CALCULATION OF STIFFNESS AND MASS ORTHOGONAL VECTORS

Transcription

1 14. CALCULATION OF STIFFNESS AND MASS ORTHOGONAL VECTORS LDR Vectors are Always More Accurate tha Usig the Exact Eigevectors i a Mode Superpositio Aalysis 14.1 INTRODUCTION The major reaso to calculate mode shapes (or eigevectors ad eigevalues) is that they are used to ucouple the dyamic equilibrium equatios for mode superpositio ad/or respose spectra aalyses. The mai purpose of a dyamic respose aalysis of a structure is to accurately estimate displacemets ad member forces i the real structure. I geeral, there is o direct relatioship betwee the accuracy of the eigevalues ad eigevectors ad the accuracy of ode poit displacemets ad member forces. I the early days of earthquake egieerig, the Rayleigh-Ritz method of dyamic aalysis was used extesively to calculate approximate solutios. With the developmet of high-speed computers, the use of exact eigevectors replaced the use of Ritz vectors as the basis for seismic aalysis. It will be illustrated i this book that Load-Depedet Ritz, LDR, vectors ca be used for the dyamic aalysis of both liear ad oliear structures. The ew modified Ritz method produces more accurate results, with less computatioal effort, tha the use of exact eigevectors.

2 14-2 DYNAMIC ANALYSIS OF STRUCTURES There are several differet umerical methods available for the evaluatio of the eigevalue problem. However, for large structural systems, oly a few methods have prove to be both accurate ad robust DETERMINATE SEARCH METHOD The equilibrium equatio, which govers the udamped free vibratio of a typical mode, is give by: 2 [ K - ω i M] vi = 0 or Ki vi = 0 (14.1) Equatio 14.1 ca be solved directly for the atural frequecies of the structure by assumig values for ad factorig the followig equatio: ω i Ki = Li Di L (14.2) i T From Appedix C the determiat of the factored matrix is defied by: Det ( ωi) = D11 D DNN (14.3) It is possible, by repeated factorizatio, to develop a plot of the determiat vs. λ, as show i Figure This classical method for evaluatig the atural frequecies of a structure is called the determiat search method [1]. It should be oted that for matrices with small badwidths the umerical effort to factor the matrices is very small. For this class of problem the determiat search method, alog with iverse iteratio, is a effective method of evaluatig the udamped frequecies ad mode shapes for small structural systems. However, because of the icrease i computer speeds, small problems ca be solved by ay method i a few secods. Therefore, the determiat search method is o loger used i moder dyamic aalysis programs.

3 EIGEN AND RITZ VECTOR EVALUATION 14-3 Det ( λ ) All Terms I D Positive Two Neg. Dii Six Neg. Dii λ 1 λ2 λ 3 λ, λ 4 5 λ 6 λ Five Neg. Dii Oe Neg. Dii Three Neg. Dii Figure 14.1 Determiat vs. Frequecy for Typical System 14.3 STURM SEQUENCE CHECK Figure 14.1 illustrates a very importat property of the sequece of diagoal terms of the factored matrix. Oe otes that for a specified value of ω i, oe ca cout the umber of egative terms i the diagoal matrix ad it is always equal to the umber of frequecies below that value. Therefore, it ca be used to check a method of solutio that fails to calculate all frequecies below a specified value. Also, aother importat applicatio of the Sturm Sequece Techique is to evaluate the umber of frequecies withi a frequecy rage. It is oly ecessary to factor the matrix at both the maximum ad miimum frequecy poits, ad the differece i the umber of egative diagoal terms is equal to the umber of frequecies i the rage. This umerical techique is useful i machie vibratio problems INVERSE ITERATION Equatio (14.1) ca be writte i a iterative solutio form as: K V (i-1) (i-1) T = λ MV or LDL V = R (14.4)

