Response of MDOF structures to ground motion 1

If damping is well-behaving, or can be approximated using equivalent viscous damping, we can decouple the equations of motion using modal decomposition: and separate the system into its natural modes. 2

becomes or when normalized with respect to modal mass 3

where, called modal participation factor for mode i. 4

For a lightly damped (underdamped) system that is initially at rest, solution can be found using the convolution/Duhamel’s integral: Once you have you can find the contribution of the i-th mode to the response of the structure. 5

Contribution of the i-th mode to the displacement at the j-th floor: Interstory drift, i.e. story distortion, in story j is given by the difference of displacements of the floor above and floor below: Using the modal response, we can find various response values in each mode. 6

To find internal forces (story shears, moments, etc.) associated with deformations convenient, we can introduce the concept of equivalent static lateral forces. Equivalent static lateral forces are external forces F which, if applied as static forces, would cause structural displacements x at given time instant. At any instant of time, the equivalent lateral forces associated with displacements due to contribution by mode i : 7

Similarly, we can use inertial forces to find the equivalent lateral forces, the velocity term is at least an order of magnitude smaller than the displacement term, and as such, neglected. 8

for underdamped structures, the equivalent static lateral force at the j-th floor can be found from Internal forces can be determined by static analysis of the structure loaded by the equivalent static lateral forces. As 9

Story shear at j-th story due to response in i-th mode may be calculated by summing the modal inertial forces above and at story j : Total shear force at the foundation level (“base shear”) due to response in i-th mode: Total overturning moment at the foundation level (“base overturning moment”) due to response in i-th mode: : elev. of story j above the base 10

We can write the base shear for i-th mode as But 11

The term is called the “effective modal mass” of mode i. 12

The overturning base moment for i-th mode could be written as 13

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“Effective modal mass” of mode i = 15

Identical mass, identical story stiffness building 16

The total response of the structure is obtained by combining the modal responses in all the modes of vibration. The displacement at the j-th floor, the lateral force at the j-th floor, the base shear, and the base moment are given by 17

MODAL DECOMPOSITION APPROACH TO ANALYSE BASE-EXCITED STRUCTURES 18

Examples 19

T 1 =1.2 sec T 2 =0.7 sec T 3 =0.4 sec Total wt=900 kip 20

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Use the response records to compute inertia forces developed in the structure. Ex: Inertial forces that develop in the structure during 1st mode response. 27

Distribution of the modal inertial forces follow the respective modeshape 28

Distribution of shear forces in the structure for the first three modes: 29

Modal base shear demand 30

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Question: Is there an easier way to estimate maximum response? YES ! use response spectra 32

For lightly damped structures, so we can approximate where For example, displacements are 33

Interested in the “maxima” – the absolute maximum quantities, such as peak displacement, peak interstory drift (story distortion), and such. Ex: Maximum displacement of floor j. First, we find the maximum story displacement for each story and in each mode. Say, we want to find, the displacement of j-th story due to response in i-th mode. The maximum displacement (relative to ground) of a single-degree-of-freedom system with period and damping ratio when excited with the given ground motion. 34

How do we find total response? 35

Square-root of Sum of Squares (SRSS) combination Absolute Sum approach (Absolute combination) 36

CAUTION: When you want to combine the effects of the modes to estimate a reasonable value for the maximum of a response parameter (story displacement, interstory drift, story force, etc.), you need to find value of the response parameter for each mode and then combine using any of the combination rules. Do not use already-combined response parameters (say, story displacement estimates that considered contributions from all modes) to estimate other response parameters (say, story forces); such an approach will result in erroneous estimates. 37

Base shear 38

How do we find total base shear? 39

Example: 7-story building 40

The spectral displacement values at the first four periods of our 7-story structure are giving 41

giving Roof displacement Note that these maxima match the maxima in the corresponding response histories. 42

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Absolute Sum approach (Absolute combination) Square-root of Sum of Squares (SRSS) combination Roof total displacement estimate 44

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Rule of thumb: Maximum displacement at the roof is 1.2~1.5 times the spectral displacement of the fundamental mode. More like 1.2 for frame and 1.5 for shearwall buildings. 46

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Base Shear Force 48

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ModeEffective Modal Mass (kip-sec 2 /ft) Effective Weight (kip) Spectral Pseudo-Acceleration, SA (g) Total : 700 kip 50

Absolute Sum approach (Absolute combination) Square-root of Sum of Squares (SRSS) combination 51

Modal base shear demand 52

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