NATIONAL TECHNICAL UNIVERSITY OF ATHENS Design of Structures for Earthquake Resistance Basic principles Ioannis N. Psycharis Lecture 3
MDOF systems Equation of motion M u + C u + K u = M r x g(t) where: M = mass matrix C = damping matrix K = stiffness matrix r = earthquake direction vector
Natural modes Eigenfrequencies They are derived from the solution of the characteristic equation: K ω 2 M = 0 Eigenvectors They are derived from the solution of the system of equations: where: K ω 2 M φ i = 0 φ i = i th eigenvector φ ji = j th component of i th eigenvector
Properties Orthogonality φ Τ i M φ j = 0 for i j φ Τ i K φ j = 0 for i j Generalized mass m i = φ Τ i M φ i Generalized stiffness k i = φ Τ i K φ i It can be shown that k i = m i ω 2
Free vibrations For arbitrary initial displacements (From Chopra, AK, Dynamics of Structures, EERI)
Free vibrations For initial displacements according to 1 st mode (From Chopra, AK, Dynamics of Structures, EERI)
Free vibrations For initial displacements according to 2 nd mode (From Chopra, AK, Dynamics of Structures, EERI)
Free vibrations For initial displacements according to 3 rd mode (From Chopra, AK, Dynamics of Structures, EERI)
Modal analysis Displacement at the j th degree of freedom: u j (t) = N n=1 u jn (t) where u jn is the displacement of the j th degree of freedom that corresponds to the n th mode. Response of n th mode: u jn (t) = Y n t φ jn Yn + 2ζ n ω n Y n + ω n 2 Y n = Γ n x g where Γ n is the participation factor of the n th mode: Γ n = φ i Τ M r φ i Τ M φ i
Modal analysis x g(t) u jn (t) = Y n t φ jn Yn + 2ζ n ω n Y n + ω n 2 Y n = Γ n x g (From Chopra, AK, Dynamics of Structures, EERI)
Use of response spectra Maximum displacement of the n th mode at the j th degree of freedom: max u jn = Γ n S d T n, ζ n φ jn T n, ζ n is the spectral displacement that where S d corresponds to period T n and damping ζ n. Maximum seismic force of the n th mode at the j th degree of freedom: max F jn = Γ n S a,d T n, ζ n m j φ jn T n, ζ n is the design spectral acceleration where S a,d that corresponds to period T n and damping ζ n.
Combination of modal responses Significant modes k < N k n=1 M n 0.90 m tot where M n is the effective mass of the n th mode: M n = Γ n φ n Τ M r The effective mass of each mode depends on the direction of the seismic action.
Combination of modal responses Let A n, A m be the maximum value of a quantity A (internal force or displacement) of the n th and the m th mode respectively. SRSS k maxa = ± A n 2 CQC n=1 k k maxa = ± ε nm A n A m n=1 m=1 ε nm = 8 ζ 2 1+r r 3/2 1 r 2 2 +4 ζ 2 r 1+r 2 with r = T n T m
Simplified formulas For planar motion in the plane of the seismic action and for the n th mode: Γ n = N j=1 N j=1 m j φ jn 2 m j φ jn M n = Γ n N j=1 m j φ jn
Design procedure Define structural properties Compute mass and stiffness matrices M and K Estimate modal damping coefficients ζ n Solve the eigen-problem to determine the k lower natural frequencies ω n and modes φ n Compute the corresponding natural periods T n = 2π ω n For a given direction of the seismic action: Compute the participation factors Γ n Compute effective modal masses M n and check that their sum is larger than 90% of the total mass. If not, increase the value of k and repeat the procedure
Design procedure For a given direction of the seismic action, compute the maximum response for each mode n by repeating the following steps: On the design response spectrum read the spectral acceleration S a,d that corresponds to period T n and damping ζ n Compute the seismic force F j,n at each degree of freedom j Perform static analysis of the structure subjected to forces F j,n and determine internal forces and displacements Combine the modal responses using SRSS or CQC for each direction of the seismic action
Spatial combination Let A x, A y, A z be the estimated maximum values of a quantity A that correspond to two horizontal orthogonal directions x, y and the vertical direction z of the seismic action. The maximum value of A for simultaneous action of the earthquake in all directions x, y, z can be estimated as: 1 st Method A = ± A x 2 + A y 2 + A z 2 2 nd Method A = ±A x ± 0.3A y ± 0.3A z A = ±0.3A x ± A y ± 0.3A z or or A = ±0.3A x ± 0.3A y ± A z
Remarks For typical buildings, the vertical component of the seismic action can be neglected. Displacements If d E are the displacements from the above analysis, the actual displacements are calculated as d = q d E where q is the value of the behavior factor used in the design response spectrum. The elastic response of a structure to a specific earthquake can also be computed using the above procedure by substituting the design response spectrum with the elastic spectrum of the ground motion.