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# Why Should we Drop R?

Jinal Doshi, P.E.- Project Manager at DCI Engineers, USA

Seismic design of structures is seen as one of the most challenging parts of structural engineering. It may not seem as difficult once you start learning, or even if you have practiced it for some years. It may just appear to be some simple mathematical equations that anyone can follow through. But, as you start understanding the underlying fundamentals and assumptions that are involved in building these equations, it will not be long before we start looking at ourselves in the mirror. In every single building code, seismic design of buildings is very simplified, the reason being, the times in which it was developed. It, in some form or the other is based on a similar underlying principle.

History of Seismic Design

It was during the early 1900s when the first seismic design philosophy was introduced to the world. What happened back then? Frequent high magnitude earthquakes started hitting modern cities around the world. Engineers observed that buildings that were designed for wind lateral forces performed better under seismic demands than those without specified lateral design. Initially, people started designing buildings for 10% of the building weight, regardless of the building height or building period. As the understanding of the building dynamics became better understood in the mid-1900, we started developing period dependent design lateral forces that reduced in proportion to building period.

Later on in the early 1960s as the understanding of inelastic time history analysis became prominent to us, we perceived from our calculations that many of the buildings should have collapsed, as it had already developed inertial forces of several magnitudes higher than their lateral strength during the past earthquakes. But surprisingly they didn’t and we were introduced to the concept of ductility. Engineers then built relationships between ductility factors and force reduction factors using equal energy approximations which were more appropriate for short period structures. This was the beginning of our understanding of the current principles of the “Force-Based Seismic Design” of buildings. We do use a check of displacements, but it is done at the final stages of design. During the same era, the concept of “Capacity-Based Design” was introduced which helped in safeguarding against undesirable modes of failure such as shear under lateral forces. The current code-based design has improved significantly over the past methods, but it is still not good enough.

Our current practice makes it hard for people to understand the interdependency between stiffness and seismic inertia. Seismic demands are not technically a group of external forces, even though we try to explain it as such; but instead, they are inertial forces that are generated by the building’s own dynamics. The term “inertial forces” is important, because inertia is a member’s own self mass multiplied by the magnitude of acceleration. This acceleration is dependent on the natural frequencies of the building, which is determined by the stiffness and mass of the building. Determining the mass of the building is an easier task, but with the underlying assumptions, it is the stiffness that is difficult and at risk here. Hence, we will be focusing on stiffness and its relation to strength in this article.

## Problems with Force-Based Design

Interdependency between Strength and Stiffness

When designing a reinforced concrete structure, the selection of appropriate member stiffness is the key. Assumptions must be made based on member sizes even before the seismic forces have been determined. If a structure is designed based on the seismic demands from initial member stiffnesses, it could lead to a change in stiffnesses of the members and thus also make the calculated design forces invalid. Even the effective stiffness of members as suggested by building codes are dependent on the member sectional dimensions rather than the strength of the member. For a reinforced concrete structure, the equation to calculate the yield stiffness of the member is as follows:

The equation above reflects that the stiffness of a member increases as the flexural capacity of the member increases because the yield curvature of the reinforced concrete member does not change. The following graphs reflect the difference between the assumed stiffness independent of the strength as followed by traditional design approaches vs realistic conditions.

Because of these assumptions, it is not possible to perform an accurate analysis of either the elastic structural periods or of the correct elastic distribution of required strength throughout the structure until the member strengths have been determined. Although the design iteration can be simple, it still does not solve the additional problems associated with initial stiffness representations. Since the member stiffnesses are also approximate, the correct estimation of the time period which in turn is utilized to calculate the base shear, also comes out to be an approximate value which varies up to 300% between different methods that are laid out by different building codes across the world.

Instead of calculating the force proportional to building period, we can calculate the target displacement, and based on target displacement, calculate the stiffness of a structure, which in turn can help in determining more accurate forces. These forces can be properly distributed between different members based on the flexural demands and stiffness proportional to member yield curvature. This method is called Displacement – Force Based Seismic Design.

Ductility Capacity and Force Reduction Factors

Ductility capacity of a member is its ability to handle the deformations in inelastic realms. We know that it is the ratio of the ultimate to yield deformation of a member. Now the yield point can be associated with the deformation at nominal strength of member, a point at secant stiffness of member or somewhere in between. Similarly, the ultimate deformation capacity of a member/system can be associated to the displacement at peak strength including strain hardening factors or displacement corresponding to 20% degradation from peak strength or displacement at bar rupture, implying imminent failure. All this leads to significant variations in the ductility capacity of members, neither the ultimate point nor the yield point is properly defined for any sections.

