2C9 Design for seismic and climate changes Jiří Máca
List of lectures 1. Elements of seismology and seismicity I 2. Elements of seismology and seismicity II 3. Dynamic analysis of single-degree-of-freedom systems I 4. Dynamic analysis of single-degree-of-freedom systems II 5. Dynamic analysis of single-degree-of-freedom systems III 6. Dynamic analysis of multi-degree-of-freedom systems 7. Finite element method in structural dynamics I 8. Finite element method in structural dynamics II 9. Earthquake analysis I 2
Finite element method in structural dynamics I Free vibration analysis 1. Finite element method in structural dynamics 2. Free vibration 3. Rayleigh Ritz method 4. Vector iteration techniques 5. Inverse vector iteration method + examples 6. Subspace iteration method 7. Examples
1. FINIE ELEMEN MEHOD IN SRUCURAL DYNAMICS Statics Kr f K A Ke N e e1 V K e B DB Dynamics dv Mr ( t) Kr( t) 0 Mr ( t) Cr ( t) Kr( t) f( t) Forces do not vary with time Structural stiffness K considered only N A e Operator e 1 denotes the direct assembly procedure for assembling the stiffness matrix for each element, e = 1 to N e, where N e is the number of elements Free vibration No external forces present Structural stiffness K and mass M considered Forced vibration External forces (usually time dependent) present Structural stiffness K, mass M and possibly damping C considered
Finite Element Method in structural dynamics Inertial force (per unit volume) ρ mass density Damping force (per unit volume) ξ damping coefficient f u f udv ˆi f u f udv ˆd i d V V Principle of virtual displacements WI W E internal virtual work = external virtual work W I ε σ dv σ stress vector ε strain vector V ˆ W u f dv u fˆ dv E i d V V
Finite Element Method in structural dynamics Finite Element context u Nr u Nr unr u r N u r N r nodal displacements N interpolation (shape) functions Strain displacement relations ε u Nr Br ε r B Constitutive relations (linear elastic material) σ D ε DBr D constitutive matrix
Finite Element Method in structural dynamics Internal virtual work ε σ r B DB r W I dv dv V V External virtual work W W E I 0 ˆ W u f dv u fˆ dv E i d V V dv V dv V u u u u V V dv r N N r r N N dvr r N N dv r N N dvr B DB dv r V V V Dynamic equations of equilibrium Mr Cr Kr 0 0 Mr Cr Kr f f nodal forces (external)
Finite Element Method in structural dynamics Mass matrix (consistent formulation) M N N V dv Damping matrix C N N V dv Stiffness matrix K B DB V dv Mass matrix - other possible formulations Diagonal mass matrix equal mass lumping lumped-mass idealization with equal diagonal terms special mass lumping - diagonal mass matrix, terms proportional to the consistent mass matrix
Finite Element Method in structural dynamics Beam element length L, modulus of elasticity E, cross-section area A moment of inertia (2nd moment of area) I z, uniform mass m Starting point approximation of displacements u uv, Exact solution (based on the assumptions adopted) - Approximation of axial displacements x x ux ( ) 1 u u huhu L L 1 2 1 1 2 2 u 1 left-end axial displacement u 2 right-end axial displacement
Beam element - Approximation of deflections 2 3 2 3 x x x x x vx ( ) 13 2 v1 2 L 1 L L L L L 2 3 2 3 x x x x 3 2 v2 L 2 L L L L vx ( ) hvh hv h 3 1 4 1 5 2 6 2 1 v1 1 v 1 h 3 v 2 2 v 1 left-end deflection v 2 right-end deflection φ 1 left-end rotation φ 2 right-end rotation 1 1 h 4 h 5 v2 1 2 1 h 6