4 14-4 DYNAMIC ANALYSIS OF STRUCTURES The computatioal steps required for the solutio of oe eigevalue ad eigevector ca be summarized as follows: 1. Factor stiffess matrix ito triagularized LD L T form durig static load solutio phase. (1) 2. For the first iteratio, assume R to be a vector of radom umbers ad (1) solve for iitial vector V. 3. Iterate with i = 1, 2... a. Normalize vector so that V b. Estimate eigevalue λ T = V M V T c. Check λ for covergece - if coverged, termiate d. i = i + 1 ad calculate R = λ R (i-1) MV T e. Solve for ew vector LD L V = R f. Repeat Step 3 It ca easily be show that this method will coverge to the smallest uique eigevalue. = I (i-1) 14.5 GRAM-SCHMIDT ORTHOGONALIZATION Additioal eigevectors ca be calculated usig the iverse iteratio method if, after each iteratio cycle, the iteratio vector is made orthogoal to all previously calculated vectors. To illustrate the method, let us assume that we have a approximate vector V that eeds to be made orthogoal to the previously calculated vector. Or, the ew vector ca be calculated from: V = V - V αv (14.5) Multiplyig Equatio (14.3) by V T M, we obtai:

5 EIGEN AND RITZ VECTOR EVALUATION 14-5 T T T V MV = V MV - α V MV = 0 (14.6) Therefore, the orthogoality requiremet is satisfied if: T V MV α = T V MV = V T MV (14.7) If the orthogoalizatio step is iserted after Step 3.e i the iverse iteratio method, additioal eigevalues ad vectors ca be calculated BLOCK SUBSPACE ITERATION Iverse iteratio with oe vector may ot coverge if eigevalues are idetical ad the eigevectors are ot uique. This case exists for may real threedimesioal structures, such as buildigs with equal stiffess ad mass i the priciple directios. This problem ca be avoided by iteratig with a block of orthogoal vectors [2]. The block subspace iteratio algorithm is summarized i Table 14.1 ad is the method used i the moder versios of the SAP program. Experiece has idicated that the subspace block size b should be set equal to the square root of the average badwidth of the stiffess matrix, but, ot less tha six. The block subspace iteratio algorithm is relatively slow; however, it is very accurate ad robust. I geeral, after a vector is added to a block, it requires five to te forward reductios ad back-substitutios before the iteratio vector coverges to the exact eigevector.

6 14-6 DYNAMIC ANALYSIS OF STRUCTURES Table 14.1 Subspace Algorithm for the Geeratio of Eigevectors I. INITIAL CALCULATIONS A. Triagularize Stiffess Matrix. (0) B. Use radom umbers to form a block of b vectors V. II. GENERATE L EIGENVECTORS BY ITERATION i = 1,2... (i-1) A. Solve for block of vectors, X i, K X = M V. B. Make block of vectors, X, stiffess ad mass orthogoal, eigevalues ad correspodig vectors i ascedig order. V. Order C. Use Gram-Schmidt method to make orthogoal to all previously T calculated vectors ad ormalized so that V MV = I. D. Perform the followig checks ad operatios: 1. If first vector i block is ot coverged, go to Step A with i = i Save Vector φ o Disk. V 3. If equals L, termiate iteratio. 4. Compact block of vectors. 5. Add radom umber vector to last colum of block. Retur to Step D.1 with = SOLUTION OF SINGULAR SYSTEMS For a few types of structures, such as aerospace vehicles, it is ot possible to use iverse or subspace iteratio directly to solve for mode shapes ad frequecies. This is because there is a miimum of six rigid-body modes with zero frequecies ad the stiffess matrix is sigular ad caot be triagularized. To

