R and D Factors: What Does a Building Trade with an Earthquake?

1. Why the elastic demand is not carried

The design spectrum S_ae(T) gives the acceleration a building would experience if it remained fully elastic, with no damage at all. Designing a reinforced concrete building for that force is theoretically possible but both very costly and unnecessary: the design earthquake (DD-2) is rare enough that a building will probably never experience it in its lifetime.

Seismic codes therefore set a different goal: under the design earthquake the building may be damaged but must not collapse. Members are allowed to yield — to deform permanently — dissipating energy while the structure stays standing. The reduced design acceleration is:

2. How R_a(T) is calculated

TBDY 2018 Clause 4.2.1, Equation (4.1) defines the load reduction factor in two ranges:

The table below was computed with the site's TBDY engine: S_DS = 1.2 and S_D1 = 0.45 (T_B = S_D1/S_DS = 0.375 s), for a high-ductility reinforced concrete frame — system A11 in TBDY Table 4.1, with R = 8 and D = 3.

The table shows two things. Beyond T_B the reduction is constant and equals R/I. As the period approaches zero, the reduction drops to D: R_a = 5.67 at T = 0.2 s and R_a = 3 at T = 0. For a very stiff building the design force is one third of the elastic force, not one eighth.

3. Why short-period buildings get less reduction

The displacement cost of yielding depends on the period. For medium- and long-period systems, the peak displacement of a yielding structure is roughly the same as that of an elastic structure with the same period (the "equal displacement" rule). Reducing the force by R is then matched by a ductility of about R.

For short-period, stiff structures this rule breaks down. The same force reduction demands much larger plastic deformation, so the structure would have to spend far more ductility than it has. Because of this behaviour, known since the classic work of Newmark and Hall (1982), TBDY tapers the reduction linearly in the short-period range down to D at T = 0. Low-rise, wall-braced school buildings sit exactly in this range.

4. R versus D

The two factors appear in the same equation but represent different things:

• D is overstrength. A building is stronger than its design calculation assumes, thanks to material safety factors, minimum-reinforcement rules and the redistribution of forces from the first member to yield. D measures this "hidden" strength and requires no ductility.
• R combines overstrength and ductility. The ratio R/D is roughly the part attributed to ductility: 8/3 ≈ 2.7 for system A11.

That is why the reduction falls to D at T = 0: in a very stiff structure ductility cannot be relied on, and only overstrength remains.

5. The importance factor I

TBDY Table 3.1 classifies buildings by use and assigns an importance factor to each class. Buildings needed after an earthquake and buildings where people gather in large numbers — hospitals and schools, for example — are in class BKS 1 with I = 1.5. The site's three-storey school block example is in this class.

I works by reducing the reduction: beyond T_B, R_a = R/I = 8/1.5 = 5.33. A school designed with the same structural system as a residential building is therefore designed for a 50% larger seismic force. The last columns of the table show the difference.

6. The price of the reduction

Choosing R = 8 is not a preference but a commitment. The value assumes full compliance with the high-ductility reinforced concrete rules of TBDY Chapter 7: confinement reinforcement in columns, the strong-column/weak-beam principle, and shear safety in beams and joints. If any of these is missing, the calculation has taken R = 8 but the building's ductility cannot honour it: the reduction has been taken, the price has not been paid.

The code also sets a floor: in the equivalent lateral force method the base shear may not be taken less than 0.04 · m_t · g · I · S_DS (Equation 4.19).