# Displacement Based Design of Steel Frames

Direct Displacement Based Design of multi story buildings is based on the generation of equivalent SDOF system or substitute structure concept. For this purpose, it is assumed that the structure vibrates in a pre-defined harmonic displaced shape. The base shears and the works developed by lateral external forces are also assumed the same for both equivalent and main structures [7-15]. Consider the relative displacement vector {8(h, t)} for the multistory building with total height of H expressed in a decomposed form of displacement and time and assume a harmonic response with amplitude A for the system. We can write,

{S(h, t) = A. Sin(a>.t).{ф(^)} , 0 < h < H (1)

which results in an acceleration vector {(h, t)} proportional to the assumed normalized displacement vector Ф^) as follows,

{(h, t)} = – A. w2 .Sin(at ).{h)} = – a2 {(h, t)} (2)

In order to obtain the equivalent system parameters, we define the normalized displacement vector {(h, t)} as,

{(h, t)} = 8- {(h, t)}

8eff

where Seff is called the effective displacement. From equation 2 and 3 we may have,

in which n stands for number of stories, aeff is called the effective acceleration of the equivalent

SDOF system and 8t, at are the story displacement and acceleration respectively. Using equation 4,

the base shear can now be determined in terms of the multi story structure and the equivalent system parameters as,

i=1

which leads to the definition for the effective mass as m

eff

fi may also be determined using equations 4 and 5 as,

= mi. S, у

Jin Vb m

The effect of story ductility may be considered substituting St defined in equation 4 with the following relationship,

Si =V, S (9)

where fti is the story ductility demand and Syi is the story yield displacement. The problem is now how to determine these two parameters. The story yield displacement Syi may be obtained by

defining the story yield mechanism and has been discussed later in section 3. Determination of the ductility distribution through the height of the structure has also been discussed in section 3. Finally for detail design of the structure the base shear is obtained as Vb = Kf. Sf and then the story forces

fi are computed using equation 6. Then the capacity design of the structure can be started

considering the ductility capacities. This capacity-designed structure may then be verified using the time history or static push over analyses.