ASSESSMENT OF MATERIALS NONLINEARITY IN FRAMED STRUCTURES OF REINFORCED CONCRETE AND COMPOSITES

Transcription

1 VIII Iteratioal Coferee o Frature ehais of Corete ad Corete Strutures FraCoS-8 J.G.. Va ier, G. Rui, C. Adrade, R.C. Yu ad X.X. Zhag (Eds) ASSESSENT OF ATERIALS NONLINEARITY IN FRAED STRUCTURES OF REINFORCED CONCRETE AND COPOSITES ERNESTO FENOLLOSA * AND ADOLFO ALONSO * Uiversidad Politéia de Valeia (UPV) Co. De Vera s/, 460 Valeia, Spai efeollo@mes.upv.es, Uiversidad Politéia de Valeia (UPV) Co. De Vera s/, 460 Valeia, Spai aalosod@mes.upv.es, Ke words: Aalsis, Stiffess, Noliear, Crakig, Corete, Composites. Abstrat: The assessmet of the fatigue loadig of a struture demads the osideratio of the oliearit effets derived from the geometri variatios ad the bars stiffess hages. The resolutio through matrixes methods frequetl uses a iremetal treatmet for load appliatio, beig osidered at eah loadig step, the solutio researh b meas of a iterative proess whih result is expeted to be overget. I eah ireasig ad iteratios a variat of the equatio of equilibrium a be brought up betwee loadig ad displaemets differetials, that variat is solved i the liear field. The taget stiffess matrix used has to osider a elasti ad liear ompoet ad the other ompoet that takes ito aout the geometri variatios ad mehaial oliearit of materials. The preset work deals with the ompoet of the affeted stiffess matrix due to the material mehai oliearit. Its quatifiatio i reifored orete ad omposites strutures is espeiall omplex beause of the presee of several materials with differet resistat behaviour eah oe ad to oliearit betwee stresses ad strais espeiall i the orete ase ad to the iertia redutio of the setio produed b orete rakig. The developig of a iterative proess based o bisetio method, whih provides the equilibrium positio of the setio i frot of kow requests, takig ito aout the oliear ostituet relatioships of materials. Oe the equilibrium has bee obtaied, the setio stiffess modulus a be dedued from the expressio E I/. INTRODUCTION The frame struture desig a be doe uder several hpotheses; urretl the most importat two of them are the assessmet of geometri oliearit of the struture ad the osideratio of a elasti or plasti behaviour of materials. The iterestig omparative stud about differet desigig models made b Netherot [] a be osulted. First approah to the strutures desig b usig a matrix method was made b Livesle []. At this, the odes deformatios o framed strutures are related to the loads applied b the amed stiffess matrix. Nowadas the essee of his mathematial model is still preserved, ad ma of later advaes i it are liked to the omputer ad software developmet.

2 Eresto Feollosa, Adolfo Aloso Studies about the assessmet of geometri oliearit of struture started with Jeigs [], beig his work simplified b Yag ad Guire [4]. The approah to geometri oliear desig is usuall made b usig a ireasig set out of load appliatio, usig Ziekiewi [5] termiolog a be expressed b: [ K K + K ][ u] [ F ] L + 0 () G Where [K L ] is the liear stiffess matrix, [K G ] ad [K 0 ] are the matrixes that take ito aout, respetivel, iitial stresses ad strais. [ u] ad [ F] are the vetors that represet, at the same time, the ireasig displaemets ad fores. I otrast to geometri oliearit, the oliear behaviour of the materials is usuall take ito aout ol for setios measuremet. However, a aurate proess must take it ito osideratio i the struture aalsis proess itself. I this ase, the liear stiffess matrix [K L ] of equatio () is replaed b the amed taget matrix [Kt]. The mehaial oliearit ifluees partiularl at orete strutures, beause of their elastiit module that depeds o the load applied value ad i the evaluatio of the setio iertia, the orete rakig ad the amout ad reiforig distributio must be take ito aout. From them, are the sleder ports, the elemets that i additio to the setio stregth, is espeiall eessar to take ito aout the seod order effets ad the oliear stress-strai behaviour of materials. There is a lot of studies about beam-olum behaviour amog them the oes made b avihak ad Furlog [6], Al-Nour ad Che [7], Virdi ad Dowlig [8], Wag ad Hsu [9] or Roik ad Bergma [0] must be quoted. The preset work develops a iterative proedure i order to obtai the stiffess modulus E I of the setio at eah loadig step takig ito aout the applied soliitatios, the oliear ostituet equatios of materials, the orete rakig ad variatio o modulus of elastiit. This alterative proedure to the wellkow quasi-newto method proposed b Ye [], is based o bisetio method i order to get the setio equilibrium. Oe ahieved this, the setio modulus of stiffess a be dedued from the expressio E I/. Subsequetl this proedure has bee implemeted i a omputer program about oliear aalsis of strutures that uses a ireasig treatmet i load appliatio. The bar stiffess is orreted i eah iteratio for both reasos geometrial ad mehaial of the materials. I this wa the assessmet of mehai oliearit of materials, although it affets ompressed bars i a speial wa, this will make it appliable i the whole framed struture. CALCULATION ETHOD. Assumed hpothesis Followig hpothesis are aepted at the alulatio proess:. Plae surfaes remai plae after deformatio (Beroulli hpothesis). The deformatio proportioall varies with the distae to the eutral fibre.. Betwee steel ad orete a ompatibilit of deformatios exists i their otat surfaes.. Creep ad shrikage effets of orete are egleted. 4. Effets produed b tagetial stress are egleted. 5. Tesile stregth of orete ad the stress ireasig due to the steel hardeig are egleted.. Costitutive laws of materials The stress-strai relatioships used i this work are provided b the Euroodes regulatios EC [] ad EC [], whih are based o relatioships uder observatio experimetall tested i test tubes. I the orete ase, the amed parabolaretagle diagram has bee used, whih relatioship betwee stresses ad strais a be expressed b the futio:

3 Eresto Feollosa, Adolfo Aloso 0,00 0,05 ε 0,00 σ < ε < 0 f σ 000 ε k ( 50 ε + ) () aximum strai ε u take i this diagram is about 0,5 %. f k (<0) -0,00 f k -0,005 ε (<0) Figure : Corete stress-strai diagram For steel ase, a simplified stress-strai diagrams ostituted b two brahes will be adopted, first brah starts from the origi, with a slope equal to E s (that is reahed at 0kN/ mm ) util f sk (the speified value of the harateristi elasti limit, depedig o the steel tpe). A seod horiotal brah will be adopted for a omputer alulatio with a slope of E s / 0000, takig a maximum value of % as the limit maximum of the uitar strai. σ f sk s - -4 ta (0 E s) - ta Es 0,0 Figure : Steel stress-strai diagram. STRUCTURAL ANALYSIS At strutural desig i elasti sstem, the equilibrium relatioship is defied i a liear wa betwee fores ad displaemets, where the matrix stiffess values of the struture are ostat values. Its resolutio osists o ε s gettig the displaemets vetor of odes { } from the relatioship: [ K e ]{ } { P} Where [K e ] is the elasti ad liear stiffess matrix ad {P} is the loads vetor. ai differee while dealig the seod order effets osist o that the values that built up the struture stiffess matrix deped o displaemets ad the loads applied. The solutio researh use to be brought up i a iremetal wa b the loads appliatio. At eah loadig step it is solved b a iterative proess, a variatio of the equilibrium equatio: [ K t ]{ d } { dp } [K t ] is the taget stiffess matrix ad {d } ad {dp} respetivel represet the vetors of both displaemet ad loads differetials. The taget stiffess matrix is formed b a liear ad elasti ompoet [K e ], ad other oliear ompoet, [K g ] [K m ], aordig to Guire termiolog []. The first oe, [K g ], takes ito osideratio geometri variatios of the bar ad the seod oe [K m ], osiders the mehaial oliearit of materials. The differet elemets of the stiffess matrix a be obtaied, aordig to the proedure exposed b Pa [], through the expressio: k L '' '' E I ψ ( x) ψ ( x dx () ij i j ) o Where ψ i (x), equatio (4), is the defiig futio of the bar deformatios (x) whe a uitar displaemet δ I takes plae ad the rest of displaemets are ull (see Figure ). δ δ δ δ 4 ψ ( x) ψ ( ) x x ψ ( x) x L x ψ 4 ( x) L x L x L x L x L + x L (4)

