Truncation Errors Numerical Integration Multiple Support Excitation

Transcription

1 Errors Numerical Integration Multiple Support Excitation Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 10, 2018

2 Part I How many eigenvectors? Modal partecipation Dynamic magnification Static Correction Modal partecipation Dynamic magnification Static Correction

3 How many eigenvectors? Modal partecipation To understand how many eigenvectors we have to use in a modal analysis, we must consider two s, the loading shape and the excitation frequency. Dynamic magnification Static Correction

4 In the following, we ll consider only external loadings whose dependance on time and space can be separated, as in p(x, t) = r f(t), Modal partecipation Dynamic magnification Static Correction so that we can regard separately the two aspects of the problem.

5 It is worth noting that earthquake loadings are precisely of this type: p(x, t) = M r ü g where the vector r is used to choose the structural dof s that are excited by the ground motion component under consideration. r is an incidence vector, often simply a vector of ones and zeroes where the ones stay for the inertial forces that are excited by a specific component of the earthquake ground acceleration. Modal partecipation Dynamic magnification Static Correction

6 It is worth noting that earthquake loadings are precisely of this type: p(x, t) = M r ü g where the vector r is used to choose the structural dof s that are excited by the ground motion component under consideration. r is an incidence vector, often simply a vector of ones and zeroes where the ones stay for the inertial forces that are excited by a specific component of the earthquake ground acceleration. Modal partecipation Dynamic magnification Static Correction Multiplication of M and division of ü g by g, acceleration of gravity, serves to show a dimensional load vector multiplied by an adimensional function. p(x, t) = g M r üg(t) g = r g f g (t)

7 Modal partecipation Under the assumption of separability, we can write the i-th modal equation of motion as q i + 2ζ i ω i q i + ω 2 i q i = { ψ T i r f(t) M i g ψi T M ˆr M i f g (t) = Γ i f(t) with the modal mass M i = ψ T i Mψ i. It is apparent that the modal response amplitude depends on the characteristics of the time dependency of loading, f(t), on the so called modal partecipation Γ i, Modal partecipation Dynamic magnification Static Correction Γ i = ψ T i r/m i or Γ i = g ψ T i M ˆr/M i = ψ T i r g /M i

8 Modal partecipation Under the assumption of separability, we can write the i-th modal equation of motion as q i + 2ζ i ω i q i + ω 2 i q i = { ψ T i r f(t) M i g ψi T M ˆr M i f g (t) = Γ i f(t) with the modal mass M i = ψ T i Mψ i. It is apparent that the modal response amplitude depends on the characteristics of the time dependency of loading, f(t), on the so called modal partecipation Γ i, Modal partecipation Dynamic magnification Static Correction Γ i = ψ T i r/m i or Γ i = g ψ T i M ˆr/M i = ψ T i r g /M i Note that both the definitions of modal partecipation give it the dimensions of an acceleration.

9 Partecipation Factor Amplitudes For a given loading r the modal partecipation Γ i is proportional to the work done by the modal displacement q iψ T i for the given loading r: if the mode shape and the loading shape are approximately equal (equal signs, component by component), the work (dot product) is maximized, if the mode shape is significantly different from the loading (different signs), there is some amount of cancellation and the value of the Γ s will be reduced. Modal partecipation Dynamic magnification Static Correction

10 Example Modal partecipation Dynamic magnification Static Correction gm ˆr r ψ 1 ψ 2 ψ 3 Consider a shear type building, its first 3 eigenvectors as sketched above, with mass distribution approximately constant over its height and its earthquake load shape vector ˆr = {1, 1,..., 1} T g M ˆr mg{1, 1,..., 1} T, consider also an external, assigned load shape vector r.

11 Example, cont. Modal partecipation Dynamic magnification Static Correction gm ˆr r ψ 1 ψ 2 ψ 3 For EQ loading, Γ 1 is relatively large for the first mode, as loading components and displacements have the same sign, with respect to other Γ i s, where the oscillating nature of the higher eigenvectors will lead to increasing cancellation. On the other hand, consider the external loading, whose peculiar shape is similar to the 3rd mode. Γ 3 will be more relevant than Γ i s for lower or higher modes.

