Note
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A square patch made of two triangular plate elements
- Basic implementation test with applied loads.
Testing the tangent stiffness computation for a
Triangle6()(using quadratic shape functions).
Using
elements.linear.Triangle6(see Triangle6 class)materials.PlaneStress(see PlaneStress material class)
free free
^ ^
| |
3--6--2 -> free
|\ b | >
| \ | >
7 8 5 > (w = 1.0)
| \ | >
| a \| >
0--4--1 -> free
width: 10.
height: 10.
Material parameters: St. Venant-Kirchhoff, plane stress
E = 10.0
nu = 0.30
t = 1.0
Element loads:
node 0: [ 0.0, 0.0]
node 1: [ 10./6, 0.0]
node 2: [ 10./6, 0.0]
node 3: [ 0.0, 0.0]
node 4: [ 0.0, 0.0]
node 5: [ 10.*2/3, 0.0]
node 6: [ 0.0, 0.0]
node 7: [ 0.0, 0.0]
node 8: [ 0.0, 0.0]
2nd Piola-Kirchhoff stress:
S_XX = w = 1.000
S_YY = S_XY = S_YX = S_ZZ = 0.000
Green Lagrange strain:
eps_XX = (1/E) ((1.000) - (0.30)(0.000)) = 0.100
eps_YY = (1/E) ((0.000) - (0.30)(1.000)) = -0.030
eps_XY = eps_YX = 0.000
eps_ZZ = -nu * (eps_XX + eps_YY) = -0.021
Stretches:
lam_X = sqrt(1 + 2 eps_XX) = 1.095
lam_Y = sqrt(1 + 2 eps_YY) = 0.9695
Displacements:
ux = (lam_X - 1) * x, uy = (lam_Y - 1) * y
node 0: [ 0.000, 0.000 ]
node 1: [ 0.954, 0.000 ]
node 2: [ 0.954, -0.305 ]
node 3: [ 0.000, -0.305 ]
node 4: [ 0.477, 0.000 ]
node 5: [ 0.954, -0.1525 ]
node 6: [ 0.477, -0.305 ]
node 7: [ 0.954, -0.1525 ]
node 8: [ 0.477, -0.1525 ]
Author: Peter Mackenzie-Helnwein
from femedu.examples import Example
from femedu.domain import System, Node
from femedu.solver import NewtonRaphsonSolver
from femedu.elements.linear import Triangle6
from femedu.materials import PlaneStress
class ExamplePlate02b(Example):
def problem(self):
params = dict(
E = 10., # Young's modulus
nu = 0.3, # Poisson's ratio
t = 1.0, # thickness of the plate
fy = 1.e30 # yield stress
)
a = 10. # length of the plate in the x-direction
b = 10. # length of the plate in the y-direction
model = System()
model.setSolver(NewtonRaphsonSolver())
nd0 = Node( 0.0, 0.0)
nd1 = Node( a, 0.0)
nd2 = Node( a, b)
nd3 = Node( 0.0, b)
nd4 = Node( a/2, 0.0)
nd5 = Node( a, b/2)
nd6 = Node( a/2, b)
nd7 = Node( 0.0, b/2)
nd8 = Node( a/2, b/2)
nd0.fixDOF('ux', 'uy')
nd1.fixDOF('uy')
nd3.fixDOF('ux')
model.addNode(nd0, nd1, nd2, nd3, nd4, nd5, nd6, nd7, nd8)
elemA = Triangle6(nd0, nd1, nd3, nd4, nd8, nd7, PlaneStress(params))
elemB = Triangle6(nd2, nd3, nd1, nd6, nd8, nd5, PlaneStress(params))
model.addElement(elemA, elemB)
elemA.setSurfaceLoad(face=2, pn=1.0)
elemB.setSurfaceLoad(face=2, pn=1.0)
model.setLoadFactor(1.0)
nd0.setDisp([0.0, 0.0])
nd1.setDisp([5.0, 0.0])
nd2.setDisp([5.0, -5.0])
nd3.setDisp([0.0, -5.0])
nd4.setDisp([2.5, 0.0])
nd5.setDisp([5.0, -2.5])
nd6.setDisp([2.5, -5.0])
nd7.setDisp([0.0, -2.5])
nd8.setDisp([2.5, -2.5])
elemA.updateState()
elemB.updateState()
model.report()
model.plot(factor=1.0, filename="plate02b_deformed.png")
model.setLoadFactor(0.0)
# model.solver.assemble()
# model.solver.showKt()
#
model.solve()
model.report() # activate this line for lots of debug info
model.plot(factor=0.0, title="Undeformed system", filename="plate02b_undeformed.png", show_bc=1)
model.setLoadFactor(1.0)
model.solve(verbose=1)
model.plot(factor=1.0, filename="plate02b_deformed.png")
model.report()
Run the example by creating an instance of the problem and executing it by calling Example.run()
if __name__ == "__main__":
ex = ExamplePlate02b()
ex.run()
System Analysis Report
=======================
Nodes:
---------------------
Node_0:
x: [0. 0.]
fix: ['ux', 'uy']
u: [0. 0.]
