What is a Response Reduction factor?

In our previous blogs we discussed about Response Spectrum Analysis, Earthquake and Energy Dissipation as well as Ductility demand in structures during seismic loading. In response spectrum analysis topics like mode shapes, modal mass participation factors, derivation of response spectrum we discussed. In earthquake vs energy dissipation blog, we talked about energy dissipated from buildings through strain energy, inelastic energy, hysteresis, damping and ductility. In ductility demand we discussed about importance of ductile detailing and how it helps a building to work during earthquakes just like a marathon runner during long runs. 

Generally inelastic energy dissipation, damping energy, ductility demand and ductility capacity, hysteresis loops are all captured when a nonlinear model is built, and time history analysis is performed for the structure. But to do nonlinear time history analysis, it takes a long time to build a model. The performance evaluation and result verification should be carried out by multiple sets of experienced eyes. It is never easy to identify the errors. It is not worth every single time to perform nonlinear analysis just because the project timeline does not give you the option to do so or the budget of the project is not big enough or sometimes code level linear analysis suffice the requirements based on the structure type.

If one wants to do linear elastic analysis of a structure under full earthquake loading and calculate forces acting on the structure, the resultant forces will be significantly large, and the design of the structure will be uneconomical and not feasible at all. Under such conditions we are not taking advantage of the fundamental concept of ductility and inelastic energy dissipation of materials. Any structure is permanently loaded under dead and live loads. To make sure that the structure does not crack significantly under these gravity loads, we make sure that static demand is always than the yield capacity of the members. We make sure that the deflection criteria are always satisfied, and floor vibrations are not exceeding certain limits. But earthquake loading is not something that is applied to a building permanently. Because a building is designed to make sure that it does not collapse under such an extreme event, we can let the building crack and let it move past the yield deformation.

Designing a structure that is supposed to stay elastic under these seismic demands, which are generally significant, does not make sense. After all what is the probability that the building will even see the seismic loading that we just designed for? Life expectancy of residential and office buildings is 100 years max and an earthquake with significant shaking has a probability of occurring every 500 years (in most regions). There are few other regions where the frequency of earthquakes is as small as 40 years and places like Mexico and Taiwan are experiencing significant shaking every 20 years or so.

How did we come up with a Response reduction factor? 

Equal Area Approximation

Well, it started with something called equal energy approach. People tried to calculate how much energy under the curve is dissipated by a building when analyzed for an earthquake. How did they do that? Simple, we can generate a pushover curve for a building which is a curve of base shear vs displacement of the structure. We know that the energy applied will be equal to the base shear times the displacement of the building. That is the area under the curve. Now, let us say instead of creating a nonlinear model, what if we take the linear model and push the structure to an extent till which it dissipates the energy equal to that of a nonlinear model? We will get a higher force value for base shear and some displacement so that the area under the curve of triangle will be equal to an area under the rectangle.

Image (A1) : Area under an elastic perfectly plastic model tested under a pushover curve

Image (A2) : Area under a linear model without any plasticity modeled which is same as the area under a pushover curve

If we look at the second image talking about a linear elastic model, we can see that because it does not go in plastic state, to capture the energy same as inelastic model, the base shear must increase, that means the external for increases as the displacement increases. (Remember in linear analysis, force is always proportional displacement). Now at certain force and displacement value, the area under the curve in this elastic model reaches the same as area under EPP curve. The base shear corresponding to this value is called Ve and the base shear at which the structure yields is called V_se.

Thus, the Force reduction factor can be represented as:

R = Ve/V_se

The fundamental problem with the way we derived the force reduction factor above is that we are not accounting for the actual energy dissipation from the structure and the other reason is, we are checking the area under the curve, but we are not checking the displacement capacity of the structure. If you carefully check in the two images above, we can see that in EPP curve we are looking at a displacement that is different than the displacement in elastic force displacement curve. This violates the fundamentals of seismic analysis where it is important to check the displacement capacity of the structure. So, we moved to a different system for calculating the force reduction factor.

We moved to a new method of estimating the force reduction factor where we use the equal displacement approach.

Equal Displacement and System over-strength factors

In equal displacement approach, a nonlinear model of a structure is created and tested under seismic loading where system's over-strength factor, hysteresis energy dissipation, level of ductility till life safety as well as extent of significant development of plastic hinge through the entire structure is measured. Once the building satisfies criteria of life safety, then a system over-strength is calculated as well as a level of ductility is also calculated before computing the force reduction factor. How is it done?

To calculate system over-strength, a structure with material over-strength as well as system over-strength is modeled. What do I mean by material over-strength? Material over-strength represents the actual strength of material per testing and not the specified strength. Members in structural systems are designed with a specified strength which is used along with a strength reduction factor. This leads to a capacity of material that is under-estimated compared to the actual capacity.