7 EIGEN AND RITZ VECTOR EVALUATION 14-7 solve this problem, it is oly ecessary to itroduce the followig eigevalue shift, or chage of variable: λ = λ ρ (14.8) Hece, the iterative eigevalue problem ca be writte as: K V (i-1) (i-1) T = λ MV or LDL V = R (14.9) The shifted stiffess matrix is ow o-sigular ad is defied by: K = K + ρm (14.10) The eigevectors are ot modified by the arbitrary shift ρ. The correct eigevalues are calculated from Equatio (14.8) GENERATION OF LOAD-DEPENDENT RITZ VECTORS The umerical effort required to calculate the exact eige solutio ca be eormous for a structural system if a large umber of modes are required. However, may egieers believe that this computatioal effort is justifiable if accurate results are to be obtaied. Oe of the purposes of this sectio is to clearly illustrate that this assumptio is ot true for the dyamic respose aalyses of all structural systems. It is possible to use the exact free-vibratio mode shapes to reduce the size of both liear ad oliear problems. However, this is ot the best approach for the followig reasos: 1. For large structural systems, the solutio of the eigevalue problem for the free-vibratio mode shapes ad frequecies ca require a sigificat amout of computatioal effort. 2. I the calculatio of the free-vibratio mode shapes, the spatial distributio of the loadig is completely disregarded. Therefore, may of the mode shapes that are calculated are orthogoal to the loadig ad do ot participate i the dyamic respose.

8 14-8 DYNAMIC ANALYSIS OF STRUCTURES 3. If dyamic loads are applied at massless degrees-of-freedom, the use of all the exact mode shapes i a mode superpositio aalysis will ot coverge to the exact solutio. I additio, displacemets ad stresses ear the applicatio of the loads ca be i sigificat error. Therefore, there is o eed to apply the static correctio method as would be required if exact eigevectors are used for such problems. 4. It is possible to calculate a set of stiffess ad mass orthogoal Ritz vectors, with a miimum of computatioal effort, which will coverge to the exact solutio for ay spatial distributio of loadig [2]. It ca be demostrated that a dyamic aalysis based o a uique set of Load Depedet Vectors yields a more accurate result tha the use of the same umber of exact mode shapes. The efficiecy of this techique has bee illustrated by solvig may problems i structural respose ad i wave propagatio types of problems [4]. Several differet algorithms for the geeratio of Load Depedet Ritz Vectors have bee published sice the method was first itroduced i 1982 [3]. Therefore, it is ecessary to preset i Table 14.2 the latest versio of the method for multiple load coditios. Table 14.2 Algorithm for Geeratio of Load Depedet Ritz Vectors I. INITIAL CALCULATIONS A. Triagularize Stiffess Matrix K = L T DL. B. Solve for block of b static displacemet vectors us resultig from spatial load patters F ; or, K u s = F. C. Make block of vectors u, stiffess ad mass orthogoal,. s V1 II. GENERATE BLOCKS OF RITZ VECTORS i = 2,...N A. Solve for block of vectors, X i, K Xi = M Vi-1. B. Make block of vectors, X,stiffess ad mass orthogoal,. i Vi

9 EIGEN AND RITZ VECTOR EVALUATION 14-9 Table 14.2 Algorithm for Geeratio of Load Depedet Ritz Vectors C. Use Modified Gram-Schmidt method (two times) to make orthogoal to all previously calculated vectors ad ormalized so that T V MVi = I. i III. MAKE VECTORS STIFFNESS ORTHOGONAL 2 A. Solve Nb by Nb eigevalue problem [ K - Ω I] Z = 0 where K = V T KV. B. Calculate stiffess orthogoal Ritz vectors, Φ = VZ. Vi 14.9 A PHYSICAL EXPLANATION OF THE LDR ALGORITHM The physical foudatio for the method is the recogitio that the dyamic respose of a structure will be a fuctio of the spatial load distributio. The udamped, dyamic equilibrium equatios of a elastic structure ca be writte i the followig form: M u&& (t) + Ku(t) = R(t) (14.11) I the case of earthquake or wid, the time-depedet loadig actig o the structure, R(t), Equatio (13.1), ca be writte as: J R( t ) = f jg( t) j = FG( t) j= 1 (14.12) Note that the idepedet load patters F are ot a fuctio of time. For costat earthquake groud motios at the base of the structure three idepedet load patters are possible. These load patters are a fuctio of the directioal mass distributio of the structure. I case of wid loadig, the dowwid mea wid pressure is oe of those vectors. The time fuctios G(t) ca always be expaded ito a Fourier series of sie ad cosie fuctios. Hece, eglectig

