, -0.0, -1.0)
     (0.0, 1.0, 1.3333333333333333)

Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, -0.0, -1.0)
     (0.0, 1.0, 1.3333333333333333)

Abr.subdivision (0, 0)

[               1.0               -0.0               -1.0]
[               0.0                1.0 1.3333333333333333]
[               0.0                0.0                0.0]

[               1.0               -0.0               -1.0]
[               0.0                1.0 1.3333333333333333]
[               0.0                0.0                0.0]

print_subspace (rk)

Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (1.0, -1.3333333333333333, 1.0)

Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (1.0, -1.3333333333333333, 1.0)

bar_forces_ss = dict_bar_force (bars, rk['basis'][0])
print_dict (bar_forces_ss)

0:   1.0
1:   -1.3333333333333333
2:   1.0

0:   1.0
1:   -1.3333333333333333
2:   1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss)

print_subspace (lk)

Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (-0.5, 1.0, 0.0)

Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (-0.5, 1.0, 0.0)

plot_vectors3d (cs['basis']) + plot_vectors3d (lk['basis'], color = 'red')

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][0])

b = load_vector (loads, free_joints)
b

(100.0, 50.0, 0.0)

(100.0, 50.0, 0.0)

is_in_subspace (b, cs['basis'])

True

True

c = vector ([-50, 100, 0])
c

(-50, 100, 0)

(-50, 100, 0)

is_in_subspace (c, cs['basis'])

False

False

is_in_subspace (c, lk['basis'])

True

True

c2 = vector ([100, 0, 0])
c2

(100, 0, 0)

(100, 0, 0)

is_in_subspace (c2, cs['basis'])

False

False

is_in_subspace (c2, lk['basis'])

False

False

has_component_in_subspace (c2, lk['basis'])

True

True

loads2 = { 3: (-50.0, 100.0, 0.0) }

plot_truss3d (joints, bars, supports, loads2)

Ab12 = augmented_equil_matrix (joints, bars, loads2, free_joints)
show (Ab12)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} -0.6666666666666666 & 0.0 & 0.6666666666666666 & 50.0 \\ -0.3333333333333333 & 0.0 & 0.3333333333333333 & -100.0 \\ -0.6666666666666666 & -1.0 & -0.6666666666666666 & -0.0 \end{array}\right)$

Abr12, pivots12 = ref_wpp (Ab12)
show (Abr12)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} 1.0 & -0.0 & -1.0 & -75.0 \\ 0.0 & 1.0 & 1.3333333333333333 & 50.0 \\ 0.0 & 0.0 & 0.0 & 1.0 \end{array}\right)$

loads3 = { 3: (100.0, 0.0, 0.0) }

plot_truss3d (joints, bars, supports, loads3)

Ab13 = augmented_equil_matrix (joints, bars, loads3, free_joints)
show (Ab13)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} -0.6666666666666666 & 0.0 & 0.6666666666666666 & -100.0 \\ -0.3333333333333333 & 0.0 & 0.3333333333333333 & -0.0 \\ -0.6666666666666666 & -1.0 & -0.6666666666666666 & -0.0 \end{array}\right)$

Abr13, pivots13 = ref_wpp (Ab13)
show (Abr13)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} 1.0 & -0.0 & -1.0 & 150.0 \\ 0.0 & 1.0 & 1.3333333333333333 & -100.0 \\ 0.0 & 0.0 & 0.0 & 1.0 \end{array}\right)$

bars2 = { 0: (0, 3), 1: (2, 3), 2: (1, 3) }
print 'bars:' 
print_dict (bars2, indent = '   ')
print

bars:
   0:   (0, 3)
   1:   (2, 3)
   2:   (1, 3)

bars:
   0:   (0, 3)
   1:   (2, 3)
   2:   (1, 3)

Ab2 = augmented_equil_matrix (joints, bars2, loads, free_joints)
show (Ab2)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} -0.6666666666666666 & 0.6666666666666666 & 0.0 & -100.0 \\ -0.3333333333333333 & 0.3333333333333333 & 0.0 & -50.0 \\ -0.6666666666666666 & -0.6666666666666666 & -1.0 & -0.0 \end{array}\right)$

show (Ab)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} -0.6666666666666666 & 0.0 & 0.6666666666666666 & -100.0 \\ -0.3333333333333333 & 0.0 & 0.3333333333333333 & -50.0 \\ -0.6666666666666666 & -1.0 & -0.6666666666666666 & -0.0 \end{array}\right)$

Abr2, p2 = ref_wpp (Ab2)
show (Abr2)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} 1.0 & -1.0 & -0.0 & 150.0 \\ 0.0 & 1.0 & 0.75 & -75.0 \\ 0.0 & 0.0 & 0.0 & 0.0 \end{array}\right)$

bar_forces2 = dict_bar_force (bars2, ref_solution (Abr2, p2))
print_dict (bar_forces2)

0:   75.0
1:   -75.0
2:   0.0

0:   75.0
1:   -75.0
2:   0.0

plot_truss_with_forces3d (joints, bars2, supports, bar_forces2)

A2 = equilibrium_matrix (joints, bars2, free_joints)
show (A2)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} -0.6666666666666666 & 0.6666666666666666 & 0.0 \\ -0.3333333333333333 & 0.3333333333333333 & 0.0 \\ -0.6666666666666666 & -0.6666666666666666 & -1.0 \end{array}\right)$

rs2, cs2, rk2, lk2 = subspaces (A2)

print_subspace (cs2)

Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (-0.6666666666666666, -0.3333333333333333, -0.6666666666666666)
     (0.6666666666666666, 0.3333333333333333, -0.6666666666666666)

Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (-0.6666666666666666, -0.3333333333333333, -0.6666666666666666)
     (0.6666666666666666, 0.3333333333333333, -0.6666666666666666)

plot_vectors3d (cs['basis']) + plot_vectors3d (cs2['basis'], scale = 0.725, color = 'orange')

print_subspace (rs2)

Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, -1.0, -0.0)
     (0.0, 1.0, 0.75)

Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, -1.0, -0.0)
     (0.0, 1.0, 0.75)

print_subspace (rk2)

Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (-0.75, -0.75, 1.0)

Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (-0.75, -0.75, 1.0)

bar_forces_ss2 = dict_bar_force (bars2, rk2['basis'][0])
print_dict (bar_forces_ss2)

0:   -0.75
1:   -0.75
2:   1.0

0:   -0.75
1:   -0.75
2:   1.0

plot_truss_with_forces3d (joints, bars2, supports, bar_forces_ss2)

print_subspace (lk2)

Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (-0.5, 1.0, 0.0)

Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (-0.5, 1.0, 0.0)

plot_vectors3d (cs2['basis']) + plot_vectors3d (lk2['basis'], scale = 0.725, color = 'orange')

plot_vectors3d (lk['basis']) + plot_vectors3d (lk2['basis'], scale = 0.725, color = 'orange')

joints = { 0: (0.0, 0.0, 0.0), 1: (2.0, 0.0, 0.0), 2: (0.0, 0.0, 2.0) }
print 'joints:'
print_dict (joints, indent = '   ')
print

bars = { 0: (0, 2), 1: (1, 2) }
print 'bars:' 
print_dict (bars, indent = '   ')
print

supports = [0, 1]
print 'supports:', supports
print

loads = { 2: (100.0, 0.0, 0.0) }
print 'loads:' 
print_dict (loads, indent = '   ')

joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 2)
   1:   (1, 2)

supports: [0, 1]

loads:
   2:   (100.000000000000, 0.000000000000000, 0.000000000000000)

joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 2)
   1:   (1, 2)

supports: [0, 1]

loads:
   2:   (100.000000000000, 0.000000000000000, 0.000000000000000)

plot_truss3d (joints, bars, supports, loads, with_axes = True)

free_joints = others (supports, joints)
free_joints

[2]

[2]

maxwells_rule (joints, supports, bars)

3 * 1 - 2  ==  1  !=  0

3 * 1 - 2  ==  1  !=  0

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
show (Ab)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr|r} 0.0 & 0.7071067811865475 & -100.0 \\ 0.0 & 0.0 & -0.0 \\ -1.0 & -0.7071067811865475 & -0.0 \end{array}\right)$

Abr, pivots = ref_wpp (Ab)
show (Abr)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr|r} 1.0 & 0.7071067811865475 & 0.0 \\ 0.0 & 1.0 & -141.4213562373095 \\ 0.0 & 0.0 & 0.0 \end{array}\right)$

bar_forces = dict_bar_force (bars, ref_solution (Abr, pivots))
print_dict (bar_forces)

0:   100.0
1:   -141.4213562373095

0:   100.0
1:   -141.4213562373095

pivots

[0, 1]

[0, 1]

plot_truss_with_forces3d (joints, bars, supports, bar_forces)

A = equilibrium_matrix (joints, bars, free_joints)
show (A)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 0.0 & 0.7071067811865475 \\ 0.0 & 0.0 \\ -1.0 & -0.7071067811865475 \end{array}\right)$

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (0.0, 0.0, -1.0)
     (0.7071067811865475, 0.0, -0.7071067811865475)

Column space:
   Vector space of degree 3 and dimension 2
   Basis:
     (0.0, 0.0, -1.0)
     (0.7071067811865475, 0.0, -0.7071067811865475)

b = load_vector (loads, free_joints)
show (b)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(100.0,\,0.0,\,0.0\right)$

is_in_subspace (b, cs['basis'])

True

True

print_subspace (rs)

Row space:
   Vector space of degree 2 and dimension 2
   Basis:
     (1.0, 0.7071067811865475)
     (0.0, 1.0)

Row space:
   Vector space of degree 2 and dimension 2
   Basis:
     (1.0, 0.7071067811865475)
     (0.0, 1.0)

Abr.subdivision (0, 0)

[               1.0 0.7071067811865475]
[               0.0                1.0]
[               0.0                0.0]

[               1.0 0.7071067811865475]
[               0.0                1.0]
[               0.0                0.0]

print_subspace (rk)

Kernel:
   Vector space of degree 2 and dimension 0

Kernel:
   Vector space of degree 2 and dimension 0

print_subspace (lk)

Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (0.0, 1.0, 0.0)

Left kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (0.0, 1.0, 0.0)

plot_vectors3d (cs['basis']) + plot_vectors3d (lk['basis'], color = 'red')

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][0], displ_scale = 1.25)

joints = { 0: (-1.0, 0.0), 1: (0.0, 0.0), 2: (4.0, 0.0), 3: (0.0, 2.0) }
print 'joints:'
print_dict (joints, indent = '   ')
print

bars = { 0: (0, 3), 1: (1, 3), 2: (2, 3) }
print 'bars:' 
print_dict (bars, indent = '   ')
print

supports = [0, 1, 2]
print 'supports:', supports
print

loads = { 3: (100.0, 0.0) }
print 'loads:' 
print_dict (loads, indent = '   ')

joints:
   0:   (-1.00000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000)
   2:   (4.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 0.000000000000000)

joints:
   0:   (-1.00000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000)
   2:   (4.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)

supports: [0, 1, 2]

loads:
   3:   (100.000000000000, 0.000000000000000)

plot_truss2d (joints, bars, supports, loads)

free_joints = others (supports, joints)
free_joints

[3]

[3]

maxwells_rule (joints, supports, bars)

2 * 1 - 3  ==  -1  !=  0

2 * 1 - 3  ==  -1  !=  0

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
show (Ab)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} -0.4472135954999579 & 0.0 & 0.8944271909999159 & -100.0 \\ -0.8944271909999159 & -1.0 & -0.4472135954999579 & -0.0 \end{array}\right)$

Abr, pivots = ref_wpp (Ab)
show (Abr)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr|r} 1.0 & 1.118033988749895 & 0.5 & 0.0 \\ 0.0 & 1.0 & 2.23606797749979 & -200.0 \end{array}\right)$

primary_system_bars = pivots
primary_system_bars

[0, 1]

[0, 1]

redundant_bars = others (pivots, bars)
redundant_bars

[2]

[2]

x = ref_solution (Abr, pivots)
x

(223.60679774997897, -200.0)

(223.60679774997897, -200.0)

abf = all_bar_forces (primary_system_bars, redundant_bars, x)
bar_forces = dict_bar_force (bars, abf)
print_dict (bar_forces)

0:   223.60679774997897
1:   -200.0
2:   0.0

0:   223.60679774997897
1:   -200.0
2:   0.0

plot_truss_with_forces2d (joints, bars, supports, bar_forces)

A = equilibrium_matrix (joints, bars, free_joints)
show (A)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} -0.4472135954999579 & 0.0 & 0.8944271909999159 \\ -0.8944271909999159 & -1.0 & -0.4472135954999579 \end{array}\right)$

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 2 and dimension 2
   Basis:
     (-0.4472135954999579, -0.8944271909999159)
     (0.0, -1.0)

Column space:
   Vector space of degree 2 and dimension 2
   Basis:
     (-0.4472135954999579, -0.8944271909999159)
     (0.0, -1.0)

print_subspace (rs)

Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, 1.118033988749895, 0.5)
     (0.0, 1.0, 2.23606797749979)

Row space:
   Vector space of degree 3 and dimension 2
   Basis:
     (1.0, 1.118033988749895, 0.5)
     (0.0, 1.0, 2.23606797749979)

print_subspace (rk)

Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (2.0000000000000004, -2.23606797749979, 1.0)

Kernel:
   Vector space of degree 3 and dimension 1
   Basis:
     (2.0000000000000004, -2.23606797749979, 1.0)

bar_forces_ss = dict_bar_force (bars, rk['basis'][0])
print_dict (bar_forces_ss)

0:   2.0000000000000004
1:   -2.23606797749979
2:   1.0

0:   2.0000000000000004
1:   -2.23606797749979
2:   1.0

plot_truss_with_forces2d (joints, bars, supports, bar_forces_ss)

print_subspace (lk)

Left kernel:
   Vector space of degree 2 and dimension 0

Left kernel:
   Vector space of degree 2 and dimension 0

joints = { 0: (-2.0, 0.0, 0.0), 1: (0.0, 0.0, 0.0), 2: (2.0, 0.0, 0.0), 3: (0.0, -2.0, 0.0), 4: (0.0, 2.0, 0.0), 5: (0.0, 0.0, 2.0) }
print 'joints:'
print_dict (joints, indent = '   ')
print

bars = { 0: (0, 5), 1: (1, 5), 2: (2, 5), 3: (3, 5), 4: (4, 5) }
print 'bars:' 
print_dict (bars, indent = '   ')
print

supports = [0, 1, 2, 3, 4]
print 'supports:', supports
print

loads = { 5: (100.0, 100.0, 0.0) }
print 'loads:' 
print_dict (loads, indent = '   ')

joints:
   0:   (-2.00000000000000, 0.000000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   3:   (0.000000000000000, -2.00000000000000, 0.000000000000000)
   4:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   5:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 5)
   1:   (1, 5)
   2:   (2, 5)
   3:   (3, 5)
   4:   (4, 5)

supports: [0, 1, 2, 3, 4]

loads:
   5:   (100.000000000000, 100.000000000000, 0.000000000000000)

joints:
   0:   (-2.00000000000000, 0.000000000000000, 0.000000000000000)
   1:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   3:   (0.000000000000000, -2.00000000000000, 0.000000000000000)
   4:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   5:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 5)
   1:   (1, 5)
   2:   (2, 5)
   3:   (3, 5)
   4:   (4, 5)

supports: [0, 1, 2, 3, 4]

loads:
   5:   (100.000000000000, 100.000000000000, 0.000000000000000)

plot_truss3d (joints, bars, supports, loads, with_axes = True)

free_joints = others (supports, joints)
free_joints

[5]

