Nonlinear Analysis of Spatial Trusses

Size: px

Start display at page:

Download "Nonlinear Analysis of Spatial Trusses"

Rodger McCoy
6 years ago
Views:

1 Noninear Anaysis of Spatia Trusses João Barrigó October 14 Abstract The present work addresses the noninear behavior of space trusses A formuation for geometrica noninear anaysis is presented, which incudes severa options to estabish an eastic constitutive reation, that differ in the adopted strain and stress measures An efficient path foowing technique is described, aowing the computation of equiibrium paths featuring snap-through and snap-back phenomena A mode for the noninear behavior of each individua member is aso deveoped This mode is based on a simpe assembage of rigid bars and eastic or eastopastic springs, but is abe to account for phenomena such as: eastic and pastic bucking, pastic unoading or cycic oading The mode depends ony on physica meaningfu parameters and is vaidated by comparing its response with the resuts obtained with a pane stress finite eement anaysis The paper ends with a set of numerica tests, where severa benchmark probems are soved, and its resuts are compared with reference soutions Keywords: space truss, geometrica noninear anaysis, bucking, pasticity 1 Introduction A space truss is a type of structura system widey used in buidings, bridges, offshore patforms and transmission towers A truss is an assembage of individua members which are pin connected at their ends, where the number and organization of members is sufficient to ensure that the structure behaves as a rigid body (ie, is not a mechanism) For space structures this usuay eads to structures composed with tetrahedrons with common edges, but other forms are possibe ike geodesic domes The inear behavior of space trusses is straightforward Negecting the sma weight of the members, a common reasonabe assumption, the pin jointed members are ony submitted to axia forces, and the equiibrium equations written at the structura nodes are the key equations to sove the structure and evauate the members axia forces Staticay indeterminate trusses of course require the knowedge of the member constitutive reations and the use of the dispacement method The very efficient inear behavior of a truss, eads to the danger of designing very sender members which then become susceptibe to severa geometrica noninear vunerabiities Exampes are the snap through behavior of curved, dome-type structure or the coapse of a ayered grid due to progressive faiure of bucked members It can thus be argued that the evauation of the structura safety of these structures can ony be propery achieved if a the reevant noninear phenomena are taken into account, incuding geometric noninearity and member faiure (yieding or bucking) In the present work, one addresses the noninear behavior of space trusses First, a geometrica noninear structura mode is formuated, where a the members remain eastic Next, the numerica strategies required to determine the noninear equiibrium paths are discussed 1

2 The main contribution of this work is the formuation of a simpe member mode, based on rigid bars and eastic or eastopastic springs The mode parameters are caibrated on the basis of fundamenta bar properties, such as its ength, initia imperfection, pastic stress resutants and bucking oad The mode is vaidated by comparing its response to the one obtained in a finite eement anaysis of an individua member It was found that the mode is remarkaby abe to describe a the foowing aspects: yieding, bucking, axia force-bending moment pastic interaction, rigid pastic unoading, arbitrary oadunoading cyces The fina section presents numerica resuts obtained with the present formuation, which are compared to known benchmark tests Geometrica noninear anaysis of space trusses 1 Kinematics Kinematics is the study of cassica mechanics which describes the motion of points, bodies and systems of bodies without consideration of the causes of motion In this case a ine is considered where the variation between the two extreme points is important B A A u B u A x B B x A x A x B O Figure 1: Kinematics - Initia and Current configuration of a bar Consider a generic bar between points A and B, which move from their initia positions, x A and xb, to their current positions x A and x B The ength vector is defined as = x B x A and the dispacement of point A is u A = x A x A (simiar definitions for and u B ) As shown in the figure the fina body ength just depends on the dispacements of points A and B and is given by = + u B u A (1) and its variation is given by = du B du A = Equiibrium d du (e) du (e) () Equiibrium is achieved when the resutant force in a system is zero In a truss, without span oads, a pin jointed bars are two-force members: [ f (e) i = f A = I 3 f B I 3 f int ] (3) with the interior force f int aigned with the current configuration of the bar given by f int = N (4) The force variation of each eement can be written as K (e) = df (e) i = K (e) du (e) with the eementa stiffness matrix given by df int df int df int df int (5) (6) A the eement contributions for the goba forces and stiffness matrix are assembed in the standard way of the finite eement method Therefore, the goba interna forces are F int = n e A f (e) e=1 i (7) and the goba equiibrium equations are written as R = F int F ext = (8) where vector F ext gathers the externa forces, appied ony at the nodes Assuming that the externa forces are configuration independent, the corresponding goba stiffness matrix is then F int K = d R d u = d d u ne = A e=1 K(e) (9)