10 14-10 DYNAMIC ANALYSIS OF STRUCTURES dampig, a typical dyamic equilibrium equatio to be solved is of the followig form: M u& ( t) + Ku( t) = F si ωt (14.13) Therefore, the exact dyamic respose for a typical loadig frequecy the followig form: ϖ is of Ku = F + ϖ 2 Mu (14.14) This equatio caot be solved directly because of the ukow frequecy of the loadig. However, a series of stiffess ad mass orthogoal vectors ca be calculated that will satisfy this equatio usig a perturbatio algorithm. The first block of vectors is calculated by eglectig the mass ad solvig for the static respose of the structure. Or: Ku = F 0 (14.15) From Equatio (14.14) it is apparet that the distributio of the error i the solutio, due to eglectig the iertia forces, ca be approximated by: F Mu 0 1 (14.16) Therefore, a additioal block of displacemet error, or correctio, vectors ca be calculated from: Ku 1 = F 1 (14.17) I calculatig u 1 the additioal iertia forces are eglected. Hece, i cotiuig this thought process, it is apparet the followig recurrece equatio exists: Ku i Mu (14.18) = i 1 A large umber of blocks of vectors ca be geerated by Equatio (14.18). However, to avoid umerical problems, the vectors must be stiffess ad mass orthogoal after each step. I additio, care should be take to make sure that all vectors are liearly idepedet. The complete umerical algorithm is summarized i Table After careful examiatio of the LDR vectors, oe ca coclude that dyamic aalysis is a simple extesio of static aalysis

11 EIGEN AND RITZ VECTOR EVALUATION because the first block of vectors is the static respose from all load patters actig o the structure. For the case where loads are applied at oly the mass degrees-of-freedom, the LDR vectors are always a liear combiatio of the exact eigevectors. It is of iterest to ote that the recursive equatio, used to geerate the LDR vectors, is similar to the Laczos algorithm for calculatig exact eigevalues ad vectors, except that the startig vectors are the static displacemets caused by the spatial load distributios. Also, there is o iteratio ivolved i the geeratio of Load Depedet Ritz vectors COMPARISON OF SOLUTIONS USING EIGEN AND RITZ VECTORS The fixed-ed beam show i Figure 14.1 is subjected to a poit load at the ceter of the beam. The load varies i time as a costat uit step fuctio = 240 Modulus of Elasticity = 30,000,000 Momet of Iertia = 100 Mass per Uit Legth = 0.1 Dampig Ratio = 0.01 All uits i Pouds ad Iches Figure 14.1 Dimesios, Stiffess ad Mass for Beam Structure The dampig ratio for each mode was set at oe percet ad the maximum displacemet ad momet occur at secod, as show i Table The results clearly idicate the advatages of usig load-depedet vectors. Oe otes that the free-vibratio modes 2, 4, 6 ad 8 are ot excited by the loadig because they are osymmetrical. However, the load depedet algorithm

12 14-12 DYNAMIC ANALYSIS OF STRUCTURES geerates oly the symmetrical modes. I fact, the algorithm will fail for this case, if more tha five vectors are requested. Table 14.3 Results from Dyamic Aalyses of Beam Structure Number of Vectors Free-Vibratio Mode Shapes Load-Depedet Ritz Vectors Displacemet Momet Displacemet Momet (-2.41) (-2.41) (-0.46) (-0.46) (-0.08) (-0.04) (0.00) 4178 (-22.8) 4178 (-22.8) 4946 (-8.5) 4946 (-8.5) 5188 (-4.1) 5304 (-2.0) 5411 (0.0) (+0.88) (-2.00) (+0.08) (+0.06) (0.00) 5907 (+9.2) 5563 (+2.8) 5603 (+3.5) 5507 (+1.8) 5411 (0.0) Note: Numbers is paretheses are percetage errors. Both methods give good results for the maximum displacemet. The results for maximum momet, however, idicate that the load-depedet vectors give sigificatly better results ad coverge from above the exact solutio. It is clear that free-vibratio mode shapes are ot ecessarily the best vectors to be used i mode-superpositio dyamic respose aalysis. Not oly is the calculatio of the exact free-vibratio mode shapes computatioally expesive, it requires more vectors, which icreases the umber of modal equatios to be itegrated ad stored withi the computer CORRECTION FOR HIGHER MODE TRUNCATION I the aalysis of may types of structures, the respose of higher modes ca be sigificat. I the use of exact eigevectors for mode superpositio or respose

