Page 174 - Dynamic Loading and Design of Structures

Page 150

where Fi is the horizontal force acting on storey i, si, sj are the displacements of the masses mi,
mj in the fundamental mode shape, and Wi, Wj are the weights corresponding to the previous
masses. It is permitted by the code to avoid the calculation of the fundamental mode shape
and assume instead that it is increasing linearly with the height of the building, hence s in eqn
k
(4.24) are substituted by z , the heights of the masses m . (typically the heights of the storeys)
k
k
above the foundation level. The forces Fi are then used for a standard static analysis of the
building, which can be based on two planar models.
In order to cover uncertainties in the distribution of mass and stiffness (of ‘non-structural’
elements), as well as the spatial variability of ground motion, an accidental eccentricity of the
loads F with respect to the mass centre CM of the storey has to be introduced in the analysis;
i
this is equal to

(4.25)

where L is the floor dimension perpendicular to the direction of force F. The eccentricity e 1
i
i
is additional to any existing eccentricity e0 between the stiffness centre Cs and the mass centre
CM at any storey. Instead of applying the forces at an eccentricity from CM, it is usually more
convenient to consider a torsional moment M =Fi(e +e ), or simply F ie1 if a three-dimensional
1
t
0
model is used, acting at the mass centre.
While the aforementioned eccentricities e and e are present in both static and dynamic
1
0
analysis, an additional complication arises when the former is used. It is known that static
analysis underestimates dynamic torsion effects (Chopra, 1995), hence EC8 requires
consideration of an additional eccentricity e to account for the dynamic effect of
2
simultaneous translational and torsional vibrations. Appropri-ate (rather complicated)
expressions for e as a function of the geometry and the stiffness of a storey are given in EC8
2
1–2 (CEN, 1994b).
The load combination involving the seismic loading is

(4.26)

where ‘+’means ‘to be combined with’, ∑implies the combined effect of several actions of
the same type (permanent or ‘dead’ G, variable or imposed Q), Gkj is the characteristic (upper
5 per cent fractile) value of the permanent action j, is the ‘quasi-permanent’ value of
the variable action, γ I the importance factor (Section 4.3.2), and Ed the design value of the
seismic action.

The UBC 1997 procedure
The method is applicable to all buildings in the low seismicity zone (Zone 1) and usual
structures in seismic Zone 2, regular structures with a height up to 73 m, and irregular
structures having no more than five storeys.