12. DYNAMIC ANALYSIS. Force Equilibrium is Fundamental in the Dynamic Analysis of Structures 12.1 INTRODUCTION

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1 12. DYNAMIC ANALYSIS Force Equilibrium is Fundmentl in the Dynmic Anlysis of Structures 12.1 INTRODUCTION { XE "Newton's Second Lw" }All rel physicl structures behve dynmiclly when subjected to lods or displcements. The dditionl inerti forces, from Newton s second lw, re equl to the mss times the ccelertion. If the lods or displcements re pplied very slowly, the inerti forces cn be neglected nd sttic lod nlysis cn be justified. Hence, dynmic nlysis is simple extension of sttic nlysis. In ddition, ll rel structures potentilly hve n infinite number of displcements. Therefore, the most criticl phse of structurl nlysis is to crete computer model with finite number of mssless members nd finite number of node (joint) displcements tht will simulte the behvior of the rel structure. The mss of structurl system, which cn be ccurtely estimted, is lumped t the nodes. Also, for liner elstic structures, the stiffness properties of the members cn be pproximted with high degree of confidence with the id of experimentl dt. However, the dynmic loding, energy dissiption properties nd boundry (foundtion) conditions for mny structures re difficult to estimte. This is lwys true for the cses of seismic input or wind lods. To reduce the errors tht my be cused by the pproximtions summrized in the previous prgrph, it is necessry to conduct mny different dynmic nlyses using different computer models, loding nd boundry conditions. It is

2 12-2 STATIC AND DYNAMIC ANALYSIS not unrelistic to conduct 20 or more computer runs to design new structure or to investigte retrofit options for n existing structure. Becuse of the lrge number of computer runs required for typicl dynmic nlysis, it is very importnt tht ccurte nd numericlly efficient methods be used within computer progrms. Some of those methods hve been developed by the uthor nd re reltively new. Therefore, one of the purposes of this book is to summrize those numericl lgorithms, their dvntges nd limittions DYNAMIC EQUILIBRIUM The force equilibrium of multi-degree-of-freedom lumped mss system s function of time cn be expressed by the following reltionship: F (t) I + F(t) + F(t) = F(t) (12.1) D in which the force vectors t time t re: S F (t) I is vector of inerti forces cting on the node msses F (t) D is vector of viscous dmping, or energy dissiption, forces F (t) S is vector of internl forces crried by the structure F (t) is vector of externlly pplied lods Eqution (12.1) is bsed on physicl lws nd is vlid for both liner nd nonliner systems if equilibrium is formulted with respect to the deformed geometry of the structure. For mny structurl systems, the pproximtion of liner structurl behvior is mde to convert the physicl equilibrium sttement, Eqution (12.1), to the following set of second-order, liner, differentil equtions: M u& (t) + Cu& (t) + Ku(t) = F(t) (12.2) in which M is the mss mtrix (lumped or consistent), C is viscous dmping mtrix (which is normlly selected to pproximte energy dissiption in the rel

3 FUNDAMENTALS OF DYNAMIC ANALYSIS 12-3 structure) nd K is the sttic stiffness mtrix for the system of structurl elements. The time-dependent vectors u (t), u& (t) nd u&(t) & re the bsolute node displcements, velocities nd ccelertions, respectively. Mny books on structurl dynmics present severl different methods of pplied mthemtics to obtin the exct solution of Eqution (12.2). Within the pst severl yers, however, with the generl vilbility of inexpensive, high-speed personl computers (see Appendix H), the exct solution of Eqution (12.2) cn be obtined without the use of complex mthemticl techniques. Therefore, the modern structurl engineer who hs physicl understnding of dynmic equilibrium nd energy dissiption cn perform dynmic nlysis of complex structurl systems. A strong engineering mthemticl bckground is desirble; however, in my opinion, it is no longer mndtory. { XE "Erthquke Loding" }For seismic loding, the externl loding F (t) is equl to zero. The bsic seismic motions re the three components of free-field ground displcements u(t) ig tht re known t some point below the foundtion level of the structure. Therefore, we cn write Eqution (12.2) in terms of the displcements u (t), velocities u& (t) nd ccelertions u& & (t) tht re reltive to the three components of free-field ground displcements. Therefore, the bsolute displcements, velocities nd ccelertions cn be eliminted from Eqution (12.2) by writing the following simple equtions: u (t) = u(t) + I x u(t) xg + Iyu(t) yg + I z u(t) zg u & (t) = u& (t) + Ix u(t) & + Iyu(t) & + I u& (t) (12.3) xg yg z zg u & (t) = u&& (t) + I x u(t) && xg + Iyu(t) && yg + I z u& (t) zg where Ii is vector with ones in the i directionl degrees-of-freedom nd zero in ll other positions. The substitution of Eqution (12.3) into Eqution (12.2) llows the node point equilibrium equtions to be rewritten s: M u& (t) + Cu& (t) + Ku(t) = -Mx u(t) && - Myu(t) && - Mzu& (t) (12.4) where M = MI. i i xg yg zg