Similarly, response reduction factors are calculated based on the ductility capacity and overstrength of a lateral system. The overstrength factor is also based on numerous assumptions. We notice that in a complex building, not all the structural elements yield at once and this can be proven by time history analysis. Because they do not yield at once, an overstrength factor that is assumed for the entire frame, which in turn determines the R-factor (Force Reduction Factor or Response Modification Factor), cannot be used for the entire frame. So, if a system is complex, or structure has a significant impact because of higher mode effects and is demanding concentrated ductility, then the response reduction factor should be reduced to effectively concentrate all the ductility demands to these localized zones. But neither do we have a way to do it nor does any elastic analysis correctly reflect it.

Ductility of Structural Systems

A response reduction factor for any structural system is a constant value that is prescribed by building codes regardless of the distribution of strength and stiffness of the individual elements in the structural system. Imagine a cantilevered building with unequal wall lengths providing seismic resistance in each direction. If we follow the Force-Based Design, the strongest wall will be taking the maximum magnitude of load regardless of the reinforcement ratio in either of the walls. All the walls are assumed to have similar ductility demand as well as capacity regardless of the configuration of the structural system.

As we already know that the yield curvature of a structure is essentially constant, regardless of its strength. For example, the yield curvature of a rectangular structural wall is given by the following equation:

This reflects the limitation of current code-based design, where different walls should have different ductility factors to provide a more reliable and safer seismic resistance to the structural system.

Relationship between Strength and Ductility Demand

There is a common belief amongst structural engineers that increasing the force/reducing the response reduction factor will lead to safer structures. But it is not exactly what we expect. Per Fig. 1 what we expect is, as we increase the flexural capacity of a section, we are increasing the yield displacement of the member, while the ultimate displacement capacity of the section will stay the same regardless of reinforcement. But we know that the yield displacement does not change per the stress-strain relationship of a beam, thus our argument that increasing the strength will reduce the ductility demand is invalid.

The second problem with our assumptions is that the ultimate displacement capacity of the structure will remain the same regardless of the stress in the section. Again, we know this is incorrect and a great example of it is ASCE 41’s equation for the column’s ultimate displacement capacity. As the axial and shear demand of the column increases, the ultimate displacement capacity of the section reduces. This is based on experimental evidences. But it is surprising that we do not follow it closely for new building constructions and just assume that the columns will work fine if we increase the reinforcement in a section.

Thus, if the ultimate displacement capacity reduces with an increase in reinforcement, and since the yield displacement of the structure does not increase with an increase in reinforcement, we can quite easily conclude that the ductility capacity of the member reduces as we increase the reinforcement in the section. Now in terms of the actual displacement demand, our structure may not see a significant decrease in displacement demand, even if we design it for a reduced R, since displacement demand is dependent on the inertial force and stiffness of the structure. Therefore, increasing the reinforcement in the member leads to an unconservative design from the displacement capacity perspective, as the displacement demand does not reduce but displacement/ductility capacity certainly does.

Let us compare, what we think is true to what is the reality in seismic design.

Concluding Remarks

Force-Based Design has been in practice for several years now. As we are experimenting more with different members as well as materials, we have learned many problems with the Force-Based Seismic Design. Instead of fixing these problems through band aid approaches, it is time that structural engineers think about a more concrete, logical and scientifically accurate approach to design structures for seismic demands. We should be designing our buildings for target displacements and demands, which are generated because of these target deformations based on accurate member stiffnesses instead of checking the structure for an “approximate” force that is causing more underlying problems than solutions. I think the Displacement-Based Seismic Design of structures is the way to go to design structural systems for seismic demands that can and will perform much more reliably compared to current practices. Dr. M. J. N. Priestley has already proven to us the advantages of Displacement + Capacity Based Seismic Design that will lead to a much reliable structural performance.

## I personally think that it is high time to Drop R!

References

1. “Displacement Based Seismic Design” – M. J. N. Priestley, C. M. Calvi, M. J. Kowalsky.

2. “Seismic Design of Reinforced Concrete and Masonry Structures” – T. Paulay and M. J. N. Priestley.

3. ASCE 7-16, ASCE 41-17, Structural Madness

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