Engineering Structures 5 (003) 951 964 www.elsevier.com/locate/engstruct Optimal esign o reinorce concrete T-sections in ening C.C. Ferreira a,, M.H.F.M. Barros a, A.F.M. Barros a Department o Civil Engineering, Faculty o Sciences an Technology, University o Coimra, Polo II, 3030 Coimra, Portugal IDMEC/Instituto Superior Técnico, Av. Rovisco Pais, 1000 Lison, Portugal Receive 9 August 00; receive in revise orm 3 Feruary 003; accepte 4 Feruary 003 Astract The optimisation o the steel area an the steel localisation in a T-eam uner ening is carrie out in the present work. The expressions giving the equilirium o a singly or ouly reinorce T-section in the ierent stages eine y the non-linear ehaviour o steel an concrete are erive ones. The ultimate material ehaviour is eine accoring to the esign coes such as EC an Moel Coe 1990. The purpose o this work is to otain the analytical optimal esign o the reinorcement o a T- section, in terms o the ultimate esign. In the last section the expressions evelope are applie to examples an esign aacus are elivere. A comparison is mae with current practice metho as inicate in CEB. 003 Elsevier Science Lt. All rights reserve. Keywors: Reinorce concrete; Doule reinorcement; T-sections; Optimisation o the reinorcement; Design aacus 1. Introuction Methos ase on optimality criteria can e applie to reinorce concrete structures in orer to otain the minimum cost esign. The costs to e minimise are generally ivie into those o concrete, steel an ormwork. This means that they inclue one or more o the ollowing variales: the geometry o the section o eams an columns; the area an topology o the steel reinorcement. Methos ase on this kin o analysis are oun in reerences [1 5]. Nevertheless, these optimisations consier a linear elastic analysis o the gloal structure. This type o linear analysis is relevant or the serviceaility limit state esign an it is important or the einitive imensions o the sections. In terms o the ultimate limit esign o concrete structures, this analysis is not correct ue to the non-linear ehaviour o reinorce concrete. Consiering esign variales such as section imensions, steel area an topology at the same time, the nonlinear optimisation o reinorce concrete structures is a very complex an yet to e solve prolem. Optimising Corresponing author. Tel.: +351-39-797-100; ax: +351-39- 797-13. E-mail aress: carla@ec.uc.pt (C.C. Ferreira). the imensions o the sections within serviceaility conitions an then optimising the steel area an location in the ultimate limit esign within a non-linear analysis, appears to e a goo solution. As a matter o act, the esign coes [6,7] suggest that non-linear analysis can e replace y linear analysis with reistriution o the moments an the esign o the reinorce concrete memers eing mae or the critical sections, where the ening moment an shear attain their maximum value. The optimisation o the reinorcement in a section is evelope within this context. The optimal esign o the critical sections is known only or rectangular sections. For other geometries, research has een mae in terms o the iaxial interaction iagrams [8 10]. These iagrams attempt to make an optimisation y a trial an error proceure. The importance o the evelopment o the optimal esign o T-sections is ue to the act that it is currently a requently use section in common structures. Another relevant aspect is that the methoology use can e extene to other sections an inclue in the computer coes with minimum programming. The esign variales consiere in the optimisation o the reinorce eam with a T-section are the steel area an the steel localisation, either in the tension or in the compression zones. The equilirium equations o a reinorce concrete T-section uner Limit State Design, 0141-096/03/$ - see ront matter 003 Elsevier Science Lt. All rights reserve. oi:10.1016/s0141-096(03)00039-7
