Directional Combination Best Practice
Best practice for combining seismic responses from multiple directions, covering SRSS vs 100-40-40, the order of modal and directional combination, and signed response recovery.
Order of combination
The standard sequence is: first, perform modal combination within each direction (SRSS or CQC); second, perform directional combination across directions (SRSS or 100-40-40). This order is important because modal combination assumes statistical independence of modes, while directional combination assumes statistical independence of directions. Reversing the order is not correct and can produce different results.
SRSS directional combination
SRSS directional combination is: R = sqrt(Rx^2 + Ry^2 + Rz^2). It assumes that the peak responses in different directions do not occur simultaneously. It gives a lower result than 100-40-40. Some standards permit SRSS directional combination (e.g. ASCE 4 for nuclear structures); others require 100-40-40 (e.g. ASCE 7 for buildings). The standard must be confirmed before selecting the method.
100-40-40 directional combination
The 100-40-40 method is: R = max(Rx, Ry, Rz) + 0.40 * mid(Rx, Ry, Rz) + 0.40 * min(Rx, Ry, Rz). It approximates the case where the peak response in one direction occurs simultaneously with 40% of the peak in the other two directions. It is more conservative than SRSS. The 100-30-30 variant uses 30% instead of 40%; the coefficient is specified by the standard.
Vertical direction
The vertical direction is sometimes combined differently from the horizontal directions. Some standards require the vertical response to be combined with the horizontal response by SRSS: R = sqrt(R_horizontal^2 + Rz^2), where R_horizontal is the 100-40-40 combination of the two horizontal directions. Other standards treat all three directions equally in 100-40-40. The standard must be confirmed.
Signed response recovery
SRSS and 100-40-40 produce positive (unsigned) results. For some design checks, the sign of the response matters (e.g. tension vs compression in an anchor, or positive vs negative bending in a beam). The standard approach is to determine the maximum positive and maximum negative response by considering the contributions from each direction with appropriate signs. Some solvers provide this as a post-processing option; others require manual calculation.
Common errors
- Performing directional combination before modal combination
- Using SRSS directional combination when the standard requires 100-40-40
- Not treating the vertical direction differently when the standard requires it
- Losing the sign of the response and not recovering it for design checks that require signed values
- Using the same combination method for forces and displacements when the standard requires different methods