[5]

maxwells_rule (joints, supports, bars)

3 * 1 - 5  ==  -2  !=  0

3 * 1 - 5  ==  -2  !=  0

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
show (Ab)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrr|r} -0.7071067811865475 & 0.0 & 0.7071067811865475 & 0.0 & 0.0 & -100.0 \\ 0.0 & 0.0 & 0.0 & -0.7071067811865475 & 0.7071067811865475 & -100.0 \\ -0.7071067811865475 & -1.0 & -0.7071067811865475 & -0.7071067811865475 & -0.7071067811865475 & -0.0 \end{array}\right)$

Abr, pivots = rref_wpp (Ab)
show (Abr)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrr|r} 1.0 & 0.0 & -1.0 & 0.0 & 0.0 & 141.4213562373095 \\ 0.0 & 1.0 & 1.414213562373095 & 0.0 & 1.414213562373095 & -200.0 \\ 0.0 & 0.0 & 0.0 & 1.0 & -1.0 & 141.4213562373095 \end{array}\right)$

primary_system_bars = pivots
primary_system_bars

[0, 1, 3]

[0, 1, 3]

redundant_bars = others (pivots, bars)
redundant_bars

[2, 4]

[2, 4]

abf = all_bar_forces (primary_system_bars, redundant_bars, rref_solution (Abr))
bar_forces = dict_bar_force (bars, abf)
print_dict (bar_forces)

0:   141.4213562373095
1:   -200.0
2:   0.0
3:   141.4213562373095
4:   0.0

0:   141.4213562373095
1:   -200.0
2:   0.0
3:   141.4213562373095
4:   0.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces)

A = equilibrium_matrix (joints, bars, free_joints)
show (A)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrr} -0.7071067811865475 & 0.0 & 0.7071067811865475 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & -0.7071067811865475 & 0.7071067811865475 \\ -0.7071067811865475 & -1.0 & -0.7071067811865475 & -0.7071067811865475 & -0.7071067811865475 \end{array}\right)$

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 3 and dimension 3
   Basis:
     (-0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.0, -1.0)
     (0.0, -0.7071067811865475, -0.7071067811865475)

Column space:
   Vector space of degree 3 and dimension 3
   Basis:
     (-0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.0, -1.0)
     (0.0, -0.7071067811865475, -0.7071067811865475)

print_subspace (rs)

Row space:
   Vector space of degree 5 and dimension 3
   Basis:
     (1.0, -0.0, -1.0, -0.0, -0.0)
     (0.0, 1.0, 1.414213562373095, 0.7071067811865475, 0.7071067811865475)
     (0.0, 0.0, 0.0, 1.0, -1.0)

Row space:
   Vector space of degree 5 and dimension 3
   Basis:
     (1.0, -0.0, -1.0, -0.0, -0.0)
     (0.0, 1.0, 1.414213562373095, 0.7071067811865475, 0.7071067811865475)
     (0.0, 0.0, 0.0, 1.0, -1.0)

Abr.subdivision (0, 0)

[              1.0               0.0              -1.0               0.0        
0.0]
[              0.0               1.0 1.414213562373095               0.0
1.414213562373095]
[              0.0               0.0               0.0               1.0        
-1.0]

[              1.0               0.0              -1.0               0.0               0.0]
[              0.0               1.0 1.414213562373095               0.0 1.414213562373095]
[              0.0               0.0               0.0               1.0              -1.0]

print_subspace (rk)

Kernel:
   Vector space of degree 5 and dimension 2
   Basis:
     (1.0, -1.414213562373095, 1.0, -0.0, 0.0)
     (0.0, -1.414213562373095, 0.0, 1.0, 1.0)

Kernel:
   Vector space of degree 5 and dimension 2
   Basis:
     (1.0, -1.414213562373095, 1.0, -0.0, 0.0)
     (0.0, -1.414213562373095, 0.0, 1.0, 1.0)

redundant_bars

[2, 4]

[2, 4]

bar_forces_ss0 = dict_bar_force (bars, rk['basis'][0])
print_dict (bar_forces_ss0)

0:   1.0
1:   -1.414213562373095
2:   1.0
3:   -0.0
4:   0.0

0:   1.0
1:   -1.414213562373095
2:   1.0
3:   -0.0
4:   0.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss0)

bar_forces_ss1 = dict_bar_force (bars, rk['basis'][1])
print_dict (bar_forces_ss1)

0:   0.0
1:   -1.414213562373095
2:   0.0
3:   1.0
4:   1.0

0:   0.0
1:   -1.414213562373095
2:   0.0
3:   1.0
4:   1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss1)

print_subspace (lk)

Left kernel:
   Vector space of degree 3 and dimension 0

Left kernel:
   Vector space of degree 3 and dimension 0

joints, bars, supports = schwedler_1 (5., [3.725, 5.], 4)

print 'joints:'
print_dict (joints, indent = '   ')
print
print 'bars:' 
print_dict (bars, indent = '   ')
print
print 'supports:', supports

joints:
   0:   (5.00000000000000, 0.000000000000000, 0.000000000000000)
   1:   (3.06161699786838e-16, 5.00000000000000, 0.000000000000000)
   2:   (-5.00000000000000, 6.12323399573676e-16, 0.000000000000000)
   3:   (-9.18485099360514e-16, -5.00000000000000, 0.000000000000000)
   4:   (3.33532232325453, 0.000000000000000, 3.72500000000000)
   5:   (2.04229590364918e-16, 3.33532232325453, 3.72500000000000)
   6:   (-3.33532232325453, 4.08459180729837e-16, 3.72500000000000)
   7:   (-6.12688771094755e-16, -3.33532232325453, 3.72500000000000)

bars:
   0:   (0, 4)
   1:   (1, 5)
   2:   (2, 6)
   3:   (3, 7)
   4:   (4, 5)
   5:   (5, 6)
   6:   (6, 7)
   7:   (7, 4)
   8:   (0, 5)
   9:   (2, 7)
   10:  (0, 7)
   11:  (2, 5)

supports: [0, 1, 2, 3]

joints:
   0:   (5.00000000000000, 0.000000000000000, 0.000000000000000)
   1:   (3.06161699786838e-16, 5.00000000000000, 0.000000000000000)
   2:   (-5.00000000000000, 6.12323399573676e-16, 0.000000000000000)
   3:   (-9.18485099360514e-16, -5.00000000000000, 0.000000000000000)
   4:   (3.33532232325453, 0.000000000000000, 3.72500000000000)
   5:   (2.04229590364918e-16, 3.33532232325453, 3.72500000000000)
   6:   (-3.33532232325453, 4.08459180729837e-16, 3.72500000000000)
   7:   (-6.12688771094755e-16, -3.33532232325453, 3.72500000000000)

bars:
   0:   (0, 4)
   1:   (1, 5)
   2:   (2, 6)
   3:   (3, 7)
   4:   (4, 5)
   5:   (5, 6)
   6:   (6, 7)
   7:   (7, 4)
   8:   (0, 5)
   9:   (2, 7)
   10:  (0, 7)
   11:  (2, 5)

supports: [0, 1, 2, 3]

plot_truss3d (joints, bars, supports, with_axes = True) . rotateZ (pi/9)

maxwells_rule (joints, supports, bars)

3 * 4 - 12  ==  0

3 * 4 - 12  ==  0

load_gen = [(4, 4, (0.0, 0.0, -100.0))]
loads = make_loads (load_gen)
print_dict (loads)

4:   (0.000000000000000, 0.000000000000000, -100.000000000000)
5:   (0.000000000000000, 0.000000000000000, -100.000000000000)
6:   (0.000000000000000, 0.000000000000000, -100.000000000000)
7:   (0.000000000000000, 0.000000000000000, -100.000000000000)

4:   (0.000000000000000, 0.000000000000000, -100.000000000000)
5:   (0.000000000000000, 0.000000000000000, -100.000000000000)
6:   (0.000000000000000, 0.000000000000000, -100.000000000000)
7:   (0.000000000000000, 0.000000000000000, -100.000000000000)

plot_truss3d (joints, bars, supports, loads, pull = False, load_scale = 0.025) . rotateZ (pi/9)

bar_forces = truss (joints, bars, supports, loads)
print_dict (bar_forces)

0:   -109.53144086499151
1:   -109.53144086499151
2:   -109.53144086499151
3:   -109.53144086499151
4:   -31.60013083802923
5:   -31.60013083802923
6:   -31.600130838029237
7:   -31.600130838029237
8:   -2.5121479338940403e-15
9:   5.02429586778808e-15
10:  -2.5121479338940415e-15
11:  -5.0242958677880805e-15

0:   -109.53144086499151
1:   -109.53144086499151
2:   -109.53144086499151
3:   -109.53144086499151
4:   -31.60013083802923
5:   -31.60013083802923
6:   -31.600130838029237
7:   -31.600130838029237
8:   -2.5121479338940403e-15
9:   5.02429586778808e-15
10:  -2.5121479338940415e-15
11:  -5.0242958677880805e-15

print_dict (map_dict (chop(), bar_forces))

0:   -109.53144086499151
1:   -109.53144086499151
2:   -109.53144086499151
3:   -109.53144086499151
4:   -31.60013083802923
5:   -31.60013083802923
6:   -31.600130838029237
7:   -31.600130838029237
8:   0.0
9:   0.0
10:  0.0
11:  0.0

0:   -109.53144086499151
1:   -109.53144086499151
2:   -109.53144086499151
3:   -109.53144086499151
4:   -31.60013083802923
5:   -31.60013083802923
6:   -31.600130838029237
7:   -31.600130838029237
8:   0.0
9:   0.0
10:  0.0
11:  0.0

free_joints = others (supports, joints)
free_joints

[4, 5, 6, 7]

[4, 5, 6, 7]

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
show (Ab.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrr|r} 0.4080 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 100.0 \\ 0.0000 & 2.498 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & -0.0000 \\ 0.0000 & 0.4080 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & -0.4717 & 0.0000 & 0.0000 & -0.4717 & -0.0000 \\ 0.0000 & -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5268 & 0.0000 & 0.0000 & -0.5268 & 100.0 \\ 0.0000 & 0.0000 & -0.4080 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & 4.997 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 100.0 \\ 0.0000 & 0.0000 & 0.0000 & -7.495 \times 10^{-17} & 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.4080 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.4717 & 0.4717 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5268 & -0.5268 & 0.0000 & 100.0 \end{array}\right)$

show (Ab.apply_map (chop()).apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrr|r} 0.4080 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 100.0 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 \\ 0.0000 & 0.4080 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & -0.4717 & 0.0000 & 0.0000 & -0.4717 & 0.0000 \\ 0.0000 & -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5268 & 0.0000 & 0.0000 & -0.5268 & 100.0 \\ 0.0000 & 0.0000 & -0.4080 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 100.0 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.4080 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.4717 & 0.4717 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5268 & -0.5268 & 0.0000 & 100.0 \end{array}\right)$

print_dict (dict_equation_index (free_joints))

print_dict (dict_unknown_index (bars))

Abr, piv = ref_wpp (Ab)

show (Abr.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrr|r} 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -109.5 \\ 0.0000 & 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & 0.5770 & -0.0000 & -0.0000 & 0.5770 & -109.5 \\ 0.0000 & 0.0000 & 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -109.5 \\ 0.0000 & 0.0000 & 0.0000 & 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & 0.5770 & 0.5770 & -0.0000 & -109.5 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 1.000 & -0.0000 & -0.0000 & 1.000 & -0.0000 & -0.0000 & 1.000 & -63.20 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & 1.000 & -31.60 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & 1.000 & -31.60 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -1.000 & -0.0000 & -0.0000 & -0.0000 & -31.60 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -0.0000 & -0.0000 & -0.0000 & -2.512 \times 10^{-15} \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 1.000 & -1.000 & 7.536 \times 10^{-15} \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 7.850 \times 10^{-17} & -2.512 \times 10^{-15} \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -5.024 \times 10^{-15} \end{array}\right)$

show (Abr.apply_map (chop()).apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrr|r} 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -109.5 \\ 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.5770 & 0.0000 & 0.0000 & 0.5770 & -109.5 \\ 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -109.5 \\ 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.5770 & 0.5770 & 0.0000 & -109.5 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 1.000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 1.000 & -63.20 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -31.60 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -31.60 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -1.000 & 0.0000 & 0.0000 & 0.0000 & -31.60 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 1.000 & -1.000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 \end{array}\right)$

bar_forces = dict_bar_force (bars, ref_solution (Abr, piv))
print_dict (bar_forces)

0:   -109.53144086499151
1:   -109.53144086499151
2:   -109.53144086499151
3:   -109.53144086499151
4:   -31.60013083802923
5:   -31.60013083802923
6:   -31.600130838029237
7:   -31.600130838029237
8:   -2.5121479338940403e-15
9:   5.02429586778808e-15
10:  -2.5121479338940415e-15
11:  -5.0242958677880805e-15

0:   -109.53144086499151
1:   -109.53144086499151
2:   -109.53144086499151
3:   -109.53144086499151
4:   -31.60013083802923
5:   -31.60013083802923
6:   -31.600130838029237
7:   -31.600130838029237
8:   -2.5121479338940403e-15
9:   5.02429586778808e-15
10:  -2.5121479338940415e-15
11:  -5.0242958677880805e-15

bar_forces = dict_bar_force (bars, map (chop(), ref_solution (Abr, piv)))
print_dict (bar_forces)

0:   -109.53144086499151
1:   -109.53144086499151
2:   -109.53144086499151
3:   -109.53144086499151
4:   -31.60013083802923
5:   -31.60013083802923
6:   -31.600130838029237
7:   -31.600130838029237
8:   0.0
9:   0.0
10:  0.0
11:  0.0

0:   -109.53144086499151
1:   -109.53144086499151
2:   -109.53144086499151
3:   -109.53144086499151
4:   -31.60013083802923
5:   -31.60013083802923
6:   -31.600130838029237
7:   -31.600130838029237
8:   0.0
9:   0.0
10:  0.0
11:  0.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces) . rotateZ (pi/9)

