Many small buildings are so stiff that they can be assumed to be rigid in a first estimate of earthquake forces. If a horizontal shaking occurs, the forces on each element of the building can be found by assuming that it is static, but has a horizontal force acting on it (through its centre of gravity) proportional to the ground acceleration and to the mass of the element, but in the opposite direction. This is what is referred to as the inertia force. The effect of vertical shaking is similar. The resistance of this stiff building is principally determined by the ability of the structure to transmit these large and rapidly varying inertia forces to the ground without failure.
Consider first a single-storey building, consisting of four walls (with window and door openings) and a flexible roof which sits on two of the walls, but does not tie them together, see diagram A in Figure 1. The effect of a primarily vertical ground shaking will be to increase or decrease the vertical forces, but as the structure is capable of carrying substantial vertical gravitational forces under normal conditions, it can usually accept extra vertical forces without difficulty.
|Figure 1: Response of single-storey masonry building to earthquake ground shaking|
The effect of a horizontal shaking parallel to two of the walls will be to set up horizontal inertia forces on each wall in proportion to their mass: the forces on the walls parallel to the direction of the shaking (the in-plane walls) will be along their length, while those on the perpendicular walls (the out-of-plane walls) will be at right angles to them. The force on the roof will also cause an additional horizontal force to be transmitted on to whichever wall supports it. The principal effect of out-of-plane forces is to cause the walls to bend (i.e. deform out of their plane), which can cause damage to brittle masonry structures even under low levels of loading. Wall elements tend to be stronger under in-plane forces: these cause in-plane shear forces which are easier to resist in a solid wall or can be provided for by bracing or other means.
In the same building the effect of a horizontal force in the direction perpendicular to that just described would be to exchange the responses of the walls, those previously out-of-plane becoming in-plane and vice versa. Thus under a real earthquake shaking, with horizontal shaking in all directions, all walls are subjected to both out-of-plane bending and in-plane shear simultaneously. This type of building tends to have little resistance to earthquake forces.
If instead the roof is constructed in such a way as to tie the tops of the walls together as a rigid diaphragm, the behaviour will be different, as in diagram B in Figure 1. The unresisted out-of-plane bending of diagram A will be prevented, as the out-of-plane wall will be connected to the roof diaphragm member, which is then able to transfer the forces involved to the tops of the stiffer in-plane walls, and then to the ground. In addition the continuity of the roof will also tie the corners together, inhibiting corner cracking. Under shaking in the other plane, the behaviour is the same in reverse.
Thus these elements – the stiff vertical shear wall in each direction, to carry the loads to the ground, and the stiff horizontal diaphragm, to transfer the earthquake forces at this level to the appropriate wall – form the basis of an effective earthquake-resistant structural system. The same system can be used as effectively in multi-storey construction, in which case the horizontal loads to be transmitted by the shear walls increase (as do the vertical gravitational loads) from top to bottom of the building, so that the ground floor walls are required to transmit to the ground the horizontal forces acting on the whole building.
|Figure 2: Alternative earthquake-resistant structural forms: shear wall structures,|
moment-resisting frames and braced frames
However, the use of extensive shear walls can often create serious limitations on the planning of a building, and the equivalent shear strength can also, in some cases, be achieved by means of alternative vertical elements such as braced frames and moment-resisting frames (Figure 2).
In the braced frame, the bracing members transmit the horizontal forces in tension and compression; such frames can be very stiff but are often appropriate only on the external walls of a building. In the moment-resisting frame, the horizontal forces are transmitted by bending moments in the columns and in their framing beams. The moment-resisting frame can be designed (using steel or reinforced concrete) to be as strong as required, but frame structures will tend to be rather more flexible than braced or shear wall structures.
Similarly it is not always necessary (especially in a small building) for a fully rigid diaphragm to be provided at each level. Cross-bracing of a framed floor (steel or timber or trusses), along with the provision of a ringbeam in concrete or even timber, may in some cases be an adequate alternative.
Where a flexible, moment-resisting frame is to be used, care also needs to be taken with the additional bending moments in the columns which arise from the relative displacement of their ends. This so-called P-delta effect can be the cause of rapid material breakdown and collapse if adequate provision has not been made for it.