4 i Eresto Feollosa, Adolfo Aloso u (<0) Y Z Z Z i eutral axis Figure 4 : Setio respose assessmet Figure : Bar deformatios Whe the material oliearit is wated to be evaluated, we aot take the E I term i the equatio () as a ostat value. I this ase, the bar stiffess will be obtaied from the E I term evaluatio i a set of disrete setios alog the bar. Below is developed a iterative proess to obtai the E I modulus of stiffess i reifored orete ad omposites setios while uder axial load ad biaxial momets. 4 EQUILIBRIU RESOLUTION OF THE SECTION 4. Setio respose. I the geeral ase of axial load ad biaxial bedig, the relatio urvature-axial loadmomet ivolves six variables, that i relatio to the referee sstem represeted at Figure 4, are: the P axial fore, ad Y bedig momets, distae from eutral axis to the setio etre of gravit, the plae urvature ad the θ agle formed b eutral ad Y axes. Amog the most kow methods to determie the setio respose, must be quoted the method that use a setio disretiatio model for segmets or ells [8]. Both segmets ad ells have a ostat stregth assiged, whih is applied to their etre of gravit. At preset work has bee used a disretiatio model of the setio based o segmets, aordig to the proess desribed below. A speifi value is assiged to the eutral axis positio (θ ad ) ad deformatio plae urvature (), the proess begis to hage all the poits oordiates that defies the setio (perimeter vertex, strutural setio vertex, ad the etre of gravit of reiforig bars), to express them with regard to the referee sstem defied b a eutral axis. Followig oordiates trasformatio expressio is used: ' os θ ' si θ si θ os θ (5) Where (, ) are the oordiates i the ew referee sstem, θ is the eutral axis rotatio agle i relatio to Y axis ad (, ) are the poit oordiates i relatio with Z-Y sstem. The the setio is divided i a redued thikess segmets set parallels to the eutral axis. The medium lie deformatio of eah 4

5 Eresto Feollosa, Adolfo Aloso segmet is determied b: ε i (6) Eah segmet tesio is obtaied b the materials ostitutive law appliatio. Whe a segmet ivolves differet materials, it is subdivided ito portios for area determiatio purposes ad the stress orrespodet to eah oe of them. The setio respose is obtaied b umerial itegratio: σ σ ( ε ( )) da N ( ε ( )) d da 4. Approahig to the solutio of the eutral axis positio ad rotatio. (7) Oe we kow the axial load ad biaxial bedig momets, the setio equilibrium positio determiatio requires a iterative proess. Although the method developig overgee is guarateed, the umber of iteratios required is highl sesitive to the iitial values of eutral axis positio ad rotatio. Beause of that to get startig values ear eough to the fial solutio is advisable. Whe a biaxial momet is atig over a setio, whih vetor oiides with oe of his mai iertia axis, the eutral axis oiides with the vetor before metioed. If two biaxial bedig momets are atig but the setio have the same iertia with respet to its two mai axes, the the eutral axis oiides with the vetor defied b those two momets. The agle formed with Y axis a obtaied through the expressio: θ ta (8) As usuall happes, if the setio iertias are differet i their two axes, the eutral axis will ot oiides with the mometum vetor, beig eessar to orret its positio takig ito aout the iertias. θ ta I I (9) We will take as erior ad iferior limits i eutral axis rotatio agle to appl the proedure, will be used: θ θ if θ θ θ + θ (0) The distae to the setio etre of gravit ad eutral axis, will deped o the eetriit of the axial load ad area ad iertias of the setio. Its value a be estimated b the expressio: e I Ω + e I () As erior ad iferior limits of eutral axis will used:, if, + () It is rather more ompliated to approah the edig urvature of the deformed plae. Beause of this, ull urvature is adopted as the iferior limit ad the urvature orrespodet to the setio failure surfae as the erior limit, for previousl defied rotatio ad positio. edium value is used as a iitial value. if 0 if ε u + 4. Equilibrium positio loatio () The iterative proess that is proposed to loate the equilibrium positio, based o bisetio method, it osists of three loops i hai. Whe rotatio ad positio of eutral axis are determied b meas of the equatios (9) ad (), the first loop loates the urvature of the deformed plae due to the equilibrium betwee axial load ad iteral fore of the setio. After setio respose has bee obtaied, 5

