Civil Engineering Reference
In-Depth Information
general forces reveal that the internal forces cannot balance the external
forces due to the effects of nonlinearities. The displacements have to be
adjusted by Equation 3.37. Tangential stiffness
K
T
will first be formed at
current displacements (
a
0
). Taking
ψ
a
( )
as
d
ψ in Equation 3.37, the dis-
placement adjustment can be solved. Once an adjustment is obtained, new
displacements
a
1
are established. The iteration process will keep looping till
the unbalanced general forces
ψ
a
( )
become significantly small. To ensure
the convergence of this iteration process, external loads are often loaded
incrementally, with each step containing only a fraction of the total loads.
3.2.7 Frame element
Frame components, which work as both beams in bending/shearing and
also as truss members in axial tension/compression, are very common in
structural engineering, and the development of a frame element is fun-
damental in FEM. This section will briefly introduce its displacement
functions, elastic stiffness matrix, and initial stress matrix.
The total strain energy of a frame element is the sum of the axial tension/
compression strain energy and the bending strain energy. Therefore, when
developing the elastic stiffness matrix, the axial tension/compression and the
bending behaviors can be separated. The beam-bending theory assumes that
the cross section at any point along the beam axis will remain a plane after
bent. Based on this assumption, bending strain energy along a cross section
can be expressed as the product of bending moment and rotation angle of a
cross section or the second-order derivative of vertical deflection. For a two-
node frame element as shown in Figure 3.8, according to the requirements in
Section 3.2.3, the displacement functions can only be linear. It is not enough
to describe the bending deflection, as the second-order derivative does not
exist. Two additional rotational displacements (
φ
1
and
φ
2
) have to be added.
Although they belong to the same nodes (nodes 1 and 2, respectively), a two-
node beam element has four independent nodal displacements and is truly
working as a four-node line element.
y
ϕ
1
ϕ
2
1
l
2
x
u
1
u
2
x
l
−
x
v
1
v
2
Figure 3.8
Two-node frame element.
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