13 EIGEN AND RITZ VECTOR EVALUATION spectra aalyses, approximate methods of aalysis have bee developed to improve the results. The purpose of those approximate methods is to accout for missig mass or to add static respose associated with higher mode trucatio. Those methods are used to reduce the umber of exact eigevectors to be calculated, which reduces computatio time ad computer storage requiremets. The use of Load Depedet Ritz, LDR, vectors, o the other had, does ot require the use of those approximate methods because the static respose is icluded i the iitial set of vectors. This is illustrated by the time history aalysis of a simple catilever structure subjected to earthquake motios show i Figure This is a model of a light-weight superstructure built o a massive foudatio supported o stiff piles that are modeled usig a sprig. Com puter Model Figure 14.2 Catilever Structure o Massive Stiff Foudatio Oly eight eige or Ritz vectors ca be used because the model has oly eight masses. The computed periods, usig the exact eige or Ritz method, are summarized i Table It is apparet that the eighth mode is associated with the vibratio of the foudatio mass ad the period is very short: secods.

14 14-14 DYNAMIC ANALYSIS OF STRUCTURES Table 14.4 Periods ad Mass Participatio Factors MODE NUMBER PERIOD (Secods) MASS PARTICIPATION (Percetage) The maximum foudatio force usig differet umbers of eige ad LDR vectors is summarized i Table I additio, the total mass participatio associated with each aalysis is show. The itegratio time step is the same as the earthquake motio iput; therefore, o errors are itroduced other tha those resultig from mode trucatio. Five percet dampig is used i all cases. Table 14.5 Foudatio Forces ad Total Mass Participatio NUMBER OF FOUNDATION FORCE (Kips) MASS PARTICIPATION (Total Percetage) VECTORS EIGEN RITZ EIGEN RITZ 8 1,635 1, , , , , The solutio for eight eige or LDR vectors produces the exact solutio for the foudatio force ad 100 percet of the participatig mass. For seve eigevectors, the solutio for the foudatio force is oly 16 percet of the exact

15 EIGEN AND RITZ VECTOR EVALUATION value a sigificat error; whereas, the LDR solutio is almost idetical to the exact foudatio force. It is of iterest to ote that the LDR method overestimates the force as the umber of vectors is reduced a coservative egieerig result. Also, it is apparet that the mass participatio factors associated with the LDR solutios are ot a accurate estimate the error i the foudatio force. I this case, 90 percet mass participatio is ot a requiremet if LDR vectors are used. If oly five LDR vectors are used, the total mass participatio factor is oly 16.2 percet; however, the foudatio force is over-estimated by 2.2 percet VERTICAL DIRECTION SEISMIC RESPONSE Structural egieers are required for certai types of structures, to calculate the vertical dyamic respose. Durig the past several years, may egieers have told me that it was ecessary to calculate several hudred mode shapes for a large structure to obtai the 90 percet mass participatio i the vertical directio. I all cases, the "exact" free vibratio frequecies ad mode shapes were used i the aalysis. To illustrate this problem ad to propose a solutio, a vertical dyamic aalysis is coducted of the two dimesioal frame show i Figure The mass is lumped at the 35 locatios show; therefore, the system has 70 possible mode shapes. Usig the exact eigevalue solutio for frequecies ad mode shapes, the mass participatio percetages are summarized i Table Oe otes that the lateral ad vertical modes are ucoupled for this very simple structure. Oly two of the first te modes are i the vertical directio. Hece, the total vertical mass participatio is oly 63.3 percet.