3 3 Constitutive Reation Constitutive reation is a reation between two physica quantities, in this case strain and stress In a inear anaysis, it is we known one can write the reation between force and strain as N = EAε (1) Engineering Strain If the variation of the bar ength is =, measured aigned with the eement, one can simpe get the strain as ε E = (15) where E is Young s moduus, A is the cross-section area, and the axia the strain ε is given by ε = (11) For a geometrica noninear anaysis, the above reationships can be generaized in severa ways, that are reviewed next The main issue is the choice of a non-inear strain measure, which is adopted as the basis of an eastic constitutive reation For each case, one determines the vaue of N, and obtains the df derivative int, essentia for the eementa stiffness expression, given by equation (6) Linear anaysis For comparising purposes, one begins with a inear anaysis Accordingy, the variation of the bar ength shoud be measured aong the initia configuration Considering the possibiity of a bar rotation, expression (11) shoud be written as ε = [ ] = 1 (1) In addition, for a geometrica inear anaysis, the equiibrium shoud be written in the undeformed configuration Hence, equation (13) shoud be repaced with f int = N (13) with N = EAε In this case the stiffness does not depend on body deformation, df int = EA( ) 3 (14) which is a non-inear function of Equation (13) hods, together with N = EAε E, eading to df int = EA ( + N I 3 ) 3 (16) which means that the stiffness matrix is now ceary dependent on the body deformation Green-Lagrange One notes that the engineering strain is very convenient to characterize the ongitudina strain state, but it is not easiy generaized to strain states where severa components pay an important roe So one considers aso the Green-Lagrange strain, which is the most used in the context of genera arge deformations For the ongitudina strain, one gets df int ε G = = EA + N (17) Keeping equation (13) and assuming N = EAε G, one has now ( I 3 ) 3 (18) Green-Lagrange A better comparison with other noninear formuations is obtained if the force featured in the constitutive reation is work-conjugated to the strain variabe In this case, this means the force N = N, which is assumed to be proportiona to ε G, by means of N = EA ε G The stiffness is now given by df int = EA 3 + N I 3 (19) 3

4 3 Equiibrium Trajectory d Q = K 1 A Q (4) The best characterization for the structura capacity is through its equiibrium trajectory Different types of structura behaviors are associated to different equiibrium paths It is important to compute the equiibrium path beyond the first maximum oad because: it may be ony a oca maximum; the structure anaysed may be ony a component; it may be important to know not just the coapse oad but whether or not this coapse is of a ductie or britte ; it confirms that we have indeed just passed a imit-point; it aows to gain insight into the mechanism or cause of the structure faiure In the foowing, one assumes that the exterior oads are proportiona to a scaar oad parameter P, where Q is a fixed vector F ext = P Q () 31 Iterative and Incrementa Method This method begins with an increase in the initia force and the cacuation of the initia stiffness matrix After that, severa iterations are performed unti equation (8) is satisfied within a given toerance This process is repeated successivey for new increments to obtain the compete equiibrium path At each iteration, from the computed vaues at a given point A, a inearized equiibrium soution is sought R B = R A + K A d P Q A + = (1) where R is the equiibrium error, d = d B d A is the dispacements variation, P = P B P A is the oad increment The soution for this probem can be written as In Figure () one can observe the geometrica interpretation of these variabes for a one dimensiona probem P O t A d R = t R R Q 1 d Q A d Figure : Geometric interpretation of the inearized equiibrium 3 Crisfied s method To obtain the compete equiibrium trajectory it is necessary to compement the equiibrium equations with a contro method Taking d as the vaues of the goba variabes at the ast known equiibrium point and d the variation of the dispacement variabes from that point d = d d (5) As a constrain surface, we wi set the discrete arcength measure L (the norm of d) to a fixed vaue L = d = d d = L (6) Now to obtain the intersection between the inearised equiibrium ine and the constraint surface we use Crisfied s method First, from () and (6) one writes ( d A + d R + P d Q ) ( d A + d R + P d Q ) = L which is a quadratic equation in P, (7) a P + b P + c = (8) d = d R + P d Q () with where d R and d Q are obtained by d R = K 1 A R A (3) a = d Q d Q b = d Q ( d A + d R ) (9) c = ( d A + d R ) ( d A + d R ) L 4

