Response of MDOF structures to ground motion

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Response of MDOF structures to ground motion
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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.
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becomes
or when normalized with respect to modal mass
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where , called modal participation factor
for mode i.
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For a lightly damped (underdamped) system that is
initially at rest, solution can be found using
the convolution/Duhamels integral
Once you have you can find the
contribution of the i-th mode to the response of
the structure.
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Using the modal response, we can find various
response values in each mode.
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
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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
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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.
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As
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.
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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
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We can write the base shear for i-th mode as
But
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is called the effective modal mass of mode i.
The term
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The overturning base moment for i-th mode could
be written as
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Effective modal mass of mode i
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Identical mass, identical story stiffness building
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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
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MODAL DECOMPOSITION APPROACH TO ANALYSE
BASE-EXCITED STRUCTURES
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Examples
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T11.2 sec T20.7 sec T30.4 sec
Total wt900 kip
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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 .
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Distribution of the modal inertial forces follow
the respective modeshape
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Distribution of shear forces in the structure for
the first three modes
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Modal base shear demand
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Question Is there an easier way to estimate
maximum response?
YES!use response spectra
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where
For lightly damped structures ,
so we can approximate
For example, displacements are
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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
.
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How do we find total response?
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Absolute Sum approach (Absolute combination)
Square-root of Sum of Squares (SRSS) combination
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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.
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Base shear
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How do we find total base shear?
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Example 7-story building
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The spectral displacement values at the first
four periods of our 7-story structure are
giving
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giving
Roof displacement
Note that these maxima match the maxima in the
corresponding response histories.
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Roof total displacement estimate
Absolute Sum approach (Absolute combination)
Square-root of Sum of Squares (SRSS) combination
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Rule of thumb Maximum displacement at the roof
is 1.21.5 times the spectral displacement of
the fundamental mode. More like 1.2 for frame
and 1.5 for shearwall buildings.
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Base Shear Force
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Mode Effective Modal Mass (kip-sec2/ft) Effective Weight (kip) Spectral Pseudo-Acceleration, SA (g)
1 18.74 603.4 0.96
2 1.96 63.1 0.84
3 0.62 20.0 0.75
4 0.26 8.4 0.65
5 0.11 3.5 -
6 0.04 1.3 -
7 0.01 0.3 -
Total 700 kip
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Absolute Sum approach (Absolute combination)
Square-root of Sum of Squares (SRSS) combination
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Modal base shear demand
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