Home | | **Structural Dynamics and Earthquake Engineering** | Modal response of Earthquake analysis of linear systems

Usually the systems are idealized as lumped-mass systems. In the first part, we will find the structural response as a function of time when the system is subjected to ground acceleration This is known as the response history analysis (RHA) procedure.

**Earthquake analysis of linear systems**

Usually the systems are idealized
as lumped-mass systems. In the first part, we will find the structural response
as a function of time when the system is subjected to ground acceleration This
is known as the response history analysis (RHA) procedure. In the second part,
we can compute peak response of a structure during an earthquake directly from
earthquake response or design spectrum without the need for response history
analysis. The values given by response spectrum analysis (RSA) are fairly
accurate.

1 RHA

Let us assume that
earthquake-induced motion *u***Ë™Ë™*** _{g}* (

Î“* _{n}* is called modal participation
factor, implying that it is a measure of the degree to which the

**Modal response**

Displacement in physical
coordinates may be obtained as

The
equivalent static force is the product of two quantities:

â€¢ *n*th mode contribution;*
*

â€¢ pseudo-acceleration
response of the *n*th mode SDF system.

The *n*th modal contribution
to any response quantity *R*(*t*) may be determined by static
analysis of structures subjected to external force *f _{n}*(

*R _{n}^{st} *may be
positive or negative and is independent of how the mode is

1 Total response

The response contributions of
some of the higher modes may under appropriate circumstances be determined by
simple static analysis instead of dynamic analysis. For short periods *T _{n}*
â‰¤ 1/33

If the period range included is the natural periods from *N _{d}*

2 Interpretation of modal analysis

At first the dynamic properties
natural frequencies and mode shapes of the structure are computed and the force
distribution vector *mi* is expanded into modal components. The rest of
the analysis procedure is shown schematically in Table 18.8.

3 Analysis of response to base rotation

The modal analysis procedure is
applicable after slight modification when the excitation is base rotation.
Consider the cantilever frame shown in Fig. 18.19.

**Multi-storey buildings with symmetrical plan**

The equation of motion for this
structure is (see Fig. 18.20)

Example18.9

The two storey shear frame shown
in Fig. 18.21 is excited by a horizontal ground motion *u***Ë™Ë™*** _{g}*
(

(a) modal
expansion of effective earthquake forces;

(b) the floor
displacement response of *D _{n}*(

(c) the
storey shear in terms of *A _{n}*(

(d) the first
floor and base overturning moments in terms of *A _{n}*(

=* D*_{2}* *(* t *)
which is same as* *[* k *]* *^{â€“1}(* F*_{2}*
*)*Ï‰** _{n}*

Combining we get

*u*_{1}(*t*)
= 0.854* D*_{1}(*t*) + 0.146* D*_{2}* *(*t*)*
u*_{2}(*t*) = 1.207* D*_{1}(*t*) â€“ 0.207* D*_{2}*
*(*t*)

(c) Storey
shear can be determined as follows (see Fig. 18.23). Substituting this we get
storey shear as

*V _{b}^{st} *= 1.4565

*V*_{1}* *(*t *)
= 0.6035* m A*_{1}* *(*t *) â€“ 0.1035* m A*_{2}*
*(*t *)

(d) Static
analysis of the structure for external floor stress *F _{N}* gives
static responses

*M _{b}^{st}
*(

*M*_{1}(*t*)
= 0.604* mh A*_{1}(*t*) â€“ 0.104* mh A*_{2}(*t*)

Example 18.10

Figure 18.24 shows a two storey
frame with flexural rigidity *EI* for beams and columns (span of the beam
= 2*h*). Determine the dynamic response of the structure to horizontal
ground motion *u***Ë™Ë™*** _{g}* (

(a) floor displacement and joint rotations in terms of Dn (t); the bending moments in a first storey
column and in the second floor

(b) beam in terms of An(t).

18.12.1 Modal responses

The relative lateral displacement *U _{in}*(

*U _{jn}*(

The storey drift is

*Dj _{n}*(

= Î“* _{n}*(

The equivalence static force for the *n*th mode *f _{n}*(

*f _{n}*(

*f _{jn}*(

where *f _{jn}*(

*R _{n}*(

*R _{n}
*(

The modal static response *R _{n}*(

2 Total response

Combining the response
contribution of the entire mode gives the earthquake response of the
multi-storey building

The steps
of analysis are given below:

1. Define
ground acceleration *u***Ë™Ë™*** _{g}* (

2. Define
structural properties:

(a) determine
mass and stiffness matrix,

(b) estimate
modal damping ratio.

3. Determine
natural frequencies (*T _{n}* = 2

4. Determine
modal components *R _{N}* of the effective earthquake force
distribution.

5. Compute
the response contribution of *n*th mode by following steps:

(a) Perform
static analysis of building subjected to *F _{n}* forces.

(b) Determine
pseudo-acceleration response *A _{n}*(

(c) Determine
*A _{n}*(

6. Combine
modal contributions *R _{n}*(

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