12 Modal Loads Expansion We define the modal load contribution as r i = M ψ ia i and express the load vector as a linear combination of the modal contributions r = M ψ ia i = r i. i i Modal partecipation Dynamic magnification Static Correction Premultiplying by ψj T the above equation we have a relation that enables the computation of the coefficients a i: ψ T i r = ψ T i M ψ ja j = j j δ ijm ja j = a im i a i = ψt i r M i

13 Modal Loads Expansion 1. A modal load component works only for the displacements associated with the corresponding eigenvector, ψ T j r i = a i ψ T j Mψ i = δ ija im i. 2. Comparing ψj T r = ψj T i M ψiai = δijmiai with the definition of Γ i = ψi T r/m i, we conclude that a i Γ i and finally write Modal partecipation Dynamic magnification Static Correction r i = Γ im ψ i. 3. The modal load contributions can be collected in a matrix: with Γ = diag Γ i we have R = M Ψ Γ.

14 Equivalent Static Forces For mode i, the equation of motion is q i + 2ζ i ω i q i + ω 2 i q i = Γ i f(t) with q i = Γ i D i, we can write, to single out the dependency on the modulating function, D i + 2ζ i ω i Ḋ i + ω 2 i D i = f(t) The modal contribution to displacement is Modal partecipation Dynamic magnification Static Correction x i = Γ i ψ i D i (t) and the modal contribution to elastic forces f i = K x i can be written (being Kψ i = ω 2 i Mψ i) as f i = K x i = Γ i K ψ i D i = ω 2 i (Γ i M ψ i )D i = r i ω 2 i D i

15 Equivalent Static Forces For mode i, the equation of motion is q i + 2ζ i ω i q i + ω 2 i q i = Γ i f(t) with q i = Γ i D i, we can write, to single out the dependency on the modulating function, D i + 2ζ i ω i Ḋ i + ω 2 i D i = f(t) The modal contribution to displacement is Modal partecipation Dynamic magnification Static Correction x i = Γ i ψ i D i (t) and the modal contribution to elastic forces f i = K x i can be written (being Kψ i = ω 2 i Mψ i) as f i = K x i = Γ i K ψ i D i = ω 2 i (Γ i M ψ i )D i = r i ω 2 i D i D is usually named pseudo-displacement.

16 Equivalent Static The response can be determined by superposition of the effects of these pseudo-static forces f i = r iωi 2 D i(t). If a required response quantity (be it a nodal displacement, a bending moment in a beam, the total shear force in a building storey, etc etc) is indicated by s(t), we can compute with a static calculation (usually using the FEM model underlying the dynamic analysis) the modal static contribution s st i and write s(t) = s st i (ω 2 i D i(t)) = s i(t), Modal partecipation Dynamic magnification Static Correction where the modal contribution to response s i(t) is given by 1. static analysis using r i as the static load vector, 2. dynamic amplification using the ω 2 i D i(t).

17 Equivalent Static The response can be determined by superposition of the effects of these pseudo-static forces f i = r iωi 2 D i(t). If a required response quantity (be it a nodal displacement, a bending moment in a beam, the total shear force in a building storey, etc etc) is indicated by s(t), we can compute with a static calculation (usually using the FEM model underlying the dynamic analysis) the modal static contribution s st i and write s(t) = s st i (ω 2 i D i(t)) = s i(t), Modal partecipation Dynamic magnification Static Correction where the modal contribution to response s i(t) is given by 1. static analysis using r i as the static load vector, 2. dynamic amplification using the ω 2 i D i(t). This formulation is particularly apt to our discussion of different contributions to response components.

18 Modal Contribution Factors Say that the static response due to r is denoted by s st, then s i (t), the modal contribution to response s(t), can be written s i (t) = s st i ωi 2 st sst i D i (t) = s s st ω2 i D i (t) = s i s st ωi 2 D i (t). Modal partecipation Dynamic magnification Static Correction We have introduced s i = sst i s st, the modal contribution, the ratio of the modal static contribution to the total static response. The s i are dimensionless, are indipendent from the eigenvector scaling procedure and their sum is unity, s i = 1.