4 12-4 STATIC AND DYNAMIC ANALYSIS The simplified form of Eqution (12.4) is possible since the rigid body velocities nd displcements ssocited with the bse motions cuse no dditionl dmping or structurl forces to be developed. It is importnt for engineers to relize tht the displcements, which re normlly printed by computer progrm, re reltive displcements nd tht the fundmentl loding on the structure is foundtion displcements nd not externlly pplied lods t the joints of the structure. For exmple, the sttic pushover nlysis of structure is poor pproximtion of the dynmic behvior of three-dimensionl structure subjected to complex time-dependent bse motions. Also, one must clculte bsolute displcements to properly evlute bse isoltion systems. There re severl different clssicl methods tht cn be used for the solution of Eqution (12.4). Ech method hs dvntges nd disdvntges tht depend on the type of structure nd loding. To provide generl bckground for the vrious topics presented in this book, the different numericl solution methods re summrized below STEP-BY-STEP SOLUTION METHOD { XE "Dynmic Anlysis by:direct Integrtion" }The most generl solution method for dynmic nlysis is n incrementl method in which the equilibrium equtions re solved t times t, 2 t, 3 t, etc. There re lrge number of different incrementl solution methods. In generl, they involve solution of the complete set of equilibrium equtions t ech time increment. In the cse of nonliner nlysis, it my be necessry to reform the stiffness mtrix for the complete structurl system for ech time step. Also, itertion my be required within ech time increment to stisfy equilibrium. As result of the lrge computtionl requirements, it cn tke significnt mount of time to solve structurl systems with just few hundred degrees-of-freedom. In ddition, rtificil or numericl dmping must be dded to most incrementl solution methods to obtin stble solutions. For this reson, engineers must be very creful in the interprettion of the results. For some nonliner structures subjected to seismic motions, incrementl solution methods re necessry.

5 FUNDAMENTALS OF DYNAMIC ANALYSIS 12-5 For very lrge structurl systems, combintion of mode superposition nd incrementl methods hs been found to be efficient for systems with smll number of nonliner members. This method hs been incorported into the new versions of SAP nd ETABS nd will be presented in detil lter in this book MODE SUPERPOSITION METHOD { XE "Dynmic Anlysis by:mode Superposition" }The most common nd effective pproch for seismic nlysis of liner structurl systems is the mode superposition method. After set of orthogonl vectors hve been evluted, this method reduces the lrge set of globl equilibrium equtions to reltively smll number of uncoupled second order differentil equtions. The numericl solution of those equtions involves gretly reduced computtionl time. It hs been shown tht seismic motions excite only the lower frequencies of the structure. Typiclly, erthquke ground ccelertions re recorded t increments of 200 points per second. Therefore, the bsic loding dt does not contin informtion over 50 cycles per second. Hence, neglecting the higher frequencies nd mode shpes of the system normlly does not introduce errors RESPONSE SPECTRA ANALYSIS { XE "Dynmic Anlysis by:response Spectrum" }The bsic mode superposition method, which is restricted to linerly elstic nlysis, produces the complete time history response of joint displcements nd member forces becuse of specific ground motion loding [1, 2]. There re two mjor disdvntges of using this pproch. First, the method produces lrge mount of output informtion tht cn require n enormous mount of computtionl effort to conduct ll possible design checks s function of time. Second, the nlysis must be repeted for severl different erthquke motions to ensure tht ll the significnt modes re excited, becuse response spectrum for one erthquke, in specified direction, is not smooth function. There re significnt computtionl dvntges in using the response spectr method of seismic nlysis for prediction of displcements nd member forces in