Beam element Vector of nodal displacements r u, v,, u, v, Matrix of interpolation functions N 1 1 1 2 2 2 h1 0 0 h2 0 0 0 h3 h4 0 h5 h 6 u Nr Constitutive relations Strain displacement relations du Nx EA 0 dx 2 M z 0 EI z dv 2 dx σ D ε ε u Nr Br B dh1 dh2 0 0 0 0 dx dx dh dh dh dh 0 0 dx dx dx dx 2 2 2 2 3 4 5 6 2 2 2 2
Beam element Element stiffness matrix K L B DB 0 dx K EA EA 0 0 0 0 L L 12EI z 6EI z 12EI z 6EI z 0 0 3 2 3 2 L L L L 6EI z 4EI z 6EI z 2EI z 0 0 2 2 L L L L EA EA 0 0 0 0 L L 12EI 6EI 12EI 6EI 0 0 L L L L 6EI z 2EI z 6EI z 4EI z 0 0 2 2 L L L L z z z z 3 2 3 2
Beam element Element mass matrix consistent formulation L M N mn 0 dx M 140 0 0 70 0 0 156 22 L 0 54 13L 2 2 ml 4L 0 13L 3L 420 140 0 0 156 22 L 2 sym 4 L
Beam element Element mass matrix other possible approximation of deflections x x vx ( ) 1 v v hvhv L L Element mass matrix diagonal mass matrix vx ( ) hv hv 1 1 2 2 1 2 1 1 2 2 where h 1 =1 for x l/2 h 1 =0 for x>l/2 h 2 =0 for x < l/2 h 2 =1 for x l/2 140 0 0 70 0 0 140 0 0 70 0 ml 0 0 0 0 M 420 140 0 0 140 0 sym 0 1 0 0 0 0 0 1 0 0 0 0 ml 0 0 0 0 M 2 1 0 0 1 0 sym 0
Comparison of Finite Element and exact solution
Comparison of Finite Element and exact solution Accuracy of FE analysis is improved by increasing number of DOFs
2. FREE VIBRAION natural vibration frequencies and modes Mr ( t) Kr( t) 0 equation of motion for undamped free vibration of MDOF system r ( t) ( A cos t B sin t) r n n n n n 2 ( t) nr( t) motions of a system in free vibrations are simple harmonic K M 0 2 n n 2 det n 0 free vibration equation, eigenvalue equation - natural mode ω n - natural frequency K M nontrivial solution condition n frequency equation (polynomial of order N) not a practical method for larger systems
3. RAYLEIGH RIZ MEHOD general technique for reducing the number of degrees of freedom K M 0 2 n n K M 0 K M K M Rayleigh s quotient Properties 1. When is an eigenvector n, Rayleigh s quotient is equal to the 2 corresponding eigenvalue n 2. Rayleigh s quotient is bounded between the smallest and largest eigenvalues 2 2 2 1 1 2 2... n n
Rayleigh Ritz method u ( t) Ψ z ( t) Ψ z displacements or modal shapes are expressed as a linear combination of shape vectors j, j = 1,2 J <N Ritz vectors make up the columns of N x J matrix Ψ z is a vector of J generalized coordinates substituting the Ritz transformation in a system of N equations of motion Ku Cu Ku sp() t we obtain a system of J equations in generalized coordinates mz cz kz sp () t m Ψ M Ψ c Ψ C Ψ k Ψ K Ψ s Ψ s Ritz transformation has made it possible to reduce original set of N equations in the nodal displacement u to a smaller set of J equations in the generalized coordinates z
Rayleigh Ritz method substituting the Ritz transformation in a Rayleigh s quotient K z Ψ K Ψ z M z Ψ M Ψ z ( z) z i 0 i1,2,... J ( ) z and using Rayleigh s stationary condition we obtain the reduced eigenvalue problem kz mz m Ψ M Ψ k Ψ K Ψ, z solution of all eigenvalues and eigenmodes e.g. Jacobi s method of rotations 2 2 2 1 1 2 2... J J i izi
Rayleigh Ritz method Selection of Ritz vectors Selection of Ritz vectors
4. VECOR IERAION ECHNIQUES Inverse iteration (Stodola-Vianello) Algorithm to determine the lowest eigenvalue Kx Mx x K Mx x 1 k1 k k1 k x k 1 k 1 1/2 ( xk 1Mxk1) k 1 1 as k x 1
Vector iteration techniques Forward iteration (Stodola-Vianello) Algorithm to determine the highest eigenvalue Mx Kx x M Kx x 1 k1 k k1 k k 1 k 1 1/2 ( xk 1Mxk1) n xk n as k 1 x