19 Maximum Denote by D i0 the maximum absolute value (or peak) of the pseudo displacement time history, It will be D i0 = max{ D i (t) }. t s i0 = s i s st ω 2 i D i0. The dynamic response for mode i, R di is defined by Modal partecipation Dynamic magnification Static Correction R di = D i0 D st i0 where Di0 st is the peak value of the static pseudo displacement D st i = f(t) ωi 2, Di0 st = f 0 ωi 2.

20 Maximum With f 0 = max{ f(t) } the peak pseudo displacement is D i0 = R di f 0 /ωi 2 and the peak of the modal contribution is s i0 (t) = s i s st ωi 2 D i0 (t) = f 0 s st s i R di Modal partecipation Dynamic magnification Static Correction

21 Maximum With f 0 = max{ f(t) } the peak pseudo displacement is D i0 = R di f 0 /ωi 2 and the peak of the modal contribution is s i0 (t) = s i s st ωi 2 D i0 (t) = f 0 s st s i R di Modal partecipation Dynamic magnification Static Correction The first two terms are independent of the mode, the last are independent from each other and their product is the that influences the modal contributions. Note that this product has the sign of s i, as the dynamic response is always positive.

22 MCF s example The following table (from Chopra, 2nd ed.) displays the s i and their partial sums for a shear-type, 5 floors building where all the storey masses are equal and all the storey stiffnesses are equal too. The response quantities chosen are x 5n, the MCF s to the top displacement and V n, the MCF s to the base shear, for two different load shapes.

23 MCF s example The following table (from Chopra, 2nd ed.) displays the s i and their partial sums for a shear-type, 5 floors building where all the storey masses are equal and all the storey stiffnesses are equal too. The response quantities chosen are x 5n, the MCF s to the top displacement and V n, the MCF s to the base shear, for two different load shapes. r = {0, 0, 0, 0, 1} T r = {0, 0, 0, 1, 2} T Top Displacement Base Shear Top Displacement Base Shear n or J x 5n J x 5i Vn J Vi x 5n J x 5i Vn J Vi

24 MCF s example The following table (from Chopra, 2nd ed.) displays the s i and their partial sums for a shear-type, 5 floors building where all the storey masses are equal and all the storey stiffnesses are equal too. The response quantities chosen are x 5n, the MCF s to the top displacement and V n, the MCF s to the base shear, for two different load shapes. r = {0, 0, 0, 0, 1} T r = {0, 0, 0, 1, 2} T Top Displacement Base Shear Top Displacement Base Shear n or J x 5n J x 5i Vn J Vi x 5n J x 5i Vn J Vi Note that for any given r, the base shear is more influenced by higher modes, and 2. for any given reponse quantity, the second, skewed r gives greater modal contributions for higher modes.

25 Dynamic Ratios Dynamic Ratios are the same that we have seen for SDOF systems. Next page, for an undamped system, harmonically excited, solid line, the ratio of the modal elastic force F S,i = K iq i sin ωt to the harmonic applied modal force, P i sin ωt, plotted against the frequency ratio β = ω/ω i. For β = 0 the ratio is 1, the applied load is fully balanced by the elastic resistance. For fixed excitation frequency, β 0 for high modal frequencies. dashed line,the ratio of the modal inertial force, F I,i = β 2 F S,i to the load. Modal partecipation Dynamic magnification Static Correction

26 Dynamic Ratios Dynamic Ratios are the same that we have seen for SDOF systems. Next page, for an undamped system, harmonically excited, solid line, the ratio of the modal elastic force F S,i = K iq i sin ωt to the harmonic applied modal force, P i sin ωt, plotted against the frequency ratio β = ω/ω i. For β = 0 the ratio is 1, the applied load is fully balanced by the elastic resistance. For fixed excitation frequency, β 0 for high modal frequencies. dashed line,the ratio of the modal inertial force, F I,i = β 2 F S,i to the load. Modal partecipation Dynamic magnification Static Correction Note that for steady-state motion the sum of the elastic and inertial force ratios is constant and equal to 1, as in (F S,i + F I,i) sin ωt = P i sin ωt.

27 Modal resistance ratios F S /P i F I /P i Frequency ratio, β=ω/ω i For a fixed excitation frequency and high modal frequencies the frequency ratio β 0.

28 Modal resistance ratios F S /P i F I /P i Frequency ratio, β=ω/ω i For a fixed excitation frequency and high modal frequencies the frequency ratio β 0. For β 0 the response is quasi-static.