6 Eresto Feollosa, Adolfo Aloso equatio (7), with iitial urvature, the routie orrets its value at the axial load futio. If the setio respose is bigger tha the axial fore, the to redue urvature is eessar, but i the opposite ase to irease it. Next iteratio value is obtaied b the expressios: N > N d N < N d if if if m + + (4) Seod loop determies eutral axis distae to etre of gravit of the setio beause of the equilibrium of Z axis bedig momet with iteral reatio of the setio. This loop iludes the first oe, so whe overgee is obtaied, the eutral axis positio will be kow, as well as the urvature of deformed plae. After gettig setio respose with iitial positio of eutral axis ( ), the routie orrets its value depedig o Z axis bedig momet. If the setio respose is bigger tha Z axis bedig momet, the to move awa the eutral axis will be eessar, i the opposite ase to move loser. As the setio respose is obtaied i relatio to the referee sstem defied b the eutral axis rotatio, to hage the referee sstem to Z-Y axes is eessar. The ew value of the eutral axis for the followig iteratio is obtaied b the expressios: Z Z > < Zd Zd,if,,if,if + +,, (5) Third loop determies eutral axis rotatio beause of the equilibrium betwee bedig momet i Y axis ad the iteral momet of the setio, keepig a fixed distae from eutral axis to setio etre of gravit. This loop iludes the two previous loops, so whe overgee is obtaied, as well the eutral axis rotatio ad positio as the urvature of the deformed plae will be kow. After the setio respose with the iitial rotatio (θ) of eutral axis is obtaied, the routie orrets its value i the Y axis bedig momet futio. If the setio respose is bigger tha Y axis bedig momet, the irease of the eutral axis rotatio is eessar, i the opposite ase, to redutio it. As setio respose is obtaied i relatio to the referee sstem defied b the rotatio of the eutral axis, the hage from a referee sstem to Z-Y axes is eessar. The ew rotatio value of the eutral axis for ext iteratio is obtaied b the expressios: Y Y > Yd θ θ if + θ θ < θ θ Yd θ θ if if + θ θ (6) Eah loops overgee is ahieved whe the error betwee soliitatio ad the iteral setio respose is osidered worthless. 5 CALCULUS ORGANIZATION CHART Desribed proedure is below represeted b a flow diagram (Figure 5). 6 APPLICATION TO A COPOSITE SECTION I order to show the workig proedure desribed before to obtai equilibrium positio, a alulatio for a 45x45 setio has bee made with a HEB-60 steel profile eased ad 8 as reiforig framework plaed with a m laer. The stregth of used materials have bee f k 5 N/mm for orete, f k 75 N/mm for strutural steel ad f sk 400 N/mm for reiforig steel. As overgee riterio a maximum error oeffiiet of % over the axial load ad bedig momets has bee aepted. 6

7 Eresto Feollosa, Adolfo Aloso DATES: Nd, xd, d Seió, ateriales Table shows the homogeied iertias of the setio, together with its axial load ad bedig momets. Set θ (equatio 9) Set (equatio ),if,,, θ if, θ, if, Referee sstem hage (equatio 5) +, if, Table : Iitial values Ix I N d x d d (0 - m 4 ) (0 - m 4 ) (kn) (m kn) (m kn) 4,46504, Table shows the iteratios that have bee doe to get the overgee betwee axial load (N d ) ad the setio respose value (desribed as loop ). θ if + θ θ if + Setio respose for Z, θ, Equatio 7 N, if Table : Axial load equilibrium Iter (x0-5 rad) N (kn) Error %, ,88 5,89 0, ,, 0,5.676,49,68 4 0, , 0,86 5 0, ,0,7 6 0,045.06,7,06 7 0, ,74 0,96 8 0,064.0,80,0 N>N d,if > d θ if θ > d Covergee riterio N-N d Udo referee sstem hage Compariso d Compariso - d N<N d, < d θ θ < d The umber of iteratios is bigger for the first etrae i the loop, due to ot to have a urvature value ear eough to the solutio. For followig etraes, where the equilibrium urvature i the previous le is alread kow, the umber of iteratios is redued to a medium value of 4. Table ad Table 4 show the iteratios that have bee doe i order to obtai the overgee betwee the biaxial bedig momets (-) ad the respose momets of the setio whe are takig part respetivel, i the positio ( ) ad i the eutral axis rotatio (θ) (desribed before as ad loops). Equilibrium solutio: sol, θ sol θ, sol E I sol Table : Equilibrium of the momet Iter (m) (m kn) Error % 66, ,88,58 76,8089.8,97 0,9 7, ,0 Figure 5: Flow diagram 7

