Frame

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Added in version v2025.2.0: After workshop 2

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Given is the following beam [HansWfDUoTechnology22]:

With:

• \(EI = 3000\)

• \(q = 12\)

• \(EA = \infty\)

Exercise (Frame)

Solve this problem by simplifying the stiffness matrix first.

import matplotlib as plt
import numpy as np
sys.path.insert(1, '/matrixmethod_solution')
import matrixmethod_solution as mm
%config InlineBackend.figure_formats = ['svg']

import numpy as np
import matplotlib as plt
import matrixmethod as mm
%config InlineBackend.figure_formats = ['svg']

#YOUR_CODE_HERE

Solution to Exercise (Frame)

mm.Node.clear()
mm.Element.clear()

EI = 3000
q = -12

nodes = []

nodes.append(mm.Node(0,0))
nodes.append(mm.Node(0,-2))
nodes.append(mm.Node(0,-5))
nodes.append(mm.Node(4,0))
nodes.append(mm.Node(4,-2))

elems = []

elems.append(mm.Element(nodes[0], nodes[1]))
elems.append(mm.Element(nodes[1], nodes[2]))
elems.append(mm.Element(nodes[3], nodes[4]))
elems.append(mm.Element(nodes[4], nodes[1]))
elems.append(mm.Element(nodes[4], nodes[2]))

section = {}
section['EI'] = EI
for elem in elems:
    elem.set_section (section)

elems[4].add_distributed_load([0,q])

con = mm.Constrainer()

con.fix_dof (nodes[0], 0)
con.fix_dof (nodes[0], 1)
con.fix_dof (nodes[1], 0)
con.fix_dof (nodes[2], 0)
con.fix_dof (nodes[3], 0)
con.fix_dof (nodes[3], 1)

print(con)
for elem in elems:
    print(elem)

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

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

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

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

print(con.support_reactions(global_k,u,global_f))

This constrainer has constrained the degrees of freedom: [0, 1, 3, 6, 9, 10] with corresponding constrained values: [0, 0, 0, 0, 0, 0])
Element connecting:
node #1:
 This node has:
 - x coordinate=0,
 - z coordinate=0,
 - degrees of freedom=[0, 1, 2],
 - load vector=[0. 0. 0.]
with node #2:
 This node has:
 - x coordinate=0,
 - z coordinate=-2,
 - degrees of freedom=[3, 4, 5],
 - load vector=[0. 0. 0.]
Element connecting:
node #1:
 This node has:
 - x coordinate=0,
 - z coordinate=-2,
 - degrees of freedom=[3, 4, 5],
 - load vector=[0. 0. 0.]
with node #2:
 This node has:
 - x coordinate=0,
 - z coordinate=-5,
 - degrees of freedom=[6, 7, 8],
 - load vector=[-18.  24. -25.]
Element connecting:
node #1:
 This node has:
 - x coordinate=4,
 - z coordinate=0,
 - degrees of freedom=[9, 10, 11],
 - load vector=[0. 0. 0.]
with node #2:
 This node has:
 - x coordinate=4,
 - z coordinate=-2,
 - degrees of freedom=[12, 13, 14],
 - load vector=[-18.  24.  25.]
Element connecting:
node #1:
 This node has:
 - x coordinate=4,
 - z coordinate=-2,
 - degrees of freedom=[12, 13, 14],
 - load vector=[-18.  24.  25.]
with node #2:
 This node has:
 - x coordinate=0,
 - z coordinate=-2,
 - degrees of freedom=[3, 4, 5],
 - load vector=[0. 0. 0.]
Element connecting:
node #1:
 This node has:
 - x coordinate=4,
 - z coordinate=-2,
 - degrees of freedom=[12, 13, 14],
 - load vector=[-18.  24.  25.]
with node #2:
 This node has:
 - x coordinate=0,
 - z coordinate=-5,
 - degrees of freedom=[6, 7, 8],
 - load vector=[-18.  24. -25.]
[-2.02922078e-04  3.83964131e-19  4.05844156e-04  8.44244743e-19
 -4.59956710e-03 -1.51064214e-03 -6.05949804e-19  5.76035869e-19
  3.02128427e-03]
[ -0.91314935 -19.19820656  24.44934034  19.26169862  -6.79788961
 -28.80179344]

for elem in elems:
    u_elem = con.full_disp(u)[elem.global_dofs()]
    elem.plot_displaced(u_elem,num_points=51,global_c=True,scale=40)

for elem in elems:
    u_elem = con.full_disp(u)[elem.global_dofs()]
    elem.plot_moment_diagram(u_elem,num_points=51,global_c=True,scale=0.08)