95 C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 as eine y the non-linear ehaviour o the concrete an the steel, are evelope or this purpose. The examples presente consist o the optimisation o the reinorcement o a T-section or ierent geometric einitions. The results are compare with the esign otaine in current practice methos such as the one inicate in CEB [11].. Ultimate esign o reinorce concrete section uner ening.1. Geometry o the T-section The T-section geometry o the reinorce concrete structure is eine y the ollowing parameters, as shown in Fig. 1: lower steel area in the tensile zone, A s ; upper steel area near the compression zone, A s ; the lange epth, h ; the eective lange with, ; the we with, w ; the istance rom the centroi o the reinorcement to opposite ace o the section, ; the concrete cover, a. In the present work, the esign variale in the optimisation process is the ratio A s /A s, since, or a given total area o reinorcement A s + A s, the ojective unction is to maximise the ening moment. As a matter o act, or a given ening moment, the gloal area o reinorcement is only a unction o the ratio A s /A s, which, or this reason, is the only esign variale in terms o the limit state esign o the section. The cover o steel is not consiere as a esign variale ecause it is ixe, in each case, y uraility conitions. The equations are evelope as a unction o the non-imensional variales a/, w / an h /... Constitutive laws The ultimate esign o reinorce concrete sections uner ening moment is eine in terms o non-linear constitutive laws o the concrete an the steel. The stress strain equation or concrete use in the esign coes is eine y a paraola ollowe y a constant Fig.. Design stress strain iagrams. value, usually calle paraola rectangle law an represente in Fig. (a). The maximum stress is equal to 0.85 c, where c is the esign strength o the concrete uner compression. It is consiere that concrete in ultimate esign oes not stan or tensile stresses. The equation or the paraola is unction o the concrete eormation e c, as ollows: s c 850 c (e c 50e c)or e c 0.00 (1) The esign stress strain law or steel is elastic-perectly plastic with a maximum value sy, which is the esign value or the strength o steel, equal in tension an compression. This law is represente in Fig. (). The steel has initial elastic ehaviour with the elasticity moulus E s. The elastic omain is vali until the maximum esign stress sy is reache, corresponing to the maximum elastic strain e sy..3. Rupture conitions The limit esign means that rupture is consiere when the strains in the section reach maximum values, which are ierent or concrete an steel. Concrete is consiere to crush at e c = 0.35%. The steel rupture uner compression is limite to 0.35% ue to the crush o the concrete, an to 1% uner tension. These conitions are represente in Fig. 3, where the section eore eormation is represente y the vertical line an ater Fig. 1. Geometry o the ouly reinorce T-section. Fig. 3. Rupture o the section.
C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 953 Fig. 4. (a) T-section; () stresses in concrete; (c) resulting loa in concrete. Tale 1 Values o m 1 0 e s sy E s m 1 E s e s 1 sy e s sy E s eormation y the incline line. In oth ruptures, y the steel [Fig. 3(a) an y the concrete [Fig. 3()], the neutral axis is locate at the istance a rom the upper zone, an a is given y: e c () e c e s The paraola rectangle transition is locate at the istance a rom the upper zone an correspons to the concrete strain e c = 0.%. In the Fig. 3, the strain in upper reinorcement, e s,is also inicate, an is compute y:.4. Resulting loa in the concrete The resulting loa in concrete, F c, is otaine y the integration o the stresses in the concrete, that is the secon egree Eq. (1), calle paraola, an the constant value, calle rectangle. These stresses may e applie either in the lange or in the we. Consiering the variation o neutral axis a an the lange epth h, the rupture conitions are also variale. The consieration o all possiilities originates the several cases escrie in Fig. 5. The value o the resulting loa, in each case i, is enote y F i c an the istance to the upper ege o the section y X i. The loa in concrete can e written in non-imensional orm, F i c re,eine y the ollowing expression: F i c re Fi c c. (4) The value o all F i c re an X i are given in Appenix A. There are VIII cases to e consiere. In cases I an II rupture occurs in steel an concrete which has a stress e s (e c e s )a a (3) The stresses in the concrete uner compression, corresponing to the strains represente in Fig. 3, are inicate in Fig. 4(). The stresses upon the steel in the upper reinorcement o the compression zone are terme m 1 an are etaile in Tale 1. The stresses upon the steel in the lower reinorcement o the tensile zone, are terme m an are presente in Tale. Tale Values o m 0 e s sy E s m E s e s 1 sy sy E s e s 1% Fig. 5. Dierent cases consiere in the computation o the resulting loa.
