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..

. ( )

2 :

2

: .

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:

+

(+ ) ( + )

( + ) (

)

.

.

, .

.

2

( ) .

,

. 2 :

(1) + +

() +

()

, ..,

.

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, , , , .

.

()()

()

()

, , ,

= 12 /3

, + ()

, + ()

:

+ + ()

+ ()

( )()

( )

()

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:

+ +

()

()

, .

, + ()

,

, ,

.

+ + ()

, , .

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The k matrix may be assembled as follows: Thejth column ofk can be obtained by

calculating the forces kij(i = 1,2,3 ..N) required to produce uj= 1 (with all otherui= 0).

The use of Direct Stiffness Method is generally used for computer solutions.

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The damping influence coefficient cijis the external force in DOF i due to a

unit velocity in DOFj. The total force fDi at DOFiassociated with velocities

:

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The mass influence coefficient mijis the external force in DOF i due to a unit

acceleration in DOFj. The total force fIi at DOF iassociated with acceleration

:

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The actual distribution of mass is continuous. However, we can use lumped mass

approximation as shown below.

. , ,

.

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Thus the mass matrix is:

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()

:

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mand k are given above.

.

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For dynamic equilibrium:

Hence,

In general,

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Therefore, the system with ground motion is equivalent to a system with stationary based

acted upon by inertia forces as shown in Fig. 9.4.2.

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Once the relative displacementsu

(t) have been determined, the element forcesneeded for structural design can be determined by static analysis at each instant

(i.e., no additional dynamic analysis is necessary. This may be done in one of

two ways:

1. At each time instant the nodal displacement are known from u(t).

From the known displacements and rotations of the nodes of each element

(beams and columns), the element forces (shears and bending moments) can

be determined through the element stiffness properties.

2. The second approach is to introduce equivalent static forces ; at any instant

of timet these forces fS are the external forces that will produce the

displacements u at the same t in the stiffness component of the structure.

Thus

fS(t) = ku(t)

Element forces or stresses can be determined at each time instant by static

analysis of the structure subjected to the forces fS.