29 Modal resistance ratios F S /P i F I /P i Frequency ratio, β=ω/ω i For a fixed excitation frequency and high modal frequencies the frequency ratio β 0. For β 0 the response is quasi-static. Hence, for higher modes the response is pseudo-static.

30 Modal resistance ratios F S /P i F I /P i Frequency ratio, β=ω/ω i For a fixed excitation frequency and high modal frequencies the frequency ratio β 0. For β 0 the response is quasi-static. Hence, for higher modes the response is pseudo-static. On the contrary, for excitation frequencies high enough the lower modes respond with purely inertial forces.

31 Static Correction The preceding discussion indicates that higher modes contributions to the response could be approximated with the static response, leading to a Static Correction of the dynamic response. Modal partecipation Dynamic magnification Static Correction

32 Static Correction The preceding discussion indicates that higher modes contributions to the response could be approximated with the static response, leading to a Static Correction of the dynamic response. For a system where q i (t) p i(t) K i for i > n dy, n dy being the number of dynamically responding modes, we can write Modal partecipation Dynamic magnification Static Correction n dy x(t) x dy (t) + x st (t) = ψ i q i (t) + 1 N n dy +1 ψ i p i (t) K i where the response for each of the first n dy modes can be computed as usual.

33 Static Modal Components The static modal displacement component x j, j > n dy can be written x j (t) = ψ j q j (t) ψ jψj T p(t) = F j p(t) K j The modal flexibility matrix is defined by F j = ψ jψ T j K j and is used to compute the j-th mode static deflections due to the applied load vector. The total displacements, the dynamic contributions and the static correction, for p(t) = r f(t), are then n dy x ψ j q j (t) + f(t) 1 N F j r. n dy +1 Modal partecipation Dynamic magnification Static Correction

34 Alternative Formulation Our last formula for static correction is n dy x ψ jq j(t) + f(t) 1 n dy +1 N F jr. Modal partecipation Dynamic magnification Static Correction To use the above formula all mode shapes, all modal stiffnesses and all modal flexibility matrices must be computed, undermining the efficiency of the procedure.

35 Alternative Formulation This problem can be obviated computing the total static displacements, x total st = K 1 p(t), and subtracting the static displacements due to the first n dy modes... n N dy ( n dy ) F jrf(t) = K 1 rf(t) F jrf(t) = f(t) K 1 F j r, n dy 1 1 so that the corrected total displacements have the expression n dy ( n dy ) x ψ iq i(t) + f(t) K 1 F i r, 1 1 Modal partecipation Dynamic magnification Static Correction The constant term (a generalized displacement vector) following f(t) can be computed with the information in our posses at the moment we begin the integration of the modal equations of motion.

36 Effectiveness of Static Correction In these circumstances, few modes with static correction give results comparable to the results obtained using much more modes in a straightforward modal displacement superposition analysis. Modal partecipation Dynamic magnification Static Correction

37 Effectiveness of Static Correction In these circumstances, few modes with static correction give results comparable to the results obtained using much more modes in a straightforward modal displacement superposition analysis. An high number of modes is required to account for the spatial distribution of the loading but only a few lower modes are subjected to significant dynamic amplification. Modal partecipation Dynamic magnification Static Correction

38 Effectiveness of Static Correction In these circumstances, few modes with static correction give results comparable to the results obtained using much more modes in a straightforward modal displacement superposition analysis. An high number of modes is required to account for the spatial distribution of the loading but only a few lower modes are subjected to significant dynamic amplification. Refined stress analysis is required even if the dynamic response involves only a few lower modes. Modal partecipation Dynamic magnification Static Correction

39 Part II Numerical Integration Constant Acceleration Wilson s Theta Method Constant Acceleration Wilson s Theta Method

40 to Numerical Integration When we reviewed the numerical integration methods, we said that some methods are unconditionally stable and others are conditionally stable, that is the response blows-out if the time step h is great with respect to the natural preriod of vibration, h > Tn a, where a is a constant that depends on the numerical algorithm. For MDOF systems, the relevant T is the one associated with the highest mode present in the structural model, so for moderately complex structures it becomes impossibile to use a conditionally stable algorithm. In the following, two unconditionally stable algorithms will be analysed, i.e., the constant acceleration method, that we already know, and the new Wilson sθ method. Constant Acceleration Wilson s Theta Method

41 Constant Acceleration, preliminaries The initial conditions are known: x 0, ẋ 0, p 0, ẍ 0 = M 1 (p 0 C ẋ 0 K x 0 ). Constant Acceleration Wilson s Theta Method With a fixed time step h, compute the constant matrices A = 2C + 4 h M, B = 2M, K+ = 2 h C + 4 h 2 M.

