Steps for Dynamic Non-linear Analysis

The solutions of Eqs. (3.6), (3.7), (3.8), (3.9), (3.10), (3.11) and (3.12) require a special treatment such as under any increment of dynamic loading, stresses, strains and plasticity are obtained in steel, concrete and composites such as the liner and its anchorages and other similar materials. An additional effort is needed to evaluate the rupture of the steel or other material when cracks develop, especially in concrete beneath the liner or its anchorages. 

The dynamic coupled equations are needed to solve the impact/explosion problems and to assess the response history of the structure, using the time increment dt. If [M] is the mass and [C] and [.K] are the damping and stiffness matrices, the equation of motion may be written in incremental form as 

The constitutive law is used with the initial stress and constant stiffness approaches throughout the non-linear and the dynamic iteration. For the iteration 

Using the principle of virtual work, the change of equilibrium and nodal loads 6P t + St) i is calculated as 

The Von Mises criterion is used with the transitional factor to form the basis of the plastic state, such as shown in Fig. 3.2  

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The performance function expressed by equation (11) or (14), enables an evaluation of the dynamic structural reliability that includes parameter uncertainties can be performed using FORM. Calculation of the gradients of the performance function is an important step of the FORM. However, it is not always easy to obtain these gradients, (especially for the case where non-linearity of the structural performance is considered). Yao and Wen (1996) have introduced response surface approach (RSA) to avoid the sensitivity analysis required in FORM, where the response surface function is expressed as a polynomial of basic random variables in original space. For simplification, the performance function shown in equation (11) is approximated by the following second-order polynomial of standard normal random variables, in which the mixed terms are neglected. 

The response, shown in Fig. 6.6 is obtained for a step change of 5% in the reactor throughput. As can be seen, the reactor outlet concentration increases by about 9%, there is not much difference between the responses of the two models. It can be concluded that linearization is a powerful tool for the analysis of the response of non-linear systems and is helpful in understanding the dynamics of the process. 

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