Abstract
Once the number of degrees of freedom exceeds a certain number, it would be impossible to solve the dynamic equilibrium equation manually, hence the need to switch to a numerical resolution, whose general principle is to convert a dynamic equation into a static one. We are interested, for the dynamic analysis of the structures and the continuous media, in “one-step” algorithms rather than “multi-step” one. It is mainly because the systems to be solved are of large size and that it is important to minimize the number of operations and value to be memorized to the detriment, if necessary, of precision. A “one-step” algorithm, like that of Newmark, makes it possible to calculate the solution at time tn+1, starting from the solution at time tn. In addition to the disadvantage of requiring the storage of several steps, the “multi-step” algorithms such as that of Houbolt requires a startup procedure. This chapter allows the reader to enumerate and understand different numerical method with different examples.
TopDifferent Structural Responses Overview
The table below gives an overview of different responses of the structure corresponding to the different types of solicitations; these later can be enumerated as follows: harmonic, periodic, impulsive and any type of solicitations.
Table 1. Different solicitation structural response
For Duhamel integral there are limitations, which are:
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The assumption of the linear system
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The solutions are not always possible example earthquake
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The integral is not generalized; for each load there is a separate solution that can’t be scaled.
In order to overcome these limitations, numerical methods are used which offer several advantages, namely:
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The use of these methods in software (seismic simulation)
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Treatment of non-linear systems
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Possibility of generalizing these methods
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Programmability of methods
TopExact Method By Increments
This method consists in dividing the history of the charge into intervals, generally constant, and in admitting that the charge varies linearly over the interval. There is therefore a linear approximation of the load per interval, and the response for the linearized load is evaluated for a continuously variable load, the solution is improved by reducing the time step h = Dt (constant or variable).
Figure 1. Loading history (Hyghes,2000)
Figure 2. Response History (Hyghes,2000)
Considering a step Dt = t1-t0 and setting t = t- t0, we then have the load:
P(𝜏) =
P0 + 𝛼𝜏
(1)The equation of motion becomes:
(2) Key Terms in this Chapter
One-Step Algorithm: It an algorithm which allow us to calculate the solution at time t n + 1 , starting from the solution at time t n (we will divide the time into a time increment and we will consider equilibrium of the equation only in the interval ? t ).
Multi-Step Algorithm: It an algorithm which require the storage of several steps, the “multi-step” and, requires a startup procedure.
Wilson's q Method: It is an extension of linear acceleration method.
Algorithms Stability: A stable algorithm is when rounding errors does not cause large errors in the result.
Linear Acceleration: It is a method, in which we assume that the acceleration varies linearly in time.
Algorithms: Is a process or set of rules to be followed in calculations or opter problem-solving operations, especially by a computer.