5 This gives us two roots for both intersections P = b ± b 4ac (3) a For the predictor, one has d A = and d R = so the soution is L P = ± d Q d = ± L Q d (31) Q and the root is picked according to 1, dq d sign( P ) = 1, otherwise (3) For the correction in the iterative process the root is seected according to P = b+ b 4ac a, dq d A > b b 4ac a, otherwise 4 Deveopment of a Bar Mode (33) The geometrica noninear formuation presented before, assumed that a the truss members behaved easticay We now extend the formuation by considering the member noninear behavior The bar mode is based on a simpe physica mode comprised of rigid bars and eastic-pastic springs The mode is buit in three stages: (i) axia eastic-pastic behavior for tension and compression, (ii) eastic-pastic compression instabiity and (iii) a series association of the the two first modes The stiffness expressions presented before can sti be used provided that the vaues of EA are repaced by dn d 41 Mode 1 - Pasticity Foowing von Mises yied criterion for uni-axia tensions the yied function can be given by f = N N p (34) N E 1 ta EA 1 ε p ε εp ε Figure 3: Eastopastic constitutive reation with hardening The axia force is cacuated by N = EA(ε ε p ) (35) where ε p is the pastic strain Due to cycing charges it is necessary to use an incrementa formuation Assume ε p and ε p are the pastic strain and equivaent pastic strain in the beginning of the increment If f tria = EA(ε ε p) n N p ( ε p) > (36) it is necessary to update the vaues with ε p = ε p + λn (37) ε p = ε p + λ (38) to ensure that (34) is satisfied This eads to and λ = ( dn dε = EA f tria EA + HA (39) HA EA + HA ) (4) Otherwise, if f tria, λ is kept nu, d λ dε = and the eastic stiffness is dn dε = EA 4 Mode - Instabiity Bucking is caused by a bifurcation in the soution of the equations of static equiibrium At a certain stage under an increasing oad, further oad is abe to be sustained in one of two states of equiibrium: an undeformed state or a ateray-deformed state with N p = Aσ c or N p ( ε p ) = Aσ c ( ε p ) if hardening is present (H is the pastic moduus) 5

6 P N p Post-bucking trajectory N cr Trajectory with imperfection Fundamenta trajectory Pastic trajectory O ε Figure 4: Beam-coumn behaviour In practice, an eement is rarey requested ony by one effort It is necessary to consider the interaction between axia force and bending moment Considering the interaction curve for a rectanguar section ( ) N + M 1 (47) N p M p With N = P and using equations (4) and (47) [ ] M p = M p m m + 1 m (48) To observe this behaviour in a noninear probem it is necessary to write the equiibrium equations in the deformed shape with m = N pd M p (49) d θ 4 d Figure 5: Hinged bar mode subjected to compression Considering the mode, shown in Figure 5, formed by two rigid bars and an eastic rotation spring, one can write d = d + ( ) P ( ) (41) Anayzing the equiibrium in a deformed shape, the equiibrium equation can be written as M = P d P = M d the compatibiity equation θ = 4 d (4) (43) and the constitutive reation for eastic materia as or for eatopastic materias as M = k(θ θ ) (44) M = k(θ t θ p ) (45) Then in eastic case dn d and in pastic case dn d = N p 8d 43 Mode 3 - Compete = k d + 4k M 4d 3 (5) [ 4m ] + m + 1 4m M 4d 3 (51) In the fu mode the previous two modes of pasticity and instabiity are combined in a series association 1 k = EA + 1 N < N p k = Pcr 4 + M < M p Figure 6: scheme of seria association of bar mode It is assumed that the tota deformation of the bar is obtained by summing the effects of previous modes = 1 + (5) N = N 1 ( 1 ) = N ( ) The new stiffness corresponds to the stiffness of an association of springs in series N To have equivaence of critica oads one sets k = π EI 4 (46) dn d = dn 1 dn d 1 d dn 1 + dn (53) d 1 d 6

7 5 Mode evauation 51 Finite eement anaysis program To evauate the performance of the mode deveoped in the previous section, its resuts were compared to those obtained by a finite eement anaysis A finite eement anaysis was performed with the program ADINA, which is abe to perform physica and geometricay noninear anaysis in pane stress probems The tests consisted in imposing dispacements at the end of the bar First, an eastic anaysis of the bar was carried out In the foowing figure it can be seen the deformed shape and the stress distribution Figure 7: Eastic deformation toward y y after an imposed dispacement For a mode section with m ( λ = 1) the trajectories between the two modes fairy agree In this case, due to the unitary senderness, the critica oad and the yied norma force are amost the same, causing the eement to bucke and yied at the same time It appears that in the finite eement program the traction yied occurs to a arger dispacement, there is a sight hardening during the traction and an increase in the maxima norma force in the second compression P (N) d (m) -,15 -,1 -,5, -1,5,1, Mode Finite eement mode In the anaysis next shown, an eastopastic bar behavior was considered The deformed configuration dispays a formation of a pastic hinge at mid span, between two amost straight sections Predictaby, the zones of pastic deformation are concentrated in a very ocaized region in the midde of the bar Figure 9: Trajectory P d of a bar with senderness λ = 1 Finay, a bar with λ = 1, 5 is anaysed with various cyces with increasing dispacement each cyce In the mode the vaue of the oad for which the bar buckes tends to decrease over the charge-discharge cyce and in the finite eement program the vaue increases after the first compression but from that moment it seems to remain unchanged Figure 8: Pastic deformation toward y y after an imposed dispacement 5 Comparison of resuts To understand if the design of the bar is conditioned by pasticity or bucking, it is usefu to consider the use of the normaized senderness λ, parameter cacuated from λ = Np P cr (54) P (N) d (m) -,1 -,5,,5, Mode Figure 1: Charge-discharge cyces in the mode 7