42 Constant Acceleration, stepping Starting with i = 0, compute the effective force increment, ˆp i = p i+1 p i + Aẋ i + Bẍ i, the tangent stiffness K i and the current incremental stiffness, ˆK i = K i + K +. Constant Acceleration Wilson s Theta Method For linear systems, it is x i = ˆK 1 i ˆp i, for a non linear system x i is produced by the modified Newton-Raphson iteration procedure. The state vectors at the end of the step are x i+1 = x i + x i, ẋ i+1 = 2 x i h ẋ i

43 Constant Acceleration, new step Constant Acceleration Increment the step index, i = i + 1. Wilson s Theta Method Compute the accelerations using the equation of equilibrium, ẍ i = M 1 (p i C ẋ i K x i ). Repeat the substeps detailed in the previous slide.

44 Modified Newton-Raphson Initialization y 0 = x i R 1 = ˆp i For j = 1, 2,... K T y j = R j y j = y j 1 + y j, f S,j = f S (updated system state) f S,0 = f S (system state) K T = ˆK i y j (test for convergence) ẏ j = ẏ j = ẏ j 1 + ẏ j Constant Acceleration Wilson s Theta Method f S,j = f S,j f S,j 1 (K T K i) y j R j+1 = R j f S,j Return the value x i = y j x i A suitable convergence test is R T j y j ˆp T i xi,j tol

45 Wilson s Theta Method The linear acceleration method is significantly more accurate than the constant acceleration method, meaning that it is possible to use a longer time step to compute the response of a SDOF system within a required accuracy. On the other hand, the method is not safely applicable to MDOF systems due to its numerical instability. Constant Acceleration Wilson s Theta Method

46 Wilson s Theta Method The linear acceleration method is significantly more accurate than the constant acceleration method, meaning that it is possible to use a longer time step to compute the response of a SDOF system within a required accuracy. On the other hand, the method is not safely applicable to MDOF systems due to its numerical instability. Professor Ed Wilson demonstrated that simple variations of the linear acceleration method can be made unconditionally stable and found the most accurate in this family of algorithms, collectively known as Wilson s θ methods. Constant Acceleration Wilson s Theta Method

47 Wilson s θ method Wilson s idea is very simple: the results of the linear acceleration algorithm are good enough only in a fraction of the time step. Wilson demonstrated that his idea was correct, too... The procedure is really simple, Constant Acceleration Wilson s Theta Method

48 Wilson s θ method Wilson s idea is very simple: the results of the linear acceleration algorithm are good enough only in a fraction of the time step. Wilson demonstrated that his idea was correct, too... The procedure is really simple, 1. solve the incremental equation of equilibrium using the linear acceleration algorithm, with an extended time step Constant Acceleration Wilson s Theta Method ĥ = θ h, θ 1,

49 Wilson s θ method Wilson s idea is very simple: the results of the linear acceleration algorithm are good enough only in a fraction of the time step. Wilson demonstrated that his idea was correct, too... The procedure is really simple, 1. solve the incremental equation of equilibrium using the linear acceleration algorithm, with an extended time step Constant Acceleration Wilson s Theta Method ĥ = θ h, θ 1, 2. compute the extended acceleration increment ˆ ẍ at ˆt = t i + ĥ,

50 Wilson s θ method Wilson s idea is very simple: the results of the linear acceleration algorithm are good enough only in a fraction of the time step. Wilson demonstrated that his idea was correct, too... The procedure is really simple, 1. solve the incremental equation of equilibrium using the linear acceleration algorithm, with an extended time step Constant Acceleration Wilson s Theta Method ĥ = θ h, θ 1, 2. compute the extended acceleration increment ˆ ẍ at ˆt = t i + ĥ, 3. scale the extended acceleration increment under the assumption of linear acceleration, ẍ = 1 θ ˆ ẍ,

