base shear modal mass or brevity effective modal mass.

**Effective modal mass and modal height**

The base shear for â€˜*n*â€™
mode is calculated as

*M _{n}*

Example 18.11

Determine the effective modal
mass and the effective modal height for the frame shown in Fig. 18.29.

Solution

The modal distribution of
effective earthquake force is given in Fig. 18.30. Taking moment at base

The effective modal mass and modal height are indicated in
Fig. 18.31.

Example 18.12

Consider a five storey building
(Example 18.1) (see Fig. 18.32) whose properties are given. Calculate effective
modal mass and height.

The effective modal mass and modal height are shown in Fig.
18.34.

The ground acceleration *u***Ë™Ë™*** _{g}*
(

Base shear = *m* (7.0926*A*_{1}(*t*) +
0.6471*A*_{2}(*t*) + 0.2*A*_{3}(*t*)

+ 0.0528*A*_{4}(*t*)
+ 0.0003*A*_{5}(*t*))

Base moment = *mh* (7.0927 Ã— 3.079*A*_{1}(*t*)
+ 0.6471 Ã— (â€“1.563)*A*_{2}(*t*)

+ 0.2*A*_{3}(*t*)
+ 0.0528 Ã— (â€“0.587)*A*_{4}(*t*)

0.0003 Ã— 7.67*A*_{5}(*t*))

**Multiple support excitation**

There are certain examples in
which the ground motion generated by an earthquake is different from support to
support. For example the Golden Gate Bridge is 1965 m in length and the ground
motion is expected to vary significantly over the length of the base at the two
ends of the bridge.

Example18.13

A uniform two span continuous
bridge shown in Fig. 18.35 with flexural stiffness *EI* idealized as
lumped mass. Let us formulate the equation of motion subjected to vertical
motion at 1, 2, 3 as *u _{g}*

Formulation stiffness matrix 10 Ã— 10. Assuming translational displacement as master and other
degrees of freedom as slaves we get reduced stiffness matrix of size 5 Ã— 5.

**Apply unit load at 3 (to get
influence vector i_{1})**

Moment at 4 = L

Calculate *u*_{2} when *u*_{3} = 1

The â€“ sign shown the deflection at 2 is downward.

Now calculate deflection at 1 due
to unit load at 3= deflection at 3 due to unit load at 1.

**Apply unit load at 1**

Both deflections at 3 and 1 are
upwards and hence positive. If *u*_{3} = 1 let us calculate what
is *u*_{1}.

To find the influence vector *i*_{2},
apply unit load at 4 and find the deflection at 1 and 4. The deflection at 4
due to unit load at 4 is given by *L*^{3}/6*EI*. If the load
is at a distance of Ã—2 and the deflection is to
be calculated at Ã—1, then deflection at Ã—1 is given

When the deflection at 4 is equal
to 1 what is the deflection at 1 and 2 which may be calculated as

**Symmetric plan
buildings: translational ground motion**

Consider an *N*-storey
symmetric plan building having rigid floor displacement and several frames in
each *x* and *y* direction as shown in Fig. 18.36.

1 One storey, two way unsymmetric system

Consider the idealized one storey
frame shown in Fig. 18.37. Assume the diaphragm is rigid. Assume frame *A*
is located at a distance of *e*:

2 Equation of motion

Considering earthquake excitation defined by *u***Ë™Ë™*** _{gx}*
(

The above equations are coupled.
Thus the response of the system to *x* and *y *components of ground
motion is not restricted to lateral displacement* x *and* y *directions
but will also include lateral motions in tranverse directions and* *the
torsion of the roof diaphragm about the vertical axis.

In Fig. 18.36, if frame *A* passes through the centre of
mass â€˜*O*â€™, then (*e* = 0) (*k _{xB}* =

All three equations are uncoupled and solved.

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Civil : Structural dynamics of earthquake engineering : Effective modal mass and modal height |

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