displacement relationship processing when assembling global stiffness matrix

As discussed in the Section 3.2.4, an element stiffness matrix will be assembled into a global stiffness matrix. The assembly is done by matching element nodes with their global order. For example, an element has two nodes, i and j, and its element stiffness matrix is shown in Figure 3.7b.

When assembling, each submatrix in Figure  3.7b will be added into its corresponding submatrix in the global matrix in Figure 3.7a. It should be noted that the element stiffness matrix must be transformed into the global coordinate system before adding it into the global matrix. The element stiffness matrix is established in its local coordinate system, which is often different from the global coordinate system. Because stiffness of a degree of freedom is a vector in space, the transformation of the stiffness matrix can be taken as a simple standard space transformation process.

All displacements at the connection of adjacent elements are continuous by default, or the connection is rigid from element to element as shown in Figure 3.7c. It is obvious that the global stiffness of node j will be the sum of submatrices of both elements e

(

1 ande2

)

. These results are due to one-to-one mapping of element stiffness and global stiffness during assembling matrices. However, the relationship between element stiffness and global stiffness does not have to be one to one. When this happens, a matrix-pro-cessing technique, the displacement relationship, will be used. Taking the simulation of commonly used joints as example, the principle of displace-ment relationship is discussed briefly next in this section.

As shown in Figure 3.7d, two beam elements are connected with a joint.

Four nodes, i j k, , ,and , in the global matrix will be needed to have enough l degrees of freedom to represent the extra rotation at the joint. If each node is assumed to have six (6) degrees of freedom, node jand will be sharing k five (5) of them and each node has one rotation independent of one another.

The relationships of displacements between nodes jand will be that the k five (5) shared displacements of node k are mapped to those of node j, and their rotation is separated. When assembling e1, it is a usual summing process. When assembling e2, matrix elements corresponding to shared dis-placements at node k will be added to node j instead, rather than to node k as is normally done. Only the rotation matrix elements will be added to its own position, node k in global. This type of relationship is often called the master–slave relationship.

i

i

i

i

i

j

e1 j e2

k

e1 j k e2 l (b) Element stiffness matrix

(a) Global stiffness matrix

(c) Default rigid connection

(d) Connected with joint i j

j

k

l

j k l

Figure 3.7 (a–d) Assembling global stiffness matrix and processing displacement relationship.

The displacement relationship and its processing are an important part of FEM. In addition to beam joints mentioned earlier, this process can be used to simulate many other complicated mechanics situations, such as spring or rigid body connections.

3.2.6 Nonlinearities

In the prior derivations of the global equilibrium equation, both the geome-try relationship (Equation 3.5) and the material relationship (Equation 3.13) are in linear forms. When displacements are small and strains are within the linear range with stresses, as for most engineering problems, linear solutions (Equation 3.13) are accurate and adequate. However, large dis-placements and/or nonlinear constitutive material problems widely exist in engineering practices. The geometric nonlinearity of long-span cable bridges, discussed in Chapter 11, and the plastic behavior of middle- and short-span bridges, discussed in Chapters 14, 15, and 17, are two typical examples of these problems in bridge structural analyses. The approach to the respective geometric nonlinear and material nonlinear problems is an important part of FEM.

In general, when material nonlinearity is considered, the stresses and strains relationship (Equation 3.13) would be

σ σ ε= ( ) (3.34)

When geometric nonlinearity is considered, the strains will contain the sec-ond order of displacement derivatives as

εε =

Thus, the strains and displacements relationship (Equation 3.28) becomes

ε =^Ba=

(

^B0+^{B a aL}

[ ] )

^(3.36)

where B0 is the same matrix as when geometric nonlinearity is not consid-ered and B aL

( )

is due to the second order of displacement derivatives and relates to current displacements.

When nonlinearities are considered, the solution of Equation 3.32 has to be approached by incremental method, in which changes of ψψ a

( )

respective to a small increment of a are to be noted.

d d

d dv d

d dv d d

T T

ψ T

ψ= σσ + σσ

 

 =

∫

^Ba

∫

B

a a K a (3.37)

In Equation 3.37, KT is the tangential stiffness, respective to small incre-ment of displaceincre-ments. Taking the geometric nonlinearity as an example, the tangential stiffness can be derived as

KT =K₀+Kσ+KL (3.38)

where:

K0=

∫

B DB0^T 0 represents the usual stiffness when displacements are small Kσ is the first term in Equation 3.37, which reflects the stiffness due

to the existence of stresses, that is, the initial stress or geometric matrix:

K B

a

B

σ=

∫

^dd dv=

∫

d a

d dv

T LT

σσ σσ (3.39)

KL is the stiffness due to large displacements:

K_L =

∫ (

B DB^0T _L+B DB^T_{L L}+B DB_L^{T 0}

)

dv ^(3.40) When material nonlinearity is considered as well, the elastic matrix D should be evaluated at strains due to current displacements.

The solutions of nonlinear problems can be reached by iterations on Equations 3.33 and 3.37. Given initial estimated displacements a0, which are obtained as linear solution, their corresponding internal strains can be computed. Furthermore, the internal stresses can be obtained by either linear or nonlinear stress and strain relationship. As shown in Equation 3.33, the initial unbalanced general forces ψψ a

( )

0 can be determined. The unbalanced

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 KT will first be formed at current displacements (a0). Taking ψψ a

( )

0 ^{as dψ}ψ in Equation 3.37, the dis-placement adjustment can be solved. Once an adjustment is obtained, new displacements a1 are established. The iteration process will keep looping till the unbalanced general forces ψ a

( )

n 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.

In document Chung C. Fu, Computational _ysis and Design of Bridge Structures, 2015 (Page 96-100)