6 - SDOF - General Loading

Structural Analysis-

spring 2013

(single degree of freedom – general Dynamic

Loading)

Fawad Muzaffar M.Sc. Structures (Stanford University)

Ph.D. Structures (Stanford University)

Civil Engineering

Department

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• Response to Unit Impulse:

– For a unit magnitude of impulse

p approaches ∞ as 𝜀 → 0

– From Newton’s 2nd Law

Rate of change of momentum = Force

or

For constant mass

Equation -1

or Magnitude of Impulse = Rate of Change of Momentum

Response to General Dynamic Loading – Superposition Methods

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–Note: spring and damper effects are neglected because duration of impulse is

short and these components don’t have time to respond.

– For a freely vibrating dampless system, the response is given by

---Equation 2

–𝑣 (0) can be calculated using Equation-1

–𝑣(0) is calculated by noting that

–Plugging initial conditions into Equation 2 results in

ℎ 𝑡 − 𝜏 ≡ 𝑣 𝑡 =1

𝑚𝜔sin[𝜔(𝑡 − 𝜏)]

– For a freely vibrating damped system, the response is given by

----Equation 3


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• Plugging in the initial values into equation 3

ℎ 𝑡 − 𝜏 ≡ 𝑣 𝑡 =1

𝑚𝜔𝐷sin 𝜔𝐷 𝑡 − 𝜏 𝑒−𝜉𝜔(𝑡−𝜏)

• Response to Arbitrary Loading:

• An arbitrary load can be represented as a series of infinitesimally short impulses.

• The response to any one of the impulses is given by

𝑑𝑣 𝑡 = 𝑝 𝜏 𝑑𝜏 ℎ 𝑡 − 𝜏 𝑡 > 𝜏

• The response of the system at time t is the sum of responses upto this time i.e.

𝑣 𝑡 = 𝑝 𝜏 𝑑𝜏 ℎ 𝑡 − 𝜏 𝑑𝜏𝑡

0

• Note: Above Integral is known as Convolution Integral.

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• The convolution integral for SDOF damped system is called Duhamel’s Integral i.e.

𝑣 𝑡 = 𝑡 =1

𝑚𝜔𝐷 𝑝(𝜏) sin 𝜔𝐷 𝑡 − 𝜏 𝑒−𝜉𝜔(𝑡−𝜏)𝑡

0

𝑑𝜏

• For an undamped system

𝑣 𝑡 =1

𝑚𝜔 𝑝(𝜏)sin[𝜔(𝑡 − 𝜏)]𝑡

0

𝑑𝜏

• Suitability:

Can be used to evaluate response of SDOF systems due to arbitrary forces.

• Limitations:

This technique can only be applied to linear systems.

For an arbitrary 𝑝 𝑡 ,the Duhamel Integrals need to be calculated

numerically 5

Response to General Dynamic Loading – Numerical Methods

• The Problem Statement

– Variation of applied force with time is given.

– The duration of applied force is divided into segments, each Δ𝑡 long.

– The value of applied force, 𝑝𝑖 at the end of each discrete time interval is evaluated.

– 𝒖𝒊, 𝒖 𝒊 and 𝒖 𝒊 at the start of each discrete interval are known and satisfy

– Calculate 𝒖𝒊+𝟏, 𝒖 𝒊+𝟏 and 𝒖 𝒊+𝟏 such that

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• Requirements of Numerical Procedures Three essential requirements are i. Convergence – Numerical solution should approach exact solution with decrease

of time step, Δ𝑡 ii. Stability – Numerical solution should be stable despite of round off errors. iii. Accuracy – Numerical solution should be close enough to exact solution.


• Types of Numerical Solution

• Three types of numerical procedures are commonly used

i. Methods based on interpolation of excitation function.

ii. Methods based on finite difference of velocity and acceleration.

iii. Methods based on assumed variation of acceleration.

• Methods Based on Interpolation of Excitation Function (Piecewise Exact Method)

Calculation Procedure:

i. The excitation function is divided

into discrete intervals, usually

defined by significant changes of

slope.

ii. For notation, refer to adjacent figure

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• The assumed linearly varying loading

during time step is given by

The equation of motion of a structural

system leads to

The total response consists of 3 components (for a dampless system).

i. Free response due to initial displacement 𝑢𝑖(0) and velocity 𝑢 𝑖(0).

ii. Response due to step force 𝑝𝑖 with at rest initial conditions.

iii.Response due to ramp force Δ𝑃𝑖 Δ𝑡𝑖 with at rest initial conditions.

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• The displacement response of the system is given by

• Differentiating the above equation yields

• At 𝜏 = Δ𝑡; 𝑢 𝜏 = 𝑢𝑖+1 and 𝑢 𝜏 = 𝑢 𝑖+1, the above equations can be written as

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• The above equations can be written in recursive form as

• For an under-critically damped system, the coefficients of above equations becomes

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Example 1 - Accuracy of Piecewise Linear Solution Algorithm

Properties of the Structure:

Approximation of Loading Function:

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• Calculated Response:

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• Advantages:

i. The calculation procedure is highly efficient.

ii. The only restriction on step length is the slope of the loading function.

• The Central Difference Method

Problem Statement

The 𝑣 𝑡 , 𝑣 (𝑡) and 𝑣 (𝑡) at time

𝑡1 needs to be evaluated such that

𝒎𝒗 𝟏 + 𝒄𝒗 𝟏 + 𝒌𝒗𝟏 = 𝒑𝟏 --- Equation 3

Step # 1: Calculate approximate values

of 𝑣1 and 𝑣 1.

This is done by first approximating 𝑣 −1/2

and 𝑣 1/2

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• The acceleration at 𝑡0 can then be evaluated

or

Plugging this value of 𝑣0 into equation 3

Solving for the displacement at the end of

the time step

𝑣−1 still needs to be evaluated and this is done by assuming that

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• From this expression

• Substituting the above value of 𝑣−1, the displacement at 𝑡1is given by

• To calculate velocity at 𝑡1, the following approximate relationship is used

From which

Advantage:

• This is an explicit method.

Disadvantage:

• The method is only conditionally stable. The condition will blow up if

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• Integration Methods

• Euler-Gauss Method (Constant Acceleration Method)

− Acceleration is assumed to remain constant during time step.

− Algorithm:

i. Acceleration at start of step is known.

ii. The acceleration at end of step is assumed.

iii. Calculate 𝑣 1 and 𝑣1

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For constant acceleration assumption


iv. Using 𝑣 1 and 𝑣1 calculated in step iii, calculate 𝑣 1 𝑚𝑣 1 + 𝑐𝑣 1 + 𝑘𝑣1 = 𝑝1 ⇒ 𝑣 1 = 1 𝑚 × (𝑝1 − 𝑘𝑣1 − 𝑐𝑣 1)

iv. Compare 𝑣 1 calculated in step iv with assumed value. Iterate if necessary.

Advantage:

i. The method is unconditionally stable.

Linear Acceleration Method:

This calculation algorithm is identical to previous algorithm, except that variation of acceleration is assumed to be linear in this case.

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6 - SDOF - General Loading

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