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CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC

RESPONSE

Analytical solution is usually not possible when excitation varies arbitrarily with time or if the system is nonlinear. Such problems can be solved by numerical time-stepping methods for integration of differential equations. Time-Stepping Method The equation to be solved is

( )mu cu ku p t+ + =&& & for linearly elastic system or

( ) ( ),Smu cu f u u p t+ + =&& & & for inelastic system

with initial condition ( ) 00u u= ( ) 00u u=& &

The applied force is given by a set of discrete values ( )i ip p t= where i =0 to N . The time interval

1i i it t t+Δ = −

is usually constant, although this is not necessary.

The response is determined at discrete time instant it . The displacement, velocity and acceleration at time it , denoted by iu , iu& , and iu&& , respectively, are assumed to be known and satisfy the equation

i i i imu cu ku p+ + =&& &

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The numerical procedure to be presented will enable us to determine the response quantities 1iu + , 1iu +& , and 1iu +&& at time

1i + which satisfy the equation

1 1 1 1i i i imu cu ku p+ + + ++ + =&& &

We first apply the procedure to time 0i = to determine response at time 1i = and repeat the procedure again to determine response at time 2i = and so on. Therefore, this progressive calculation is called “time-stepping method.”

The response at time 1i + determined from response at time i is usually not exact. Many approximate procedures implemented numerically are possible. The requirements for a numerical procedure are

(1) Convergence—the numerical solution should approach the exact solution as the time step decreases

(2) Stability—the numerical solution should be stable even if there is some round-off error or approximation.

(3) Accuracy—the numerical solution should provide results that are close enough to the exact solution.

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These issues are very important in numerical methods of solving equations. They will govern the limitation of time-stepping procedures. Three types of methods will be discussed:

1) Method based on interpolation of excitation 2) Method based on finite difference expression of velocity

and acceleration 3) Method based on assumed variation of acceleration.

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Method Based on Interpolation of Excitation

This method is highly efficient by interpolation excitation during a time step as a linearly varying function.

( ) ii

i

pp pt

τ τΔ= +

Δ

where 1i i ip p p+Δ = −

and the time variable τ varies from 0 to itΔ .

For simplicity, we will show derivation of this procedure

for a system without damping, although this procedure can be extended to damped systems. The equation to be solved is

ii

i

pmu ku ptτΔ

+ = +Δ

&&

The response ( )u τ over time 0 itτ≤ ≤ Δ is the sum of three parts: 1) Free vibration due to initial displacement iu and velocity iu&

at 0τ = 2) Response to step force ip with zero initial condition

3) Response to ramp force i

i

ptτΔ

Δ with zero initial condition

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Analytical solution derived in Chapter 3 can be used to determined the above three parts of responses and we will get

These formulae are derived from exact solution of the equation of motion. Therefore the result is exact if the excitation is actually varies linearly during each time step as usually assumed for earthquake ground excitation which is recorded at closely spaced time intervals.

The exact solution used in deriving this procedure is available only if the system is linear.

The only restriction on the size of time step is that it permits a close approximation to the excitation function and it provides response results at closely spaced time intervals so that the response peaks are not missed.

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If the time step tΔ is constant, the coefficients A, B , … 'D in this procedure need to be computed only once.

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Central Difference Method

This method is based on a finite difference approximation of the time derivatives of displacements, which are velocity and acceleration. Suppose itΔ is constant tΔ . The central difference expression for velocity and acceleration at time i are

1 1

2i i

iu uu

t+ −−

=Δ

& and ( )

1 12

2i i ii

u u uut

+ −− +=

Δ&&

Substituting these in the equation of motion at time i , we get

( )1 1 1 1

2

22

i i i i ii i

u u u u um c ku ptt

+ − + −− + −+ + =

ΔΔ

We assume that iu and 1iu − are known from previous steps. Transferring known quantities to the right hand side, we get

( ) ( ) ( )1 12 2 2

22 2i i i i

m c m c mu p u k ut tt t t

+ −

⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ = − − − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥

Δ ΔΔ Δ Δ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

or 1

ˆ ˆi iku p+ = where

( )2ˆ

2im ck

tt= +

ΔΔ

( ) ( )12 2

2ˆ2i i i i

m c mp p u k utt t

−

⎡ ⎤ ⎡ ⎤= − − − −⎢ ⎥ ⎢ ⎥

ΔΔ Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

The unknown 1iu + is then given by 1

ˆˆi

ipuk+ =

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Note that 1iu + is obtained without using equation of motion at time 1i + but from equation of motion at time i . And 1iu + can be computed explicitly from the known displacement iu and 1iu − . Such method is called an “explicit method.”

When 0i = , 1u− is needed for computing 1u , so we consider

1 10 2

u uut−−

=Δ

& and ( )

1 0 10 2

2u u uut

−− +=

Δ&&

Using the first equation to eliminate 1u in the second equation, we then have

( ) ( )2

1 0 02o

tu u t u u−

Δ= − Δ +& &&

And consider equation of motion at time 0i =

0 0 0 0mu cu ku p+ + =&& & we get

0 0 00

p cu kuum

− −=

&&&

to be used for determining 1u− The procedure is summarized next

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This central difference method will give meaningless results, called “unstable”, if the time step is not short enough. The requirement for stability of this procedure is that

1

n

tT πΔ

<

However, this requirement is never a constraint because the time step needs to be much shorter, typically / 0.1nt TΔ ≤ , to obtain acceptable accuracy of results. In analysis of earthquake response, a time step about 0.005 sec up to 0.02 sec is chosen to define ground excitation.

