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.

FE - More Engineering Content

Okay, so by now I've mostly posted some really, really nice design content.  And I'm going to keep doing that.  After all, this is an architecture website, focused on everything that is designed and built.

But this is also an information website, and as such, I'm going to post some information regarding what any junior engineer should know.  This will include, among other things, the following:

• Fluids
• Mechanics
• Statics
• Dynamics
• Chemistry
• Thermodynamics
• Material Properties
• Mathematics
• Biology

These are not design words.  That's a given.  But are they important to understanding the language of architecture?  Absolutely.  Any LEED accredited building has to incorporate a management system for fluids; any structural engineer knows that a building has to absorb both external and time-dependent displacements and forces to mitigate failure or collapse.  

In short, these are the unspoken tools of any practicing master builder who wants to absorb his/her profession to the best of his/her abilities.

SS - More on Stiffness

The Stiffness Method of Analysis is the basis of all commercial structural analysis programs. Unlike before, where we described stiffness as the unit displacement over a net force, here the focus of this blog will be the development of equations that only take into account bending deformations, i.e., ignore axial member deformations. Within the assumptions, the stiffness method for beam and frame structures is “exact”.

In the stiffness method of analysis, we write equilibrium equations in terms of unknown joint (node) displacements. The number of unknowns in the stiffness method of analysis is known as the degree of kinematic indeterminacy, which refers to the number of node/joint displace-ments that are unknown and are needed to describe the displaced shape of the structure.

One major advantage of the stiffness method of analysis is that the kinematic degrees of freedom are well-defined.

Definitions and Terminology

Positive Sign Convention: Counterclockwise moments and rotations along with transverse forces and displacements in the positive y-axis direction.

Fixed-End Forces: Forces at the “fixed” supports of the kinematically restrained structure.

Member-End Forces: Calculated forces at the end of each element/member resulting from the applied loading and deformation of the structure.

Stiffness Analysis Procedure

The steps to be followed in performing a stiffness analysis can be summarized as:

1. Below is a portal frame with column height "h" and beam length "2h".  The beam has a gross moment of inertia half that of the column.  Determine the needed displacement unknowns at the nodes/joints and label them u1, u2, …, un in sequence where n = the number of displacement unknowns or degrees of freedom.

2. Modify the structure such that only one displacement (u) or rotational displacement is applied.  Let the other translations/rotations equal zero.  This is virtual work that must be analyzed, in other words.

3. Indicate the locations of individual stiffness acting in the frame.  Note: this happens at each degree of freedom.  There are no degrees of freedom at the base, since the base is fixed, but there would be two more degrees of freedom at the base were the columns pinned instead of fixed.



4. Indicate the locations of individual stiffness acting in the frame. Note: this happens at each degree of freedom. There are no degrees of freedom at the base, since the base is fixed, but there would be two more degrees of freedom at the base were the columns pinned instead of fixed.  Below is a diagram which shows what stiffness coefficient is used for either a rotational displacement of 1 radian or a translational displacement of 1.  Note how the end conditions don't exactly match the end conditions in the portal frame as drawn above.  The diagram is just for acquiring stiffnesses, not for replicating end conditions in a structure.



5. Next, after having found all of the stiffness coefficients, place them into a 3x3 matrix and multiply them by the elastic modulus and the moment of inertia of the column (recall the inertia of the beam is half that of the column).  The following matrix should look as follows:

5. Once the matrix is set, divide the matrix into fourths, which will respond to (A) the stiffness caused by translation (k11) and (B) the stiffness caused by rotation at the joints (k12, k21 and k22). 

6.  Lastly, solve for the effective stiffness by putting each structural stiffness component together in the following equation:

So what is the point of all this?  The dynamic movement of a building is a relation of the mass, the damping (to be posted later on the blog) and the effective stiffness of the structure.  The first two are generally investigated or already known.  The latter has to be analyzed.  This blog was to show how that can be done. 

Finally, by taking all three components, a universal equation governing displacement and force can be derived as follows:

This equation is essentially the most important governing equation in all of structural analysis.  Dynamically, it helps predict stiffness, displacement and force all in one unified equation.  By understanding why this equation works the way it does, an architect not only begins to speak the language of structures - rather, he becomes a master of structures entirely. 

SS - And Now For Something Completely Different

As of last month, I started taking courses to become a structural engineer. This, I thought, would be a nice complement to being an architect already.

If I'm to admit the truth, after taking several ARE exams, I've sort of reluctantly fallen in love with the idea of being structurally trained. The structures exams are widely considered the most difficult on the ARE, therefore one should (I believe) overcompensate in this realm of design. Reading the diaries of many architects, such as Frank Lloyd Wright and Mies van der Rohe, who enjoined their pupils to take on the science of the profession, further illustrates the truth that great architecture lies in great construction.

So this is a gift to you - namely, the homework problems that I've taken in my structural analysis courses. Enjoy.