954 C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 e c lower than 0.% (see Fig. 5). In cases III V rupture is also in the steel ut concrete strain e c is larger than 0.%. In cases VI VIII rupture is in concrete with strain equal to 0.35%. The cases can also e groupe in terms o the lange epth. Cases I, III, an VI can e calle the cases or high lange epth. The neutral axis is in the lange an the section is equivalent to a rectangular section in terms o the rupture conition. Cases II, V an VIII can e relate y the act that the lange epth is small. Cases IV an VII are intermeiate to the others an correspon to meium epth lange..5. Equilirium equations in the section The equilirium equations o all the loas acting on the section, when zero axial orce is applie an a ening moment is impose, estalishes the expressions (5) an (8). The irst one is: F i c re A s wm A 1 wm 0 (5) s where m 1 an m were previously eine (Tales 1 an ), w is the percentile o the lower reinorcement, calculate, through: w A s sy. (6) c Consiering a ening moment M s applie to the section an the reuce ening moment as: m M s, (7) c the secon equilirium conition is written as: F i c re 1 Xi A s w1 a m A s 1 m 0 (8) It must e remarke that there are as many equations as the numer o cases enote in Fig. 5. 3. Minimum area o the reinorcement 3.1. Ojective unction The optimisation prolems concern the achievement o the est solution or a given ojective unction satisying certain conitions. Currently, the esign is mae or the most stresse section that elimits the cross section imensions an the amount o steel. Optimisation o the steel area, in a preeine cross section, is similar to computing the minimum rate o steel. In practice, the computation o ierent rates with ierent locations, using esign tales, approaches the optimal solution y trial. For the esigner, the purpose is to estalish the expression giving the maximum resistant ening moment, unction o the area an location o the reinorcement with the relation A s /A s. In the present prolem the system o Eqs (5) an (8) are the equilirium equations o the section. Sustituting into these expressions the einitions o F i c re, X i, m 1 an m, they ecome perectly eine in terms o the variales a, A s, h A s,a,, sy an can e written respectively in w c the orm: w ga, A s A s, h,a, w, sy c (9) m a, A s, h A s,a,, sy w c (10) In practical terms, the purpose is to maximise the ening moment m, Eq. (10), with the constraint eine y Eq. (9). 3.. Design variales The variales eining the position o the steel in the section are use in the optimisation process in the present work. This is the ratio o steel areas A s /A s. The total area o reinorcement is given y the sum A s an A s an is consiere in the constrain Eq. (9). In nonimensional terms, it is eine y w t, that is the percentile reinorcement given y: w t A s A s sy 1 A s (11) c A sw This total reinorcement can also e written in unction o all variales, y using Eq. (9): w t 1 A s A s ga, A s, h A s,a,, sy w c (1) The arguments a an sy in Eqs (9) an (10) are not c consiere esign variales since a is usually impose y uraility an other construction requirements. The strength ratio sy is a known value ater the choice o c the materials. The geometry parameters h an can w e use as esign variales in the case o shape optimisation o the section. Ater eliminating the escrie constant parameters, Eqs (1) an (10) ecome respectively: w t 1 A s A s ga, A s A s (13)
C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 955 m a, A s A s (14) Choosing m as the ojective unction an w t as an impose variale, Eq. (13) can e solve in terms o a, that is: a h A s A s (15) The value a is turne into the ojective unction, Eq. (14), that ecomes: m h A s A s, A s A s (16) This equation shows that, in the present work, there is only one esign variale, A s /A s, ut the ojective unction is eine as a piecewise unction, giving as many ierent expressions as the cases o Fig. 5. As the constant characteristics o the arguments are accepte, a,, h w an sy, the analytical optimal solution is always c written in terms o those arguments. 3.3. Optimal neutral axis epth In the present work, the choice o the ormula or the mathematical optimisation prolem oes not allow the calculus o the sensiilities ue to the characteristic iscontinuity o the ojective unction. The optimal solution is otaine y consiering an increment o the esign variale A s /A s, that is A s /A s, giving the ollowing value o the ojective unction: m h A s A s A s A s, A s A s A s A s (17) The unction has a stationary value since a maximum or a minimum point exists an Eqs (16) an (17) shoul e equal when A s /A s approaches zero, such as: a, A s A s lim A s A s 0 h A s A s A s A s, A s A s A s A s (18) This equation is solve or each o the cases in Fig. 5, giving or high lange epth (cases I, III an VI) the optimal value o neutral axis epth, a, given y: a 119 1981 a (19) With this optimal value, the minimum lange epth availale or these cases I, III an VI can e estalishe y: h a 119 1981 a (0) The optimal solution otaine rom Eq. (18), or the cases o small lange epth (cases II, V an VIII), is the same value a o expression (19). The limit value o the lange epth, h, is otaine y imposing the limits o these cases, as shown in Fig. 6, an ecomes: h 17 661 a (1) For the intermeiate values o the lange epth, corresponing to cases IV an VII, the optimal solution can not e otaine analytically through Eq. (18) in unction o the parameters. The computation o the optimal neutral axis epth using Eq. (18) is possile only when iscrete values o the ierent parameters are impose. For this reason an approximate solution or cases IV an VII is propose in section 3.6. 3.4. Ductility limitation The position o neutral axis aects the eormation o the steel, that can e in the elastic or in the plastic omain. In the esign o reinorce concrete, it is important to guarantee a certain uctility. The conition or the plasticity o the lower steel A s,eine y a limit value o a, terme a p, epens on the esigne yiel strain sy, that varies or each class o steel. This limit or the plastic zone o the steel a p is the ollowing: 7 a p. () 7 000e sy The plastic limits a p Eq. (), eine or three classes o steel, namely S35, S400 an S500, are represente in Fig. 7. The optimal value a Eq. (19), is also represente in Fig. 7. Since the conition o applicaility o a is the yieling o steel A s, as impose in cases I VIII, the corresponing equation is only vali when it satisies the ollowing inequality: a a p (3) Fig. 6. Limit zone or h in case VIII.