51 Wilson s θ method Wilson s idea is very simple: the results of the linear acceleration algorithm are good enough only in a fraction of the time step. Wilson demonstrated that his idea was correct, too... The procedure is really simple, 1. solve the incremental equation of equilibrium using the linear acceleration algorithm, with an extended time step Constant Acceleration Wilson s Theta Method ĥ = θ h, θ 1, 2. compute the extended acceleration increment ˆ ẍ at ˆt = t i + ĥ, 3. scale the extended acceleration increment under the assumption of linear acceleration, ẍ = 1 θ ˆ ẍ, 4. compute the velocity and displacements increment using the reduced value of the increment of acceleration.

52 Wilson s θ method description Using the same symbols used for constant acceleration. First of all, for given initial conditions x 0 and ẋ 0, initialise the procedure computing the constants (matrices) used in the following procedure and the initial acceleration, Constant Acceleration Wilson s Theta Method ẍ 0 = M 1 (p 0 C ẋ 0 K x 0 ), A = 6M/ĥ + 3C, B = 3M + ĥc/2, K + = 3C/ĥ + 6M/ĥ2.

53 Wilson s θ method description Starting with i = 0, 1. update the tangent stiffness, K i = K(x, ẋ i ) and the effective stiffness, ˆKi = K i + K +, compute ˆ ˆp i = θ p i + Aẋ i + Bẍ i, with p i = p(t i + h) p(t i ) Constant Acceleration Wilson s Theta Method

54 Wilson s θ method description Starting with i = 0, 1. update the tangent stiffness, K i = K(x, ẋ i ) and the effective stiffness, ˆKi = K i + K +, compute ˆ ˆp i = θ p i + Aẋ i + Bẍ i, with p i = p(t i + h) p(t i ) 2. solve ˆK i ˆ x = ˆ ˆpi, compute ˆ ẍ = 6 ˆ x ĥ 2 6ẋi ĥ 3ẍ i ẍ = 1 θ ˆ ẍ Constant Acceleration Wilson s Theta Method

55 Wilson s θ method description Starting with i = 0, 1. update the tangent stiffness, K i = K(x, ẋ i ) and the effective stiffness, ˆKi = K i + K +, compute ˆ ˆp i = θ p i + Aẋ i + Bẍ i, with p i = p(t i + h) p(t i ) 2. solve ˆK i ˆ x = ˆ ˆpi, compute 3. compute ˆ ẍ = 6 ˆ x ĥ 2 6ẋi ĥ 3ẍ i ẍ = 1 θ ˆ ẍ Constant Acceleration Wilson s Theta Method ẋ = (ẍ i ẍ)h x = ẋ i h + ( 1 2ẍi ẍ)h2

56 Wilson s θ method description Starting with i = 0, 1. update the tangent stiffness, K i = K(x, ẋ i ) and the effective stiffness, ˆKi = K i + K +, compute ˆ ˆp i = θ p i + Aẋ i + Bẍ i, with p i = p(t i + h) p(t i ) 2. solve ˆK i ˆ x = ˆ ˆpi, compute 3. compute ˆ ẍ = 6 ˆ x ĥ 2 6ẋi ĥ 3ẍ i ẍ = 1 θ ˆ ẍ Constant Acceleration Wilson s Theta Method ẋ = (ẍ i ẍ)h x = ẋ i h + ( 1 2ẍi ẍ)h2 4. update state, x i+1 = x i + x, ẋ i+1 = ẋ i + ẋ, i = i + 1, iterate restarting from 1.

57 A final remark Constant Acceleration Wilson s Theta Method The Theta Method is unconditionally stable for θ > 1.37 and it achieves the maximum accuracy for θ = 1.42.

58 Part III Multiple Support Excitation The Equation of Motion The Equation of Motion An Example Analysis Analysis Example An Example Analysis Analysis Example

59 Definitions Consider the case of a structure where the supports are subjected to assigned displacements histories, u i = u i (t). To solve this problem, we start with augmenting the degrees of freedom with the support displacements. We denote the superstructure DOF with x T, the support DOF with x g and we have a global displacement vector x, { } xt x =. x g The Equation of Motion An Example Analysis Analysis Example

60 The Equation of Motion Damping effects will be introduced at the end of our manipulations. The equation of motion is [ ] } [ ] { } { } M Mg {ẍt K Kg xt 0 + M gg ẍ g Kg T = K gg x g p g M T g The Equation of Motion An Example Analysis Analysis Example where M and K are the usual structural matrices, while M g and M gg are, in the common case of a lumped mass model, zero matrices.

