The RM mode superposition method is a special superposition method to get the maximum of all components of the result vector considering all corresponding forces and/or deformations. The total result vector for each mode uses factorized natural modes with the element start and end points treated independently. This method complexity is directly related to the selected mode superposition rule.
All standard mode superposition methods in literature aim at getting the supposable maxima of all components of the result vector independently from each other. The total result vector for each mode (factorized natural mode) consists of sub-vectors related to the result points of the structure (element start and end points) which can in fact be treated independently. We use in the following the terms Rk,i, Rk,j for such sub-vectors related to the modes i and j of the result point k. Within such sub-vector the different internal force components and/or displacement components are always related to each other, and for design we usually need the corresponding values to each maximum or minimum component (e.g. bending moments and shear forces concomitant with maximum normal force, normal and shear forces concomitant with minimum bending moment etc.). Therefore, we implemented a special method for considering all corresponding forces and/or deformations. The complexity of this method is dependent on the selected mode superposition rule and is rather simple for ABS and SRSS rules. In the following we denote Rl,i, Rl,j as the leading components of the contributions from modes i and j, and Rk,i, Rk,j as concomitant components.
For ABS rule we have basically Rtot = Σ|Ri|: To get corresponding values we just must check the sign of the leading value and to decide, whether the Eigenvector shall be directly superimposed or with reversed sign in order to get the maximum or minimum of the leading value. When denoting Rl,i as result value of the leading component and Rk,i the result values of the concomitant components we get for the total response Rtot:
Rtot,max = Σ (Rl,i*sgn(Rl,i)) Rtot,min = - Σ (Rl,i*sgn(Rl,i))
Rtot,con = Σ (Rk,j*sgn(Rl,i)) Rtot,con = - Σ (Rk,j*sgn(Rl,i))
For SRSS rule it is also relatively simple. The SRSS rule corresponds to a “white noise” behavior, i.e. the different modes are completely uncorrelated. The modes are superimposed in a Pythagorean manner. The standard formula for getting the max/min total response values is
Rtot,max = √ Σ ( n, i=1 ) Ri2
Rtot,min = - √ Σ ( n, i=1 ) Ri2
For concomitant values we have similarly to the ABS rule, in the case of the SRSS rule:
Rtot,con = sgn(Σ)*√(|Σ(Rk,i2*sgn(Rl,i))|) for values concomitant with max values and,
Rtot,con = - sgn(Σ)*√(|Σ(Rk,i2*sgn(Rl,i))|) for values concomitant with min values,
with sgn(Σ) denoting the sign of (Σ(Rk,i2*sgn(Rl,i)).
In matrix form we can write this in the same form than given in the textbooks for DSC and CQC rules, with the correlation matrix being the unit matrix (ρij=1.0 for i=j, ρij=0.0 for i≠j):
Rtot,max = √([R]T*[ρ]*[R])
Brought down to component level and introducing sign adjustment we get for the maximum/minimum values:
Rk,tot,max = √( ΣΣ(Rk,i*sgn(Rk,i)*ρij*Rk,j*sgn(Rk,j)) ) and
Rk,tot,min = - √( ΣΣ(Rk,i*sgn(Rk,i)*ρij*Rk,j*sgn(Rk,j)) )
with sgn(Rk,i) and sgn(Rk,j) denoting the sign of the respective value.
For concomitant values there is the only difference that we have to use the sign of the leading value for sign adjustment, rather than using the sign of the actual component. Denoting l as the leading component we get for concomitant values:
Rk,tot,con = sgn(Σ)*√( |ΣΣ (Rk,i*sgn(Rl,i)*ρij*Rk,j*sgn(Rl,j))| ) and
Rk,tot,con = - sgn(Σ)*√( |ΣΣ (Rk,i*sgn(Rl,i)*ρij*Rk,j*sgn(Rl,j( |
with sgn(Σ) denoting the sign of (ΣΣ(Rk,i*sgn(Rl,i)*ρij*Rk,j*sgn(Rl,j)).
For DSC and CQC we have an interaction between the different modes. The interaction is defined by the so-called correlation matrix [ρij] which now has also off-diagonal terms.
Rtot = √ ( | ΣΣ RTi*ρij*Rj | )
The idea for getting relevant concomitant values is to decompose the correlation matrix and to transform the equation such that it can later be treated like SRSS. We decompose [ρ] into 2 parts: [ρ] = [a]T*[a] and calculate the modified result matrix [Rm] = [a]* [R]. Then we apply the standard SRSS procedure to this modified matrix as described above.
For the decomposition of the matrix [ρ] we use currently a Cholesky procedure, but this decomposition in the real space is not always possible. In rare cases (considering many modes over a wide frequency range) the Cholesky decomposition fails due to negative pivots arising in the calculation process.
More technical information about the superposition rules can be found at the link below.