Mode Superposition Rules

Note: Selection of the required superposition rules for the amplitude values contributed by the individual natural modes. Note that the ABS rule gives an upper limit and is usually very conservative. SRSS is a lower limit assuming completely uncorrelated modes. Results of DSC and CQC rules are between the two limits and usually only slightly higher than SRSS results.

Setting	Description
ABS rule - Sum of absolute values	The absolute values are added together algebraically.
SRSS rule - Square Root of Sum of Squares (Pythagorean addition)	Combination as square root of the sum of squares.
DSC rule - Double Sum Combination	Combination with accounting for correlation between the modes (Newmark, Rosenblueth 1969, 1971). R_max = Sqrt (DoubleSum of R_iR_jrho_ij). For sign adjustment modified to R_max = Sqrt (DoubleSum of abs(R_iR_j)rho_ij). R_min=-R_max. rho_ij are the correlation factors depending on the specified damping ratio and the duration of the seismic event. See CQC for sign adjustment justification.
CQC rule - Complete Quadratic Combination	Combination with accounting for correlation between the modes (Der Kiureghian, 1979, 1980). R_max = Sqrt (DoubleSum of R_iR_jrho_ij). For sign adjustment modified to R = Sqrt (DoubleSum of abs(R_iR_j)rho_ij). rho_ij are the correlation factors depending on the specified damping ratio and the duration of the seismic event.
CQCX - Variant of CQC (without sign adjustment)	Variant of the CQC rule. CQCX is the original CQC as described in literature (without sign adjustment). Sign adjustment corresponds to limiting phase shift considerations to 180° whereas the original CQC essentially considers phase shifting constraints over 360°. Using this approach effects in fully considering any mass damper effects, i.e. the result values can drop down below SRSS results. Note: Due to scientific discussions in 2012 we came to the conclusion that CQCX is the correct approach and the CQC solution with sign adjustment is too conservative.