61 Static Components We decompose the vector of displacements into two contributions, a static contribution and a dynamic contribution, attributing the given support displacements to the static contribution. { xt x g } = { xs x g } + { x 0 } The Equation of Motion An Example Analysis Analysis Example where x is the usual relative displacements vector.

62 Determination of static components Because the x g are given, we can write two matricial equations that give us the static superstructure displacements and the forces we must apply to the supports, From the first equation we have Kx s + K g x g = 0 K T g x s + K gg x g = p g The Equation of Motion An Example Analysis Analysis Example and from the second we have x s = K 1 K g x g p g = (K gg K T g K 1 K g )x g

63 Determination of static components Because the x g are given, we can write two matricial equations that give us the static superstructure displacements and the forces we must apply to the supports, From the first equation we have Kx s + K g x g = 0 K T g x s + K gg x g = p g The Equation of Motion An Example Analysis Analysis Example and from the second we have x s = K 1 K g x g p g = (K gg K T g K 1 K g )x g The support forces are zero when the structure is isostatic or the structure is subjected to a rigid motion.

64 Going back to the EOM We need the first row of the two matrix equation of equilibrium, [ ] } [ ] { } { } M Mg {ẍt K Kg xt 0 + = M T g M gg ẍ g K T g K gg substituting x T = x s + x in the first row Mẍ + Mẍ s + M g ẍ g + Kx + Kx s + K g x g = 0 x g p g The Equation of Motion An Example Analysis Analysis Example by the equation of static equilibrium, Kx s + K g x g = 0 we can simplify Mẍ+Mẍ s +M g ẍ g +Kx = Mẍ+(M g MK 1 K g )ẍ g +Kx = 0.

65 Influence matrix The equation of motion is Mẍ + (M g MK 1 K g )ẍ g + Kx = 0. We define the influence matrix E by E = K 1 K g, The Equation of Motion An Example Analysis Analysis Example and write, reintroducing the damping effects, Mẍ + Cẋ + Kx = (ME + M g )ẍ g (CE + C g )ẋ g

66 Simplification of the EOM The Equation of Motion For a lumped mass model, M g = 0 and also the efficace forces due to damping are really small with respect to the inertial ones, and with this understanding we write An Example Analysis Analysis Example Mẍ + Cẋ + Kx = MEẍ g.

67 Significance of E E can be understood as a collection of vectors e i, i = 1,..., N g (N g being the number of DOF associated with the support motion), E = [ e 1 e 2 e Ng ] The Equation of Motion An Example Analysis Analysis Example where the individual e i collects the displacements in all the DOF of the superstructure due to imposing a unit displacement to the support DOF number i.

68 Significance of E This understanding means that the influence matrix can be computed column by column, in the general case by releasing one support DOF, applying a unit force to the released DOF, computing all the displacements and scaling the displacements so that the support displacement component is made equal to 1, or in the case of an isostatic component by examining the instantaneous motion of the 1 DOF rigid system that we obtain by releasing one constraint. The Equation of Motion An Example Analysis Analysis Example

69 A First Example m m A x 2 B v B = v B (t) We want to determine the influence matrix E for the structure in the figure above, subjected to an assigned motion in B. x 1 The Equation of Motion An Example Analysis A x 2 x 3 x 1 Analysis Example First step, put in evidence another degree of freedom x 3 corresponding to the assigned vertical motion of the support in B and compute, using e.g. the PVD, the flexibility matrix: F = L3 3EJ

70 Example, cont. The stiffness matrix is found by inversion, K = 3EJ L We are interested in the partitions K xx and K xg : K xx = 3EJ 13L 3 [ ] , K xg = 3EJ 13L 3 [ ] The Equation of Motion An Example Analysis Analysis Example The influence matrix is E = K 1 xx K xg = 1 16 [ ] , please compare E with the last column of the flexibility matrix, F.