956 C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 approximate value consiere in cases IV an VII is a, Eq. (19), satisying the ollowing conition: Fig. 7. Optimal neutral axis epth: a or a p. As a result, or a given a/, the optimum value or a is either a or a p whichever the smaller value might e. 3.5. Optimal solution an esign The optimal solution o the esign variale (A s/a s ) is otaine y solving Eq. (15) in terms o A s /A s an sustituting in it a or a p, that is: A s A s h 1 (a or a p ) (4) For a small ening moment there is no reinorcement in the compression zone, meaning that A s = 0. The limit value o the ening moment can e terme as economic moment m e an is otaine y replacing a or a p an A s /A s =0 into Eq. (14), which ecomes: m e (a or a p ) (5) This value is relevant or the practical esign o reinorce concrete sections as well as or the optimal area o reinorcement wt. The optimal area o reinorcement is otaine y replacing the values a an (A s/ A s ) in Eq. (13), that is: wt 1 A s A s ga or a p, A s A s (6) I only the area A s is consiere, the optimum percentile lower reinorcement w, Eq. (9), ecomes: s (7) w ga or a p, A A s 3.6. Approximate solution in cases IV an VII For the cases IV an VII the optimal solution is only otaine i values or h / an w / are prescrie, preventing a generally analytical solution. In practical terms, it may e relevant to have esign expressions since they have small errors. For this reason, the a apr 119 1981 a a p (8) In orer to etermine the error introuce y the approximate neutral axis epth a apr in the optimal esign o the section, the computation o optimal neutral axis epth a was reache, accoring to section 3.3, or a representative numer o prescrie values o h / an w / an a/. Replacing these optimal a successively in expressions (14) an (13), the corresponing percentile o reinorcement w real was oun. The values o the percentile o reinorcement w apro, otaine when a apr is consiere in expressions (14) an (13), were also compute. The error introuce in the optimal esign was evaluate y the ollowing expression: w apro w real 100 (9) w real Tale 3 gives the values o this einition o error or / w =, 3, 4, 5, 6, 7, 8, 9, 10, with a/ = 0.05 an h / varying rom 0.8 to 0.37. Tale 4 exempliies the errors that can e otaine y the variation o parameter a/. InTale 4, the values o expression (3) are plotte or a/ = 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, with / w = 4 an h / varying rom 0.7 to 0.41. The lank cells in Tales 3 an 4 correspon to the optimal value given y a p. Other calculations, similar to the ones relate aove were mae, namely ixing the parameter / w an computing the error with variale h / an a/, as mae in Tale 4. The results are misse out ut they are always less than 0.14%. This is the reason why expression () was consiere a goo approximation to the optimal solution. 4. Examples The optimal neutral axis epth an the corresponing expressions o the esign, otaine accoring to section 3.5, was oun or a general T-section. The expressions otaine in these computations are evelope in Appenix B an they are presente in Tale 5, with their respective limitations. 4.1. Aacus or optimal esign The evelope expressions are applie in a T-section or a/ = 0.1 with variale geometric einitions o / w :/ w equal 1/10, 1/8, 1/6, 1/4 an 1/. The consiere steel classes are S400 or S500. The values o the economic reuce ening moment m e are plotte in Fig. 8. Fig. 8 emphasises the limits o expressions (6a), (7a)
