A highly geometrically nonlinear problem is solved with the geometrically exact frame element formulation.

Corotational frame elements are used to approximate Euler's buckling load.

This example demonstrates how to orient beam elements in 3D.

A finite element analysis is performed of a plane tapered cantilever using constant-strain triangles.

Bilinear quadrilateral elements are used to solve the infinitesimal pure bending problem.

In this example we investigate the offset argument for frame geometric transformations. Av=[−dX/LedY/Le0dX/Le−dY/Le2−dX/Le20dY/Le2−dY/Le2−dX/Le21dY/Le2] \boldsymbol{A}_v = \begin{bmatrix} -dX/L_e & dY/L_e & 0 & dX/L_e \\ -dY/L_e^2 & -dX/L_e^2 & 0 & dY/L_e^2 \\ -dY/L_e^2 & -dX/L_e^2 & 1 & dY/L_e^2 \\ \end{bmatrix} Ao=[01010−Lo/200100Lo/2] \boldsymbol{A}_o = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & -L_o/\sqrt{2} \\ 0 & 0 & 1 \\ 0 & 0 & L_o/\sqrt{2} \\ \end{bmatrix} Ke=[EA/Le0004EI/Le2EI/Le02EI/Le4EI/Le] \boldsymbol{K}_e = \begin{bmatrix} EA/L_e & 0 & 0 \\ 0 & 4 EI/L_e & 2 EI/L_e \\ 0 & 2 EI/L_e & 4 EI/L_e \\ \end{bmatrix} diamond.py (collapsed) """ ^ P | .<|o . . y ^ . | . _._ -> x o o """ import opensees.openseespy as ops import numpy as np import veux def diamond_solution(EI, EA, L, off=0): import numpy as np Le = L - 2*off dX = dY = Le/np.sqrt(2) Ag = np.array([dX/Le , dY/Le , 0, dX/Le ], dY/Le**2, -dX/Le**2, 0, dY/Le**2], dY/Le**2, -dX/Le**2, 1, dY/Le**2]]) Ao = np.array([[0, 1, 0], [1, 0,-off/np.sqrt(2)], [0, 0, 1], [0, 0, off/np.sqrt(2)]]) A = Ag@Ao Ke = np.array([[EA/Le, 0, 0], [ 0, 4*EI/Le, 2*EI/Le], [ 0, 2*EI/Le, 4*EI/Le]]) K = A.T@Ke@A kuw = K[1:,0] return 1/np.linalg.solve(K, [1,0,0]) # return K[0][0] - kuw.dot(np.linalg.solve(K[1:,1:], kuw)) def create_diamond(): m = ops.Model(ndm=3, ndf=6) L = 5 off = 0.1*L X = L / np.sqrt(2) E = 30 I = 1 A = 500*I m.node(1, (0, 0, 0)) m.node(2, (X, X, 0)) m.fix(1, (0, 1, 1, 1, 1, 1)) m.fix(2, (1, 0, 1, 1, 0, 0)) if False: m.geomTransf("Linear", 1, (0, 0, 1), #offset_local=True, offset={ 1: ( off,), 2: off,), } ) else: m.geomTransf("Linear", 1, (0, 0, 1), offset={ 1: ( off/np.sqrt(2), off/np.sqrt(2), 0), 2: off/np.sqrt(2),-off/np.sqrt(2), 0), } ) m.section("ElasticFrame", 1, E=E, A=A, Iy=I, Iz=I, G=1, J=3/2*I) m.element("MixedFrame", 1, (1, 2), transform=1, section=1, shear=0) m.pattern("Plain", 1, "Linear", loads={2: (0, 1, 0, 0, 0, 0)}) P = -4*L**2/(E*I) m.integrator("LoadControl", P) m.analysis("Static") m.analyze(1) # m.print(json=True) print(P/diamond_solution(E*I, E*A, L, off=off)/L) print(m.nodeDisp(2, 2)/L) return m if __name__ == "__main__": model = create_diamond() a = veux.create_artist(model) a.draw_outlines() a.draw_origin() a.draw_outlines(state=model.nodeDisp) a.draw_axes() veux.serve(a) ...

Geometrically nonlinear analysis of a cantilever rolling up under the action of a point moment, performed with shell finite elements.

This example uses the getTangent method to perform static condensation.

The Laplace problem is solved for St. Venant's warping function.