71 Analysis Consider the vector of support accelerations, ẍ g = { ẍ gl, l = 1,..., N g } and the effective load vector p eff = MEẍ g = Me l ẍ gl (t). We can write the modal equation of motion for mode number n N g l=1 The Equation of Motion An Example Analysis Analysis Example where N g q n + 2ζ n ω n q n + ωnq 2 n = Γ nl ẍ gl (t) Γ nl = ψt n Me l M n l=1

72 Analysis, cont. The solution q n (t) is hence, with the notation of last lesson, N g q n (t) = Γ nl D nl (t), l=1 The Equation of Motion An Example Analysis Analysis Example D nl being the response function for ζ n and ω n due to the ground excitation ẍ gl.

73 Analysis, cont. The total displacements x T are given by two contributions, x T = x s + x, the expression of the contributions are and finally we have N g x s = Ex g (t) = e l x gl (t), x = l=1 N N g ψ n Γ nl D nl (t), n=1 l=1 The Equation of Motion An Example Analysis Analysis Example N g x T = e l x gl (t) + l=1 n=1 l=1 N N g ψ n Γ nl D nl (t).

74 in terms of Forces For a computer program, the easiest way to compute the nodal forces is a) compute, element by element, the nodal displacements by x T and x g, b) use the element stiffness matrix compute nodal forces, c) assemble element nodal loads into global nodal loads. That said, let s see the analytical development... The Equation of Motion An Example Analysis Analysis Example

75 Forces, cont. The forces on superstructure nodes due to deformations are N N g f s = Γ nl Kψ n D nl (t) n=1 l=1 N N g f s = (Γ nl Mψ n )(ωnd 2 nl (t)) = r nl A nl (t) n=1 l=1 the forces on support The Equation of Motion An Example Analysis Analysis Example or, using x s = Ex g l=1 f gs = K T g x T + K gg x g = K T g x + p g N g f gs = ( Kg T e l + K gg,l )x gl + N N g Γ nl Kg T ψ n D nl (t) n=1 l=1

76 Forces The Equation of Motion The structure response components must be computed considering the structure loaded by all the nodal forces, { } fs f =. f gs An Example Analysis Analysis Example

77 Example 3,6 1,7 4,8 2,9 5,10 m L L L L m The Equation of Motion An Example Analysis Analysis Example The dynamic DOF are x 1 and x 2, vertical displacements of the two equal masses, x 3, x 4, x 5 are the imposed vertical displacements of the supports, x 6,..., x 10 are the rotational degrees of freedom (removed by static condensation).

78 Example The stiffness matrix for the 10x10 model is K = EJ L L 6L L 0 6L L 0 6L L 0 6L L 6L 6L 6L L 2 2L L 0 6L 0 0 2L 2 8L 2 2L L 0 6L 0 0 2L 2 8L 2 2L L 0 6L 0 0 2L 2 8L 2 2L L 6L L 2 4L 2 The Equation of Motion An Example Analysis Analysis Example

79 Example, cont. The first product of the static condensation procedure is the linear mapping between translational and rotational degrees of freedom, given by φ = 1 56L [ ] x The Equation of Motion An Example Analysis Analysis Example

80 Example, cont. Following static condensation and reordering rows and columns, the partitioned stiffness matrices are The influence matrix is K = EJ [ L ], K g = EJ [ L 3 K gg = EJ [ L 3 ] , ] The Equation of Motion An Example Analysis Analysis Example E = K 1 K g = 32[ ]

81 Example, cont. The eigenvector matrix is the matrix of modal masses is Ψ = [ ] M = Ψ T MΨ = m[ ] the matrix of the non normalized modal partecipation coefficients is [ ] L = Ψ T ME = m The Equation of Motion An Example Analysis Analysis Example and, finally, the matrix of modal partecipation s, [ ] Γ = (M ) 1 L =

82 Example, cont. Denoting with D ij = D ij (t) the response function for mode i due to ground excitation ẍ gj, the response can be written x = = ( ψ 11( 1 4 D D 13)+ψ 12( 5 32 D D D 22) ψ 21( 1 4 D D 13)+ψ 22( 5 32 D D D 22) ( 1 4 D D D D D D D D D D 22 ). ) The Equation of Motion An Example Analysis Analysis Example