The governing dynamic equilibrium equation for a single-degree-of-freedom (SDOF) structure is defined as: \[K×u+C×dot{u}+M×ddot{u}=-M×ddot{u}_g(t)\]
A design response spectrum—a graph showing the relationship between natural periods and maximum acceleration for a given region and soil type—is used to find the spectral acceleration for each mode.
The governing dynamic equilibrium equation for a multi-degree-of-freedom (MDOF) structure is defined as: \[{K}u + {C}dot{u} + {M}ddot{u} = -{M}ddot{u}_g(t)\]
This system is decoupled into individual Multi-Degree-of-Freedom (MDOF) equations using the mode shape vector \[(φ_n)\]: \[ddot{q}_n + 2*ζ_n ω_n dot{q}_n+ω_n^2 q_n=-Γ_n ddot{u}_g(t)\]
\[({K}-ω_n^2*{M})φ_n=0\] \[|{K}-ω_n^2*{M}|=0\] Modal participation factor: \[Γ_n = (φ_n^T*{M}*r)/(φ_n^T*{M}*φ_n)\]
For the n-th mode, the maximum displacement (Sd,n) is read directly from the design response spectrum \[u_{n,max}=Γ_n*φ_n*S_{d,n}\] \[S_{d,n} = S_{a,n}/ω_n^2\] The peak modal inertia forces are evaluated as: \[F_n=M*Γ_n*φ_n*S_{a,n}\] Modal mass: \[M_n^* = Γ_n^2*(φ_n^T*M*φ_n)\] Modal base shear: \[V_n=M_n^**S_a(T_n)\] Modal forces: \[F_n=Γ_n*M*φ_n*S_a(T_n)\] Mass Participation Ratio: \[|M_perc| = M_n^*/(Σm_i)*100\]
Modal Displacement \[(D_n)\]: \[D_n=S_d(T_n)=S_a(T_n)/ω_n^2\] Modal Base Shear \[(Vn)\]: \[V_{n,max}=M_n^**S_a(T_n)\]
SRSS (Square Root of the Sum of Squares): Best for widely separated modal frequencies. SRSS Storey forces: \[R = sqrt(sum_j^n×R_n^2)\] CQC (Complete Quadratic Combination): Accounts for modes that have closely spaced natural frequencies by including cross-modal correlation coefficients. Cross-Modal Correlation \[(ρ_{ij})\]: \[ρ_ij=(8*ζ^2*(1+r)*r^1.5)/((1-r^2)^2*4*ζ^2*r*(1+r)^2)\] \[r=ω_i/ω_j\] \[ζ=0.05\] CQC Storey forces: \[F_i = sqrt(sum_j^n×sum_k^n×F_{i,j}*F_{i,k}*ρ_ij)\]