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Numerical Evaluation of Dynamic Response
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9/18/2014
1
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
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• Almost all structural analysis software use
numerical integration.
• Numerical solutions can accommodate
nonlinear systems
• Solution can be generalized and computerized
Numerical Evaluation of Dynamic Response
• 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.
Numerical Evaluation of Dynamic Response
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Numerical Evaluation of Dynamic Response
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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Numerical Evaluation of Dynamic Response
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
Numerical Evaluation of Dynamic Response
• Method based on interpolation of excitation function
Note:
Linear interpolation is
satisfactory if the time
intervals are short
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Numerical Evaluation of Dynamic Response
• 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)
Numerical Evaluation of Dynamic Response
Evaluate at τ = ∆�� gives
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Numerical Evaluation of Dynamic Response
• 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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Numerical Evaluation of Dynamic Response
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3. Compute the theoretical response.
4. Check the accuracy of the numerical results.
Numerical Evaluation of Dynamic Response
• Central Difference Method
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Numerical Evaluation of Dynamic Response
• Central Difference Method
Numerical Evaluation of Dynamic Response Central Difference Method
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Numerical Evaluation of Dynamic Response
• Central Difference Method
Numerical Evaluation of Dynamic Response
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Numerical Evaluation of Dynamic Response
• Central Difference Method
Numerical Evaluation of Dynamic Response
• Central Difference Method
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Numerical Evaluation of Dynamic Response
• Newmark’s Method
– Family of Integration Methods all focused on acceleration
• Constant ��
• Average ��
• Linear ��
Numerical Evaluation of Dynamic Response
• Newmark’s Method
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
Numerical Evaluation of Dynamic Response
• Newmark’s Method
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Numerical Evaluation of Dynamic Response
Newmark’s Method
Define the incremental quantities:
Eq. 5.4.1 can be rewritten as:
Numerical Evaluation of Dynamic Response
• Newmark’s Method
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Numerical Evaluation of Dynamic Response
• Newmark’s Method
Numerical Evaluation of Dynamic Response
• Newmark’s Method
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Numerical Evaluation of Dynamic Response
• Newmark’s Method
Numerical Evaluation of Dynamic Response
• Newmark’s Method
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Numerical Evaluation of Dynamic Response
• Newmark’s Method
Numerical Evaluation of Dynamic Response
• Newmark’s Method
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Numerical Evaluation of Dynamic Response
• Newmark’s Method
Numerical Evaluation of Dynamic Response
• Newmark’s Method
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Numerical Evaluation of Dynamic Response
• Newmark’s Method
Numerical Evaluation of Dynamic Response
• Newmark’s Method
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Numerical Evaluation of Dynamic Response