6 12-6 STATIC AND DYNAMIC ANALYSIS structurl systems. The method involves the clcultion of only the mximum vlues of the displcements nd member forces in ech mode using smooth design spectr tht re the verge of severl erthquke motions. In this book, we will recommend the CQC method to combine these mximum modl response vlues to obtin the most probble pek vlue of displcement or force. In ddition, it will be shown tht the SRSS nd CQC3 methods of combining results from orthogonl erthquke motions will llow one dynmic nlysis to produce design forces for ll members in the structure SOLUTION IN THE FREQUENCY DOMAIN { XE "Dynmic Anlysis by:frequency Domin" }The bsic pproch used to solve the dynmic equilibrium equtions in the frequency domin is to expnd the externl lods F (t) in terms of Fourier series or Fourier integrls. The solution is in terms of complex numbers tht cover the time spn from - to. Therefore, it is very effective for periodic types of lods such s mechnicl vibrtions, coustics, se-wves nd wind [1]. However, the use of the frequency domin solution method for solving structures subjected to erthquke motions hs the following disdvntges: 1. The mthemtics for most structurl engineers, including myself, is difficult to understnd. Also, the solutions re difficult to verify. 2. Erthquke loding is not periodic; therefore, it is necessry to select long time period so tht the solution from finite length erthquke is completely dmped out before ppliction of the sme erthquke t the strt of the next period of loding. 3. For seismic type loding, the method is not numericlly efficient. The trnsformtion of the result from the frequency domin to the time domin, even with the use of Fst Fourier Trnsformtion methods, requires significnt mount of computtionl effort. 4. The method is restricted to the solution of liner structurl systems. 5. The method hs been used, without sufficient theoreticl justifiction, for the pproximte nonliner solution of site response problems nd soil/structure

7 FUNDAMENTALS OF DYNAMIC ANALYSIS 12-7 interction problems. Typiclly, it is used in n itertive mnner to crete liner equtions. The liner dmping terms re chnged fter ech itertion to pproximte the energy dissiption in the soil. Hence, dynmic equilibrium within the soil is not stisfied SOLUTION OF LINEAR EQUATIONS { XE "Solution of Equtions" }The step-by-step solution of the dynmic equilibrium equtions, the solution in the frequency domin, nd the evlution of eigenvectors nd Ritz vectors ll require the solution of liner equtions of the following form: AX = B (12.5) Where A is n 'N by N' symmetric mtrix tht contins lrge number of zero terms. The 'N by M' X displcement nd B lod mtrices indicte tht more thn one lod condition cn be solved t the sme time. The method used in mny computer progrms, including SAP2000 [5] nd ETABS [6], is bsed on the profile or ctive column method of compct storge. Becuse the mtrix is symmetric, it is only necessry to form nd store the first non-zero term in ech column down to the digonl term in tht column. Therefore, the sprse squre mtrix cn be stored s one-dimensionl rry long with N by 1 integer rry tht indictes the loction of ech digonl term. If the stiffness mtrix exceeds the high-speed memory cpcity of the computer, block storge form of the lgorithm exists. Therefore, the cpcity of the solution method is governed by the low speed disk cpcity of the computer. This solution method is presented in detil in Appendix C of this book UNDAMPED HARMONIC RESPONSE { XE "Hrmonic Loding" }The most common nd very simple type of dynmic loding is the ppliction of stedy-stte hrmonic lods of the following form: F (t) = f sin ( ωt) (12.6)