C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 957 Tale 3 Error or ierent values o h / an / w / w h / 3 4 5 6 7 8 9 10 0.8 0.000001 0.000003 0.000005 0.000007 0.000009 0.000011 0.000013 0.000015 0.000017 0.9 0.00001 0.000055 0.000089 0.00013 0.000156 0.000189 0.0001 0.0005 0.00080 0.30 0.000110 0.00078 0.000449 0.000616 0.000775 0.00097 0.00107 0.00110 0.001341 0.31 0.000346 0.000865 0.001385 0.001881 0.00347 0.00786 0.003198 0.003585 0.003950 0.3 0.00083 0.00059 0.00365 0.004396 0.005445 0.006416 0.007317 0.008155 0.008937 0.33 0.001689 0.004133 0.006496 0.008675 0.010668 0.0149 0.014165 0.015707 0.017133 0.34 0.003041 0.007374 0.011489 0.0158 0.018604 0.01660 0.04439 0.06977 0.09309 0.35 0.00500 0.01056 0.01869 0.0451 0.0978 0.034498 0.038751 0.04609 0.04619 0.36 0.007747 0.01848 0.0856 0.036956 0.044645 0.051479 0.057597 0.37 0.01135 0.06694 0.040635 0.05836 0.06350 0.38 0.015834 0.036997 0.055939 0.07348 0.39 0.013 0.049409 0.07441 0.40 0.07800 0.063917 0.095489 0.41 0.03535 0.080418 Tale 4 Error or ierent values o h / an a/ a/h / 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.7 0.00000 0.000000 0.000000 0.8 0.000056 0.00009 0.000014 0.000010 0.000000 0.000000-0.000000 0.9 0.000335 0.0006 0.000146 0.000089 0.000051 0.00006 0.00001 0.30 0.00111 0.000846 0.00064 0.000449 0.000313 0.00011 0.000135 0.31 0.00770 0.0030 0.001771 0.001385 0.001065 0.00080 0.000591 0.3 0.005680 0.004768 0.003966 0.00365 0.00589 0.00139 0.001697 0.33 0.01055 0.008868 0.007618 0.006496 0.005497 0.00461 0.003835 0.34 0.01675 0.01493 0.013130 0.011488 0.009995 0.008641 0.00741 0.35 0.05870 0.0385 0.00873 0.01869 0.01655 0.014636 0.01876 0.36 0.037498 0.03441 0.031160 0.0856 0.0559 0.0975 0.00594 0.37 0.0519 0.047990 0.0445 0.040634 0.0370 0.033983 0.38 0.069199 0.06469 0.06007 0.055940 0.051836 0.39 0.08966 0.084143 0.079130 0.07441 0.069488 0.40 0.11199 0.106387 0.100899 0.095488 0.41 0.136864 0.131085 0.1590 Tale 5 Resume o equations in Appenix B, or the ierent cases Case o Fig. 5 II, V, VII IV, VII I, III, VI Optimal sol. a a a p a Equations (7a), (7), (7c) (8a), (8), (8c) (9a), (9), (9c) (6a), (6), (6c) Limit values 0 h / 17 661 + a 17 661 + a h /a 17 661 + a h /a p h / 17 661 + a an (8a), given in Appenix B, epening on the epth o neutral axis that changes with the values o h /. The other involve variales, such as a/; w / an steel class lea to ierent curves. Fig. 9(a) is an aacus giving the optimal total area o reinorcement (A s + A s )asα unction o the applie reuce ening moment m, an the ratio h /, or a/ = 0.1 an / w = 10. The istriution o this area o steel etween the upper an lower aces, is otaine rom Fig. 9()). Figs. 10 13, have the same interpretation o Fig. 9, an correspon to the ierent values o the geometric parameters an a/=0.1. These igures are resume in Tale 6 an cover some current sections permitting the irect calculation rom the igures or employing linear interpolation. Design aacuses or other values o a/ are in [1].
958 C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 4.. Numerical results Fig. 8. Economic m e or a/ = 0.1. In this section a comparison is mae etween the esign otaine in the present ormulation an the approximate solutions given y CEB [11]. The percentile o total reinorcement, the amount o upper steel area an the errors in percentile are etaile in Tale 7. The percentile o error is calculate y: Error Re.[11] mo el mo el 100 As can e oserve in Tale 7, the total area o reinorcement has a maximum ierence o.05% when compare with the CEB solution. Generally the ierences in the istriution A s /A s. is always larger with a maximum o 10.4%. Although the ierences or the require reinorcement o not excee 10.4% in the stuie case, it seems that in repeate structures, such as in prearication, this ierence can e relevant in econ- Fig. 9. (a) Increasing in µ with ω t, or / w =10 () µ variation with A s /A s, or / w =10. Fig. 10. (a) Increase in m with w t, or / w = 8; () m variation with A s /A s, or / w = 8.
C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 959 Fig. 11. (a) Increase in m with w t, or / w = 6; () m variation with A s /A s, or / w = 6. Fig. 1. (a) Increase in m with w t, or / w = 4; () m variation with A s /A s, or / w = 4. Fig. 13. (a) Increase in m with w t, or / w = ; () m variation with A s /A s, or / w =.
