Simple triangular truss.

Study of snap-through behavior using finite deformation truss elements.

We shall be using load stepping to illustrate the limitation of this control technique.

Author: Peter Mackenzie-Helnwein

Setup

import numpy as np
import matplotlib.pyplot as plt

from femedu.examples import Example

from femedu.domain import System, Node
from femedu.elements.finite import Truss
from femedu.materials import FiberMaterial
from femedu.solver import NewtonRaphsonSolver

Create the example by subclassing the Example

class ExampleTruss05(Example):

    def problem(self):
        # initialize a system model
        model = System()
        model.setSolver(NewtonRaphsonSolver())

        # create notes
        x1=Node(0.0,0.0)
        x2=Node(5.5,0.5)
        x3=Node(9.5,0.0)

        model.addNode(x1,x2,x3)

        params = dict(
            E = 2100.,   # MOE
            A = 1.       # cross section area
        )

        # create elements
        elemA = Truss(x1,x2, FiberMaterial(params))
        elemB = Truss(x3,x2, FiberMaterial(params))

        model += elemA
        model += elemB

        # apply boundary conditions
        x1.fixDOF(['ux','uy'])
        x3.fixDOF(['ux','uy'])

        # build reference load
        x2.addLoad([-1.],['uy'])

        # write out report
        model.report()

        # create plots
        model.plot(factor=1., filename="truss05_deformed.png")

        #
        # performing the analysis
        #
        model.resetDisp()

        # setting target load levels
        levels = [0., .2, .4, .6, .7, .8, .9, .95, .975,  1.05, 1.1, 1.2]

        # set up data collection
        data_list = []

        # reset the analysis
        model.resetDisp()

        # apply all load steps
        for lam in levels:

            model.setLoadFactor(lam)
            model.solve()

            # collect data
            data_list.append(x2.getDisp())

            # adding a plot
            if np.isclose(lam, .975):
                # plot the deformed shape
                model.plot(factor=1.0, show_loads=False, show_reactions=False)

        # plot the deformed shape
        model.plot(factor=1.0, show_loads=False, show_reactions=False)

        data = np.array(data_list)

        plt.figure()
        plt.plot(levels, data)
        plt.grid(True)
        plt.xlabel('load factor $ \lambda $')
        plt.ylabel('displacements $ u_i $')
        plt.legend(['$ u_x $','$ u_y $'])
        plt.show()

        #
        #  adding some unloading steps
        #

        # setting additional target load levels
        more_levels = [1.1, 1., .9, .8, .6, .4, .2, 0.0 ]

        # apply all load steps
        for lam in more_levels:

            model.setLoadFactor(lam)
            model.solve()

            # collect data
            data_list.append(x2.getDisp())

        # plot the deformed shape
        model.plot(factor=1.0, show_loads=False, show_reactions=False)

        data = np.array(data_list)

        plt.figure()
        plt.plot(levels + more_levels, data, linestyle=':', marker='o')
        plt.grid(True)
        plt.xlabel('load factor $ \lambda $')
        plt.ylabel('displacements $ u_i $')
        plt.legend(['$ u_x $','$ u_y $'])
        plt.savefig("load-displacement.png")
        plt.show()

Run the example by creating an instance of the problem and executing it by calling Example.run()

if __name__ == "__main__":
    ex = ExampleTruss05()
    ex.run()

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System Analysis Report
=======================

Nodes:
---------------------
  Node_45:
      x:    [0. 0.]
      fix:  ['ux', 'uy']
      u:    None
  Node_46:
      x:    [5.5 0.5]
      P:    [ 0. -1.]
      u:    None
  Node_47:
      x:    [9.5 0. ]
      fix:  ['ux', 'uy']
      u:    None

Elements:
---------------------
  Truss: Node_45 to Node_46:
      material properties: FiberMaterial(Material)({'E': 2100.0, 'A': 1.0, 'nu': 0.0, 'fy': 1e+30})  strain:0.0   stress:{'xx': 0.0, 'yy': 0.0, 'zz': 0.0, 'xy': 0.0}
      internal force: 0.0
  Truss: Node_47 to Node_46:
      material properties: FiberMaterial(Material)({'E': 2100.0, 'A': 1.0, 'nu': 0.0, 'fy': 1e+30})  strain:0.0   stress:{'xx': 0.0, 'yy': 0.0, 'zz': 0.0, 'xy': 0.0}
      internal force: 0.0

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