A = equilibrium_matrix (joints, bars, free_joints)
show (A.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrr} 0.4080 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 2.498 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 \\ 0.0000 & 0.4080 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & -0.4717 & 0.0000 & 0.0000 & -0.4717 \\ 0.0000 & -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5268 & 0.0000 & 0.0000 & -0.5268 \\ 0.0000 & 0.0000 & -0.4080 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 4.997 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -7.495 \times 10^{-17} & 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & -0.7071 & 0.7071 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.4080 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.4717 & 0.4717 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.9130 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5268 & -0.5268 & 0.0000 \end{array}\right)$

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 12 and dimension 12
   Basis:
     (0.4080046172220934, 0.0, -0.9129798641401973, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 2.4983077425918857e-17, 0.4080046172220934,
-0.9129798641401973, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4080046172220934, 4.9966154851837714e-17,
-0.9129798641401973, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -7.494923227775656e-17,
-0.4080046172220934, -0.9129798641401973)
     (-0.7071067811865476, 0.7071067811865476, 0.0, 0.7071067811865476,
-0.7071067811865476, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865476, -0.7071067811865475, 0.0,
0.7071067811865476, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865475, -0.7071067811865476,
0.0, -0.7071067811865475, 0.7071067811865476, 0.0)
     (-0.7071067811865475, -0.7071067811865474, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.7071067811865475, 0.7071067811865474, 0.0)
     (0.0, 0.0, 0.0, 0.7071067811865475, -0.4716858064432297,
-0.5267945519839778, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475,
0.4716858064432298, -0.526794551983978)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865476,
0.4716858064432297, -0.5267945519839778)
     (0.0, 0.0, 0.0, -0.7071067811865476, -0.47168580644322966,
-0.526794551983978, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

Column space:
   Vector space of degree 12 and dimension 12
   Basis:
     (0.4080046172220934, 0.0, -0.9129798641401973, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 2.4983077425918857e-17, 0.4080046172220934, -0.9129798641401973, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4080046172220934, 4.9966154851837714e-17, -0.9129798641401973, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -7.494923227775656e-17, -0.4080046172220934, -0.9129798641401973)
     (-0.7071067811865476, 0.7071067811865476, 0.0, 0.7071067811865476, -0.7071067811865476, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865476, -0.7071067811865475, 0.0, 0.7071067811865476, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865475, -0.7071067811865476, 0.0, -0.7071067811865475, 0.7071067811865476, 0.0)
     (-0.7071067811865475, -0.7071067811865474, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865475, 0.7071067811865474, 0.0)
     (0.0, 0.0, 0.0, 0.7071067811865475, -0.4716858064432297, -0.5267945519839778, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475, 0.4716858064432298, -0.526794551983978)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865476, 0.4716858064432297, -0.5267945519839778)
     (0.0, 0.0, 0.0, -0.7071067811865476, -0.47168580644322966, -0.526794551983978, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

print_subspace (rs)

Row space:
   Vector space of degree 12 and dimension 12
   Basis:
     (1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5770056631863276, -0.0,
-0.0, 0.5770056631863278)
     (0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5770056631863278,
0.5770056631863276, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 0.9999999999999998, -0.0, -0.0,
0.9999999999999998, -0.0, -0.0, 0.9999999999999998)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, 1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0,
0.9999999999999998)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.9999999999999997,
-0.9999999999999997)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0,
7.850462293418876e-17)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0)

Row space:
   Vector space of degree 12 and dimension 12
   Basis:
     (1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5770056631863276, -0.0, -0.0, 0.5770056631863278)
     (0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5770056631863278, 0.5770056631863276, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 0.9999999999999998, -0.0, -0.0, 0.9999999999999998, -0.0, -0.0, 0.9999999999999998)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, 1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, 0.9999999999999998)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.9999999999999997, -0.9999999999999997)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 7.850462293418876e-17)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0)

print_subspace (rk)

Kernel:
   Vector space of degree 12 and dimension 0

Kernel:
   Vector space of degree 12 and dimension 0

print_subspace (lk)

Left kernel:
   Vector space of degree 12 and dimension 0

Left kernel:
   Vector space of degree 12 and dimension 0

joints, bars, supports = schwedler_1 (10., [4.25, 7.75, 9.25, 10.], 8)

plot_truss3d (joints, bars, supports) . rotateZ (pi/42)

maxwells_rule (joints, supports, bars)

3 * 24 - 72  ==  0

3 * 24 - 72  ==  0

free_joints = others (supports, joints)
len (free_joints)

A = equilibrium_matrix (joints, bars, free_joints)
A

72 x 72 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

72 x 72 dense matrix over Real Double Field (use the '.str()' method to see the entries)

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 72 and dimension 72

Column space:
   Vector space of degree 72 and dimension 72

print_subspace (rs)

Row space:
   Vector space of degree 72 and dimension 72

Row space:
   Vector space of degree 72 and dimension 72

print_subspace (rk)

Kernel:
   Vector space of degree 72 and dimension 0

Kernel:
   Vector space of degree 72 and dimension 0

print_subspace (lk)

Left kernel:
   Vector space of degree 72 and dimension 0

Left kernel:
   Vector space of degree 72 and dimension 0

load_gen = [(8, 8, (0.0, 0.0, -70.0)), 
            (8, 16, (0.0, 0.0, -80.0)),
            (8, 24, (0.0, 0.0, -50.0))]
loads = make_loads (load_gen)

plot_truss3d (joints, bars, supports, loads, pull = False, load_scale = 0.0325) . rotateZ (pi/42)

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
Ab

72 x 73 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

72 x 73 dense matrix over Real Double Field (use the '.str()' method to see the entries)

Abr, piv = ref_wpp (Ab)

bar_forces = dict_bar_force (bars, ref_solution (Abr, piv))
print_dict (bar_forces)

0:   -204.91580904761287
1:   -204.91580904761233
2:   -204.915809047613
3:   -204.91580904761224
4:   -204.91580904761292
5:   -204.91580904761236
6:   -204.91580904761287
7:   -204.91580904761256
8:   74.3059609475446
9:   74.30596094754473
10:  74.30596094754475
11:  74.30596094754468
12:  74.30596094754466
13:  74.30596094754462
14:  74.30596094754459
15:  74.30596094754458
16:  -2.344791028008332e-13
17:  -4.123438810097339e-13
18:  -3.107228441586106e-13
19:  -1.6891368726487216e-13
20:  -1.8474111129762605e-13
21:  -3.789786770149045e-13
22:  -3.484466443801332e-13
23:  -2.723070371319736e-13
24:  -164.92256296788312
25:  -164.9225629678829
26:  -164.92256296788318
27:  -164.92256296788287
28:  -164.92256296788312
29:  -164.92256296788293
30:  -164.9225629678831
31:  -164.922562967883
32:  -22.84939741307861
33:  -22.84939741307855
34:  -22.84939741307855
35:  -22.849397413078606
36:  -22.849397413078613
37:  -22.849397413078623
38:  -22.849397413078638
39:  -22.849397413078655
40:  -7.64356351572866e-14
41:  -1.623481864654965e-13
42:  -8.511261034077713e-14
43:  -3.266854609446906e-14
44:  -2.8421709430403998e-14
45:  -1.2372766761210908e-13
46:  -1.0513433228822938e-13
47:  -8.518831645185593e-14
48:  -97.75311369041621
49:  -97.75311369041613
50:  -97.75311369041621
51:  -97.7531136904161
52:  -97.75311369041621
53:  -97.75311369041616
54:  -97.7531136904162
55:  -97.75311369041619
56:  -109.7487321707687
57:  -109.74873217076869
58:  -109.74873217076869
59:  -109.74873217076872
60:  -109.74873217076869
61:  -109.74873217076869
62:  -109.7487321707687
63:  -109.7487321707687
64:  -7.675729386064e-14
65:  -9.210875263276802e-14
66:  -4.605437631638405e-14
67:  -2.5962967559471267e-14
68:  1.5351458772128003e-14
69:  -9.210875263276803e-14
70:  -6.140583508851202e-14
71:  -5.0794326301168806e-14

0:   -204.91580904761287
1:   -204.91580904761233
2:   -204.915809047613
3:   -204.91580904761224
4:   -204.91580904761292
5:   -204.91580904761236
6:   -204.91580904761287
7:   -204.91580904761256
8:   74.3059609475446
9:   74.30596094754473
10:  74.30596094754475
11:  74.30596094754468
12:  74.30596094754466
13:  74.30596094754462
14:  74.30596094754459
15:  74.30596094754458
16:  -2.344791028008332e-13
17:  -4.123438810097339e-13
18:  -3.107228441586106e-13
19:  -1.6891368726487216e-13
20:  -1.8474111129762605e-13
21:  -3.789786770149045e-13
22:  -3.484466443801332e-13
23:  -2.723070371319736e-13
24:  -164.92256296788312
25:  -164.9225629678829
26:  -164.92256296788318
27:  -164.92256296788287
28:  -164.92256296788312
29:  -164.92256296788293
30:  -164.9225629678831
31:  -164.922562967883
32:  -22.84939741307861
33:  -22.84939741307855
34:  -22.84939741307855
35:  -22.849397413078606
36:  -22.849397413078613
37:  -22.849397413078623
38:  -22.849397413078638
39:  -22.849397413078655
40:  -7.64356351572866e-14
41:  -1.623481864654965e-13
42:  -8.511261034077713e-14
43:  -3.266854609446906e-14
44:  -2.8421709430403998e-14
45:  -1.2372766761210908e-13
46:  -1.0513433228822938e-13
47:  -8.518831645185593e-14
48:  -97.75311369041621
49:  -97.75311369041613
50:  -97.75311369041621
51:  -97.7531136904161
52:  -97.75311369041621
53:  -97.75311369041616
54:  -97.7531136904162
55:  -97.75311369041619
56:  -109.7487321707687
57:  -109.74873217076869
58:  -109.74873217076869
59:  -109.74873217076872
60:  -109.74873217076869
61:  -109.74873217076869
62:  -109.7487321707687
63:  -109.7487321707687
64:  -7.675729386064e-14
65:  -9.210875263276802e-14
66:  -4.605437631638405e-14
67:  -2.5962967559471267e-14
68:  1.5351458772128003e-14
69:  -9.210875263276803e-14
70:  -6.140583508851202e-14
71:  -5.0794326301168806e-14

bar_forces = dict_bar_force (bars, map (chop(), ref_solution (Abr, piv)))
print_dict (bar_forces)

0:   -204.91580904761287
1:   -204.91580904761233
2:   -204.915809047613
3:   -204.91580904761224
4:   -204.91580904761292
5:   -204.91580904761236
6:   -204.91580904761287
7:   -204.91580904761256
8:   74.3059609475446
9:   74.30596094754473
10:  74.30596094754475
11:  74.30596094754468
12:  74.30596094754466
13:  74.30596094754462
14:  74.30596094754459
15:  74.30596094754458
16:  0.0
17:  0.0
18:  0.0
19:  0.0
20:  0.0
21:  0.0
22:  0.0
23:  0.0
24:  -164.92256296788312
25:  -164.9225629678829
26:  -164.92256296788318
27:  -164.92256296788287
28:  -164.92256296788312
29:  -164.92256296788293
30:  -164.9225629678831
31:  -164.922562967883
32:  -22.84939741307861
33:  -22.84939741307855
34:  -22.84939741307855
35:  -22.849397413078606
36:  -22.849397413078613
37:  -22.849397413078623
38:  -22.849397413078638
39:  -22.849397413078655
40:  0.0
41:  0.0
42:  0.0
43:  0.0
44:  0.0
45:  0.0
46:  0.0
47:  0.0
48:  -97.75311369041621
49:  -97.75311369041613
50:  -97.75311369041621
51:  -97.7531136904161
52:  -97.75311369041621
53:  -97.75311369041616
54:  -97.7531136904162
55:  -97.75311369041619
56:  -109.7487321707687
57:  -109.74873217076869
58:  -109.74873217076869
59:  -109.74873217076872
60:  -109.74873217076869
61:  -109.74873217076869
62:  -109.7487321707687
63:  -109.7487321707687
64:  0.0
65:  0.0
66:  0.0
67:  0.0
68:  0.0
69:  0.0
70:  0.0
71:  0.0

0:   -204.91580904761287
1:   -204.91580904761233
2:   -204.915809047613
3:   -204.91580904761224
4:   -204.91580904761292
5:   -204.91580904761236
6:   -204.91580904761287
7:   -204.91580904761256
8:   74.3059609475446
9:   74.30596094754473
10:  74.30596094754475
11:  74.30596094754468
12:  74.30596094754466
13:  74.30596094754462
14:  74.30596094754459
15:  74.30596094754458
16:  0.0
17:  0.0
18:  0.0
19:  0.0
20:  0.0
21:  0.0
22:  0.0
23:  0.0
24:  -164.92256296788312
25:  -164.9225629678829
26:  -164.92256296788318
27:  -164.92256296788287
28:  -164.92256296788312
29:  -164.92256296788293
30:  -164.9225629678831
31:  -164.922562967883
32:  -22.84939741307861
33:  -22.84939741307855
34:  -22.84939741307855
35:  -22.849397413078606
36:  -22.849397413078613
37:  -22.849397413078623
38:  -22.849397413078638
39:  -22.849397413078655
40:  0.0
41:  0.0
42:  0.0
43:  0.0
44:  0.0
45:  0.0
46:  0.0
47:  0.0
48:  -97.75311369041621
49:  -97.75311369041613
50:  -97.75311369041621
51:  -97.7531136904161
52:  -97.75311369041621
53:  -97.75311369041616
54:  -97.7531136904162
55:  -97.75311369041619
56:  -109.7487321707687
57:  -109.74873217076869
58:  -109.74873217076869
59:  -109.74873217076872
60:  -109.74873217076869
61:  -109.74873217076869
62:  -109.7487321707687
63:  -109.7487321707687
64:  0.0
65:  0.0
66:  0.0
67:  0.0
68:  0.0
69:  0.0
70:  0.0
71:  0.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces) . rotateZ (pi/42)

joints, bars, supports = schwedler_2 (10., [2.725, 4.75, 5.75, 6.25], 9)

plot_truss3d (joints, bars, supports) . rotateZ (pi/42)

joints, bars, supports = schwedler_2 (10., [2.725, 4.75, 5.75, 6.25], 9, ccw = False)

plot_truss3d (joints, bars, supports) . rotateZ (pi/42)

maxwells_rule (joints, supports, bars)

3 * 27 - 81  ==  0

3 * 27 - 81  ==  0

free_joints = others (supports, joints)
len (free_joints)

A = equilibrium_matrix (joints, bars, free_joints)
A

81 x 81 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

81 x 81 dense matrix over Real Double Field (use the '.str()' method to see the entries)

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 81 and dimension 81

Column space:
   Vector space of degree 81 and dimension 81

print_subspace (rs)

Row space:
   Vector space of degree 81 and dimension 81

Row space:
   Vector space of degree 81 and dimension 81

print_subspace (rk)