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Newmark’s Method This method is developed by Nathan M. Newmark in 1959

based on the following equations:

( ) ( )1 11i i i iu u t u t uγ γ+ += + − Δ + Δ⎡ ⎤⎣ ⎦& & && &&

( ) ( )( ) ( )2 21 10.5i i i i iu u t u t u t uβ β+ +

⎡ ⎤ ⎡ ⎤= + Δ + − Δ + Δ⎣ ⎦ ⎣ ⎦& && &&

The parameters β and γ define the variation of

acceleration over a time step and determine the stability and

accuracy characteristics of the method. Typical selection is

γ =0.5 and 1 16 4β≤ ≤ .

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Special Cases 1. Average acceleration If 1

2γ = and 14β = are chosen, the above equations for 1iu + and

1iu +& corresponds to the special case that acceleration during the time step i is constant and equal to the average of iu&& and 1iu +&& as can be shown below. 2. Linear acceleration If 1

2γ = and 16β = are chosen, the above equations for 1iu + and

1iu +& corresponds to the special case that acceleration during the time step i varies linearly between iu&& and 1iu +&& as can be shown below.

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Time-Stepping Formula

This method uses equilibrium equation at time i and time 1i + , which involves response quantities at time 1i + , i.e. 1iu + , 1iu +& , and 1iu +&& . Such method is called an “implicit method.”

Let us define the incremental form

1i i iu u u+Δ = − 1i i iu u u+Δ = −& & & 1i i iu u u+Δ = −&& && && 1i i ip p p+Δ = −

From the basis equations of Newmark

( ) ( ) ( ) ( )11i i i i iu t u t u t u t uγ γ γ+Δ = − Δ + Δ = Δ + Δ Δ⎡ ⎤⎣ ⎦& && && && && and

( ) ( )( ) ( )

( ) ( ) ( )

2 21

22

0.5

2

i i i i

i i i

u t u t u t u

tt u u t u

β β

β

+⎡ ⎤ ⎡ ⎤Δ = Δ + − Δ + Δ⎣ ⎦ ⎣ ⎦

Δ= Δ + + Δ Δ

& && &&

& && &&

Solve for iuΔ&&

( )2

1 1 12i i i iu u u u

tt β ββΔ = Δ − −

ΔΔ&& & &&

and substitute iuΔ&& in equation for iuΔ &

12i i i iu u u t u

tγ γ γβ β β

⎛ ⎞Δ = Δ − + Δ −⎜ ⎟Δ ⎝ ⎠& & &&

Then, substitute iuΔ & and iuΔ&& in the incremental form of equation of motion

i i i im u c u k u pΔ + Δ + Δ = Δ&& & It can be written as ˆ ˆi ik u pΔ = Δ

We obtain ˆˆ

ii

pukΔ

Δ =

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where ( )2

1k̂ k c mt t

γβ β

= + +Δ Δ

and 1 1ˆ 12 2i i i ip p m c u m t c u

tγ γ

β β β β⎡ ⎤⎛ ⎞ ⎛ ⎞

Δ = Δ + + + + Δ −⎢ ⎥⎜ ⎟ ⎜ ⎟Δ⎝ ⎠ ⎝ ⎠⎣ ⎦& &&

Once iuΔ is known, iuΔ & , iuΔ&& , 1iu + , 1iu +& , and 1iu +&& can be computed

1i i iu u u+ = + Δ 1i i iu u u+ = + Δ& & & 1i i iu u u+ = + Δ&& && && Alternatively, 1iu +&& can be computed from

1 1 11

i i ii

p cu kuum

+ + ++

− −=

&&&

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Newmark’s method is stable if 1 1

2 2n

tT π γ βΔ

≤−

For Average acceleration method

12γ = and 1

4β = n

tTΔ

< ∞

This implies that average acceleration method is stable for any

tΔ , although results would not be accurate for large tΔ .

For Linear acceleration method

12γ = and 1

6β = 0.551n

tTΔ

<

This requirement is not significant because a much smaller

time step is required for accurate representation of excitation

and response.

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Stability

Numerical procedures that give bounded results if time step is shorter than a certain limit are called “conditionally stable.”

Numerical procedures that give bounded results regardless of time step size, no matter how large, are called “unconditionally stable.”

Stability of the method is important for multi-degree-of-freedom system where a unconditionally stable method is sometimes necessary. Computational Error

Error is inherent in any numerical method both from round-off error and approximation of solution.

Let us consider solutions of free vibration using different procedures discussed earlier; 0.1 nt TΔ = ; and compare to the exact analytical solution.

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All numerical methods give results that have amplitude decay, implying that these procedures introduce numerical damping.

Most methods make the period of vibration longer except the central difference method, which gives result that has shorter period than the exact result.

Period shortening in the central difference method is highly significant when / nt TΔ is close to its stability limit 1/π .

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The linear acceleration Newmark’s method seems to be

most accurate in the sense of least period elongation error for

these methods considered for linear SDF system.

The choice of methods would be different for MDF

system or nonlinear response analysis.

The choice of time step also depends on the time

variation of the dynamic excitation and natural period of the

system. 0.1 nt TΔ = gives reasonably accurate results, but it also

has to be small enough to avoid distortion of the excitation

function. For earthquake excitation tΔ is usually less than

0.02 sec.

A useful technique for selecting the time step is to solve

the problem with a time step that seems reasonable and re-

solve the problem with a small time step. The time step is

deemed small enough if results from two analyses are

essentially the same, otherwise reduce the time step and repeat

such comparison until two successive solutions are close

enough.