960 C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 Tale 6 Resume o geometric einitions in Figs. 9 13 S400 S500 / w 10 8 6 4 w t (9a) (10) (11a) (1a) (13a) A s /A s (9) (10) (11) (1) (13) omical terms. It can e emphasise that L-, I- an T- sections are commonly use in uilings. Since the solution otaine in this work is an exact one, it can e note that the errors in the CEB solution have consequences in terms o: not satisying the equilirium Eqs (5) an (8); eviation o the neutral axis an error in the rotation o the section at rupture. 5. Conclusion In the present work a moel is evelope or the optimisation o reinorce concrete T-sections, in ultimate esign, uner ening moment. In the moel, the non-linear ehaviour o concrete, with paraola rectangle law or compression an no tensile strength, an elasto-plastic ehaviour or steel are consiere. In this work, the equation o maximum value o the ening moment consiering only single reinorcement, or T-section is otaine. The equations or the optimal area o reinorcement an corresponing localisation are also presente. These equations are expresse in terms o the geometry an mechanical characteristics as inicate in Appenices A an B. The equations are plotte or current T-section geometric characteristics in Figs. 9 13, or S400 an S500 steel. The comparison etween the results otaine with the optimal esign an current practice methos (CEB solution) is mae. The avantages o the present moel are: the correct solution is economic when compare to current practice solutions; the use o non-linear ehaviour o the materials; the evelopment o a methoology that can e extene to other sections; the estalishment o the optimal esign equations that can e implemente in computer coes. Acknowlegements This work was perorme uner the inancial support o the Portuguese Minister o Science an Technology y Programa Operacional o Quaro Comunitário e Apoio (POCTI) an y FEDER grant POCTI/EMC/116/1998, Fase II. Appenix A The resulting reuce compressive orce F i c re in the concrete an the location X i o the orce F i c, inicate in Fig. 4, are liste ellow, or the ierent cases (see Fig. 5): F I c re 17 a 1(a1) (38a); XI 1 9a4 48a3 a c re 17 1 1(a1) 1 h a 6 h a18 h F II 5 w h 3 w 17 X II 48F II c re (a1) 15 h h 1 w F III 1 8 w a3 3 w a 3 h a 4a 48 h a 1a8h 4 w a3 9 w a4 c re 68 17 a 75 300 ; XIII 1 171a a 0 16a1 17 F IV c re 300(a1) 16a 3 18a1 a 75 w 108 w 1 h (55a 150a) 75450a 15 h w 1 h 17 X IV F IV c re 6000(a1) (36a 1a 1) 1875 h 4 w w 1 h 1 550 w 1000 h a4 500 w a3 115 w 1 h (16a)1500a a
C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 961 Tale 7 Design o a T-section or m s = 0.5, 1.0,.0 an 3.0 h / 0.05 0.40 / w 10 10 a/ 0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15 m s =0.5 w t Moel 0.9103 0.9473 0.9881 0.9884 1.0387 1.0948 0.8171 0.8405 0.8661 0.807 0.8464 0.875 Re. [8] 0.911 0.949 0.991 0.988 1.039 1.095 0.817 0.841 0.869 0.81 0.847 0.877 %Error 0.07 0.18 0.9 0.04 0.03 0.0 0.01 0.06 0.33 0.04 0.07 0.1 A s Moel 0.5850 0.5841 0.5846 0.8473 0.8508 0.8546 0.3597 0.3596 0.3608 0.4079 0.4165 0.466 A s Re.[8] 0.593 0.606 0.619 0.850 0.855 0.86 0.366 0.397 0.393 0.411 0.44 0.438 %Error 1.37 3.75 5.88 0.3 0.49 0.86 1.75 10.40 8.9 0.76 1.80.67 m s = 1.0 moel 1.969.0584.1646.0410.1498.713 1.8697 1.9516.046 1.8733 1.9575.0516 Re. [8] 1.964.060.167.041.150.7 1.870 1.95.045 1.874 1.959.054 %Error 0.06 0.08 0.11 0.00 0.01 0.03 1.60.05 0.1 0.04 0.08 0.1 A s Moel 0.7834 0.7844 0.786 0.930 0.950 0.97 0.6587 0.667 0.6678 0.6888 0.6976 0.7073 A s Re. [8] 0.787 0.796 0.806 0.94 0.97 0.930 0.668 0.675 0.687 0.690 0.70 0.713 %Error 0.46 1.48.5 0.11 0. 0.30 1.41 1.86.88 0.17 0.63 0.81 m s =.0 Moel 4.068 4.806 4.5175 4.1463 4.370 4.64 3.9750 4.1738 4.3955 3.9786 4.1798 4.4046 Re. [8] 4.069 4.8 4.50 4.146 4.37 4.65 3.975 4.175 4.397 3.979 4.181 4.407 %Error 0.0 0.03 0.06 0.01 0.00 0.0 0.00 0.03 0.03 0.01 0.03 0.05 A s Moel 0.8893 0.890 0.8915 0.9614 0.964 0.9636 0.835 0.867 0.8306 0.8403 0.8460 0.851 A s Re. [8] 0.89 0.896 0.90 0.96 0.964 0.966 0.86 0.834 0.841 0.841 0.848 0.856 %Error 0.30 0.65 1.18 0.06 0.17 0.5 0.30 0.88 1.5 0.08 0.4 0.46 m s = 3.0 Moel 6.1734 6.509 6.8705 6.515 6.594 6.9771 6.0803 6.3961 6.7484 6.0838 6.400 6.7575 Re. [8] 6.175 6.504 6.873 6.5 6.594 6.977 6.080 6.397 6.751 6.084 6.403 6.759 %Error 0.03 0.0 0.04 0.01 0.00 0.00 0.00 0.01 0.04 0.00 0.0 0.0 A s Moel 0.956 0.963 0.973 0.974 0.9749 0.9557 0.8810 0.8834 0.8863 0.896 0.8967 0.9010 A s Re.[8] 0.97 0.931 0.934 0.974 0.976 0.977 0.883 0.888 0.894 0.894 0.898 0.903 %Error 0.15 0.51 0.7 0.0 0.11.3 0.3 0.5 0.87 0.16 0.14 0.