Kernel:
   Vector space of degree 81 and dimension 0

Kernel:
   Vector space of degree 81 and dimension 0

print_subspace (lk)

Left kernel:
   Vector space of degree 81 and dimension 0

Left kernel:
   Vector space of degree 81 and dimension 0

loads = { 17 : (-25, -25, -50),  23 : (-25, 25, -50),  31 : (25, 15, -50), 34 : (25, -5, -50) }

plot_truss3d (joints, bars, supports, loads, pull = False, load_scale = 0.0425) . rotateZ (pi/42)

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
Ab

81 x 82 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

81 x 82 dense matrix over Real Double Field (use the '.str()' method to see the entries)

Abr, piv = ref_wpp (Ab)

bar_forces = dict_bar_force (bars, map (chop(), ref_solution (Abr, piv)))
print_dict (bar_forces)

0:   38.31709601066699
1:   -11.820854133014523
2:   148.71154384434323
3:   -478.0987061578426
4:   461.3094526157981
5:   -175.7980321365149
6:   -319.95626928659163
7:   371.00935110582776
8:   -305.6897740958585
9:   0.0
10:  -24.442817402623973
11:  209.25602071615936
12:  -370.21985726545563
13:  236.69341356679251
14:  122.76595440294201
15:  -270.02323811939914
16:  188.5775368128822
17:  -64.39599275802256
18:  150.6825660326815
19:  -20.163519154402916
20:  25.0025003629832
21:  -239.04999119562868
22:  536.0684402408183
23:  -424.63077733973887
24:  -1.8244508669559565
25:  375.85768989202825
26:  -340.13976812572355
27:  0.0
28:  0.0
29:  47.22459414143603
30:  -250.01773494571543
31:  181.27448470928414
32:  -135.12078475542737
33:  -166.6193036116831
34:  170.65837961037107
35:  -107.88696717718597
36:  0.0
37:  0.0
38:  80.68695925095594
39:  -198.60405856306593
40:  91.4545127516499
41:  55.15322303693489
42:  -128.4435257346175
43:  163.09957429928346
44:  0.0
45:  66.01561809617336
46:  0.0
47:  0.0
48:  -81.858504102034
49:  283.34622025206005
50:  -170.95080753882408
51:  -37.44470182110243
52:  186.26250576245593
53:  -211.48181761235386
54:  0.0
55:  0.0
56:  0.0
57:  -134.21901968049764
58:  -6.050451908071864
59:  0.0
60:  -91.74483208883017
61:  -16.29797221743867
62:  0.0
63:  0.0
64:  0.0
65:  0.0
66:  -205.64145509232745
67:  0.0
68:  0.0
69:  -140.5653298083926
70:  0.0
71:  0.0
72:  0.0
73:  0.0
74:  0.0
75:  0.0
76:  207.39181591315503
77:  -183.0874341335986
78:  0.0
79:  141.76178139911323
80:  -167.2532263542124

0:   38.31709601066699
1:   -11.820854133014523
2:   148.71154384434323
3:   -478.0987061578426
4:   461.3094526157981
5:   -175.7980321365149
6:   -319.95626928659163
7:   371.00935110582776
8:   -305.6897740958585
9:   0.0
10:  -24.442817402623973
11:  209.25602071615936
12:  -370.21985726545563
13:  236.69341356679251
14:  122.76595440294201
15:  -270.02323811939914
16:  188.5775368128822
17:  -64.39599275802256
18:  150.6825660326815
19:  -20.163519154402916
20:  25.0025003629832
21:  -239.04999119562868
22:  536.0684402408183
23:  -424.63077733973887
24:  -1.8244508669559565
25:  375.85768989202825
26:  -340.13976812572355
27:  0.0
28:  0.0
29:  47.22459414143603
30:  -250.01773494571543
31:  181.27448470928414
32:  -135.12078475542737
33:  -166.6193036116831
34:  170.65837961037107
35:  -107.88696717718597
36:  0.0
37:  0.0
38:  80.68695925095594
39:  -198.60405856306593
40:  91.4545127516499
41:  55.15322303693489
42:  -128.4435257346175
43:  163.09957429928346
44:  0.0
45:  66.01561809617336
46:  0.0
47:  0.0
48:  -81.858504102034
49:  283.34622025206005
50:  -170.95080753882408
51:  -37.44470182110243
52:  186.26250576245593
53:  -211.48181761235386
54:  0.0
55:  0.0
56:  0.0
57:  -134.21901968049764
58:  -6.050451908071864
59:  0.0
60:  -91.74483208883017
61:  -16.29797221743867
62:  0.0
63:  0.0
64:  0.0
65:  0.0
66:  -205.64145509232745
67:  0.0
68:  0.0
69:  -140.5653298083926
70:  0.0
71:  0.0
72:  0.0
73:  0.0
74:  0.0
75:  0.0
76:  207.39181591315503
77:  -183.0874341335986
78:  0.0
79:  141.76178139911323
80:  -167.2532263542124

plot_truss_with_forces3d (joints, bars, supports, bar_forces) . rotateZ (pi/42)

joints, bars, supports = schwedler_kind (10., [7.5, 10.], 4)

print 'joints:'
print_dict (joints, indent = '   ')
print
print 'bars:' 
print_dict (bars, indent = '   ')
print
print 'supports:', supports

joints:
   0:   (10.0000000000000, 0.000000000000000, 0.000000000000000)
   1:   (6.12323399573676e-16, 10.0000000000000, 0.000000000000000)
   2:   (-10.0000000000000, 1.22464679914735e-15, 0.000000000000000)
   3:   (-1.83697019872103e-15, -10.0000000000000, 0.000000000000000)
   4:   (6.61437827766148, 0.000000000000000, 7.50000000000000)
   5:   (4.05013859304395e-16, 6.61437827766148, 7.50000000000000)
   6:   (-6.61437827766148, 8.10027718608790e-16, 7.50000000000000)
   7:   (-1.21504157791318e-15, -6.61437827766148, 7.50000000000000)

bars:
   0:   (0, 4)
   1:   (1, 5)
   2:   (2, 6)
   3:   (3, 7)
   4:   (4, 5)
   5:   (5, 6)
   6:   (6, 7)
   7:   (7, 4)

supports: [0, 1, 2, 3]

joints:
   0:   (10.0000000000000, 0.000000000000000, 0.000000000000000)
   1:   (6.12323399573676e-16, 10.0000000000000, 0.000000000000000)
   2:   (-10.0000000000000, 1.22464679914735e-15, 0.000000000000000)
   3:   (-1.83697019872103e-15, -10.0000000000000, 0.000000000000000)
   4:   (6.61437827766148, 0.000000000000000, 7.50000000000000)
   5:   (4.05013859304395e-16, 6.61437827766148, 7.50000000000000)
   6:   (-6.61437827766148, 8.10027718608790e-16, 7.50000000000000)
   7:   (-1.21504157791318e-15, -6.61437827766148, 7.50000000000000)

bars:
   0:   (0, 4)
   1:   (1, 5)
   2:   (2, 6)
   3:   (3, 7)
   4:   (4, 5)
   5:   (5, 6)
   6:   (6, 7)
   7:   (7, 4)

supports: [0, 1, 2, 3]

plot_truss3d (joints, bars, supports) . rotateZ (pi/15)

maxwells_rule (joints, supports, bars)

3 * 4 - 8  ==  4  !=  0

3 * 4 - 8  ==  4  !=  0

free_joints = others (supports, joints)
free_joints

[4, 5, 6, 7]

[4, 5, 6, 7]

A = equilibrium_matrix (joints, bars, free_joints)
show (A.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrr} 0.4114 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & -0.7071 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 \\ -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 2.519 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 \\ 0.0000 & 0.4114 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 \\ 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 \\ 0.0000 & 0.0000 & 5.039 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 \\ 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -7.558 \times 10^{-17} & 0.0000 & 0.0000 & -0.7071 & 0.7071 \\ 0.0000 & 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 \\ 0.0000 & 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \end{array}\right)$

A.nrows(), A.ncols()

(12, 8)

(12, 8)

print_dict (dict_equation_index (free_joints))

print_dict (dict_unknown_index (bars))

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 12 and dimension 8
   Basis:
     (0.4114378277661477, 0.0, -0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 2.5193300941097603e-17, 0.4114378277661477,
-0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4114378277661477, 5.0386601882195206e-17,
-0.9114378277661477, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -7.557990282329279e-17,
-0.4114378277661477, -0.9114378277661477)
     (-0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475,
-0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865476, -0.7071067811865475, 0.0,
0.7071067811865476, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865474, -0.7071067811865476,
0.0, -0.7071067811865474, 0.7071067811865476, 0.0)
     (-0.7071067811865476, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.7071067811865476, 0.7071067811865475, 0.0)

Column space:
   Vector space of degree 12 and dimension 8
   Basis:
     (0.4114378277661477, 0.0, -0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 2.5193300941097603e-17, 0.4114378277661477, -0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4114378277661477, 5.0386601882195206e-17, -0.9114378277661477, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -7.557990282329279e-17, -0.4114378277661477, -0.9114378277661477)
     (-0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865476, -0.7071067811865475, 0.0, 0.7071067811865476, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865474, -0.7071067811865476, 0.0, -0.7071067811865474, 0.7071067811865476, 0.0)
     (-0.7071067811865476, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865476, 0.7071067811865475, 0.0)

print_list (map (lambda x : x.apply_map (chop()), cs['basis']))

(0.4114378277661477, 0.0, -0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 0.4114378277661477, -0.9114378277661477, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4114378277661477, 0.0, -0.9114378277661477,
0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4114378277661477,
-0.9114378277661477)
(-0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475,
-0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, -0.7071067811865476, -0.7071067811865475, 0.0,
0.7071067811865476, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865474, -0.7071067811865476, 0.0,
-0.7071067811865474, 0.7071067811865476, 0.0)
(-0.7071067811865476, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.7071067811865476, 0.7071067811865475, 0.0)

(0.4114378277661477, 0.0, -0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 0.4114378277661477, -0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4114378277661477, 0.0, -0.9114378277661477, 0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4114378277661477, -0.9114378277661477)
(-0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, -0.7071067811865476, -0.7071067811865475, 0.0, 0.7071067811865476, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865474, -0.7071067811865476, 0.0, -0.7071067811865474, 0.7071067811865476, 0.0)
(-0.7071067811865476, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865476, 0.7071067811865475, 0.0)

print_subspace (rs)

Row space:
   Vector space of degree 8 and dimension 8
   Basis:
     (1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 1.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0)

Row space:
   Vector space of degree 8 and dimension 8
   Basis:
     (1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 1.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0)

print_subspace (rk)

Kernel:
   Vector space of degree 8 and dimension 0

Kernel:
   Vector space of degree 8 and dimension 0

print_subspace (lk)

Left kernel:
   Vector space of degree 12 and dimension 4
   Basis:
     (0.0, 0.0, 0.0, 1.0, 1.0, 0.45141622964513645, 1.0, 0.9999999999999997,
-0.45141622964513645, 0.0, 0.0, 0.0)
     (1.0, 0.0, 0.4514162296451365, 1.6653345369377348e-16, -0.9999999999999998,
-0.4514162296451364, 0.0, -0.9999999999999997, -5.5282544071808195e-17, 1.0,
0.0, -8.292381610771231e-17)
     (0.9999999999999998, 0.0, 0.4514162296451364, 0.9999999999999999,
1.1102230246251565e-16, 5.551115123125783e-17, 0.0, 1.0, 5.5282544071808213e-17,
0.0, 1.0, -0.4514162296451365)
     (-0.9999999999999998, 1.0, -0.4514162296451364, -0.9999999999999999,
0.9999999999999999, 0.45141622964513645, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

Left kernel:
   Vector space of degree 12 and dimension 4
   Basis:
     (0.0, 0.0, 0.0, 1.0, 1.0, 0.45141622964513645, 1.0, 0.9999999999999997, -0.45141622964513645, 0.0, 0.0, 0.0)
     (1.0, 0.0, 0.4514162296451365, 1.6653345369377348e-16, -0.9999999999999998, -0.4514162296451364, 0.0, -0.9999999999999997, -5.5282544071808195e-17, 1.0, 0.0, -8.292381610771231e-17)
     (0.9999999999999998, 0.0, 0.4514162296451364, 0.9999999999999999, 1.1102230246251565e-16, 5.551115123125783e-17, 0.0, 1.0, 5.5282544071808213e-17, 0.0, 1.0, -0.4514162296451365)
     (-0.9999999999999998, 1.0, -0.4514162296451364, -0.9999999999999999, 0.9999999999999999, 0.45141622964513645, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

print_list (map (lambda x : x.apply_map (chop()), lk['basis']))

(0.0, 0.0, 0.0, 1.0, 1.0, 0.45141622964513645, 1.0, 0.9999999999999997,
-0.45141622964513645, 0.0, 0.0, 0.0)
(1.0, 0.0, 0.4514162296451365, 0.0, -0.9999999999999998, -0.4514162296451364,
0.0, -0.9999999999999997, 0.0, 1.0, 0.0, 0.0)
(0.9999999999999998, 0.0, 0.4514162296451364, 0.9999999999999999, 0.0, 0.0, 0.0,
1.0, 0.0, 0.0, 1.0, -0.4514162296451365)
(-0.9999999999999998, 1.0, -0.4514162296451364, -0.9999999999999999,
0.9999999999999999, 0.45141622964513645, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

(0.0, 0.0, 0.0, 1.0, 1.0, 0.45141622964513645, 1.0, 0.9999999999999997, -0.45141622964513645, 0.0, 0.0, 0.0)
(1.0, 0.0, 0.4514162296451365, 0.0, -0.9999999999999998, -0.4514162296451364, 0.0, -0.9999999999999997, 0.0, 1.0, 0.0, 0.0)
(0.9999999999999998, 0.0, 0.4514162296451364, 0.9999999999999999, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, -0.4514162296451365)
(-0.9999999999999998, 1.0, -0.4514162296451364, -0.9999999999999999, 0.9999999999999999, 0.45141622964513645, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][0], displ_scale = 1.5) . rotateZ (pi/12)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][1], displ_scale = 1.5) . rotateZ (pi/9)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][2], displ_scale = 1.5) . rotateZ (pi/9)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][3], displ_scale = 1.5) . rotateZ (pi/9)

plot_truss_with_displacements3d (joints, bars, supports, sum (lk['basis']), displ_scale = 1.5) . rotateZ (pi/9)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][1] + lk['basis'][3], displ_scale = 2.5) . rotateZ (pi/9)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][1] + lk['basis'][2], displ_scale = 2.5) . rotateZ (pi/9)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][0] + lk['basis'][3], displ_scale = 2.5) . rotateZ (pi/9)

loads = { 4: (0.0, 0.0, -100.0), 5: (0.0, 0.0, -100.0), 6: (0.0, 0.0, -100.0), 7: (0.0, 0.0, -100.0),  }
print_dict (loads)

