# جلسه یازدهم: آموزش جامع روش های پیشرفته آنالیز دینامیکی غیر خطی (دکتر حشمدار)

AN EFFICIENT METHOD FOR NONLINEAR DYNAMIC ANALYSIS OF 3D SPACE STRUCTURES

Prof.Dr.Hashamdar

The linear acceleration method

This method is based on the step-by-step time integration of the equations of motion. A method is linear change of acceleration during each time step is assumed. Equilibrium of the dynamic forces is established at the beginning and at the end of each time interval. The nonlinear nature of a structure is in accordance with the deformed state at the beginning of each time increment.

Accuracy and stability

According to hypothesis of acceleration method that acceleration varies linearly during any time increment, the accuracy of the method depends on the size of ∆t. The time step must be short enough to justify this assumption and also short enough to ensure correct representation of the loading history. The method is only conditionally stable and will diverge if ∆t is greater than approximately half the period of vibration. The time step must, however, be considerably smaller than this value to provide accurate result.

#### The linear acceleration method to multi degree of freedom systems

When the above method is employed for the analysis of multi degree of freedom systems the derivation of the incremental equations of motion can be carried out exactly as the one for single degree of freedom systems  By assuming constant stiffness and damping during the time interval.

In 1973, Wilson presented a general solution scheme for the dynamic analysis of an arbitrary assemblage of structural elements with both physical and geometrical nonlinearity. The scheme is unconditionally stable and therefore relatively large time steps can be used.

The Wilson-θ method

The Wilson-θ method is a modification of the standard linear acceleration method. The modification is based on the assumption that acceleration varies linearly over an extended computational time و this assumption leads to a set of new equations for the dynamic equilibrium at the end of the extended time interval τ. This method is a set of equations relating the accelerations, velocities and the displacement at the end of the actual time step . Time step is also changed by variation of displacement at the end of the extended time step τ. The above requires the introduction of a third subscript t + τ in addition to the subscripts n and n+1 in order to identify the variable parameters at time.

The Wilson-θ method for MDOF system

The condition of dynamic equilibrium at the end of an extended time increment τ, assuming damping and stiffness remain constant during the time increment .

Calculation of  involves the solution of a set of simultaneous equations as implicit methods. The method is only conditionally stable and as a rule of thumb the size of the time step should be equal to or less than half the smallest natural period of a system to avoid instability. This condition implies the calculation of a large number of steps to cover the required range of analysis. Thus unconditionally stable methods which permit the use of large time steps are likely to be more advantageous when analysing multi degree of freedom system.

#### Unconditionally stable linear acceleration of Wilson θ -method

Several different unconditionally stable step-by-step methods have been developed for dynamic analysis of linear and nonlinear structural system. In linear system solution a recurrence matrix solution is used. Many researchers such as Hughes (1976) investigated about Wilson-θ method by assumption of linear acceleration. For the nonlinear system, however, most of the investigations have been concerned with a particular type of structure and nonlinearity.

Different calculations procedures are given for the analysis of linear and nonlinear systems together with the following values of  to ensure unconditionally stable algorithms:

Linear systems

Nonlinear system

For nonlinear system and for any time step (n+1) where the values of  and  are known either from the initial conditions or from the calculation of the previous step.

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