8 Eresto Feollosa, Adolfo Aloso Table 4: Equilibrium of the momet Y Iter θ Error (degree) (m kn) % 54, ,94,05 It a be observed that iitial values, espeiall rotatio (θ) are ear eough to the edig solutio, due to that the iteratios umber is small. 7 OENT-CURVATURE DIAGRA Remaiig axial load ostat ad progressivel ireasig biaxial bedig momets, several urvatures of the deformed plae that lead to the equilibrium of the setio a be obtaied. The graphial represetatio of the pair of values -, are kow as momet-urvature diagram. As a example, ad usig the setio desribed at poit 6, has bee elaborated the momet-urvature diagram (Figure 6) orrespodet to a ostat axial load of.000 kn ad the relatio betwee two bedig momets is /,66. - (x0 m kn) (x0 rad) Figure 6 : omet-urvature diagram (N.000kN) The biggest ireasig of the urvature i relatio to the momet is due to the oliearit betwee both of them ad makes the progressive stiffess loss (E I) lear i the setio as the bedig momet is beig ireased. overgee has bee exposed, whe startig from the appropriate startig values that allow gettig the positio of equilibrium of a reifored orete or omposite setio that has bee requested for the axial load ad biaxial bedig momets. I that proedure are take ito aout the differet stregth behaviours of materials that built them up through their oliear ostitutive laws. Its implemetatio o software about oliear aalsis of strutures allows to modif the member stiffess i framed strutures takig ito aout the oliear mehaial of the materials, i eah loadig step. Although the osideratio of the mehaial oliearit a be take ito aout at a material, it affets sigifiatl to orete strutures due to the variatio of its elastiit module depedig o the stregth applied ad the iertia of the setio, affeted b the material rakig. Its iidee aquires a speial importae i fatigue load of ompressed bars desig, its measuremet is oditioed b the seod order effets. As a example a appliatio of the developed proedure has bee show to obtai the equilibrium of a steel orete eased setio. The tables show the small umber of iteratios eeded to obtai the setio equilibrium. This together with the urret powerful omputers allows quikl eough the approahig to mehaial oliear desig of orete strutures. Fiall, the momet-urvature diagram of the setio for a ostat axial load of.000 kn has bee represeted. The show proedure represets a powerful ad quik method to strutural desig takig ito osideratio the mehai oliear of materials ad a effetive tool to the future researhes developig. 8 CONCLUSIONS A iterative proedure of quik 8

9 Eresto Feollosa, Adolfo Aloso REFERENCES [] Netherot, D.A., 000. Frame strutures: global performae, stati ad stabilit behavior. Geeral Report. Joural of Costrutioal Steel Researh, 55: [] Livesle, R.K., 956. The appliatio of a eletroi digital omputer to some problems of strutural aalsis. The Strutural Egieer, 4: -. [] Jeigs, A., 968. Frame aalsis iludig hage of geometr. Joural Strutural Divisio, ASCE, 94(): [4] Yag, Y.B. ad Guire, W., 986. Joit rotatio ad geometri oliear aalsis. Joural Strutural Egieerig, (4): [0] Roik, K. ad Bergma, R. 99. Composite olum. Costrutioal Steel Desig: A iteatioal Guide. Elsevier Applied Siee. Lodo. Uited Kigdom. [] Ye, J.Y.R., 99. Quasi-Newto method for reifored orete olum aalsis ad desig. ASCE Joural Strutural Egieerig, 7(): [] Guire, W., Gallagher R. ad Ziemia, R atrix strutural aalsis. Joh Wile & Sos, I. USA. [] Pa,. 00. Diámia estrutural. Teoría álulo. Editorial Reverté, S.A. [5] Ziekiewi, O.C., 977. The fiite elemet method, rd ed. Graw-Hill, New York. USA. [6] avihak, V. ad Furlog, R.W., 976. Stregth ad stiffess of reifored orete olums uder biaxial bedig. Res. Rep. 7-F, Ctr. for Hw. Res., Uiversit of Texas at Austi, Texas. [7] Al-Nour, S.L. ad Che, W.F., 98. Fiite segmet method for biaxial loaded RC olums. Joural Strutural Divisio, ASCE, 08(4): [8] Virdi, K.S., Dowlig, P.J., 97. The ultimate stregth of omposite olums i biaxial bedig. I: Proeedigs of the istitutio of ivil egieers, Part ; 55: 5-7. [9] Wag, G.G. ad Hsu, C.T., 99. Complete biaxial load-deformatio behavior of RC olums. Joural Strutural Egieerig, ASCE, 8(9):