4:   (0.000000000000000, 0.000000000000000, -100.000000000000)
5:   (0.000000000000000, 0.000000000000000, -100.000000000000)
6:   (0.000000000000000, 0.000000000000000, -100.000000000000)
7:   (0.000000000000000, 0.000000000000000, -100.000000000000)

4:   (0.000000000000000, 0.000000000000000, -100.000000000000)
5:   (0.000000000000000, 0.000000000000000, -100.000000000000)
6:   (0.000000000000000, 0.000000000000000, -100.000000000000)
7:   (0.000000000000000, 0.000000000000000, -100.000000000000)

plot_truss3d (joints, bars, supports, loads, pull = False, load_scale = 0.0325)

b = load_vector (loads, free_joints)
b

(0.0, 0.0, -100.0, 0.0, 0.0, -100.0, 0.0, 0.0, -100.0, 0.0, 0.0, -100.0)

(0.0, 0.0, -100.0, 0.0, 0.0, -100.0, 0.0, 0.0, -100.0, 0.0, 0.0, -100.0)

is_in_subspace (b, cs['basis'])

True

True

has_component_in_subspace (b, lk['basis'])

False

False

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
show (Ab.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrr|r} 0.4114 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & -0.7071 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & -0.0000 \\ -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 100.0 \\ 0.0000 & 2.519 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 0.4114 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 100.0 \\ 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & 5.039 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 100.0 \\ 0.0000 & 0.0000 & 0.0000 & -7.558 \times 10^{-17} & 0.0000 & 0.0000 & -0.7071 & 0.7071 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 100.0 \end{array}\right)$

Abr, piv = ref_wpp (Ab)
show (Abr.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrr|r} 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -109.7 \\ 0.0000 & 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -109.7 \\ 0.0000 & 0.0000 & 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -109.7 \\ 0.0000 & 0.0000 & 0.0000 & 1.000 & -0.0000 & -0.0000 & -0.0000 & -0.0000 & -109.7 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 1.000 & -0.0000 & -0.0000 & -63.84 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -0.0000 & -0.0000 & -31.92 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -0.0000 & -31.92 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -31.92 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -7.105 \times 10^{-15} \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 3.553 \times 10^{-15} \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 3.553 \times 10^{-15} \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 3.553 \times 10^{-15} \end{array}\right)$

show (Abr.apply_map (NDC(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrr|r} 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -109.7 \\ 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -109.7 \\ 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -109.7 \\ 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -109.7 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 1.000 & 0.0000 & 0.0000 & -63.84 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & -31.92 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & -31.92 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & -31.92 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \end{array}\right)$

bar_forces = dict_bar_force (bars, ref_solution (Abr, piv))
print_dict (bar_forces)

0:   -109.7167540709727
1:   -109.7167540709727
2:   -109.7167540709727
3:   -109.7167540709727
4:   -31.91994771197398
5:   -31.91994771197398
6:   -31.919947711973986
7:   -31.919947711973975

0:   -109.7167540709727
1:   -109.7167540709727
2:   -109.7167540709727
3:   -109.7167540709727
4:   -31.91994771197398
5:   -31.91994771197398
6:   -31.919947711973986
7:   -31.919947711973975

plot_truss_with_forces3d (joints, bars, supports, bar_forces)

loads2 = { 4: (100.0, 0.0, 0.0), 5: (0.0, 100.0, 0.0), 6: (-100.0, 0.0, 0.0), 7: (0.0, -100.0, 0.0),  }
print_dict (loads2)

4:   (100.000000000000, 0.000000000000000, 0.000000000000000)
5:   (0.000000000000000, 100.000000000000, 0.000000000000000)
6:   (-100.000000000000, 0.000000000000000, 0.000000000000000)
7:   (0.000000000000000, -100.000000000000, 0.000000000000000)

4:   (100.000000000000, 0.000000000000000, 0.000000000000000)
5:   (0.000000000000000, 100.000000000000, 0.000000000000000)
6:   (-100.000000000000, 0.000000000000000, 0.000000000000000)
7:   (0.000000000000000, -100.000000000000, 0.000000000000000)

plot_truss3d (joints, bars, supports, loads2, load_scale = 0.0325)

b2 = load_vector (loads2, free_joints)
b2

(100.0, 0.0, 0.0, 0.0, 100.0, 0.0, -100.0, 0.0, 0.0, 0.0, -100.0, 0.0)

(100.0, 0.0, 0.0, 0.0, 100.0, 0.0, -100.0, 0.0, 0.0, 0.0, -100.0, 0.0)

is_in_subspace (b2, cs['basis'])

True

True

Ab2 = augmented_equil_matrix (joints, bars, loads2, free_joints)
show (Ab2.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrr|r} 0.4114 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & -0.7071 & -100.0 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & -0.0000 \\ -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 2.519 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 0.4114 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & -100.0 \\ 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 100.0 \\ 0.0000 & 0.0000 & 5.039 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -7.558 \times 10^{-17} & 0.0000 & 0.0000 & -0.7071 & 0.7071 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 100.0 \\ 0.0000 & 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \end{array}\right)$

Abr2, piv2 = ref_wpp (Ab2)
show (Abr2.apply_map (NDC(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrr|r} 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 1.000 & 0.0000 & 0.0000 & 141.4 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 70.71 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 70.71 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 70.71 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \end{array}\right)$

bar_forces2 = dict_bar_force (bars, ref_solution (Abr2, piv2))
print_dict (bar_forces2)

0:   0.0
1:   0.0
2:   0.0
3:   0.0
4:   70.71067811865476
5:   70.71067811865476
6:   70.71067811865474
7:   70.71067811865474

0:   0.0
1:   0.0
2:   0.0
3:   0.0
4:   70.71067811865476
5:   70.71067811865476
6:   70.71067811865474
7:   70.71067811865474

plot_truss_with_forces3d (joints, bars, supports, bar_forces2)

loads3 = { 4: (100.0, 0.0, 0.0), 5: (-100.0, 0.0, 0.0), 6: (100.0, 0.0, 0.0), 7: (-100.0, 0.0, 0.0),  }
print_dict (loads3)

4:   (100.000000000000, 0.000000000000000, 0.000000000000000)
5:   (-100.000000000000, 0.000000000000000, 0.000000000000000)
6:   (100.000000000000, 0.000000000000000, 0.000000000000000)
7:   (-100.000000000000, 0.000000000000000, 0.000000000000000)

4:   (100.000000000000, 0.000000000000000, 0.000000000000000)
5:   (-100.000000000000, 0.000000000000000, 0.000000000000000)
6:   (100.000000000000, 0.000000000000000, 0.000000000000000)
7:   (-100.000000000000, 0.000000000000000, 0.000000000000000)

plot_truss3d (joints, bars, supports, loads3, pull = False, load_scale = 0.0325)

b3 = load_vector (loads3, free_joints)
b3

(100.0, 0.0, 0.0, -100.0, 0.0, 0.0, 100.0, 0.0, 0.0, -100.0, 0.0, 0.0)

(100.0, 0.0, 0.0, -100.0, 0.0, 0.0, 100.0, 0.0, 0.0, -100.0, 0.0, 0.0)

is_in_subspace (b3, cs['basis'])

True

True

Ab3 = augmented_equil_matrix (joints, bars, loads3, free_joints)
show (Ab3.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrr|r} 0.4114 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & -0.7071 & -100.0 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & -0.0000 \\ -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 2.519 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 100.0 \\ 0.0000 & 0.4114 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & -100.0 \\ 0.0000 & 0.0000 & 5.039 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -7.558 \times 10^{-17} & 0.0000 & 0.0000 & -0.7071 & 0.7071 & 100.0 \\ 0.0000 & 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & -0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.0000 \end{array}\right)$

Abr3, piv3 = ref_wpp (Ab3, is_zero_f = is_zz_tol())
show (Abr3.apply_map (NDC(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrr|r} 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 1.000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 & -70.71 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & -70.71 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 & 70.71 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \end{array}\right)$

bar_forces3 = dict_bar_force (bars, ref_solution (Abr3, piv3))
print_dict (bar_forces3)

0:   0.0
1:   0.0
2:   0.0
3:   0.0
4:   70.71067811865476
5:   -70.71067811865476
6:   -70.71067811865474
7:   70.71067811865474

0:   0.0
1:   0.0
2:   0.0
3:   0.0
4:   70.71067811865476
5:   -70.71067811865476
6:   -70.71067811865474
7:   70.71067811865474

plot_truss_with_forces3d (joints, bars, supports, bar_forces3)

joints, bars, supports = schwedler_kind (10., [2.725, 4.75, 5.75, 6.25], 9)

plot_truss3d (joints, bars, supports)

maxwells_rule (joints, supports, bars)

3 * 27 - 54  ==  27  !=  0

3 * 27 - 54  ==  27  !=  0

free_joints = others (supports, joints)
len (free_joints)

A = equilibrium_matrix (joints, bars, free_joints)
A

81 x 54 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

81 x 54 dense matrix over Real Double Field (use the '.str()' method to see the entries)

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 81 and dimension 54

Column space:
   Vector space of degree 81 and dimension 54

print_subspace (rs)

Row space:
   Vector space of degree 54 and dimension 54

Row space:
   Vector space of degree 54 and dimension 54

print_subspace (rk)

Kernel:
   Vector space of degree 54 and dimension 0

Kernel:
   Vector space of degree 54 and dimension 0

print_subspace (lk)

Left kernel:
   Vector space of degree 81 and dimension 27

Left kernel:
   Vector space of degree 81 and dimension 27

load_gen = [(9, 9, (0.0, 0.0, -70.0)), 
            (9, 18, (0.0, 0.0, -80.0)),
            (9, 27, (0.0, 0.0, -50.0))]
loads = make_loads (load_gen)

plot_truss3d (joints, bars, supports, loads, pull = False, load_scale = 0.0325)

b = load_vector (loads, free_joints)

is_in_subspace (b, cs['basis'])

True

True

has_component_in_subspace (b, lk['basis'])

False

False

Ab = augmented_equil_matrix (joints, bars, loads, free_joints)
Ab

81 x 55 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

81 x 55 dense matrix over Real Double Field (use the '.str()' method to see the entries)

Abr, piv = ref_wpp (Ab)
len (piv)

bar_forces = dict_bar_force (bars, ref_solution (Abr, piv))
print_dict (bar_forces)

0:   -242.79669133959382
1:   -242.79669133959385
2:   -242.79669133959368
3:   -242.79669133959396
4:   -242.79669133959396
5:   -242.79669133959396
6:   -242.79669133959376
7:   -242.79669133959396
8:   -242.79669133959396
9:   37.64244088603992
10:  37.642440886039864
11:  37.642440886039886
12:  37.64244088603995
13:  37.64244088603996
14:  37.642440886039935
15:  37.64244088603992
16:  37.64244088603993
17:  37.642440886039935
18:  -208.811957197297
19:  -208.811957197297
20:  -208.81195719729686
21:  -208.81195719729703
22:  -208.81195719729703
23:  -208.81195719729703
24:  -208.81195719729692
25:  -208.8119571972971
26:  -208.8119571972971
27:  -72.13931385770061
28:  -72.13931385770064
29:  -72.13931385770064
30:  -72.13931385770069
31:  -72.13931385770069
32:  -72.13931385770064
33:  -72.13931385770061
34:  -72.13931385770059
35:  -72.13931385770063
36:  -124.54025738583246
37:  -124.5402573858324
38:  -124.54025738583235
39:  -124.54025738583246
40:  -124.54025738583243
41:  -124.54025738583243
42:  -124.54025738583243
43:  -124.54025738583248
44:  -124.54025738583248
45:  -166.74835942081702
46:  -166.74835942081702
47:  -166.748359420817
48:  -166.74835942081702
49:  -166.74835942081702
50:  -166.748359420817
51:  -166.74835942081702
52:  -166.74835942081705
53:  -166.74835942081705

0:   -242.79669133959382
1:   -242.79669133959385
2:   -242.79669133959368
3:   -242.79669133959396
4:   -242.79669133959396
5:   -242.79669133959396
6:   -242.79669133959376
7:   -242.79669133959396
8:   -242.79669133959396
9:   37.64244088603992
10:  37.642440886039864
11:  37.642440886039886
12:  37.64244088603995
13:  37.64244088603996
14:  37.642440886039935
15:  37.64244088603992
16:  37.64244088603993
17:  37.642440886039935
18:  -208.811957197297
19:  -208.811957197297
20:  -208.81195719729686
21:  -208.81195719729703
22:  -208.81195719729703
23:  -208.81195719729703
24:  -208.81195719729692
25:  -208.8119571972971
26:  -208.8119571972971
27:  -72.13931385770061
28:  -72.13931385770064
29:  -72.13931385770064
30:  -72.13931385770069
31:  -72.13931385770069
32:  -72.13931385770064
33:  -72.13931385770061
34:  -72.13931385770059
35:  -72.13931385770063
36:  -124.54025738583246
37:  -124.5402573858324
38:  -124.54025738583235
39:  -124.54025738583246
40:  -124.54025738583243
41:  -124.54025738583243
42:  -124.54025738583243
43:  -124.54025738583248
44:  -124.54025738583248
45:  -166.74835942081702
46:  -166.74835942081702
47:  -166.748359420817
48:  -166.74835942081702
49:  -166.74835942081702
50:  -166.748359420817
51:  -166.74835942081702
52:  -166.74835942081705
53:  -166.74835942081705

plot_truss_with_forces3d (joints, bars, supports, bar_forces)

joints, bars, supports = schwedler_sind (10., [7.5, 10.], 4)

print 'joints:'
print_dict (joints, indent = '   ')
print
print 'bars:' 
print_dict (bars, indent = '   ')
print
print 'supports:', supports

joints:
   0:   (10.0000000000000, 0.000000000000000, 0.000000000000000)
   1:   (6.12323399573676e-16, 10.0000000000000, 0.000000000000000)
   2:   (-10.0000000000000, 1.22464679914735e-15, 0.000000000000000)
   3:   (-1.83697019872103e-15, -10.0000000000000, 0.000000000000000)
   4:   (6.61437827766148, 0.000000000000000, 7.50000000000000)
   5:   (4.05013859304395e-16, 6.61437827766148, 7.50000000000000)
   6:   (-6.61437827766148, 8.10027718608790e-16, 7.50000000000000)
   7:   (-1.21504157791318e-15, -6.61437827766148, 7.50000000000000)

bars:
   0:   (0, 4)
   1:   (1, 5)
   2:   (2, 6)
   3:   (3, 7)
   4:   (4, 5)
   5:   (5, 6)
   6:   (6, 7)
   7:   (7, 4)
   8:   (0, 5)
   9:   (1, 6)
   10:  (2, 7)
   11:  (3, 4)
   12:  (0, 7)
   13:  (1, 4)
   14:  (2, 5)
   15:  (3, 6)

supports: [0, 1, 2, 3]

joints:
   0:   (10.0000000000000, 0.000000000000000, 0.000000000000000)
   1:   (6.12323399573676e-16, 10.0000000000000, 0.000000000000000)
   2:   (-10.0000000000000, 1.22464679914735e-15, 0.000000000000000)
   3:   (-1.83697019872103e-15, -10.0000000000000, 0.000000000000000)
   4:   (6.61437827766148, 0.000000000000000, 7.50000000000000)
   5:   (4.05013859304395e-16, 6.61437827766148, 7.50000000000000)
   6:   (-6.61437827766148, 8.10027718608790e-16, 7.50000000000000)
   7:   (-1.21504157791318e-15, -6.61437827766148, 7.50000000000000)

bars:
   0:   (0, 4)
   1:   (1, 5)
   2:   (2, 6)
   3:   (3, 7)
   4:   (4, 5)
   5:   (5, 6)
   6:   (6, 7)
   7:   (7, 4)
   8:   (0, 5)
   9:   (1, 6)
   10:  (2, 7)
   11:  (3, 4)
   12:  (0, 7)
   13:  (1, 4)
   14:  (2, 5)
   15:  (3, 6)

supports: [0, 1, 2, 3]

plot_truss3d (joints, bars, supports)

maxwells_rule (joints, supports, bars)