8 P (N) P(N) P (N) d (m) -,1 -,5,5, Finit eement program changes consideraby with the variation of the hardening One reaizes that the decrease in hardening causes a decrease in maximum oad that the structure can support whie maintaining its fundamenta form However, pastic yied aways occurs for the same vaue of P Figure 11: Charge-discharge cyces in the finite eement program In any case, in genera, it can be said that the mode performance is quite good, especiay for simpe oads,1,,3,4, v (m) Eastic response H= H=1 H=5 H=5 H= 6 Numerica appication The foowing exampes are intended to iustrate the basic functions of the program as easticity and pasticity, with and without instabiity and to compare the resuts obtained with some pubished artices 61 Cassica truss In Figure 1 it is possibe to see the geometry and properties of the two-dimensiona truss anayzed Simiar exampes have aso been studied by severa authors, incuding [Driemeier et a, 5] Figure 13: Cassica truss - Hardening effects To show the tota potentia of the mode, an anaysis with mode 3 was performed (with zero hardening) The equiibrium path, dispayed in Figure 14, shows that after the snap-through the bars are abe to withstand a much greater oad unti they yied in traction ,5,1,15,,5, v (m) Figure 1: Cassica truss The anaysis is performed using an eastic materia and biinear easto-pastic, with severa vaues of H, the hardening pastic moduus of the materia As it can be seen in Figure 13 the response Figure 14: Cassica truss - Mode 3 6 Shaow geodesic dome In this case a shaow geodesic dome shown in Figure 15 is subjected to a concentrated force at the center node, aso anayzed by [Tanaka et a, 1985] 8

9 P/EA (N) P/EA (x1-4 ) d (cm) Eastic anaysis Eastic anaysis with bucking EI/EA=,4 cm² Eastic anaysis with bucking EI/EA=,11 cm² Eastic anaysis with bucking EI/EA=,844 cm² Figure 15: Shaow geodesic dome Figure 17: Shaow geodesic dome - Member bucking effect In the first anaysis a inear materia was used with axia stiffness EA The equiibrium path was drawn way beyond the first oca maximum as shown in Figure 16,6,5,4,3 Figure 18: Shaow geodesic dome - Deformed shape,,1 7 Cosure -,1 -, d (cm) Figure 16: Shaow geodesic dome - Eastic equiibrium path Considering ony the initia portion of the trajectory, it is possibe to observe the infuence of instabiity in Figure 17 For the eastic mode without oca bucking the instabiity occurs by snap-through when reaching a vaue of oading on the order of 3, EA (N) Considering now the oca instabiity aows us to see the bucking infuence in the structure instabiity In Figure 18, one can observe the shaow geodesic dome deformed shape for an eastic anaysis after the snap-through In the present work the noninear anaysis of space trusses was considered The impemented mode aows the structure response cacuation assuming different constitutive reations The incrementa/iterative method was combined with the Crisfied method for cacuating the equiibrium path This mode aows us to consider different materia properties, incuding easticity and pasticity (with or without hardening) The bucking in compression is aso considered, taking into account the axia force and bending moment pastic interaction The resuts were compared with the resuts of a noninear finite eement program, being quite satisfactory Numerica tests were performed in some structures, evauated and compared with the anaysis made on other reference artices The deveoped mode can aso be extended to the interaction of axia force with bending moment for eements with other types of non-rectanguar section 9

10 References [Driemeier et a, 5] Driemeier, L, Baroncini Proença, S P, e Aves, M (5) A contribution to the numerica noninear anaysis of three-dimensiona truss systems considering arge strains, damage and pasticity Communications in noninear science and numerica simuation, 1(5): [Tanaka et a, 1985] Tanaka, K, Kondoh, K, e Aturi, S (1985) Instabiity anaysis of space trusses using exact tangent-stiffness matrices Finite eements in anaysis and design, 1(4):

Strain Energy in Linear Elastic Solids

Strain Energy in Linear Elastic Solids Strain Energ in Linear Eastic Soids CEE L. Uncertaint, Design, and Optimiation Department of Civi and Environmenta Engineering Duke Universit Henri P. Gavin Spring, 5 Consider a force, F i, appied gradua