16 14-16 DYNAMIC ANALYSIS OF STRUCTURES Figure 14.3 Frame Structure Subjected to Vertical Earthquake Motios Table 14.6 Mass Participatio Percetage Factors for Exact Eigevalues MODE PERIOD (Secods) LATERAL MASS PARTICIPATION VERTICAL MASS PARTICIPATION EACH MODE TOTAL EACH MODE TOTAL

17 EIGEN AND RITZ VECTOR EVALUATION The first 10 Load Depedet Ritz vectors are calculated ad the mass participatio percetages are summarized i Table The two startig LDR vectors were geerated usig static loadig proportioal to the lateral ad vertical mass distributios. Table 14.7 Mass Participatio Percetage Factors Usig LDR Vectors MODE PERIOD (Secods) LATERAL MASS PARTICIPATION VERTICAL MASS PARTICIPATION EACH MODE TOTAL EACH MODE TOTAL The te vectors produced by the LDR method more tha satisfy the 90 percet code requiremet. It would require the calculatio of 34 eigevectors for the exact eigevalue approach to obtai the same mass participatio percetage. This is just oe additioal example of why use of the LDR method is superior to the use of the exact eigevectors for seismic loadig. The reaso for the impressive accuracy of the LDR method compared to the exact eigevector method is that oly the mode shapes that are excited by the seismic loadig are calculated.

18 14-18 DYNAMIC ANALYSIS OF STRUCTURES SUMMARY There are three differet mathematical methods for the umerical solutio of the eigevalue problem. They all have advatages for certai types of problems. First, the determiat search method, which is related to fidig the roots of a polyomial, is a fudametal traditioal method. It is ot efficiet for large structural problems. The Sturm sequece property of the diagoal elemets of the factored matrix ca be used to determie the umber of frequecies of vibratio withi a specified rage. Secod, the iverse ad subspace iteratio methods are subsets of a large umber of power methods. The Stodola method is a power method. However, the use of a sweepig matrix to obtai higher modes is ot practical because it elimiates the sparseess of the matrices. Gram-Schmidt orthogoalizatio is the most effective method to force iteratio vectors to coverge to higher modes. Third, trasformatio methods are very effective for the calculatio of all eigevalues ad eigevectors of small dese matrices. Jacobi, Gives, Householder, Wilkiso ad Rutishauser are all well-kow trasformatio methods. The author prefers to use a moder versio of the Jacobi method i the ETABS ad SAP programs. It is ot the fastest; however, we have foud it to be accurate ad robust. Because it is oly used for problems equal to the size of the subspace, the computatioal time for this phase of the solutio is very small compared to the time required to form the subspace eigevalue problem. The derivatio of the Jacobi method is give i Appedix D. The use of Load Depedet Ritz vectors is the most efficiet approach to solve for accurate ode displacemets ad member forces withi structures subjected to dyamic loads. The lower frequecies obtaied from a Ritz vector aalysis are always very close to the exact free vibratio frequecies. If frequecies ad mode shapes are missed, it is because the dyamic loadig does ot excite them; therefore, they are of o practical value. Aother major advatage of usig LDR vectors is that it is ot ecessary to be cocered about errors itroduced by higher mode trucatio of a set of exact eigevectors. All LDR mode shapes are liear combiatios of the exact eigevectors; therefore, the method always coverges to the exact solutio. Also, the

19 EIGEN AND RITZ VECTOR EVALUATION computatioal time required to calculate the LDR vectors is sigificatly less tha the time required to solve for eigevectors REFERENCES 1. Bathe, K. J., ad E. L. Wilso "Large Eigevalue Problems i Dyamic Aalysis," Proceedigs, America Society of Civil Egieers, Joural of the Egieerig Mechaics Divisio, EM6. December. pp Wilso, E. L., ad T. Itoh "A Eigesolutio Strategy for Large Systems," i J. Computers ad Structures. Vol. 16, No pp Wilso, E. L., M. Yua ad J. Dickes Dyamic Aalysis by Direct Superpositio of Ritz Vectors, Earthquake Egieerig ad Structural Dyamics. Vol. 10. pp Bayo, E. ad E. L. Wilso "Use of Ritz Vectors i Wave Propagatio ad Foudatio Respose," Earthquake Egieerig ad Structural Dyamics. Vol. 12. pp

20 14-20 DYNAMIC ANALYSIS OF STRUCTURES