For example, a concrete beam is designed for a moment demand of M, but the capacity of the beam is ΦM, also apart from this the expected capacity of concrete is 1.3 x f'c while that of steel is 1.17 x fy. M x 1.17/0.9 = 1.3. This leads to an actual capacity that is 1.3 x M thus a 30% reserve strength of the member. I have ignored concrete intentionally because concrete does not significantly increase moment capacity of beam as the depth of concrete block in compression is not significantly impacted.

A design capacity is like a checking account of a bank which is in constant use, while the remaining of actual capacity or also known as material over-strength acts as a savings account for the structure which can be depleted when there is significant demands from seismic action or some other accidental loads. This savings account or the reserve strength of each individual member contributes to the total over-strength of the structure. Thus, during earthquake, any building is stronger than what it is designed for because of this reserved strength.

Similarly, any building goes into inelastic state because we want the building to perform that way. Inelastic energy dissipation and supplemental damping is the best way to dissipate seismic energy demands. So, a structure designed for ductility can go into inelastic state, yield and crack and dissipate energy in such a manner. But because we are performing linear analysis, we cannot represent that level of energy dissipation, so the only way to account for this is equal displacement approximation. How this works? After performing a time history or any other method of evaluation of building that considers actual seismic loading, we check the maximum displacement demand on the structure. Then we create another linear model and push it to a similar displacement value that we obtained from linear model and check the base shear of the structure. But we know that the structure has already yielded before at a base shear value significantly lower than this full seismic demand. This ratio is known as response reduction factor accounted for ductility.

Image (B) : Courtesy of ASCE 7-10

Now we all know what ductility demand is, it is ratio of inelastic displacement to elastic displacement.

μ = Δ_inelastic/Δ_elastic

R = R_µ+ R_Ω

R_µ = Response reduction considering the system ductility

R_Ω = Response reduction considering system over-strength
(Caution: This over-strength is different than the over-strength factor Ω_o specified in building codes)

If we look carefully at image B, V_s is the design base shear of the structure after using an R factor on the response spectrum. V_Y is the actual strength of the structure when we consider the savings account in addition to checking account. You can also say that it is the net worth of the structure. Similarly, V_E is the base shear of the structure when we displace the structure equal to that of a model where structure loses its stiffness and strength as we push the structure. Why V_E is such a large value? Let us do something, in one hand take a plastic bag and in the other take a rubber band and stretch both. After a point, it will be very easy to stretch a plastic bag and you no longer have to increase the force applied in order to do so. But in case of rubber band, the more you stretch, the higher the force you must apply. In linear elastic analysis, stress is always proportional to strain, so the more you push a structure, larger will be the forces in structure.

We know that the current structure does not actually possess this much strength, so what do we do? We reduce the forces on the structure by R_µ factor, to get to the actual strength of the structure, but if we reduce the forces only to account for ductility then what happens? Well, we design the structure for certain force demands with specified properties and phi factors. If we take the over-strength force level and design the structure for specified capacity, then actual capacity of structure will be higher as we discussed. So, we have to reduce the forces acting on the structure by R_Ω factor which accounts for structural over-strength and then design the structure with appropriate material properties and phi factors.

Thus, the combination of the two gives us the correct response reduction factors. But what happens to the deformation check of the structure? How do we check if the structure will be deforming under the drift limits? To encounter that, a factor called amplification of displacements is added into the picture called a displacement amplification factor C_d. What this displacement amplification factor does is, it takes the displacement of the structure at reduced force levels, amplifies it by a number to account for stiffness loss of the structure and in-elasticity of the structure and tries to predict the actual drifts. This helps in checking the deformation demands on non-structural elements, components and cladding, checking demands on columns because of this amplified lateral deformation.

So how do we design buildings with this R factor? Here are the steps:

1. Take the response spectrum that is specific to the building site.
2. Calculate approximate period of the building to figure out the spectral accelerations for that building.
3. Reduce the forces by the factor of R.
4. Check the structural demands and basic checks for torsional irregularity. 
5. Amplify the displacements by C_d and check drift demands which should be less than allowable drifts.
6. If strength or stiffness is not enough, increase either that is deficient and start again from step 2.

But why does R factor vary among different building codes? 

It turns out that the R factor is not consistent within the structural engineering community. There are many reasons behind that, first because it is based on many assumptions in design philosophy and analysis methodology. Second, research has consistently shown that R factor is conservative for very short structures, it underestimates demands for tall structures and the position of R factor is questionable even for midrise buildings. And hence the dispute between the community. We will keep this discussion for another post.

I hope I was able to clear your ideas on the fundamentals of response reduction factors. Do let me know if you had any confusion anywhere and I will be happy to clarify it.

Thank you