NUMERICAL SIMULATION FOR NONLINEAR STATIC & DYNAMIC STRUCTURAL ANALYSIS

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1 Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada NUMERICAL SIMULATION FOR NONLINEAR STATIC & DYNAMIC STRUCTURAL ANALYSIS ABSTRACT Xiangbai Zhang 1 and Daisy Le CAO 2 Geneal pocedue fo numeical simulaion is biefly descibed and applied o ealisic sucual poblems. Analyical sucual analysis mehods, especially he Finie Elemen Mehod (FEM, have become inceasingly popula. This pape pesens cases of numeical simulaion sysemaically on he hee-dimensional nonlinea saic and dynamic sucual analysis, including einfoced concee sucue and seel fame sysem wih shea wall sucue. By compaing wih simulaion cuves, i descibes he sucual behavio subjeced o saic loading condiion, and evaluaes on he seviceabiliy fo pefomance funcions wih dynamic esponses applied in ime domain. All analyical daa deived fom he hee-dimensional finie elemen pocedue by ANSYS Sofwae Package. KEY WORDS Finie elemen analysis, nonlinea saic & dynamic analysis, einfoced concee sucue, seel fame sysem wih shea wall sucue, hee-dimensional numeical simulaion. INTRODUCTION Geneal pocedue fo numeical simulaion is biefly descibed and applied o ealisic sucual poblems. Analyical sucual analysis mehods, especially he Finie Elemen Mehod (FEM, have become inceasingly popula. This is no supise given he fac ha FEM pogams ae now available on he enie compue hadwae plafom, anging fom PCs o mainfames, and ha FEM has been applied o all engineeing disciplines. Modeling, meshing, solving and pos-pocessing have been highly inegaed by FEM ools. Thee ae wo poposed sucual modeling issues: (I Reinfoced Concee (RC beam elasically suppoed; (II Seel fame sysem wih shea wall sucue. Boh of hem ae implemened by nonlinea saic and dynamic sucual analysis in his pape. And hee-dimensional elemens ae all defined wih ACI & ASTM specificaions. 1 Ph.D. Regiseed Pofessional Enginee, Senio Enginee, Dep. of Civil Engineeing, Tsinghua Univesiy, Beijing, 184 P.R.China, Tel: (86-( , zxb2@mails.singhua.edu.cn 2 M.ASCE, Pofessional Enginee, Dep. of Civil Engineeing, Beijing Insiue of Civil Engineeing and Achiecue, No.7 MinZu Univ. Souh Rd, Haidian Disic, Beijing, 181 P.R.China, Tel: (86-( , achiecue1@homail.com Page 2163

2 Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada NONLINEAR ANALYSIS OF FINITE ELEMENT MODEL Geomeic Nonlineaiy Linea sucual analysis is based on he assumpion of small defomaion and linea elasic behavio of maeial. As applied loads incease, he assumpion is no longe accuae, because defomaions may cause significan changes in he sucual shape. If a sucue expeiences lage defomaions, is changing geomeic configuaion can cause he sucue o espond nonlinealy. Newon-Raphson mehod is used o solve nonlinea poblems in he eseach. Figue 1 illusaes equilibium ieaions of adiional Newon-Raphson mehod in a single Degee of Feedom (DOF nonlinea analysis. Figue 1 Newon-Raphson Mehod. Figue 2 Sess-Sain elaions of RC maeials. Maeial Nonlineaiy The einfoced concee maeial, iniially assumed o be isoopic, is capable of diecional inegaed poins cacking and cushing besides incopoaing plasic and ceep behavios. The sess-sain elaionships ae given in Figue 2 above, wih he sess-sain maix defined as [D] in Equaion 1 (Eq.1 below: [ D] N N R C R (1 Vi [ D ] + Vi [ D ] i Eq.1 i 1 i 1 whee: N he numbe of einfocemen; V i R aio of he volume of einfocemen. [ D C (1 E ] (1 + (1 2 (1 (1 (1 2 (1 2 (1 2 Eq.2 Page 2164

