Staggered Displacement Method

In this method, equation 4.35 can be approximated as following:

[M ]{u K+1 = {F1}i+1 + [Q]{p}P+1 — [C ]{U }P+1 — (1 + a)[K ]{U }P+1 + a[K ]{U }

Combining equations 4.37 and 4.35 gives:

[M ]{U }i+i = [M ]{U }*+1 +ß At2[Q]{p}m—y At[C ]{U }i+i — (1+a)ßAt2 [K ]{U }i+i

Taking advantage of the lumped mass which results in a diagonal mass matrix, equation 4.38 can be modified as:

Substituting equation 4.39 into equation 4.36, then:

([G] + pPAt2[Q]T [M]-1[Q]){P}i+1 + [C ]{p }i+1 + (1 + a)[K']{p}i+1 =

In equation 4.40, the right hand side terms are known, thus, {p}i+1 can be obtained. In order to correct the approximation made in equation 4.39, {p}i+1 can be substituted in equation 4.35 to calculate {U}i+1 and its derivatives.

Therefore, the procedure of the staggered displacement method can be summarized by the following steps:

1. Solving equation 4.37 to calculate {U}*+1.

2. Substituting {U}*+1in equation 4.40 to calculate {p}i+1.

3. Substituting {p}i+1 in equation 4.35 to calculate {U}i+1 and its derivatives.

It can be shown that the method of staggered displacement is unconditionally stable for the linear coupled equations of the dam-reservoir system with structural damping when a = 0. For the nonlinear equations, the numerical solution is based on piece-wise linear solution. The solution stability depends on the length of the time steps and the introduced numerical damping. The dam-reservoir interaction representation using the staggered solution technique using a-method is suitable for nonlinear fracture analysis of concrete dams.

The nonlinear seismic analysis of concrete gravity dams includes opening and closing of the cracks due to the cyclic nature of the earthquake. When the cracks are closed, cracked elements recover their strength and therefore the structure gains stiffness. As the cracks open, the stiffness of the structure reduces. The effect of opening and closing of cracks introduce high frequency shock waves into the structure. The numerical difficulties due to opening and closing of cracks can be overcome by using the a-method (Hilber et al., 1977; Hilber and Hughes, 1978).

The a-method of time integration algorithms introduces numerical damping to the system. It is an efficient method that is accurate in lower modes and dissipate energy in the higher modes when compared with other time integration techniques. Thus, using the a-method ensures that the response of higher modes is damped out. Direct integration is used to determine the displacement and hydrodynamic pressure at the time increment i + 1. The a-method is used for discretization of both equations of the coupled field problem (implicit-implicit method). The dam-reservoir interaction representation using the staggered solution technique is introduced to the nonlinear fracture analysis of concrete dams.

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