Node_1:
x: [10. 0.]
fix: ['uy']
u: [5. 0.]
Node_2:
x: [10. 10.]
u: [ 5. -5.]
Node_3:
x: [ 0. 10.]
fix: ['ux']
u: [ 0. -5.]
Node_4:
x: [5. 0.]
u: [2.5 0. ]
Node_5:
x: [10. 5.]
u: [ 5. -2.5]
Node_6:
x: [ 5. 10.]
u: [ 2.5 -5. ]
Node_7:
x: [0. 5.]
u: [ 0. -2.5]
Node_8:
x: [5. 5.]
u: [ 2.5 -2.5]
Elements:
---------------------
Triangle6_0: nodes ( Node_0 Node_1 Node_3 Node_4 Node_8 Node_7 )
material: PlaneStress
strain 0: xx=6.250e-01 yy=-3.750e-01 xy=0.000e+00 zz=-7.500e-02
stress 0: xx=5.632e+00 yy=-2.060e+00 xy=0.000e+00 zz=0.000e+00
strain 1: xx=6.250e-01 yy=-3.750e-01 xy=0.000e+00 zz=-7.500e-02
stress 1: xx=5.632e+00 yy=-2.060e+00 xy=0.000e+00 zz=0.000e+00
strain 2: xx=6.250e-01 yy=-3.750e-01 xy=0.000e+00 zz=-7.500e-02
stress 2: xx=5.632e+00 yy=-2.060e+00 xy=0.000e+00 zz=0.000e+00
Triangle6_1: nodes ( Node_2 Node_3 Node_1 Node_6 Node_8 Node_5 )
material: PlaneStress
strain 0: xx=6.250e-01 yy=-3.750e-01 xy=-1.865e-15 zz=-7.500e-02
stress 0: xx=5.632e+00 yy=-2.060e+00 xy=-7.174e-15 zz=0.000e+00
strain 1: xx=6.250e-01 yy=-3.750e-01 xy=-2.665e-16 zz=-7.500e-02
stress 1: xx=5.632e+00 yy=-2.060e+00 xy=-1.025e-15 zz=0.000e+00
strain 2: xx=6.250e-01 yy=-3.750e-01 xy=-7.994e-16 zz=-7.500e-02
stress 2: xx=5.632e+00 yy=-2.060e+00 xy=-3.074e-15 zz=0.000e+00
+
System Analysis Report
=======================
Nodes:
---------------------
Node_0:
x: [0. 0.]
fix: ['ux', 'uy']
u: [0. 0.]
Node_1:
x: [10. 0.]
fix: ['uy']
u: [-2.2004262e-12 0.0000000e+00]
Node_2:
x: [10. 10.]
u: [-2.19815779e-12 1.08940151e-11]
Node_3:
x: [ 0. 10.]
fix: ['ux']
u: [0.00000000e+00 1.08945491e-11]
Node_4:
x: [5. 0.]
u: [-1.10086125e-12 3.39908785e-17]
Node_5:
x: [10. 5.]
u: [-2.20151180e-12 5.44645326e-12]
Node_6:
x: [ 5. 10.]
u: [-1.09955281e-12 1.08931920e-11]
Node_7:
x: [0. 5.]
u: [3.17666985e-16 5.44624201e-12]
Node_8:
x: [5. 5.]