3 xz yz xy zz yy xx i xz yz xy zz yy xx i xz yz xy zz yy xx D E ] [ Eq.3 whee: [D C ] he sess-sain maix fo concee maeial, expessed by Equaion 2 above, deived fom specializing and inveing he ohoopic sess-sain elaion o he case of an isoopic maeial; [D ] i he sess-sain maix fo einfocemen, epesened by Equaion 3 above, depiced in oienaion; E Young s modulus fo concee maeial; aio of he volume of einfocemen; E i Young s modulus fo einfocemen. As menioned peviously, he einfoced concee model can pedic elasic behavio, cacking o cushing behavio. If he elasic behavio is pediced, i is eaed as a linea elasic maeial discussed above. If he cacking o cushing behavio is pediced, he elasic sess-sain maix would be adjused o he failue mode of bile model, given by Equaion 4 below: S f F C Eq.4 whee: he funcion of pinciple sess sae ( F xp, yp, zp ; he failue suface expessed in ems of pincipal sess and maeial paamees; f S c he uniaxial cushing sengh; xp, yp, zp pincipal sesses in pincipal diecions. The sess-sain elaion fo concee maeial while cacking opens in all hee diecions is expessed by Equaion 5 below, fuhemoe, if cacks in all hee diecions eclose, he sess-sain elaion would be adjused o Equaion 6 below: (1 2(1 2(1 1 ] [ β β β CK C E R E R E D Eq.5 whee: R he secan modulus in sengh of cacked condiion. June 14-16, 26 - Monéal, Canada Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing Page 2165

4 Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada (1 1 1 β (1 2 CK E C [ D C ] + 2 Eq.6 (1 (1 2 (1 2 2 β C (1 2 2 whee:β,β c he shea ansfe coefficien (<β <β C <1. In addiion o cacking and cushing, his model may also undego plasiciy, wih he Ducke-Page failue suface being commonly used. Theefoe, he plasiciy is done befoe cacking and cushing ae checked ou. NUMERICAL SIMULATION FOR RESEARCH MODELS Issue I: Reinfoced Concee Beam Models In his issue, wo poposed pojecs (Pojec I & II ae ceaed fo numeical simulaion of hee-dimensional nonlinea saic and dynamic analysis. The algoihm of Newon-Raphson fo an efficien numeical implemenaion of nonlinea soluion saegy, calculaes effecs of lage defomaion, plasiciy, ceep, cacks, inenal foces and sesses beween diffeen elemens in seady loading condiion. I is sill exended o dynamic analysis in ime-vaying loading condiion. Models Descipion Pojec I is esed o pefom he nonlinea saic behavio of einfoced concee beam elasically suppoed unde seady loads, shown as Figue 3 below. Pojec II is esed o pefom he nonlinea dynamic esponses of he same beam model unde moving loads, shown as Figue 4 below. Maeial popeies ae given in Table 1 below. Figue 3 Full model in Pojec I. Figue 4 Full model in Pojec II. Page 2166

5 Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada Table 1: Maeials Popeies of Reinfoced Concee Beam Models. No I II Reinfocemen Poisson Young s Modulus (psi Maeial Sengh (psi Raio Concee Reinfocemen Concee f c Reinfocemen f y Diamee (inch C: Gade6 #9: 9/8 (29mm R:.3 ( Mpa (2 1 5 Mpa (28Mpa (415Mpa #3: 3/8 (9.5mm C: Gade6 #9: 9/8 (29mm R:.3 ( Mpa (2 1 5 Mpa (28Mpa (415Mpa #3: 3/8 (9.5mm Wih he same elasically suppoed syle in boh pojecs, veical exenal loads ae acing on he beam model. In Pojec I, exenal seady loads ae equally divided ino wo pas. Each pa value is defined as F1.63kip (47.3kN, symmeically disibued ino mulipoin nodal loads along he x axis, fomed in a line, locaed wih he same disance by 1.5 m nea he end of beam, deailed in Figue 3 above. In Pojec II, he consan value of aveling loads is defined as P kip (94.6kN, symmeically disibued ino mulipoin nodal loads along he x axis, fomed in a line duing each load sep, bu moving on he beam wih a consan velociy 12 km/ h along he z axis, deailed in Figue 4 above. The oal value of exenal loads in each pojec is definiely he same wih he ohe. In fac, moving loading condiion in Pojec II simulaes he ealisic complexiy of aveling vehicle loads on he einfoced concee bidge in analysis. Numeical Simulaion fo Nonlinea Saic & Dynamic Analysis Boh nonlinea saic and dynamic esponses of sucue can be deemined hough he Newon-Raphson mehod. The basic simulaneous equaion, yielded by he finie elemen discee pocess, is expessed by Equaion 7 below: a [ K ] { u} { F } Eq.7 whee: [K] he cuen siffness maix; {u} he veco of unknown DOF values; {F a } he veco of applied loads. I is an ieaive pocess in solving nonlinea poblems. The soluion obained a he end of ieaive pocess would coespond o he load veco {F a }. The final conveged soluion would be in equilibium. If he analysis included nonlineaiies such as plasiciy, hen he pocess equies ha some inemediae seps each in equilibium in ode o follow he load pah coecly. I is accomplished effecively by specifying a sep-by-sep incemenal analysis, he final load veco {F a } acually is obained by applying load incemens and pefoming Newon-Raphson ieaions in each sep. In Pojec I, wih measued sess-sain elaions of he einfoced concee beam model, geomeic and maeial popeies epoed above, and pinciples of FEM pocedues in nonlinea saic analysis, he simple skech of calculaed defomaions by concee and einfocing ba ae shown as Figue 5 below. In Pojec II, he echnical ansien dynamic analysis is employed o deemine dynamic Page 2167

Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada Figue 5 Displacemen anslaed in Y-axis by RC beam model (Pojec I.

8 whee: [M] he mass maix; [C] he mass damping maix; [K] he siffness maix; {P(} he applied load veco; {u& &} he nodal acceleaion veco; {u& } he nodal velociy veco; {u} he nodal displacemen veco.

6 Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada Figue 5 Displacemen anslaed in Y-axis by RC beam model (Pojec I. behavio of he model. The basic equaion of ansien dynamic soluion given by Equaion 8: [ M ]{ u& } + [ C]{ u& } + [ K ]{ u} { P( } Eq.8 whee: [M] he mass maix; [C] he mass damping maix; [K] he siffness maix; {P(} he applied load veco; {u& &} he nodal acceleaion veco; {u& } he nodal velociy veco; {u} he nodal displacemen veco. On he ohe hand, he full ansien mehod uns wih full maices, fuhemoe, i allows all ypes of nonlineaiies o be included, such as lage defomaions, ec. Newmak assumpions ae consideed wih Newon-Raphson mehod. Since he aveling loads of P (wih he consan velociy 12 km/ h along he z axis. excie all modes of he model, dynamic esponses have been acked. This pojec employed 15 load seps wih 5 sub-seps in each one duing compuaional simulaion pocessing. Defomaions in he 2 nd and 8 h load sep ae moe chaaceisic, shown as Figue 6 below. Figue 6 Displacemen anslaed in Y-axis by RC beam model (2 nd & 8 h load sep Pojec II Disibuions on peak values of nodal displacemen esponses ae no symmeical while aveling loads ae moving on he beam wih a consan velociy in Pojec II. Figue 7 shows wo goups of nodal displacemen esponses in ime domain. Each goup is made up of wo chaaceisic nodes locaed symmeically along he lengh of beam model. Page 2168

7 Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada Figue 7 Two goups of nodal displacemen esponses by RC beam model (Pojec II. The lages nodal displacemen and velociy esponses occued in he 8 h load sep, shown as Figue 8. Simulaion of inenal sesses and cacks ae implemened as Figue 9. Figue 8 Displacemen (L & velociy (R esponses in he 8 h load sep by RC beam model. Figue 9 Simulaion of einfocing ba elemen sess & RC beam concee cacks. Issue II: Seel Fame Sysem wih Shea Wall Sucue Models Many govenmen agencies and some pivae building ownes oday equie ha new buildings be designed and exising buildings upgaded o esis he effecs of poenial blass. Alhough he pobabiliy ha any building acually would be subjec o such hazad is low, a pefomance-based analysis wih a visible compaison beween he nonlinea saic and Page 2169

Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada dynamic behavios of building sucues has made a conibuion o opimize fuue

8 Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada dynamic behavios of building sucues has made a conibuion o opimize fuue design. Similaly, his is anohe issue of wo poposed pojecs (Pojec III & IIII ceaed fo numeical simulaion of hee-dimensional nonlinea saic and dynamic analysis. Tesing deemined on he seel fame sysem wih shea wall sucue models. Some fundamenal algoihms of nonlinea soluion saegy ae sill he same wih issue I. Howeve, wih he idenificaion by diffeen maeial popeies and diffeen elemen ypes, he finie elemen model pocedues mus be changed. Models Descipion As he oiginal fame sysem wih shea wall sucue had been designed fo gaviy load, i poved o aemp o esis in cuen dynamic blas-impulsive loading condiion. Pojec III is esed o pefom he nonlinea saic behavio of seel fame sysem wih shea wall sucue unde seady loads. Pojec IIII is esed o pefom he nonlinea dynamic esponses of he same sucual model unde dynamic blas-impulsive loads wih ansien dynamic analysis. This fame sysem is modeled by wo-soey wo-bay in each diecion. Rigid mulipoin consains ae used o enfoce he igid siffness a boom. Besides gaviy load is applied, hoizonal and veical exenal nodal loads ae defined as H5kip (222kN and F15kip (667kN. Moeove, each value of nodal loads in coesponding node in each pojec is definiely equal, bu acing wih absoluely diffeen modes, shown as Figue 1. Maeial popeies ae given in Table 2 below. Seel gides and columns ae defined as A992, einfoced concee slabs and shea wall ae defined by a compessive sengh of 6psi. Figue 1 Full model of Issue II (Pojec III & IIII. Table 2: Popeies of Seel Fame Sysem wih Shea Wall Sucue Models. No III IIII Poisson s Size of Seel Seel Young s Modulus (psi Thickness of RC (inch Raio Beam Column Seel Fame RC sysem Floo Wall S: W18 6 W12 58 C:.2 (2 1 5 Mpa ( Mpa (14 mm (127 mm S: W18 6 W12 58 C:.2 (2 1 5 Mpa ( Mpa (14 mm (127 mm Page 217

Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada Numeical Simulaion fo Nonlinea Saic & Dynamic Analysis The soluion algoihm

9 Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada Numeical Simulaion fo Nonlinea Saic & Dynamic Analysis The soluion algoihm of Newon-Raphson mehod has been used in he simulaion pocess. All loads in Pojec III ae acing on he model wihin 1 load sep fo nonlinea saic analysis. To simulae he dynamic blas-impulsive nodal loads in Pojec IIII, oally 7 load seps adoped duing he full ansien dynamic analysis wih ime inegaion scheme. The basic equaion of Eq.8 can be hough of as a se of saic equilibium equaions ha also ake ino accoun ineia foces and damping foces. Newmak ime inegaion scheme is used o solve hese equaions a discee ime poins. In he pos pocessing, Figue 11 pu ogehe wih he calculaed defomaions in Pojec III & IIII,and boh main nodal displacemen esponses of he cenal column in ime domain ae shown as Figue 12. Figue 13 pu ogehe wih he nodal sess disibuions in Pojec III & IIII, shown as below. I makes a visible compaison wih nonlinea saic and dynamic behavios of he same sucual model. Acually, when esising diec ai-blas loading can poduce high sain aes, pehaps of lage magniude, and will occu simulaneously wih lage axial ension demands. In his condiion, seel fame sysem becomes songe bu moe bile. Thee is evidence ha indicaed sandad beam-column connecion ypes used in seel fame sysem migh no be Figue 11 Nodal Displacemen Responses by Pojec III & IIII. Figue 12 Nodal Displacemen Responses in Time Domain by Pojec III & IIII. Page 2171

Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada Figue 13 Nodal Sess Responses by Pojec III & IIII.

10 Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada Figue 13 Nodal Sess Responses by Pojec III & IIII. capable of allowing sucue o develop he lage inelasic oaion and ensile sains. Wheeas, when popely configued and consuced using maeials wih appopiae oughness, seel connecions can povide ousanding duciliy and oughness. CONCLUSION A geneal goal of his pape on nonlinea soluion is o avoid sevee damage in vaious complex loading condiions. The applicaions of wo poposed modeling issues including 4 pojecs on cases of einfoced concee sucue and seel fame sysem wih shea wall sucue subjeced o saic and dynamic loads ae implemened by ANSYS Sofwae Package using finie elemen mehod in his pape. The compaison wih simulaion cuves ae epesened o opimize sucual design. Compuaional simulaion of nonlinea saic and dynamic sucual analysis based on advanced Thee-Dimensional Finie Elemen (3DFE models can be effecively used fo an efficien invesigaion of ealisic sucual behavios in diffeen loading condiions, fo a good soluion saegy o solve echnical poblems. In fac, he poweful FEM pogams have esablished a se of ules fo sucual behavios in numeical simulaion. REFERENCE Ahu H. Nilson (1997 Design of Concee Sucues, McGaw-Hill Educaion Co. ANSYS (1999 Manual fo ANSYS, Beijing Agency of ANSYS Company, Beijing Ronald Hambuge (24 Design of Seel Sucues fo Blas-Relaed Pogessive Collapse Resisance, Moden Seel Consucion, AISC, Chicago, IL Achinya Halda (23 Reliabiliy of Fame and Shea wall Sucue Sysem, J. Sucual Engineeing, ASCE, New Yok, NY Page 2172

Orthotropic Materials

Orthotropic Materials Kapiel 2 Ohoopic Maeials 2. Elasic Sain maix Elasic sains ae elaed o sesses by Hooke's law, as saed below. The sesssain elaionship is in each maeial poin fomulaed in he local caesian coodinae sysem. ε