optimal design of reinforced concrete t-sections in … design of reinforced concrete t-sections in...

Engineering Structures 25 (2003) 951–964www.elsevier.com/locate/engstruct

Optimal design of reinforced concrete T-sections in bending

C.C. Ferreiraa,∗, M.H.F.M. Barrosa, A.F.M. Barrosb

a Department of Civil Engineering, Faculty of Sciences and Technology, University of Coimbra, Polo II, 3030 Coimbra, Portugalb IDMEC/Instituto Superior Tecnico, Av. Rovisco Pais, 1000 Lisbon, Portugal

Received 9 August 2002; received in revised form 3 February 2003; accepted 4 February 2003

Abstract

The optimisation of the steel area and the steel localisation in a T-beam under bending is carried out in the present work. Theexpressions giving the equilibrium of a singly or doubly reinforced T-section in the different stages defined by the non-linearbehaviour of steel and concrete are derived ones. The ultimate material behaviour is defined according to the design codes suchas EC2 and Model Code 1990. The purpose of this work is to obtain the analytical optimal design of the reinforcement of a T-section, in terms of the ultimate design. In the last section the expressions developed are applied to examples and design abacusare delivered. A comparison is made with current practice method as indicated in CEB. 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Reinforced concrete; Double reinforcement; T-sections; Optimisation of the reinforcement; Design abacus

1. Introduction

Methods based on optimality criteria can be appliedto reinforced concrete structures in order to obtain theminimum cost design. The costs to be minimised aregenerally divided into those of concrete, steel andformwork. This means that they include one or more ofthe following variables: the geometry of the section ofbeams and columns; the area and topology of the steelreinforcement. Methods based on this kind of analysisare found in references[1–5]. Nevertheless, these optim-isations consider a linear elastic analysis of the globalstructure. This type of linear analysis is relevant for theserviceability limit state design and it is important forthe definitive dimensions of the sections. In terms of theultimate limit design of concrete structures, this analysisis not correct due to the non-linear behaviour ofreinforced concrete.

Considering design variables such as section dimen-sions, steel area and topology at the same time, the non-linear optimisation of reinforced concrete structures is avery complex and yet to be solved problem. Optimising

∗ Corresponding author. Tel.:+351-239-797-100; fax:+351-239-797-123.

E-mail address: [email protected] (C.C. Ferreira).

0141-0296/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0141-0296(03)00039-7

the dimensions of the sections within serviceability con-ditions and then optimising the steel area and locationin the ultimate limit design within a non-linear analysis,appears to be a good solution. As a matter of fact, thedesign codes[6,7] suggest that non-linear analysis canbe replaced by linear analysis with redistribution of themoments and the design of the reinforced concrete mem-bers being made for the critical sections, where the bend-ing moment and shear attain their maximum value. Theoptimisation of the reinforcement in a section isdeveloped within this context.

The optimal design of the critical sections is knownonly for rectangular sections. For other geometries,research has been made in terms of the biaxial interac-tion diagrams[8–10]. These diagrams attempt to makean optimisation by a trial and error procedure. Theimportance of the development of the optimal design ofT-sections is due to the fact that it is currently a fre-quently used section in common structures. Another rel-evant aspect is that the methodology used can beextended to other sections and included in the computercodes with minimum programming.

The design variables considered in the optimisation ofthe reinforced beam with a T-section are the steel areaand the steel localisation, either in the tension or in thecompression zones. The equilibrium equations of areinforced concrete T-section under Limit State Design,

952 C.C. Ferreira et al. / Engineering Structures 25 (2003) 951–964

as defined by the non-linear behaviour of the concreteand the steel, are developed for this purpose.

The examples presented consist of the optimisation ofthe reinforcement of a T-section for different geometricdefinitions. The results are compared with the designobtained in current practice methods such as the oneindicated in CEB [11].

2. Ultimate design of reinforced concrete sectionunder bending

2.1. Geometry of the T-section

The T-section geometry of the reinforced concretestructure is defined by the following parameters, asshown in Fig. 1: lower steel area in the tensile zone, As;upper steel area near the compression zone, A’ s; theflange depth, hf ; the effective flange width, b; the webwidth, bw; the distance from the centroid of thereinforcement to opposite face of the section, d; the con-crete cover, a.

In the present work, the design variable in the optimis-ation process is the ratio A’ s /As, since, for a given totalarea of reinforcement As + A’ s, the objective function isto maximise the bending moment. As a matter of fact,for a given bending moment, the global area ofreinforcement is only a function of the ratio A’ s /As,which, for this reason, is the only design variable interms of the limit state design of the section. The coverof steel is not considered as a design variable becauseit is fixed, in each case, by durability conditions. Theequations are developed as a function of the non-dimen-sional variables a/d, bw/b and hf /d.

2.2. Constitutive laws

The ultimate design of reinforced concrete sectionsunder bending moment is defined in terms of non-linearconstitutive laws of the concrete and the steel. Thestress–strain equation for concrete used in the designcodes is defined by a parabola followed by a constant

Fig. 1. Geometry of the doubly reinforced T-section.

