SS - Seismic Design

A Good Example of How Material Ductility Impacts Lateral Structural Performance

How do buildings survive strong earthquakes without collapse?  That is a question that has puzzled expert structural designers and analysts for ages.  To begin to understand, however, the rigors involved with earthquakes... well, that is a simpler puzzle.  Let's elucidate that a little bit:

Above is the visual depiction of structural ductility. Ductility is the ability of a structure (or element of a structure) to withstand large inelastic deformations without significant loss of strength. It is equal to the inelastic deformation capacity of a structural member.

For a simple, elastic - perfectly plastic structure subject to monotonic loading, ductility can be quantified using the ductility factor, μ. In the definition of μ, Δyield is the displacement when the structure yields (i.e. when the structure has reached its plastic lateral capacity), and Δfailure is the displacement at which the structures begins to lose lateral load carrying capacity.

A basic concept of seismic-resistant design is the trade-off between strength and ductility. This concept is illustrated (in a highly simplified manner) by this slide.

The plot shows the relationship between lateral force and lateral displacement for a simple single-degree-of-freedom structure, with an elastic-perfectly plastic response. The plot can be viewed as the force-displacement response of the structure for a half-cycle of loading during an earthquake.

The solid line represents the response of a structure that remains elastic during the earthquake. The maximum lateral force experienced by the elastic structure is Helastic and the maximum displacement experienced by the structure is Δmax. Thus, if we wanted the structure to remain elastic during the earthquake, we would need to design our structure to remain elastic under a lateral load equal to Helastic.

Say we designed the structure to have a lateral strength of only 3/4*Helastic. If the structure sees the same earthquake as above (which will generate a force of Helastic in an elastic structure), the structure will yield when the lateral force reaches 3/4*Helastic. At this point, the lateral force on the structure can no longer increase. Rather, the effect of the earthquake on the structure beyond this point will be to impose inelastic displacement. That is, the earthquake will demand ductility of the structure rather than strength. The earthquake will continue to deform the structure in the inelastic range until the total displacement is about the same as for the elastic structure. Thus, a structure with a lateral strength less than Helastic can still survive the earthquake, as long as the structure can supply the needed inelastic deformation. i.e., can supply the needed ductility.

Similarly, we can design our structure with even lower levels of lateral strength, say 1/2 or 1/4 of the lateral force that an elastic structure would see. In each case, when the lateral strength of the structure ( 1/2*Helastic or 1/4*Helastic ) is reached, the structure will be incapable of resisting any additional lateral force. As before, the effect of the earthquake beyond this point will be to impose additional displacement upon the structure, rather than additional force. In each case, the maximum displacement will be about same as for the structure that remains elastic. That is, regardless of the structure's lateral strength, Δmax will be approximately the same.

Remnants of the San Francisco Earthquake in 1906

Some observations on seismic response......
• A structure can be designed with a lateral strength significantly less than that which will be seen by an elastic structure in an earthquake. However, to survive the earthquake without collapse, the structure must supply ductility. In the plot, ductility (inelastic deformation capacity) is represented by the horizontal dotted lines).
• As illustrated by the plot, the lower the lateral strength of the structure, the greater will be the required ductility. Thus, in seismic design, we can trade strength for ductility. We can give our structure a high lateral strength, in which case we need to provide little ductility. Alternatively, we can give our structure very low lateral strength (by designing for very low lateral forces), but then we must detail our structure to supply high levels of ductility. Building codes permit us (within a limited extent) to make this trade off between strength and ductility.
• Ductility means damage. That is, when we use ductility to survive an earthquake, we have to expect damage. For a structure that is designed to yield in an earthquake (the usual case), the maximum lateral force that the structure will see during the earthquake is defined by the structure's own lateral strength. In building codes, the Amplified Seismic Load is intended to provide a rough estimate of a structure's lateral strength, and therefore is intended to provide an estimate of the maximum lateral force that can be experienced by a structure in an earthquake. (The Amplified Seismic Load will be discussed in more detail later).
• Code specified seismic lateral forces are generally much smaller than would be required for the structure to remain elastic. That is, they are usually much less than Helastic. Thus, a typical code based design uses ductility to survive an earthquake. In this sense, the code specified seismic lateral forces do not represent the actual lateral force that an earthquake would generate in an elastic structure. Thus, in cases where code specified wind forces are greater than code specified earthquake forces, it is still necessary to provide ductility. Even though the lateral strength of the structure will be larger as a result of the fact that "wind controls," the resulting lateral strength of the structure is still likely well below Helastic, and therefore ductility will be needed. Thus, even when code specified wind forces are larger than code specified earthquake forces, ductile detailing requirements in building codes must still be satisfied.

So to recap, structures must have adequate strength and ductility to resist earthquakes.  Strength is not the only important determinant of stability and impact resistance.  So the construction of heavier structures with redundancies (multiple ways of transferring load to a foundation) does not necessarily infer a wise and conservative decision by a structural designer.  And oftentimes, if these structures are swayed fiercely by lateral forces, it can quickly become a highly unwise or possibly even regretful decision.  So to all designers out there who value strength performance over stiffness: be wise.

Note above: The concept that Δmax remains the same, regardless of the lateral strength of the structure, and regardless of whether the structure responds elastically or inelastically, is a simplification. It is a useful simplification to understand the basic concept of trading strength for ductility in seismic design.