u: [-1.10032167e-12 5.44614435e-12]
Elements:
---------------------
Triangle6_0: nodes ( Node_0 Node_1 Node_3 Node_4 Node_8 Node_7 )
material: PlaneStress
strain 0: xx=-2.198e-13 yy=1.089e-12 xy=6.353e-17 zz=-2.609e-13
stress 0: xx=1.176e-12 yy=1.125e-11 xy=2.444e-16 zz=0.000e+00
strain 1: xx=-2.198e-13 yy=1.089e-12 xy=7.054e-33 zz=-2.609e-13
stress 1: xx=1.176e-12 yy=1.125e-11 xy=2.713e-32 zz=0.000e+00
strain 2: xx=-2.200e-13 yy=1.090e-12 xy=-6.353e-17 zz=-2.609e-13
stress 2: xx=1.174e-12 yy=1.125e-11 xy=-2.444e-16 zz=0.000e+00
Triangle6_1: nodes ( Node_2 Node_3 Node_1 Node_6 Node_8 Node_5 )
material: PlaneStress
strain 0: xx=-2.199e-13 yy=1.089e-12 xy=3.553e-16 zz=-2.608e-13
stress 0: xx=1.174e-12 yy=1.124e-11 xy=1.366e-15 zz=0.000e+00
strain 1: xx=-2.198e-13 yy=1.089e-12 xy=-3.553e-16 zz=-2.609e-13
stress 1: xx=1.176e-12 yy=1.125e-11 xy=-1.366e-15 zz=0.000e+00
strain 2: xx=-2.203e-13 yy=1.089e-12 xy=-3.553e-16 zz=-2.607e-13
stress 2: xx=1.171e-12 yy=1.124e-11 xy=-1.366e-15 zz=0.000e+00
/usr/local/lib/python3.11/site-packages/matplotlib/quiver.py:632: RuntimeWarning: Mean of empty slice.
amean = a.mean()
/usr/local/lib/python3.11/site-packages/numpy/core/_methods.py:129: RuntimeWarning: invalid value encountered in scalar divide
ret = ret.dtype.type(ret / rcount)
norm of the out-of-balance force: 1.2019e+01
norm of the out-of-balance force: 2.4414e+00
norm of the out-of-balance force: 5.2114e-02
norm of the out-of-balance force: 2.8292e-05
norm of the out-of-balance force: 9.9149e-12
+
System Analysis Report
=======================
Nodes:
---------------------
Node_0:
x: [0. 0.]
fix: ['ux', 'uy']
u: [0. 0.]
Node_1:
x: [10. 0.]
fix: ['uy']
u: [0.93634551 0. ]
Node_2:
x: [10. 10.]
u: [ 0.94185675 -0.30050703]
Node_3:
x: [ 0. 10.]
fix: ['ux']
u: [ 0. -0.31608555]
Node_4:
x: [5. 0.]
u: [0.4643119 0.01196108]
Node_5:
x: [10. 5.]
u: [ 0.99272035 -0.16590176]
Node_6:
x: [ 5. 10.]
u: [ 0.47573439 -0.31903817]
Node_7:
x: [0. 5.]
u: [-0.05024981 -0.14188329]
Node_8:
x: [5. 5.]
u: [ 0.47172318 -0.15330258]
Elements:
---------------------
Triangle6_0: nodes ( Node_0 Node_1 Node_3 Node_4 Node_8 Node_7 )
material: PlaneStress
strain 0: xx=1.011e-01 yy=-2.840e-02 xy=-8.894e-03 zz=-2.181e-02
stress 0: xx=1.017e+00 yy=2.126e-02 xy=-3.421e-02 zz=0.000e+00
strain 1: xx=1.028e-01 yy=-3.297e-02 xy=-8.777e-04 zz=-2.095e-02
stress 1: xx=1.021e+00 yy=-2.341e-02 xy=-3.376e-03 zz=0.000e+00
strain 2: xx=1.138e-01 yy=-3.464e-02 xy=8.745e-03 zz=-2.375e-02
stress 2: xx=1.137e+00 yy=-5.469e-03 xy=3.363e-02 zz=0.000e+00
Triangle6_1: nodes ( Node_2 Node_3 Node_1 Node_6 Node_8 Node_5 )
material: PlaneStress
strain 0: xx=1.012e-01 yy=-2.751e-02 xy=-8.782e-03 zz=-2.211e-02
stress 0: xx=1.022e+00 yy=3.139e-02 xy=-3.378e-02 zz=0.000e+00
strain 1: xx=1.033e-01 yy=-3.359e-02 xy=-9.099e-04 zz=-2.092e-02
stress 1: xx=1.025e+00 yy=-2.850e-02 xy=-3.500e-03 zz=0.000e+00
strain 2: xx=1.133e-01 yy=-3.356e-02 xy=8.834e-03 zz=-2.393e-02
stress 2: xx=1.135e+00 yy=4.784e-03 xy=3.398e-02 zz=0.000e+00
Total running time of the script: (0 minutes 1.385 seconds)