Numerical Evaluation of Dynamic Response

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Numerical Evaluation of Dynamic Response Chopra, Chapter 5

Limitations of Duhamel’s Integral

• Assumes linear function

• Closed Form solution not always possible

(specially earthquake loading)

• Not generalized solution - for each load, separate

solution; it is not scalable

Hence, we resort to Numerical Integration

Numerical Evaluation of Dynamic Response

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• Almost all structural analysis software use

numerical integration.

• Numerical solutions can accommodate

nonlinear systems

• Solution can be generalized and computerized


• Time Stepping Methods

The applied force p(t) is given by a set of discrete values

pi = p(ti), i = 0, 1,2, …N. The time interval

∆ti = ti+1 - ti

is usually taken to be constant, although this is not necessary.


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At any time interval the DEM must be satisfied:

m�� + c�� i + (fs)i = pi (5.1.3)

where (fs)i is the resisting force at time ti ; (fs)i = kui for a

linearly elastic system but would depend on the prior history

of displacement and the velocity at time ti if the system were

inelastic.

Response at time ti : �� , �� i , ui must satisfy Eq 5.1.3

• Numerical procedures enable us to determine the response quantities �

� i+1 , �� i+1 , ui+1 at time ti+1 that should satisfy:

m�� i+1 + c�� i+1 + (fs) i+1 = pi+1 (5.1.4)

The known initial conditions, uo=u(0)and �� o=�� 0 ,

provide the information to start the procedure.

• Requirements for a numerical procedure:

� Convergence – as the time step decreases, the numerical solution should

approach the exact solution � Stability – the numerical solution should be stable in the presence of

numerical round-off errors � Accuracy – should provide results that are close enough to the exact

solution


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Three types of time stepping procedure

presented in this chapter:

• Methods based on interpolation of the excitation

function

• Methods based on finite difference expressions of

velocity and acceleration

• Methods based on assumed variation of

acceleration


• Method based on interpolation of excitation function

Note:

Linear interpolation is

satisfactory if the time

intervals are short

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• Method based on interpolation of excitation function.

Consider undamped case

Eq. 5.2.2 m�� + ku = pi +

∆��

∆��

�

Response within ∆�� is the sum of 3 parts:

1. Free vibration due to initial displacement and velocity at τ = 0 (Sec. 2.1)

2. Response to step force pi without initial condition (Sec. 4.3)

3. Response to ramp force ( ∆��

∆�� )τ without initial

condition (Sec 4.4)


Evaluate at τ = ∆�� gives

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• Recurrence formulas

ui+1 = Aui + B�� i + Cpi + Dpi+1 5.2.5a

�� i+1 = A’ui + B’�� i + C’pi + D’pi+1 5.2.5b

• Coefficients A to D’ can be computed from Eq 5.2.4 a and b.

• For underdamped system (ζ<1), the equations can be modified accordingly. The coefficients for this case are summarized in Table 5.2.1.

• If the time step ∆t is constant, the coefficients need to be computed only once.


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3. Compute the theoretical response.

4. Check the accuracy of the numerical results.


• Central Difference Method

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Numerical Evaluation of Dynamic Response Central Difference Method

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• Newmark’s Method

– Family of Integration Methods all focused on acceleration

• Constant ��

• Average ��

• Linear ��



Other combinations of �and�arepossible.

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Numerical Evaluation of Dynamic Response • Newmark’s Method

The parameters � and γ define the variation of

acceleration over a time step and determine the

stability and accuracy characteristics of the method.

Typical selection for γ is ½ and

! ≤�≤

#

γ



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Newmark’s Method

Define the incremental quantities:

Eq. 5.4.1 can be rewritten as:



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Numerical Evaluation of Dynamic Response

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