3 * 4 - 16  ==  -4  !=  0

3 * 4 - 16  ==  -4  !=  0

free_joints = others (supports, joints)
free_joints

[4, 5, 6, 7]

[4, 5, 6, 7]

A = equilibrium_matrix (joints, bars, free_joints)
show (A.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrrrrrr} 0.4114 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & -0.4677 & 0.0000 & -0.4677 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.7071 & 0.0000 & 0.0000 \\ -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5303 & 0.0000 & -0.5303 & 0.0000 & 0.0000 \\ 0.0000 & 2.519 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.0000 \\ 0.0000 & 0.4114 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & -0.4677 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.4677 & 0.0000 \\ 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5303 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5303 & 0.0000 \\ 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.0000 & 0.4677 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.4677 \\ 0.0000 & 0.0000 & 5.039 \times 10^{-17} & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 \\ 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5303 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5303 \\ 0.0000 & 0.0000 & 0.0000 & -7.558 \times 10^{-17} & 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & 0.0000 & -0.7071 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.4114 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.0000 & 0.4677 & 0.0000 & 0.4677 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & -0.9114 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.5303 & 0.0000 & -0.5303 & 0.0000 & 0.0000 & 0.0000 \end{array}\right)$

print_dict (dict_equation_index (free_joints))

Ar, pivots = ref_wpp (A)
show (Ar)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrrrrrr} 1.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & 0.5818609561002115 & -0.0 & 0.5818609561002115 & -0.0 & -0.0 \\ 0.0 & 1.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & 0.5818609561002115 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & 0.5818609561002115 & -0.0 \\ 0.0 & 0.0 & 1.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & 0.5818609561002115 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & 0.5818609561002116 \\ 0.0 & 0.0 & 0.0 & 1.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & 0.5818609561002116 & -0.0 & 0.5818609561002115 & -0.0 & -0.0 & -0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 1.0 & -0.0 & -0.0 & 1.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & 1.0 & -0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & -0.0 & 1.0 & -0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & -0.0 & -0.0 & -0.9999999999999998 & -0.0 & -0.0 & -0.0 & -0.0 & 0.9999999999999998 & 1.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & -0.9999999999999998 & -0.0 & -0.0 & 1.0 & -0.0 & 0.9999999999999998 & -0.0 & -0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & -0.0 & -0.0 & -0.0 & -0.0 & -1.0 & -0.0 & -0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & 0.0 & 0.0 & 0.0 & 0.0 & -1.0 & 7.850462293418876 \times 10^{-17} \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & -0.9999999999999997 & 0.9999999999999997 & 0.0 & 0.0 & -1.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 1.0 & -1.0 & -0.0 & -0.0 & 7.850462293418876 \times 10^{-17} \end{array}\right)$

rs, cs, rk, lk, pivots = subspaces (A, is_zero_f = is_zz_tol(), return_pivots = True)

redundant_bars = others (pivots, bars)
redundant_bars

[12, 13, 14, 15]

[12, 13, 14, 15]

print_subspace (cs)

Column space:
   Vector space of degree 12 and dimension 12
   Basis:
     (0.4114378277661477, 0.0, -0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 2.5193300941097603e-17, 0.4114378277661477,
-0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4114378277661477, 5.0386601882195206e-17,
-0.9114378277661477, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -7.557990282329279e-17,
-0.4114378277661477, -0.9114378277661477)
     (-0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475,
-0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865476, -0.7071067811865475, 0.0,
0.7071067811865476, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865474, -0.7071067811865476,
0.0, -0.7071067811865474, 0.7071067811865476, 0.0)
     (-0.7071067811865476, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.7071067811865476, 0.7071067811865475, 0.0)
     (0.0, 0.0, 0.0, 0.7071067811865475, -0.46770717334674267,
-0.5303300858899106, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.4677071733467427, 0.7071067811865475,
-0.5303300858899106, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475,
0.4677071733467428, -0.5303300858899107)
     (-0.4677071733467428, -0.7071067811865475, -0.5303300858899106, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

Column space:
   Vector space of degree 12 and dimension 12
   Basis:
     (0.4114378277661477, 0.0, -0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 2.5193300941097603e-17, 0.4114378277661477, -0.9114378277661477, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.4114378277661477, 5.0386601882195206e-17, -0.9114378277661477, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -7.557990282329279e-17, -0.4114378277661477, -0.9114378277661477)
     (-0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865476, -0.7071067811865475, 0.0, 0.7071067811865476, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865474, -0.7071067811865476, 0.0, -0.7071067811865474, 0.7071067811865476, 0.0)
     (-0.7071067811865476, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865476, 0.7071067811865475, 0.0)
     (0.0, 0.0, 0.0, 0.7071067811865475, -0.46770717334674267, -0.5303300858899106, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.4677071733467427, 0.7071067811865475, -0.5303300858899106, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475, 0.4677071733467428, -0.5303300858899107)
     (-0.4677071733467428, -0.7071067811865475, -0.5303300858899106, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)

print_subspace (rs)

Row space:
   Vector space of degree 16 and dimension 12
   Basis:
     (1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0,
0.5818609561002115, -0.0, 0.5818609561002115, -0.0, -0.0)
     (0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5818609561002115, -0.0,
-0.0, -0.0, -0.0, -0.0, 0.5818609561002115, -0.0)
     (0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5818609561002115,
-0.0, -0.0, -0.0, -0.0, -0.0, 0.5818609561002116)
     (0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0,
0.5818609561002116, -0.0, 0.5818609561002115, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 1.0, -0.0, -0.0, 1.0, -0.0, -0.0, -0.0, -0.0,
-0.0, 1.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0,
-0.0, 1.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.9999999999999998, -0.0,
-0.0, -0.0, -0.0, 0.9999999999999998, 1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.9999999999999998, -0.0, -0.0,
1.0, -0.0, 0.9999999999999998, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -1.0,
-0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0,
-1.0, 7.850462293418876e-17)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0,
-0.9999999999999997, 0.9999999999999997, 0.0, 0.0, -1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, -0.0,
-0.0, 7.850462293418876e-17)

Row space:
   Vector space of degree 16 and dimension 12
   Basis:
     (1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5818609561002115, -0.0, 0.5818609561002115, -0.0, -0.0)
     (0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5818609561002115, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5818609561002115, -0.0)
     (0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5818609561002115, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5818609561002116)
     (0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 0.5818609561002116, -0.0, 0.5818609561002115, -0.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 1.0, -0.0, -0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, 1.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 1.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.9999999999999998, -0.0, -0.0, -0.0, -0.0, 0.9999999999999998, 1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.9999999999999998, -0.0, -0.0, 1.0, -0.0, 0.9999999999999998, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -1.0, -0.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, -1.0, 7.850462293418876e-17)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.9999999999999997, 0.9999999999999997, 0.0, 0.0, -1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, -0.0, -0.0, 7.850462293418876e-17)

print_subspace (rk)

Kernel:
   Vector space of degree 16 and dimension 4
   Basis:
     (-0.5818609561002115, 0.0, 0.0, -0.5818609561002115, 0.0, 0.0, 0.0, -1.0,
0.0, -0.0, 0.0, 1.0, 1.0, 0.0, 0.0, 0.0)
     (-0.5818609561002115, -0.5818609561002115, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0,
1.0, -0.0, -0.0, 0.0, 0.0, 1.0, 0.0, 0.0)
     (0.0, -0.5818609561002115, -0.5818609561002115, 0.0, 0.0, -1.0, 0.0, 0.0,
0.0, 1.0, -0.0, 0.0, 0.0, 0.0, 1.0, 0.0)
     (4.567877495877366e-17, 0.0, -0.5818609561002116, -0.5818609561002115, 0.0,
0.0, -1.0, 7.850462293418876e-17, 0.0, -7.850462293418876e-17,
0.9999999999999999, -7.850462293418876e-17, 0.0, 0.0, 0.0, 1.0)

Kernel:
   Vector space of degree 16 and dimension 4
   Basis:
     (-0.5818609561002115, 0.0, 0.0, -0.5818609561002115, 0.0, 0.0, 0.0, -1.0, 0.0, -0.0, 0.0, 1.0, 1.0, 0.0, 0.0, 0.0)
     (-0.5818609561002115, -0.5818609561002115, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, 0.0, 0.0, 1.0, 0.0, 0.0)
     (0.0, -0.5818609561002115, -0.5818609561002115, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0, -0.0, 0.0, 0.0, 0.0, 1.0, 0.0)
     (4.567877495877366e-17, 0.0, -0.5818609561002116, -0.5818609561002115, 0.0, 0.0, -1.0, 7.850462293418876e-17, 0.0, -7.850462293418876e-17, 0.9999999999999999, -7.850462293418876e-17, 0.0, 0.0, 0.0, 1.0)

print_subspace (lk)

Left kernel:
   Vector space of degree 12 and dimension 0

Left kernel:
   Vector space of degree 12 and dimension 0

bar_forces_ss0 = dict_bar_force (bars, rk['basis'][0])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss0))

0:   -0.5818609561002115
3:   -0.5818609561002115
7:   -1.0
11:  1.0
12:  1.0

0:   -0.5818609561002115
3:   -0.5818609561002115
7:   -1.0
11:  1.0
12:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss0)

bar_forces_ss1 = dict_bar_force (bars, rk['basis'][1])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss1))

0:   -0.5818609561002115
1:   -0.5818609561002115
4:   -1.0
8:   1.0
13:  1.0

0:   -0.5818609561002115
1:   -0.5818609561002115
4:   -1.0
8:   1.0
13:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss1)

bar_forces_ss2 = dict_bar_force (bars, rk['basis'][2])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss2))

1:   -0.5818609561002115
2:   -0.5818609561002115
5:   -1.0
9:   1.0
14:  1.0

1:   -0.5818609561002115
2:   -0.5818609561002115
5:   -1.0
9:   1.0
14:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss2)

bar_forces_ss3 = dict_bar_force (bars, rk['basis'][3])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss3))

2:   -0.5818609561002116
3:   -0.5818609561002115
6:   -1.0
10:  0.9999999999999999
15:  1.0

2:   -0.5818609561002116
3:   -0.5818609561002115
6:   -1.0
10:  0.9999999999999999
15:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss3)

joints, bars, supports = schwedler_sind (10., [2.725, 4.75, 5.75, 6.25], 9)

plot_truss3d (joints, bars, supports)

maxwells_rule (joints, supports, bars)

3 * 27 - 108  ==  -27  !=  0

3 * 27 - 108  ==  -27  !=  0

free_joints = others (supports, joints)
len (free_joints)

A = equilibrium_matrix (joints, bars, free_joints)
A

81 x 108 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

81 x 108 dense matrix over Real Double Field (use the '.str()' method to see the entries)

rs, cs, rk, lk, pivots = subspaces (A, is_zero_f = is_zz_tol(), return_pivots = True)

redundant_bars = others (pivots, bars)

print_subspace (cs)

Column space:
   Vector space of degree 81 and dimension 81

Column space:
   Vector space of degree 81 and dimension 81

print_subspace (rs)

Row space:
   Vector space of degree 108 and dimension 81

Row space:
   Vector space of degree 108 and dimension 81

print_subspace (rk)

Kernel:
   Vector space of degree 108 and dimension 27

Kernel:
   Vector space of degree 108 and dimension 27

bar_forces_ss0 = dict_bar_force (bars, rk['basis'][0])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss0))

0:   -0.472786879768055
8:   -0.47278687976805506
17:  -0.9776149204186074
26:  1.0
27:  1.0

0:   -0.472786879768055
8:   -0.47278687976805506
17:  -0.9776149204186074
26:  1.0
27:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss0) . rotateZ (pi/42)

bar_forces_ss1 = dict_bar_force (bars, rk['basis'][1])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss1))

0:   -0.47278687976805517
1:   -0.47278687976805517
9:   -0.9776149204186072
18:  1.0
28:  1.0

0:   -0.47278687976805517
1:   -0.47278687976805517
9:   -0.9776149204186072
18:  1.0
28:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss1) . rotateZ (pi/42)

bar_forces_ss9 = dict_bar_force (bars, rk['basis'][9])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss9))

17:  -0.6768676394036834
36:  -0.5769051689800252
44:  -0.5769051689800252
53:  -0.9856881717552798
62:  1.0
63:  1.0

17:  -0.6768676394036834
36:  -0.5769051689800252
44:  -0.5769051689800252
53:  -0.9856881717552798
62:  1.0
63:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss9) . rotateZ (pi/42)

bar_forces_ss19 = dict_bar_force (bars, rk['basis'][19])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss19))

45:  -0.5861098319452693
72:  -0.6471760666616742
73:  -0.6471760666616739
81:  -0.9915601258751666
90:  0.9999999999999999
100: 1.0

45:  -0.5861098319452693
72:  -0.6471760666616742
73:  -0.6471760666616739
81:  -0.9915601258751666
90:  0.9999999999999999
100: 1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss19) . rotateZ (pi/42)

print_subspace (lk)

Left kernel:
   Vector space of degree 81 and dimension 0

Left kernel:
   Vector space of degree 81 and dimension 0

def hypar_x2y2 (x, y, z_scale) :
    return (x^2 - y^2) * z_scale

def hypar2 (x, y) :
    return hypar_x2y2 (x, y, 0.175)

joints, bars, supports = surface_net (hypar2, 5., 6., 4, 5)

plot_truss3d (joints, bars, supports)

maxwells_rule (joints, supports, bars)

3 * 20 - 49  ==  11  !=  0

3 * 20 - 49  ==  11  !=  0

free_joints = others (supports, joints)
free_joints

[7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 26, 27, 28, 31, 32, 33, 34]