8 12-8 STATIC AND DYNAMIC ANALYSIS The node point distribution of ll sttic lod ptterns, f, which re not function of time, nd the frequency of the pplied loding, ω, re user specified. Therefore, for the cse of zero dmping, the exct node point equilibrium equtions for the structurl system re: M u& (t) + Ku(t) = f sin ( ω t) (12.7) The exct stedy-stte solution of this eqution requires tht the node point displcements nd ccelertions re given by: 2 u(t) = v sin( ω t), u& (t) = - v ω sin( ω t) (12.8) Therefore, the hrmonic node point response mplitude is given by the solution of the following set of liner equtions: [ K -ω 2 M] v = f or Kv = f (12.9) It is of interest to note tht the norml solution for sttic lods is nothing more thn solution of this eqution for zero frequency for ll lods. It is pprent tht the computtionl effort required for the clcultion of undmped stedy-stte response is lmost identicl to tht required by sttic lod nlysis. Note tht it is not necessry to evlute mode shpes or frequencies to solve for this very common type of loding. The resulting node point displcements nd member forces vry s sin (ω t). However, other types of lods tht do not vry with time, such s ded lods, must be evluted in seprte computer run UNDAMPED FREE VIBRATIONS { XE "Undmped Free Vibrtion" }Most structures re in continuous stte of dynmic motion becuse of rndom loding such s wind, vibrting equipment, or humn lods. These smll mbient vibrtions re normlly ner the nturl frequencies of the structure nd re terminted by energy dissiption in the rel structure. However, specil instruments ttched to the structure cn esily mesure the motion. Ambient vibrtion field tests re often used to clibrte computer models of structures nd their foundtions.

9 FUNDAMENTALS OF DYNAMIC ANALYSIS 12-9 After ll externl lods hve been removed from the structure, the equilibrium eqution, which governs the undmped free vibrtion of typicl displced shpe v, is: M v& + Kv = 0 (12.10) { XE "Energy:Mechnicl Energy" }{ XE "Mechnicl Energy" }At ny time, the displced shpe v my be nturl mode shpe of the system, or ny combintion of the nturl mode shpes. However, it is pprent the totl energy within n undmped free vibrting system is constnt with respect to time. The sum of the kinetic energy nd strin energy t ll points in time is constnt tht is defined s the mechnicl energy of the dynmic system nd clculted from: E M 1 T 1 T = v& Mv& + v Kv (12.11) SUMMARY Dynmic nlysis of three-dimensionl structurl systems is direct extension of sttic nlysis. The elstic stiffness mtrices re the sme for both dynmic nd sttic nlysis. It is only necessry to lump the mss of the structure t the joints. The ddition of inerti forces nd energy dissiption forces will stisfy dynmic equilibrium. The dynmic solution for stedy stte hrmonic loding, without dmping, involves the sme numericl effort s sttic solution. Clssiclly, there re mny different mthemticl methods to solve the dynmic equilibrium equtions. However, it will lter be shown in this book tht the mjority of both liner nd nonliner systems cn be solved with one numericl method. Energy is fundmentl in dynmic nlysis. At ny point in time, the externl work supplied to the system must be equl to the sum of the kinetic nd strin energy plus the energy dissipted in the system. { XE "Energy:Zero Strin Energy" }It is my opinion, with respect to erthquke resistnt design, tht we should try to minimize the mechnicl energy in the structure. It is pprent tht rigid structure will hve only kinetic energy nd zero strin energy. On the other hnd, completely bse isolted structure will

10 12-10 STATIC AND DYNAMIC ANALYSIS hve zero kinetic energy nd zero strin energy. A structure cnnot fil if it hs zero strin energy REFERENCES 1. Clough, R., nd J. Penzien Dynmics of Structures, Second Edition. McGrw-Hill, Inc. ISBN { XE "Chopr, A." }Chopr, A Dynmics of Structures. Prentice-Hll, Inc. Englewood Cliffs, New Jersey, ISBN { XE "Bthe, K. J." }Bthe, K Finite Element Procedures in Engineering Anlysis. Prentice-Hll, Inc. Englewood Cliffs, New Jersey ISBN Wilson, E. L., nd K. Bthe "Stbility nd Accurcy Anlysis of Direct Integrtion Methods," Erthquke Engineering nd Structurl Dynmics. Vol. 1. pp Computers nd Structures, Inc SAP Integrted Structurl Anlysis & Design Softwre. Berkeley, Cliforni. 6. Hbibullh, A ETABS - Three Dimensionl Anlysis of Building Systems, User's Mnul. Computers nd Structures Inc. Berkeley, Cliforni.