96 C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 F V c re 17 h 0 X V 0 F VI 150 w c re 89 40 68 w 75 1 h a17 h w 0 17 300 w 171a a1 15 w 1 h 16 w a w a; XVI c re 99 38 a a F VII c re 17 670a 45 w a3 147 h w 441 h a w 1 343 h 3 w X VIII 3 109 w 686 h a4 8 c re 17 h 0 F VIII X VIII 3 14 Appenix B a w 744 1 h 45 w a3 147 h a w 1 441 h 89 w 40 49 h a17 w 0 1 3 a w 1 401 h a w 1 343 h 3 w h 49 w h 33 w 1 7a 3 h 1 h 17 w a 1 w a 4 w 1 7a 4 1 7a 3 The optimal esign o the sections reerre to in section 3.5, with the limits reerre in Tale 5, can e represente y the ollowing expressions: m e 4913 47501 a 3 a. A s A s 14739 986 a 4913 a 4750m 4913 986 a 14739 a 4750m, (6a) (6) 4913 986 a 14739 a 4750m w 4750( a (6c) 1) A s A s w 40 17 h 1 w 4913 w 4750 1 w 14739 w a 4039 h 1 w 0196 h 986 wa 4913 w 4750m 0196 h 1 w 4913 w 986 wa 14739 w 4039 h a 1 w 4750m 1 4750 a 1 4039 a h a a 0196 h w 4913 w 1 a 14739 w a 4750m 1 1 m e 843596801 a 10574656 h 1 w w a a 7776577 w 3013108a ( 351895104 a 1034985600 ) h w 5131787 h 41 w 1091599 4096400 h 31 751689 a a 1 w 587318 h a 1 w 30509730 a 114951 h a 1 w 47496958 h 1 w 8647703 a 4 w 031808 wa 866653 a 3 w 1458866 a w 1 ( A s /A s ) K A /K B, with K A 866653 a w 1 84359680m1 a 10574656 a h 1 a w w a 4096400 a 587318 4 (7) (7c) (8a) m e 17 h 01 w 4913 w 15840 4913 3760 w a (7a) 1034985600 a h 351895104h
5131787 h 1 8647703 w 114951 w 1 h C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 47496958 a. h w 1091599 a 4 751689 30509730 a 568719360 m a 031808 w 7776577 w a 4 a 3 1458866 w a 3013108 a 3, K B 587318 w 1 h a ( h ) 84359680m1 a 3013108 a 7776577 w 16 w (8) 963 4096400 h a 1 w (587318 (8c) 47496958 ) a h 1 w 3013108 a 3 10574656 a h 3 1 w 114951 a h 1 w 30509730 a a 1 w 568719360m a 10574656h 031808 a 3 w 7776577 w 1458866 w a 17 m e 3840(7 000e sy ) 3773 w a 4 w 1 h 1954150 a h 198933h 97913a 37184 h 1 h 1 a 3 3031808 wa 1555 w 367 a 66550 866653 w 8647703 w a 4 1091599 751689 a 430509730 a 1458866 w a 1 843596801 a 351895104 h 568719360 a m a a 1034985600 1 31 w 866653 a 1 w 8647703 w 84359680 m 1 a 5131787 h 41 w 1091599 a 751689 4 16000e sy 675196 10 4 w e sy w 17836 h 14116 h 703 h 1088 10 9 h 3 856 10 6 h 48 10 1 h 4 83 10 3 h 4 e 3 sy67 10 7 h e sy64 10 1 h 3 3 3 e sy e 4 sy e 4 sy 336 10 6 h e sy 4 e sy 1090 h 3 10 h 3 e 3 sy 9408 10 h 10 3 h e sy 67 10 9 h 10 9 h Á 6997 s A s h w 4 e 3 sy 358 10 6 h 31 w 1803 e sy 35 e 3 sy 88 4 e sy 41 w 164141 w 188160m 3031h 1451 h 5479 w 188160m 139944h 31 w 1451 h 41 w 1377 9 11475367000e sy18496 10 h 3 w sy306544000 h 1e3 3 w 9 364 10 h 3 w sy11995000 h 1e3 3 6 w 1e sy347 10 h 3 w 7 1144 10 h 3 w 1e sy 159936000 h 6 w 1e sy 4855 10 h w 1999000 h 6 w 1e sy856 10 h 9 w 1e sy 1144 10 h 4 1 3 w 1e sy 1e sy w 4 w (9a) (9) 1e sy 39984000 h 1e sy 1 81610 h sye 1e4 sy