5. INVERSE VECOR IERAION MEHOD 1. Starting vector x! arbitrary choice R Mx 2. ( k 1) k1 k1 3. x Kx (Rayleigh s quotient) x Mx 4. Kx Kx 1 1 R 2 1 k1 Mx ( k1) ( k) ( k 1) k k1 k1 tol 5. If convergence criterion is not satisfied, normalize xk 1 x( k 1) 1/2 and go back to 2. and set k = k+1 x Mx k1 k1 x k1 1 K Mxk
Inverse vector iteration method 6. For the last iteration (k+1), when convergence is satisfied 2 ( k 1) 1 1 x k 1 1 ( k 1) 1/2 xk1mxk1 x Gramm-Schmidt orthogonalisation to progress to other than limiting eigenvalues (highest and lowest) evaluation of n1 correction of trial vector x x x Mx n j j1 jmj j
Inverse vector iteration method Vector iteration with shift μ the shifting concept enables computation of any eigenpair KM KKM eigenvectors of the two eigenvalue problems are the same inverse vector iteration converge to the eigenvalue closer to the shifted origin, e.g. to, see (c) 3
Inverse vector iteration method Example 3DOF frame, 1 st natural frequency k k 1,5 k k k 1,5 k K 1 flexibility matrix 2 2 2 1 δ 2 5 5 6k 2 5 8 m 0 0 M 0 m 0 0 0 m 2 2 2 m δm 2 5 5 6k 2 5 8 Kx Mx x K Mx Mx 1 k1 k k1 k k x 0 1 1 1 starting vector
Inverse vector iteration method 1 st iteration 6 1 m (1) xmx 1 0 k x1 δmx0 12 0, 489 x1 2 6k 1 1 m 15 xmx 2,5 2 nd iteration 11 1 m (2) xmx 2 1 k x2 δmx1 24,5 0, 481 x2 2, 23 6k 2 2 m 32 xmx 2,91 12, 28 1 m (3) xmx 3 2 k x3 δmx2 27,70 0,481 x3 2,26 6k 3 3 m 36, 43 xmx 2,97 (3) 1 0,694 k 1 x3 m 3 rd iteration normalization u1 1
Appendix flexibility matrix k k u u 3k 2k 2k m 1 m 2 m u 3 k 1,5 k δ k 1,5 k 2 2 2 1 2 5 5 6k 2 5 8 F 1 1 1 1 1 11 21 31 3k 3k 3k F 2 1 1 1 1 1 1 12 22 32 3k 3k 2k 3k 2k F 3 1 1 1 1 1 1 1 13 23 33 3k 3k 2k 3k 2k 2k
6. SUBSPACE IERAION MEHOD efficient method for eigensolution of large systems when only the lower modes are of interest similar to inverse iteration method - iteration is performed simultaneously on a number of trial vectors m m smaller of 2p and p+8, p is the number of modes to be determined (p is usually much less then N, number of DOF) 1. Staring vectors X 1 R MX 1 1 KX R 2 1 2. Subspace iteration a) KX k1 MX k b) Ritz transformation K k1 Xk 1KXk1 M k1xk1mxk1
Subspace iteration method 2. Subspace iteration cont. c) Reduced eigenvalue problem (m eigenvalues) K Q Ω M Q 2 k1 k1 k1 k1 k1 d) New vectors X X Q k1 k1 k1 e) Go back to 2 a) 3. Sturm sequence check verification that the required eigenvalues and vectors have been calculated - i.e. first p eigenpairs Subspace iteration method - combines vector iteration method with the transformation method
7. EXAMPLES Simple frame exact 3 6 12 solution elem elem elem f 1 [Hz] 7,270 7,282 7,270 7,270 f 2 [Hz] 28,693 34,285 28,845 28,711 EI = 32 000 knm 2 μ = 252 kgm -1 f 3 [Hz] 46,799 74,084 47,084 46,854 1. shape of vibration 2. shape of vibration 3. shape of vibration
Examples Simple frame exact 3 6 solution elem elem f 1 [Hz] 28,662 34,285 28,845 f 2 [Hz] 41,863 65,623 42,393 EI = 32 000 knm 2 μ = 252 kgm -1 f 3 [Hz] 50,653-51,474 1. shape of vibration Recommendation: it is good to divide beams into (at least) 2 elements in the dynamic FE analysis
Examples 3-D frame (turbomachinery frame foundation)
Examples 3-D frame (turbo-machinery frame foundation)
Examples 3-D frame (turbo-machinery frame foundation)
Examples 3-D frame + plate + elastic foundation
Examples cable-stayed bridge
Examples cable-stayed bridge
Examples cable-stayed bridge