Postprocessing for continuum fields

Postprocessing for continuum fields#

Up until now, we’ve only looked at discrete results: nodal displacement and support reactions. However, these results can be used to obtain the continuum field.

After solving for discrete nodal displacements, we can use the expressions derived in Force-displacement relations single extension element and Element loads to obtain continuous results. Remember than the nodal displacements are in the global coordinate system, which needs to be converted back into the local coordinate systems (\( \bar {\mathbf{u}}^e = \mathbf{T}\mathbf{u}^e\)) to use the derived expressions.

If you want to create a figure which combines the internal forces / displacements of multiple elements, you need the results in the global coordinate system again. In the provided package, this is implemented with a boolean operation global_c in the class elements.py function plot_moment_diagram and plot_displaced. For example, the frame treated in Workshop 1 - Apply:

https://raw.githubusercontent.com/ibcmrocha/public/main/vierendeel.png

gives the following displaced shape:

import matplotlib as plt
import numpy as np
sys.path.insert(1, '/matrixmethod_solution')
import matrixmethod_solution as mm
%config InlineBackend.figure_formats = ['svg']
Hide code cell source 
import numpy as np
import matplotlib as plt
import matrixmethod as mm
%config InlineBackend.figure_formats = ['svg']
Hide code cell source 
mm.Node.clear()
mm.Element.clear()

h = 1
b = 1
EIr = 10000
EIk = 1000
EA  = 1e10
H = 100

nodes = []

nodes.append(mm.Node(0,0))
nodes.append(mm.Node(b,0))
nodes.append(mm.Node(b,-h))
nodes.append(mm.Node(0,-h))

elems = []

elems.append(mm.Element(nodes[0], nodes[1]))
elems.append(mm.Element(nodes[1], nodes[2]))
elems.append(mm.Element(nodes[2], nodes[3]))
elems.append(mm.Element(nodes[0], nodes[3]))

beams = {}
columns = {}
beams['EI'] = EIr
beams['EA'] = EA
columns['EI'] = EIk
columns['EA'] = EA

elems[0].set_section (beams)
elems[1].set_section (columns)
elems[2].set_section (beams)
elems[3].set_section (columns)

con = mm.Constrainer()

con.fix_dof (nodes[0], 0)
con.fix_dof (nodes[0], 1)
con.fix_dof (nodes[1], 1)

nodes[3].add_load ([H,0,0])

global_k = np.zeros ((3*len(nodes), 3*len(nodes)))
global_f = np.zeros (3*len(nodes))

for elem in elems:
    elmat = elem.stiffness()
    idofs = elem.global_dofs()
    
    global_k[np.ix_(idofs,idofs)] += elmat

for node in nodes:
    global_f[node.dofs] += node.p

Kff, Ff = con.constrain ( global_k, global_f )
u_free = np.matmul ( np.linalg.inv(Kff), Ff )

for elem in elems:
    u_elem = con.full_disp(u_free)[elem.global_dofs()]
    elem.plot_displaced(u_elem,num_points=51,global_c=True,scale=150)
../_images/0244cc149aa37f7034056be5c43195f6a0d983144f75bdfd364d88c66c3769e1.svg