Among the various types of non-linear analyses, the plastic hinge method appears to have sufficient accuracy for practical purposes. The improvements proposed by Liew et al. (1993) and Kim et al. (1996) have made the method even more atractive for general applications. The practical refined plastic hinge method incorporates an explicit imperfection modeling, an equivalent notional load modeling, and a reduced tangent modulus modeling. These are the background for a more
specified range of structures, the planar trusses, which is implemented in this research. The chosen method can predict the strength of the truss system in addition to the strength of the individual members. Furthermore, the capacity check after the usual analysis step can be neglected and both the material and geometrical nonlinearities can be included in the analysis process.
II.1.1 The virtual work equation
Let us consider a truss element as shown in Figure 2.4. The behavior of the truss is generally nonlinear. However, the approach employed in most cases, due to its simplicity, is the linearization of the problem. In particular, all the pieces of curves can be considered as the pieces of straight lines. Similarly, by using the updated Lagrangian formulation, the curve of strain increments can be considered sufficiently small within each incremental step of the nonlinear analysis. By this manner, the virtual work equation for the truss element can be written as (Kim et al. 2001):
1 2
ijkl kl ij ij ij
V V
C e δe dV + τ δη dV+ R= R
∫ ∫ (2.4)
where Cijkl is the incremental constitutive coefficients;
τij is the initial axial stress;
ekl are the linear parts of strain increment εij; ηij are the nonlinear parts of strain increment εij;
1R and 2R are the virtual works done by the external loads acting on the body at the current configuration 2C and the last calculated configuration 1C , respectively.
Figure 2.4 The planar truss element in the global coordinates.
II.1.2 The incremental constitutive law
In the field of elasticity, the relation between the axial stress Sxxand the axial strain exxis linear; stated by the Hooke’s law Sxx =Eexx, where E denotes the modulus of elasticity. Similarly, the incremental constitutive law can be expressed as:
xx t xx
S =E e (2.5)
where Sxx is the axial stress increment;
exx is the axial strain increment; and
Et is the tangent modulus accounting for gradual yielding due to residual stresses.
By applying the relation in equation (2.5) to (2.4), we obtain:
1 2
t xx xx xx xx
V V
E e δe dV + τ δη dV + R= R
∫ ∫ (2.6)
II.1.3 The derivation of displacements
Suppose the length of a truss element in Figure 2.4 is denoted as L, the displacements of the element at a specified distance x are:
a 1 b
x x
u u u
L L
⎛ ⎞
= ⎜⎝ − ⎟⎠+ (2.7)
a 1 b
x x
v v v
L L
⎛ ⎞
= ⎜⎝ − ⎟⎠+ (2.8)
where u is the horizontal displacement;
v is the vertical displacement;
A, A
u v are the displacements at the end A of the element;
B, B
u v are the displacements at the end B of the element.
As referred to in equation (2.4), the linear and nonlinear parts of the axial strain can be determined by differentiating the displacements:
xx
u u
e x L
∂ ∆
= =
∂ (2.9)
2 2 2 2
2 2
1 1
2 2
xx
u v u v
x x L L
η = ⎡⎢⎢⎣⎛⎜⎝∂∂ ⎞⎟⎠ +⎛⎜⎝∂∂ ⎞⎟⎠ ⎤⎥⎥⎦= ⎡⎢⎣∆ +∆ ⎤⎥⎦ (2.10) where ∆ =u uB −uA
B A
v v v
∆ = −
II.1.4 The matrix-form expression
As shown in Figure 2.4, the truss element has four degrees of freedom, which form the nodal displacement vector as follows:
A A B B
u v u v
⎧ ⎫⎪ ⎪
= ⎨ ⎬⎪ ⎪
⎪ ⎪⎪ ⎪
⎩ ⎭
u (2.11)
In the local coordinates, the transverse shear forces 1FyA and 1FyB are equal to zero while the axial forces 1FxA and 1FxB are equal in magnitude but opposite in direction. The upper-left superscripts denote the equilibrium status at a specified step.
At the step-by-step equilibrium status, the forces vectors are:
1
1
1
0 0
xA
xA
F F
⎧ ⎫
⎪ ⎪
⎪ ⎪
= ⎨ ⎬
⎪− ⎪
⎪ ⎪
⎩ ⎭
f and
2
2
2
0 0
xA
xA
F F
⎧ ⎫
⎪ ⎪
⎪ ⎪
= ⎨ ⎬
⎪− ⎪
⎪ ⎪
⎩ ⎭
f (2.12)
It is also noted that the initial axial force 1Fx can be computed as the integration of τxx over the cross-sectional area A:
1
x xx
A
F =∫τ dA (2.13)
Assume that only the point loads are applied on the two ends of the truss element. The self weight is a distributed load that can be replaced by statically equivalent nodal loads. Therefore, we can represent each part of equation (2.6) in matrix form as follows:
T
t xx xx t e
V V
u u
E e e dV E dV
L L
δ = ∆ δ ∆ =
∫ ∫ δu K u (2.14)
1 T
0 L
xx xx x g
V
u u v v
dV F dx
L L L L
τ δη = ⎡∆⎢⎣ δ⎛⎜⎝∆ ⎞⎟⎠+∆ δ⎛⎜⎝∆ ⎤⎞⎟⎠⎥⎦ =
∫ ∫ δu K u (2.15)
1 T 1
i i
S
R=∫t u dSδ =δu f (2.16)
where Ke and Kg are the inelastic and geometric local stiffness matrices, respectively:
0 0
0 0 0 0
0 0
0 0 0 0
t t
e
t t
E A E A
L L
E A E A
L L
⎡ − ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
= ⎢ ⎥
⎢− ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
K (2.17)
1 1
1 1
1 1
1 1
0 0
0 0
0 0
0 0
xB xB
xB xB
g
xB xB
xB xB
F F
L L
F F
L L
F F
L L
F F
L L
⎡ ⎤
⎢ − ⎥
⎢ ⎥
⎢ − ⎥
⎢ ⎥
= ⎢ ⎥
⎢− ⎥
⎢ ⎥
⎢ ⎥
⎢ − ⎥
⎢ ⎥
⎣ ⎦
K (2.18)
The governing equation for the planar truss element in its local coordinates can be written as:
(Ke+Kg)δu+1f =2f (2.19)
Or equivalently,
(Ke+Kg)δu= 2f− =1f ∆f (2.20)
where the term 1f represents the initial force acting on the element at the equilibrium status 1C , the term 2f represents the total force acting on the element at the equilibrium status 2C , and the term ∆f on the left-hand side of the above equation denotes the incremental force between the two configurations.