chapter 5 numerical evaluation of dynamic response

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 mu&&+cu& + fS

( )

u u^,& = p t

( )

for inelastic system

with initial condition

( )

0 0

u =u u&

( )

0 =u&0

The applied force is given by a set of discrete values pi = p t

( )

i

where i=0 to N. The time interval

1

i i i

t t₊ t Δ = −

is usually constant, although this is not necessary.

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

i i i i

mu&& +cu& +ku = p

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

1

i+ which satisfy the equation

1 1 1 1

i i i i

mu&&₊ +cu&₊ +ku₊ = p₊

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

The response at time i+1 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.

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.

( )

i ⁱ i

p p p

τ = + ^Δt τ Δ

where Δ =p_i p_i+1− p_i

and the time variable τ varies from 0 to Δt_i.

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

i i

i

mu ku p p

t τ + = + Δ

&& Δ

The response ^u

( )

^τ over time 0≤ ≤ Δτ t_i is the sum of three parts:

1) Free vibration due to initial displacement u_i and velocity u&_i

at τ =0

2) Response to step force p_i with zero initial condition 3) Response to ramp force ⁱ

i

p t τ Δ

Δ with zero initial condition

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 Δt 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 Δt_i is constant Δt. The central difference expression for velocity and acceleration at time i are

1 1

2

i i

i

u u

u t

+ − −

= Δ

& and

1

( )

1

2

i 2 i i

i

u u u

u

t

+ − + −

= Δ

&&

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

1

( )

1 1 1

2

2

2

i i i i i

i i

u u u u u

m c ku p

t t

+ − + − + − −

+ + =

Δ Δ

We assume that u_i and u_i−1 are known from previous steps.

Transferring known quantities to the right hand side, we get

( )

^{2 1}

( )

^{2 1}

( )

²

2

2 ^{i i} 2 ^{i i}

m c m c m

u p u k u

t t

t ⁺ t ⁻ t

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

+ = − − − −

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

Δ Δ

Δ Δ Δ

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

or

ˆ _i 1 ˆ_i ku₊ = p

where

( )

²

ˆ

i 2

m c

k t t

= +

Δ Δ

( )

^{2 1}

( )

²

ˆ 2

i i 2 i i

m c m

p p u k u

t t ⁻ t

⎡ ⎤ ⎡ ⎤

= −⎢ − ⎥ −⎢ − ⎥

Δ Δ Δ

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

The unknown u_i+1 is then given by ₁ ^ˆ

ˆ

i i

u p

+ = k

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Note that u_i+1 is obtained without using equation of motion at time i+1 but from equation of motion at time i.

And u_i+1 can be computed explicitly from the known displacement u_i and u_i−1. Such method is called an “explicit method.”

When i =0, u₋₁ is needed for computing u₁, so we consider

1 1

0 2

u u

u t

− −

= Δ

& and

1

( )

0 1

0 2

2

u u u

u

t

− + −

= Δ

&&

Using the first equation to eliminate u₁ in the second equation, we then have

( ) ( )

²

1 0 0

o 2

u₋ u t u Δt u

= − Δ & + &&

And consider equation of motion at time i = 0

0 0 0 0

mu&& +cu& +ku = p

we get

0 0 0

0

p cu ku

u m

− −

= &

&&

to be used for determining u₋₁

The procedure is summarized next

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

t

T π

Δ <

However, this requirement is never a constraint because the time step needs to be much shorter, typically Δt T/ _n ≤ 0.1, 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 1 1

i i i i

u&₊ = +u& ⎡⎣ − Δγ t u⎤⎦&& + Δγ t u&&₊

( ) ( )( )

²

( )

²

1 0.5 1

i i i i i

u₊ = + Δu t u& +^⎡⎣ −β Δt ^⎤⎦u&& + ^⎡⎣β Δt ^⎤⎦u&&₊

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 ¹₆ ≤ ≤β ¹₄.

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Special Cases

1. Average acceleration

If γ = ¹₂ and β = ¹₄ are chosen, the above equations for u_i+1 and

1

u&i₊ corresponds to the special case that acceleration during the

time step i is constant and equal to the average of u&&_i and u&&_i+1

as can be shown below.

2. Linear acceleration

If γ = ¹₂ and β = ¹₆ are chosen, the above equations for u_i+1 and

1

u&i₊ corresponds to the special case that acceleration during the

time step i varies linearly between u&&_i and u&&_i+1 as can be shown below.

Time-Stepping Formula

This method uses equilibrium equation at time i and time i+1, which involves response quantities at timei+1, i.e. u_i+1, u&_i+1,

and u&&_i+1. Such method is called an “implicit method.”

Let us define the incremental form

1

i i i

u u₊ u

Δ = − Δ =u&_i u&_i+1−u&_i Δ =u&&i u&&i₊1−u&&i Δ =pi pi₊1− pi

From the basis equations of Newmark

(

1

) ( )

1

( ) ( )

i i i i i

u γ t u γ t u₊ t u γ t u

Δ =& ⎡⎣ − Δ ⎤⎦&& + Δ && = Δ && + Δ Δ&&

and

( ) ( )( ) ( )

( ) ( ) ( )

2 2

1 2

2

0.5

2

i i i i

i i i

u t u t u t u

t u t u t u

β β

β

⎡ ⎤ ⎡ ⎤ +

Δ = Δ + ⎣ − Δ ⎦ +⎣ Δ ⎦

= Δ + Δ + Δ Δ

& && &&

& && &&

Solve for Δu&&_i

( )

²

1 1 1

i i i 2 i

u u u u

t β t β

Δ = β Δ − −

Δ Δ

&& & &&

and substitute Δu&&_i in equation for Δu&_i 1 2

i i i i

u u u t u

t

γ γ γ

β β β

⎛ ⎞

Δ =& Δ Δ − & + Δ ⎜⎝ − ⎟⎠&&

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

i i i i

m uΔ + Δ + Δ = Δ&& c u& k u p

It can be written as k u^ˆΔ = Δ_i pˆ_i

We obtain ^ˆ

ˆ

i i

u p

k Δ = Δ

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where

( )

²

ˆ 1

k k c m

t t

γ

β β

= + +

Δ Δ

and ˆ ^{1 1} 1

2 2

i i i i

p p m c u m t c u

t

γ γ

β β β β

⎡ ⎤

⎛ ⎞ ⎛ ⎞

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

Once Δu_i is known, Δu&_i, Δu&&_i, u_i+1, u&_i+1, and u&&_i+1can be computed

1

i i i

u₊ = + Δu u u&_i+1 = + Δu&_i u&_i u&&_i+1 = + Δu&&_i u&&_i

Alternatively, u_&&i+1 can be computed from

1 1 1

1

i i i

i

p cu ku

u m

+ + +

+

− −

= ^&

&&

Newmark’s method is stable if

1 1

2 2

n

t

T π γ β

Δ ≤

−

For Average acceleration method

1

γ = 2 and β = ¹₄

n

t T

Δ < ∞

This implies that average acceleration method is stable for any

Δt, although results would not be accurate for large Δt. For Linear acceleration method

1

γ = 2 and β = ¹₆ 0.551

n

t T Δ <

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; Δ =t 0.1T_n; and compare to the exact analytical solution.

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 Δt T/ _n 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. Δ =t 0.1T_n gives reasonably accurate results, but it also has to be small enough to avoid distortion of the excitation function. For earthquake excitation Δt 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.