You can view and download this file on Github: LShapeGeomExactBeam2D.py
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
# This is an EXUDYN example
#
# Details: Test model for GeometricallyExactBeam2D, evaluating behavior of rotated beam
#
# Model: Planar model of L-shaped beams with tip load; the beam has 4 elements at each part of the L-shape:
# the element length is 0.25m, the material is steel and height is 0.05m and with 0.1m;
# the L-shape is fixed at the left end to ground and a tip load of [1e6,0,0] is applied to the tip.
#
# Author: Johannes Gerstmayr
# Date: 2021-03-25
#
# Copyright:This file is part of Exudyn. Exudyn is free software. You can redistribute it and/or modify it under the terms of the Exudyn license. See 'LICENSE.txt' for more details.
#
# *clean example*
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
## import libaries
import exudyn as exu
from exudyn.utilities import *
import numpy as np
from math import sin, cos, pi
useGraphics = True #without test
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
#you can erase the following lines and all exudynTestGlobals related operations if this is not intended to be used as TestModel:
try: #only if called from test suite
from modelUnitTests import exudynTestGlobals #for globally storing test results
useGraphics = exudynTestGlobals.useGraphics
except:
class ExudynTestGlobals:
pass
exudynTestGlobals = ExudynTestGlobals()
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
## setup system container and mbs
SC = exu.SystemContainer()
mbs = SC.AddSystem()
## define parameters for beams
a = 0.25
lElem = a # length of one finite element
E=2.1e11 # Steel; Young's modulus of beam element in N/m^2
rho=7800 # Steel; density of beam element in kg/m^3
b=0.1 # width of rectangular beam element in m
h=0.05 # height of rectangular beam element in m
A=b*h # cross sectional area of beam element in m^2
I=b*h**3/12 # second moment of area of beam element in m^4
nu = 0.3 # Poisson's ratio for steel
EI = E*I
EA = E*A
rhoA = rho*A
rhoI = rho*I
ks = 10*(1+nu)/(12+11*nu) # shear correction factor
G = E/(2*(1+nu)) # shear modulus
GA = ks*G*A # shear stiffness of beam
## create position list for nodes:
nodeList=[]
pRefList=[[0,0],
[1*a,0],
[2*a,0],
[3*a,0],
[4*a,0],
[4*a,1*a],
[4*a,2*a],
[4*a,3*a],
[4*a,4*a],
]
## create nodes
for p in pRefList:
pRef = [p[0],p[1],0] #always angle zero, which is not considered in beam for includeReferenceRotations=False
ni=mbs.AddNode(NodeRigidBody2D(referenceCoordinates = pRef,
initialCoordinates = [0,0,0],
initialVelocities= [0,0,0]))
nodeList += [ni]
## create beam elements:
for i in range(len(nodeList)-1):
mbs.AddObject(ObjectBeamGeometricallyExact2D(nodeNumbers = [nodeList[i],nodeList[i+1]],
physicsLength=lElem,
physicsMassPerLength=rhoA,
physicsCrossSectionInertia=rhoI,
physicsBendingStiffness=EI,
physicsAxialStiffness=EA,
physicsShearStiffness=GA,
includeReferenceRotations=False, #to connect beams at 90° at same node
visualization=VObjectBeamGeometricallyExact2D(drawHeight = h) ))
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
## create ground node with marker for coordinate constraints
nGround = mbs.AddNode(NodePointGround(referenceCoordinates=[0,0,0]))
mNCground = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=nGround, coordinate=0))
## add markers and constraints for fixed support
n0 = nodeList[0]
mC0 = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=n0, coordinate=0))
mC1 = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=n0, coordinate=1))
mC2 = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=n0, coordinate=2))
mbs.AddObject(CoordinateConstraint(markerNumbers=[mNCground, mC0]))
mbs.AddObject(CoordinateConstraint(markerNumbers=[mNCground, mC1]))
mbs.AddObject(CoordinateConstraint(markerNumbers=[mNCground, mC2]))
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
## add tip load
tipNodeMarker = mbs.AddMarker(MarkerNodeRigid(nodeNumber=nodeList[-1]))
mbs.AddLoad(Force(markerNumber = tipNodeMarker, loadVector = [1e6, 0, 0]))
## assemble system and define simulation settings
mbs.Assemble()
simulationSettings = exu.SimulationSettings()
tEnd = 1
steps = 2000
simulationSettings.timeIntegration.numberOfSteps = steps
simulationSettings.timeIntegration.endTime = tEnd
simulationSettings.solutionSettings.solutionWritePeriod = tEnd/steps
simulationSettings.timeIntegration.verboseMode = 1
simulationSettings.solutionSettings.writeSolutionToFile = False
#simulationSettings.solutionSettings.solutionWritePeriod = tEnd/steps
simulationSettings.linearSolverType = exu.LinearSolverType.EigenSparse
simulationSettings.timeIntegration.newton.useModifiedNewton = True
simulationSettings.staticSolver.newton.maxIterations = 50
simulationSettings.staticSolver.numberOfLoadSteps = 10
# simulationSettings.displayComputationTime = True
# simulationSettings.displayStatistics = True
simulationSettings.staticSolver.newton.relativeTolerance = 1e-6
## add some visualization settings
SC.visualizationSettings.nodes.defaultSize = 0.005
SC.visualizationSettings.bodies.beams.crossSectionFilled = False
## start graphics
if useGraphics:
exu.StartRenderer()
mbs.WaitForUserToContinue()
## start static solver
mbs.SolveStatic(simulationSettings)
## print some output
uLast = mbs.GetNodeOutput(nodeList[-1], exu.OutputVariableType.Coordinates)
exu.Print("uTip =", uLast[0:2])
## stop graphics
if useGraphics:
SC.WaitForRenderEngineStopFlag()
exu.StopRenderer() #safely close rendering window!
exu.Print('solution of LShapeGeomExactBeam2D=',uLast[1]) #use y-coordinate
exudynTestGlobals.testError = uLast[1] - (-2.2115028353806547)
exudynTestGlobals.testResult = uLast[1]