[7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 26, 27, 28, 31, 32, 33, 34]

A = equilibrium_matrix (joints, bars, free_joints)
A

60 x 49 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

60 x 49 dense matrix over Real Double Field (use the '.str()' method to see the entries)

rs, cs, rk, lk = subspaces (A, is_zero_f = is_zz_tol())

print_subspace (cs)

Column space:
   Vector space of degree 60 and dimension 48

Column space:
   Vector space of degree 60 and dimension 48

print_subspace (rs)

Row space:
   Vector space of degree 49 and dimension 48

Row space:
   Vector space of degree 49 and dimension 48

print_subspace (rk)

Kernel:
   Vector space of degree 49 and dimension 1

Kernel:
   Vector space of degree 49 and dimension 1

bar_forces_ss = dict_bar_force (bars, rk['basis'][0])
print_dict (bar_forces_ss)

0:   0.9186369279044543
1:   0.7973407128633153
2:   0.7525766947068775
3:   0.7973407128633153
4:   0.9186369279044544
5:   0.9186369279044532
6:   0.7973407128633143
7:   0.7525766947068765
8:   0.7973407128633143
9:   0.9186369279044532
10:  0.9186369279044533
11:  0.7973407128633144
12:  0.7525766947068766
13:  0.7973407128633144
14:  0.9186369279044533
15:  0.9186369279044534
16:  0.7973407128633145
17:  0.7525766947068767
18:  0.7973407128633145
19:  0.9186369279044534
20:  0.9186369279044537
21:  0.7973407128633148
22:  0.7525766947068769
23:  0.7973407128633148
24:  0.9186369279044537
25:  0.9999999999999988
26:  0.849986985845191
27:  0.7640136217105626
28:  0.7640136217105626
29:  0.8499869858451911
30:  0.9999999999999989
31:  0.9999999999999997
32:  0.8499869858451917
33:  0.7640136217105632
34:  0.7640136217105632
35:  0.8499869858451917
36:  0.9999999999999996
37:  0.9999999999999979
38:  0.8499869858451902
39:  0.7640136217105618
40:  0.7640136217105618
41:  0.8499869858451902
42:  0.9999999999999978
43:  0.9999999999999999
44:  0.849986985845192
45:  0.7640136217105635
46:  0.7640136217105635
47:  0.849986985845192
48:  1.0

0:   0.9186369279044543
1:   0.7973407128633153
2:   0.7525766947068775
3:   0.7973407128633153
4:   0.9186369279044544
5:   0.9186369279044532
6:   0.7973407128633143
7:   0.7525766947068765
8:   0.7973407128633143
9:   0.9186369279044532
10:  0.9186369279044533
11:  0.7973407128633144
12:  0.7525766947068766
13:  0.7973407128633144
14:  0.9186369279044533
15:  0.9186369279044534
16:  0.7973407128633145
17:  0.7525766947068767
18:  0.7973407128633145
19:  0.9186369279044534
20:  0.9186369279044537
21:  0.7973407128633148
22:  0.7525766947068769
23:  0.7973407128633148
24:  0.9186369279044537
25:  0.9999999999999988
26:  0.849986985845191
27:  0.7640136217105626
28:  0.7640136217105626
29:  0.8499869858451911
30:  0.9999999999999989
31:  0.9999999999999997
32:  0.8499869858451917
33:  0.7640136217105632
34:  0.7640136217105632
35:  0.8499869858451917
36:  0.9999999999999996
37:  0.9999999999999979
38:  0.8499869858451902
39:  0.7640136217105618
40:  0.7640136217105618
41:  0.8499869858451902
42:  0.9999999999999978
43:  0.9999999999999999
44:  0.849986985845192
45:  0.7640136217105635
46:  0.7640136217105635
47:  0.849986985845192
48:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss)

print_subspace (lk)

Left kernel:
   Vector space of degree 60 and dimension 12

Left kernel:
   Vector space of degree 60 and dimension 12

plot_truss_with_displacements3d (joints, bars, supports, map (chop(), lk['basis'][0]), displ_scale = 0.25)

plot_truss_with_displacements3d (joints, bars, supports, map (chop(), lk['basis'][1]), displ_scale = 0.25)

plot_truss_with_displacements3d (joints, bars, supports, map (chop(), lk['basis'][11]), displ_scale = 0.25)

plot_truss_with_displacements3d (joints, bars, supports, map (chop(), lk['basis'][10]), displ_scale = 0.25)

plot_truss_with_displacements3d (joints, bars, supports, map (chop(), lk['basis'][6]), displ_scale = 0.25)

def hypar_xy (x, y, z_scale) :
    return x * y * z_scale

def hypar1 (x, y) :
    return hypar_xy (x, y, 0.25)

joints, bars, supports = surface_net (hypar1, 5., 6., 4, 5)

plot_truss3d (joints, bars, supports)

maxwells_rule (joints, supports, bars)

3 * 20 - 49  ==  11  !=  0

3 * 20 - 49  ==  11  !=  0

free_joints = others (supports, joints)
len (free_joints)

A = equilibrium_matrix (joints, bars, free_joints)
A

60 x 49 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

60 x 49 dense matrix over Real Double Field (use the '.str()' method to see the entries)

rs, cs, rk, lk = subspaces (A, is_zero_f = is_zz_tol())

print_subspace (cs)

Column space:
   Vector space of degree 60 and dimension 40

Column space:
   Vector space of degree 60 and dimension 40

print_subspace (rs)

Row space:
   Vector space of degree 49 and dimension 40

Row space:
   Vector space of degree 49 and dimension 40

print_subspace (rk)

Kernel:
   Vector space of degree 49 and dimension 9

Kernel:
   Vector space of degree 49 and dimension 9

bar_forces_ss0 = dict_bar_force (bars, rk['basis'][0])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss0))

0:   1.0
1:   1.0
2:   1.0
3:   1.0
4:   1.0

0:   1.0
1:   1.0
2:   1.0
3:   1.0
4:   1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss0)

bar_forces_ss1 = dict_bar_force (bars, rk['basis'][1])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss1))

5:   1.0
6:   1.0
7:   1.0
8:   1.0
9:   1.0

5:   1.0
6:   1.0
7:   1.0
8:   1.0
9:   1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss1)

bar_forces_ss8 = dict_bar_force (bars, rk['basis'][8])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss8))

43:  0.9999999999999994
44:  0.9999999999999996
45:  0.9999999999999997
46:  0.9999999999999998
47:  0.9999999999999999
48:  1.0

43:  0.9999999999999994
44:  0.9999999999999996
45:  0.9999999999999997
46:  0.9999999999999998
47:  0.9999999999999999
48:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss8)

print_subspace (lk)

Left kernel:
   Vector space of degree 60 and dimension 20

Left kernel:
   Vector space of degree 60 and dimension 20

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][3], displ_scale = 0.325)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][15], displ_scale = 0.325)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][19], displ_scale = 0.325)

joints = { 0: (0.0, 0.0, 0.0), 1: (2.0, 0.0, 0.0), 2: (0.0, 2.0, 0.0), 3: (0.0, 0.0, 2.0) }
print 'joints:'
print_dict (joints, indent = '   ')
print

bars = { 0: (0, 3), 1: (1, 3), 2: (2, 3), 3: (0, 1), 4: (1, 2), 5: (2, 0) }
print 'bars:' 
print_dict (bars, indent = '   ')
print

supports = []

joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)
   3:   (0, 1)
   4:   (1, 2)
   5:   (2, 0)

joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 0.000000000000000, 2.00000000000000)

bars:
   0:   (0, 3)
   1:   (1, 3)
   2:   (2, 3)
   3:   (0, 1)
   4:   (1, 2)
   5:   (2, 0)

maxwells_rule (joints, supports, bars)

3 * 4 - 6  ==  6  !=  0

3 * 4 - 6  ==  6  !=  0

free_joints = others (supports, joints)
free_joints

[0, 1, 2, 3]

[0, 1, 2, 3]

plot_truss3d (joints, bars, supports)

A = equilibrium_matrix (joints, bars, free_joints)
show (A.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrr} 0.0000 & 0.0000 & 0.0000 & 1.000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.000 \\ 1.000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & -0.7071 & 0.0000 & -1.000 & -0.7071 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 \\ 0.0000 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 \\ 0.0000 & 0.0000 & -0.7071 & 0.0000 & -0.7071 & -1.000 \\ 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & 0.0000 \\ -1.000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 \end{array}\right)$

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 12 and dimension 6
   Basis:
     (0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.0, 0.7071067811865475, 0.0, 0.0,
0.0, 0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475,
0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475)
     (1.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.7071067811865475, 0.0,
0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0)

Column space:
   Vector space of degree 12 and dimension 6
   Basis:
     (0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.0, 0.7071067811865475, 0.0, 0.0, 0.0, 0.7071067811865475, 0.0, -0.7071067811865475)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475)
     (1.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 0.0)

print_subspace (rs)

Row space:
   Vector space of degree 6 and dimension 6
   Basis:
     (1.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, -0.0, 1.4142135623730951, 1.0, -0.0)
     (0.0, 0.0, 1.0, -0.0, 1.0, 1.4142135623730951)
     (0.0, 0.0, 0.0, 1.0, 0.7071067811865475, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0)

Row space:
   Vector space of degree 6 and dimension 6
   Basis:
     (1.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, -0.0, 1.4142135623730951, 1.0, -0.0)
     (0.0, 0.0, 1.0, -0.0, 1.0, 1.4142135623730951)
     (0.0, 0.0, 0.0, 1.0, 0.7071067811865475, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0)

print_subspace (rk)

Kernel:
   Vector space of degree 6 and dimension 0

Kernel:
   Vector space of degree 6 and dimension 0

print_subspace (lk)

Left kernel:
   Vector space of degree 12 and dimension 6
   Basis:
     (0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (1.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 1.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0)
     (0.0, -1.0, 1.0, 0.0, -1.0, 1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0)

Left kernel:
   Vector space of degree 12 and dimension 6
   Basis:
     (0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (1.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 1.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0)
     (0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0)
     (0.0, -1.0, 1.0, 0.0, -1.0, 1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][0], displ_scale = 0.325)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][1], displ_scale = 0.325)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][2], displ_scale = 0.325)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][3], displ_scale = 0.325)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][4], displ_scale = 0.325)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][5], displ_scale = 0.325)

joints = { 0: (0.0, 0.0, 0.0), 1: (2.0, 0.0, 0.0), 2: (2.0, 2.0, 0.0), 3: (0.0, 2.0, 0.0), 4: (1.0, 0.0, 1.0), 5: (2.0, 1.0, 1.0), 6: (1.0, 2.0, 1.0), 7: (0.0, 1.0, 1.0) }
print 'joints:'
print_dict (joints, indent = '   ')
print

bars = { 0: (0, 4), 1: (1, 4), 2: (1, 5), 3: (2, 5), 4: (2, 6), 5: (3, 6), 6: (3, 7), 7: (0, 7), 8: (7, 4), 9: (4, 5), 10: (5, 6), 11: (6, 7) }
print 'bars:' 
print_dict (bars, indent = '   ')
print

supports = [0, 1, 2, 3]
print 'supports:', supports
print

joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   4:   (1.00000000000000, 0.000000000000000, 1.00000000000000)
   5:   (2.00000000000000, 1.00000000000000, 1.00000000000000)
   6:   (1.00000000000000, 2.00000000000000, 1.00000000000000)
   7:   (0.000000000000000, 1.00000000000000, 1.00000000000000)

bars:
   0:   (0, 4)
   1:   (1, 4)
   2:   (1, 5)
   3:   (2, 5)
   4:   (2, 6)
   5:   (3, 6)
   6:   (3, 7)
   7:   (0, 7)
   8:   (7, 4)
   9:   (4, 5)
   10:  (5, 6)
   11:  (6, 7)

supports: [0, 1, 2, 3]

joints:
   0:   (0.000000000000000, 0.000000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   4:   (1.00000000000000, 0.000000000000000, 1.00000000000000)
   5:   (2.00000000000000, 1.00000000000000, 1.00000000000000)
   6:   (1.00000000000000, 2.00000000000000, 1.00000000000000)
   7:   (0.000000000000000, 1.00000000000000, 1.00000000000000)

bars:
   0:   (0, 4)
   1:   (1, 4)
   2:   (1, 5)
   3:   (2, 5)
   4:   (2, 6)
   5:   (3, 6)
   6:   (3, 7)
   7:   (0, 7)
   8:   (7, 4)
   9:   (4, 5)
   10:  (5, 6)
   11:  (6, 7)

supports: [0, 1, 2, 3]

plot_truss3d (joints, bars, supports, with_axes = True)

maxwells_rule (joints, supports, bars)

3 * 4 - 12  ==  0

3 * 4 - 12  ==  0

free_joints = others (supports, joints)
free_joints

[4, 5, 6, 7]

[4, 5, 6, 7]

A = equilibrium_matrix (joints, bars, free_joints)
show (A.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrr} -0.7071 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.0000 \\ -0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 \\ 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 \\ 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & -0.7071 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & -0.7071 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & 0.7071 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.7071 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \end{array}\right)$

rs, cs, rk, lk = subspaces (A)

print_subspace (cs)

Column space:
   Vector space of degree 12 and dimension 11
   Basis:
     (-0.7071067811865475, 0.0, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0)
     (0.7071067811865475, 0.0, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, -0.7071067811865475, -0.7071067811865475, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.7071067811865475, -0.7071067811865475, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865475, 0.0,
-0.7071067811865475, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475, 0.0,
-0.7071067811865475, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865475,
-0.7071067811865475)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475,
-0.7071067811865475)
     (-0.7071067811865475, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.7071067811865475, -0.7071067811865475, 0.0)
     (0.7071067811865475, 0.7071067811865475, 0.0, -0.7071067811865475,
-0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.7071067811865475, 0.0,
0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0)

Column space:
   Vector space of degree 12 and dimension 11
   Basis:
     (-0.7071067811865475, 0.0, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.7071067811865475, 0.0, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, -0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865475, 0.0, -0.7071067811865475, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475, 0.0, -0.7071067811865475, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865475, -0.7071067811865475)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, -0.7071067811865475, -0.7071067811865475)
     (-0.7071067811865475, 0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.7071067811865475, -0.7071067811865475, 0.0)
     (0.7071067811865475, 0.7071067811865475, 0.0, -0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)
     (0.0, 0.0, 0.0, -0.7071067811865475, 0.7071067811865475, 0.0, 0.7071067811865475, -0.7071067811865475, 0.0, 0.0, 0.0, 0.0)

print_subspace (rs)

Row space:
   Vector space of degree 12 and dimension 11
   Basis:
     (1.0, -1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 1.0, -1.0, -0.0, -0.0)
     (0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.5, 0.5, -0.0, -0.0)
     (0.0, 0.0, 1.0, -1.0, -0.0, -0.0, -0.0, -0.0, -0.0, 1.0, -1.0, -0.0)
     (0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.5, 0.5, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.5, 0.5)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, -1.0, 0.0, 0.0, 1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.5, -0.0, -0.0, -0.5)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0)