964 C.C. Ferreira et al. / Engineering Structures 5 (003) 951 964 1 1088 10 h 3 w 1e4 sy h 4 9 w 1(1144 109 e 3 sy 816 10 1 e 4 sy) 4896 10 h w h 4 w 1(59976 106 e sy 139944000e sy)34986 h 1 w 1536 107 e sym 1075000e sym 6 571x10 h 6 w 1e sy 59976 10 h 4 w 1e sy 139944000 h 4 w 1e sy 1536 10 7 e sym 185 a a h 3 w 1(16368153 109 e 3 sy79968 10 6 e sy1088 10 1 10 6 e 4 sy18659000e sy) 174930 h w 13330000 w e sy 10750000e sym a h w 1( 51408 106 e sy 4896 10 9 e 3 sy 17998000e sy 09916)367000 a e sy3330000 a w 571 10 6a h w 1 1e sy11660 wa a h w 1(39984000e sy 6997) 1 w 3840(000e sy 7)1 (000e a 5479 w 188160m sy 7) 1e3 sy 139944 h 31 w 1 w 1451 h 4 34986 h 364 10 9 h 3 e 3 sy59976 10 6 h 4 e sy 1377 1 w 10 6 e sy 11995000e sy ) 1 w h 11999000e sy ) 1 w h 4 10 9 e 3 sy)185 a 16368a h 31 w h 3 (856 10 6 e sy (347 (816x10 1 e 4 sy139944000e sy 1144 1088 10 1a 1 39984000 a h e sy w 1 18659000 a h 3 571 10 6h a sy e w 1 4896 10 9a w 10 9a w 1 esy h 3 3 79968 10 6a h 3 1 e 3 sy h e sy1 w 3330000 a e w sy 17998000a w 1 e sy h h 3 e sy1 w e 4 sy w 153 367000 a e sy e sy (9c) 51408 10 6a h e sy 09916 a w 1 h 11660 a w 51408 106a w h 10750000e sy m] Reerences 6997 a h w 1 e sy 1536 10 7 e 3 sym [1] Yen JYR. Optimize irect esign o reinorce concrete columns with uniaxial loas. ACI Structural Journal 1990;May- June:47 51. [] Kangasunram S, Karihaloo BL. Minimum cost esign o reinorce concrete structures. Structural Optimization 1990;:173 84. [3] Aamu A, Karihaloo BL. Minimum cost reinorce concrete eams using continuum-type optimality criteria structures. Structural Optimization 1994;7:91 10. [4] Aamu A, Karihaloo BL. Minimum cost esign o R.C. T-eams using DCOC. n International Conerence on Computational Structures Technology, Athens, Greece, 1994;151 160. [5] Al-Salloum YA, Siiqi GH. Cost-optimum minimum esign o reinorce concrete eams. ACI Structural Journal 1994;Nov- Dec:647 55. [6] Eurocoe, Design o concrete structures, CEN; 1991. [7] American Concrete Institute, Builing coe requirements or structural concrete, ACI 318-95, Detroit; 1995. [8] Roriguez JA, Ochoa JDA. Biaxial interaction iagrams or short RC columns o any cross section. Journal Structural Engineering ASCE 1999;15(6):67 83. [9] Faitis A. Interaction suraces or o reinorce concrete sections in iaxial ening. Journal Structural Engineering ASCE 001;17(7):840 6. [10] Bonet JL, Miguel PF, Romero ML, Fernanez MAA. A moiie algorithm or reinorce concrete cross sections integration. In: Proceeings o VI Int. Con. Computational Structures Technology. Scotlan: Civil-Comp Press; 00. [11] CEB/FIP, Manual on ening an compression, Committee Euro- International u Beton Construction Press, ulletin no.141; 198. [1] Ferreira CC, Melão Barros MHF, Barros AFM. Optimização a Armaura e Secções e Betão Armao em T so Flexão Simples, VII Congresso e Mecânica Aplicaa e Computacional, Aril 003, Univ. Évora, Portugal.