Civil Engineering Reference
In-Depth Information
The uniaxial stiffness matrix of rebars [
D
si
] is evaluated as follows:
⎡
⎤
ρ
si
·
E
si
00
⎢
⎢
⎣
⎥
⎥
⎦
[
D
si
]
=
(9.36)
0
0
0
0
0
0
where
E
si
is the uniaxial tangent modulus for the rebars, as determined for a particular
stress/strain state.
9.4.3 Analysis Procedures
After the tangent material constitutive matrix [
D
] is determined, the tangent element stiffness
matrix can be evaluated using the basic finite element procedure and can be expressed as:
[
B
]
T
[
D
][
B
]
d
V
[
K
]
e
=
(9.37)
V
where [
B
] is a matrix that depends on the assumed element displacement functions of the
elements.
An iterative tangent stiffness procedure was developed to perform nonlinear analyses of
reinforced concrete structures. A flow chart for an iterative analysis solution under load
increment using the Newton-Raphson method is described in Figure 9.10. Throughout the
procedure, the tangent material constitutive matrix [
D
] is determined first, and then the
tangent element stiffness matrix [
k
] and the element resisting force increment vector
}
are calculated. After that, the global stiffness matrix [
K
] and global resisting force increment
vector
{
f
are assembled. In each iteration, the material constitutive matrix [
D
], the element
tangent stiffness matrix [
k
], and the global stiffness matrix [
K
] are iteratively refined until
convergence criterion is achieved.
The procedure for establishing the material constitutive matrix using CSMM is shown
in the grey block in Figure 9.10. It is noted that an additional iterative loop is defined to
obtain the material constitutive matrix [
D
] for RC plane stress elements because the principal
stress direction
{
F
}
θ
1
is an unknown value before [
D
] is established. The procedure for the
stiffness calculation of RC plane stress elements is outlined by the outer white block in
Figure 9.10.
This flow chart shows the simple analysis procedure of RC plane stress structures using load
increment. It can be incorporated with other static integrators such as displacement control,
and dynamic integrators such as the Newmark and the Wilson methods for different kinds of
nonlinear finite element analysis. The integrator schemes for static and dynamic analyses used
in this chapter are presented in Section 9.6.
9.5 Equation of Motion for Earthquake Loading
9.5.1 Single Degree of Freedom versus Multiple Degrees of Freedom
Any concrete structure can be represented as a single-degree-of-freedom (SDOF) system in
dynamic analysis; and the dynamic response of the structure can be evaluated by the solution