Row space:
   Vector space of degree 12 and dimension 11
   Basis:
     (1.0, -1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 1.0, -1.0, -0.0, -0.0)
     (0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.5, 0.5, -0.0, -0.0)
     (0.0, 0.0, 1.0, -1.0, -0.0, -0.0, -0.0, -0.0, -0.0, 1.0, -1.0, -0.0)
     (0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.5, 0.5, -0.0)
     (0.0, 0.0, 0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -0.0, -0.0, -0.0, -0.0, -0.5, 0.5)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, -1.0, -1.0, 0.0, 0.0, 1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.5, -0.0, -0.0, -0.5)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0, -0.0)
     (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0, 1.0)

print_subspace (rk)

Kernel:
   Vector space of degree 12 and dimension 1
   Basis:
     (1.0, -1.0, -1.0, 1.0, 1.0, -1.0, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0)

Kernel:
   Vector space of degree 12 and dimension 1
   Basis:
     (1.0, -1.0, -1.0, 1.0, 1.0, -1.0, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0)

bar_forces_ss0 = dict_bar_force (bars, rk['basis'][0])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss0))

0:   1.0
1:   -1.0
2:   -1.0
3:   1.0
4:   1.0
5:   -1.0
6:   -1.0
7:   1.0
8:   -1.0
9:   1.0
10:  -1.0
11:  1.0

0:   1.0
1:   -1.0
2:   -1.0
3:   1.0
4:   1.0
5:   -1.0
6:   -1.0
7:   1.0
8:   -1.0
9:   1.0
10:  -1.0
11:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss0)

print_subspace (lk)

Left kernel:
   Vector space of degree 12 and dimension 1
   Basis:
     (0.0, -1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 1.0, 0.0, 0.0)

Left kernel:
   Vector space of degree 12 and dimension 1
   Basis:
     (0.0, -1.0, 0.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 1.0, 0.0, 0.0)

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][0], displ_scale = 0.425)

joints = { 0: (0.0, -0.25, 0.0), 1: (2.0, 0.0, 0.0), 2: (2.0, 2.0, 0.0), 3: (0.0, 2.0, 0.0), 4: (1.0, 0.0, 1.0), 5: (2.0, 1.0, 1.0), 6: (1.0, 2.0, 1.0), 7: (0.0, 1.0, 1.0) }
print 'joints:'
print_dict (joints, indent = '   ')
print

bars = { 0: (0, 4), 1: (1, 4), 2: (1, 5), 3: (2, 5), 4: (2, 6), 5: (3, 6), 6: (3, 7), 7: (0, 7), 8: (7, 4), 9: (4, 5), 10: (5, 6), 11: (6, 7) }
print 'bars:' 
print_dict (bars, indent = '   ')
print

supports = [0, 1, 2, 3]
print 'supports:', supports
print

joints:
   0:   (0.000000000000000, -0.250000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   4:   (1.00000000000000, 0.000000000000000, 1.00000000000000)
   5:   (2.00000000000000, 1.00000000000000, 1.00000000000000)
   6:   (1.00000000000000, 2.00000000000000, 1.00000000000000)
   7:   (0.000000000000000, 1.00000000000000, 1.00000000000000)

bars:
   0:   (0, 4)
   1:   (1, 4)
   2:   (1, 5)
   3:   (2, 5)
   4:   (2, 6)
   5:   (3, 6)
   6:   (3, 7)
   7:   (0, 7)
   8:   (7, 4)
   9:   (4, 5)
   10:  (5, 6)
   11:  (6, 7)

supports: [0, 1, 2, 3]

joints:
   0:   (0.000000000000000, -0.250000000000000, 0.000000000000000)
   1:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   2:   (2.00000000000000, 2.00000000000000, 0.000000000000000)
   3:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   4:   (1.00000000000000, 0.000000000000000, 1.00000000000000)
   5:   (2.00000000000000, 1.00000000000000, 1.00000000000000)
   6:   (1.00000000000000, 2.00000000000000, 1.00000000000000)
   7:   (0.000000000000000, 1.00000000000000, 1.00000000000000)

bars:
   0:   (0, 4)
   1:   (1, 4)
   2:   (1, 5)
   3:   (2, 5)
   4:   (2, 6)
   5:   (3, 6)
   6:   (3, 7)
   7:   (0, 7)
   8:   (7, 4)
   9:   (4, 5)
   10:  (5, 6)
   11:  (6, 7)

supports: [0, 1, 2, 3]

plot_truss3d (joints, bars, supports)

free_joints = others (supports, joints)
free_joints

[4, 5, 6, 7]

[4, 5, 6, 7]

A = equilibrium_matrix (joints, bars, free_joints)
show (A.apply_map (ND(4)))

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrrrrrrrr} -0.6963 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & 0.0000 \\ -0.1741 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.7071 & 0.0000 & 0.0000 \\ -0.6963 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 \\ 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & 0.7071 & 0.0000 \\ 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & -0.7071 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & -0.7071 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & -0.7071 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & 0.0000 & 0.0000 & 0.7071 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.7071 & -0.7809 & -0.7071 & 0.0000 & 0.0000 & 0.7071 \\ 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & -0.7071 & -0.6247 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \end{array}\right)$

rs, cs, rk, lk = subspaces (A)

print_subspace (rk)

Kernel:
   Vector space of degree 12 and dimension 0

Kernel:
   Vector space of degree 12 and dimension 0

print_subspace (lk)

Left kernel:
   Vector space of degree 12 and dimension 0

Left kernel:
   Vector space of degree 12 and dimension 0

hh = sqrt(2.0)
joints = { 0: (0.0, -2.0, 0.0), 1: (sqrt(2.0), -sqrt(2.0), 0.0), 2: (2.0, 0.0, 0.0), 3: (sqrt(2.0), sqrt(2.0), 0.0), 4: (0.0, 2.0, 0.0), 5: (-sqrt(2.0), sqrt(2.0), 0.0), 6: (-2.0, 0.0, 0.0), 7: (-sqrt(2.0), -sqrt(2.0), 0.0),  8: (sin(0.125*pi), -cos(0.125*pi), hh), 9: (sin(0.375*pi), -cos(0.375*pi), hh), 10: (sin(0.625*pi), -cos(0.625*pi), hh), 11: (sin(0.875*pi), -cos(0.875*pi), hh), 12: (sin(1.125*pi), -cos(1.125*pi), hh), 13: (sin(1.375*pi), -cos(1.375*pi), hh), 14: (sin(1.625*pi), -cos(1.625*pi), hh), 15: (sin(1.875*pi), -cos(1.875*pi), hh)}
print 'joints:'
print_dict (joints, indent = '   ')
print

bars = { 0: (0, 8), 1: (1, 8), 2: (1, 9), 3: (2, 9), 4: (2, 10), 5: (3, 10), 6: (3, 11), 7: (4, 11), 8: (4, 12), 9: (5, 12), 10: (5, 13), 11: (6, 13), 12: (6, 14), 13: (7, 14), 15: (7, 15), 16: (0, 15), 17: (15, 8), 18: (8, 9), 19: (9, 10), 20: (10, 11), 21: (11, 12), 22: (12, 13), 23: (13, 14), 24: (14, 15) }
print 'bars:' 
print_dict (bars, indent = '   ')
print

supports = [0, 1, 2, 3, 4, 5, 6, 7]
print 'supports:', supports
print

joints:
   0:   (0.000000000000000, -2.00000000000000, 0.000000000000000)
   1:   (1.41421356237310, -1.41421356237310, 0.000000000000000)
   2:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   3:   (1.41421356237310, 1.41421356237310, 0.000000000000000)
   4:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   5:   (-1.41421356237310, 1.41421356237310, 0.000000000000000)
   6:   (-2.00000000000000, 0.000000000000000, 0.000000000000000)
   7:   (-1.41421356237310, -1.41421356237310, 0.000000000000000)
   8:   (sin(0.125000000000000*pi), -cos(0.125000000000000*pi),
1.41421356237310)
   9:   (sin(0.375000000000000*pi), -cos(0.375000000000000*pi),
1.41421356237310)
   10:  (sin(0.625000000000000*pi), -cos(0.625000000000000*pi),
1.41421356237310)
   11:  (sin(0.875000000000000*pi), -cos(0.875000000000000*pi),
1.41421356237310)
   12:  (sin(1.12500000000000*pi), -cos(1.12500000000000*pi), 1.41421356237310)
   13:  (sin(1.37500000000000*pi), -cos(1.37500000000000*pi), 1.41421356237310)
   14:  (sin(1.62500000000000*pi), -cos(1.62500000000000*pi), 1.41421356237310)
   15:  (sin(1.87500000000000*pi), -cos(1.87500000000000*pi), 1.41421356237310)

bars:
   0:   (0, 8)
   1:   (1, 8)
   2:   (1, 9)
   3:   (2, 9)
   4:   (2, 10)
   5:   (3, 10)
   6:   (3, 11)
   7:   (4, 11)
   8:   (4, 12)
   9:   (5, 12)
   10:  (5, 13)
   11:  (6, 13)
   12:  (6, 14)
   13:  (7, 14)
   15:  (7, 15)
   16:  (0, 15)
   17:  (15, 8)
   18:  (8, 9)
   19:  (9, 10)
   20:  (10, 11)
   21:  (11, 12)
   22:  (12, 13)
   23:  (13, 14)
   24:  (14, 15)

supports: [0, 1, 2, 3, 4, 5, 6, 7]

joints:
   0:   (0.000000000000000, -2.00000000000000, 0.000000000000000)
   1:   (1.41421356237310, -1.41421356237310, 0.000000000000000)
   2:   (2.00000000000000, 0.000000000000000, 0.000000000000000)
   3:   (1.41421356237310, 1.41421356237310, 0.000000000000000)
   4:   (0.000000000000000, 2.00000000000000, 0.000000000000000)
   5:   (-1.41421356237310, 1.41421356237310, 0.000000000000000)
   6:   (-2.00000000000000, 0.000000000000000, 0.000000000000000)
   7:   (-1.41421356237310, -1.41421356237310, 0.000000000000000)
   8:   (sin(0.125000000000000*pi), -cos(0.125000000000000*pi), 1.41421356237310)
   9:   (sin(0.375000000000000*pi), -cos(0.375000000000000*pi), 1.41421356237310)
   10:  (sin(0.625000000000000*pi), -cos(0.625000000000000*pi), 1.41421356237310)
   11:  (sin(0.875000000000000*pi), -cos(0.875000000000000*pi), 1.41421356237310)
   12:  (sin(1.12500000000000*pi), -cos(1.12500000000000*pi), 1.41421356237310)
   13:  (sin(1.37500000000000*pi), -cos(1.37500000000000*pi), 1.41421356237310)
   14:  (sin(1.62500000000000*pi), -cos(1.62500000000000*pi), 1.41421356237310)
   15:  (sin(1.87500000000000*pi), -cos(1.87500000000000*pi), 1.41421356237310)

bars:
   0:   (0, 8)
   1:   (1, 8)
   2:   (1, 9)
   3:   (2, 9)
   4:   (2, 10)
   5:   (3, 10)
   6:   (3, 11)
   7:   (4, 11)
   8:   (4, 12)
   9:   (5, 12)
   10:  (5, 13)
   11:  (6, 13)
   12:  (6, 14)
   13:  (7, 14)
   15:  (7, 15)
   16:  (0, 15)
   17:  (15, 8)
   18:  (8, 9)
   19:  (9, 10)
   20:  (10, 11)
   21:  (11, 12)
   22:  (12, 13)
   23:  (13, 14)
   24:  (14, 15)

supports: [0, 1, 2, 3, 4, 5, 6, 7]

plot_truss3d (joints, bars, supports)

maxwells_rule (joints, supports, bars)

3 * 8 - 24  ==  0

3 * 8 - 24  ==  0

free_joints = others (supports, joints)
free_joints

[8, 9, 10, 11, 12, 13, 14, 15]

[8, 9, 10, 11, 12, 13, 14, 15]

A = equilibrium_matrix (joints, bars, free_joints)
A

24 x 24 dense matrix over Real Double Field (use the '.str()' method to see the
entries)

24 x 24 dense matrix over Real Double Field (use the '.str()' method to see the entries)

rs, cs, rk, lk = subspaces (A, is_zero_f = is_zz_tol())

print_subspace (cs)

Column space:
   Vector space of degree 24 and dimension 23

Column space:
   Vector space of degree 24 and dimension 23

print_subspace (rs)

Row space:
   Vector space of degree 24 and dimension 23

Row space:
   Vector space of degree 24 and dimension 23

print_subspace (rk)

Kernel:
   Vector space of degree 24 and dimension 1

Kernel:
   Vector space of degree 24 and dimension 1

bar_forces_ss0 = dict_bar_force (bars, rk['basis'][0])
print_dict (filter_dict (lambda x : not is_zz_tol()(x), bar_forces_ss0))

0:   2.1943069386526513
1:   -2.1943069386526513
2:   -2.1943069386526517
3:   2.1943069386526517
4:   2.1943069386526526
5:   -2.1943069386526526
6:   -2.194306938652653
7:   2.194306938652653
8:   2.1943069386526557
9:   -2.1943069386526557
10:  -2.194306938652659
11:  2.1943069386526592
12:  2.1943069386526624
13:  -2.194306938652662
15:  -2.1943069386526552
16:  2.1943069386526557
17:  -0.9999999999999946
18:  0.999999999999995
19:  -0.9999999999999953
20:  0.9999999999999953
21:  -0.9999999999999962
22:  0.9999999999999972
23:  -0.9999999999999994
24:  1.0

0:   2.1943069386526513
1:   -2.1943069386526513
2:   -2.1943069386526517
3:   2.1943069386526517
4:   2.1943069386526526
5:   -2.1943069386526526
6:   -2.194306938652653
7:   2.194306938652653
8:   2.1943069386526557
9:   -2.1943069386526557
10:  -2.194306938652659
11:  2.1943069386526592
12:  2.1943069386526624
13:  -2.194306938652662
15:  -2.1943069386526552
16:  2.1943069386526557
17:  -0.9999999999999946
18:  0.999999999999995
19:  -0.9999999999999953
20:  0.9999999999999953
21:  -0.9999999999999962
22:  0.9999999999999972
23:  -0.9999999999999994
24:  1.0

plot_truss_with_forces3d (joints, bars, supports, bar_forces_ss0)

print_subspace (lk)

Left kernel:
   Vector space of degree 24 and dimension 1

Left kernel:
   Vector space of degree 24 and dimension 1

plot_truss_with_displacements3d (joints, bars, supports, lk['basis'][0], displ_scale = 0.25)

GS_Truss_(v0.5.5)

1128 days ago by fresl