Fig. 2. Design stress–strain diagrams.

value, usually called parabola–rectangle law and rep-resented in Fig. 2(a). The maximum stress is equal to0.85fcd , where fcd is the design strength of the concreteunder compression. It is considered that concrete in ulti-mate design does not stand for tensile stresses. The equ-ation for the parabola is function of the concrete defor-mation ec, as follows:

sc � 850fcd(ec�250e2c)for ec�0.002 (1)

The design stress–strain law for steel is elastic-per-fectly plastic with a maximum value fsyd , which is thedesign value for the strength of steel, equal in tensionand compression. This law is represented in Fig. 2(b).The steel has initial elastic behaviour with the elasticitymodulus Es. The elastic domain is valid until themaximum design stress fsyd is reached, corresponding tothe maximum elastic strain esyd.

2.3. Rupture conditions

The limit design means that rupture is consideredwhen the strains in the section reach maximum values,which are different for concrete and steel. Concrete isconsidered to crush at ec = 0.35%. The steel ruptureunder compression is limited to 0.35% due to the crushof the concrete, and to 1% under tension. These con-ditions are represented in Fig. 3, where the section beforedeformation is represented by the vertical line and after

Fig. 3. Rupture of the section.

953C.C. Ferreira et al. / Engineering Structures 25 (2003) 951–964

Fig. 4. (a) T-section; (b) stresses in concrete; (c) resulting load in concrete.

Table 1Values of m1

0 � e’ s �fsyd

Es

e’ s�fsyd

Es

m1 1Ese’ s

fsyd

deformation by the inclined line. In both ruptures, by thesteel [Fig. 3(a) and by the concrete [Fig. 3(b)], the neu-tral axis is located at the distance ad from the upperzone, and a is given by:

ecec�es

(2)

The parabola–rectangle transition is located at the dis-tance a‘d from the upper zone and corresponds to theconcrete strain ec = 0.2%.

In the Fig. 3, the strain in upper reinforcement, e’ s, isalso indicated, and is computed by:

e’ s � (ec�es)�a�ad� (3)

The stresses in the concrete under compression, corre-sponding to the strains represented in Fig. 3, are indi-cated in Fig. 4(b). The stresses upon the steel in theupper reinforcement of the compression zone are termedm1 and are detailed in Table 1. The stresses upon thesteel in the lower reinforcement of the tensile zone, aretermed m2 and are presented in Table 2.

Table 2Values of m2

0 � �es�fsyd

Es

fsyd

Es

� �es � 1%

m2 � 1Esesfsyd

2.4. Resulting load in the concrete

The resulting load in concrete, Fcd , is obtained by theintegration of the stresses in the concrete, that is thesecond degree Eq. (1), called parabola, and the constantvalue, called rectangle. These stresses may be appliedeither in the flange or in the web. Considering the vari-ation of neutral axis ad and the flange depth hf, the rup-ture conditions are also variable. The consideration ofall possibilities originates the several cases described inFig. 5.

The value of the resulting load, in each case i, isdenoted by Fi

cd and the distance to the upper edge of thesection by X i . The load in concrete can be written innon-dimensional form, Fi

cdred, defined by the following

expression:

Ficdred

�Fi

cd

fcdb d. (4)

The value of all Ficdred

and X i are given in Appendix A.There are VIII cases to be considered. In cases I and

II rupture occurs in steel and concrete which has a stress

Fig. 5. Different cases considered in the computation of theresulting load.


ec lower than 0.2% (see Fig. 5). In cases III–V ruptureis also in the steel but concrete strain ec is larger than0.2%. In cases VI–VIII rupture is in concrete with strainequal to 0.35%. The cases can also be grouped in termsof the flange depth. Cases I, III, and VI can be calledthe cases for high flange depth. The neutral axis is inthe flange and the section is equivalent to a rectangularsection in terms of the rupture condition. Cases II, V andVIII can be related by the fact that the flange depth issmall. Cases IV and VII are intermediate to the othersand correspond to medium depth flange.

2.5. Equilibrium equations in the section

The equilibrium equations of all the loads acting onthe section, when zero axial force is applied and a bend-ing moment is imposed, establishes the expressions (5)and (8). The first one is:

Ficdred

�A’ s

Asw m1 � w m2 � 0 (5)

where m1 and m2 were previously defined (Tables 1 and2), w is the percentile of the lower reinforcement, calcu-lated, through:

w �Asfsyd

bd fcd

. (6)

Considering a bending moment Msd applied to the sec-tion and the reduced bending moment as:

m �Msd

d2b fcd, (7)

the second equilibrium condition is written as:

Ficdred� 1 �

Xi

d � �A’ s

Asw�1�

ad� m1�m � 0 (8)

It must be remarked that there are as many equations asthe number of cases denoted in Fig. 5.

3. Minimum area of the reinforcement

3.1. Objective function

The optimisation problems concern the achievementof the best solution for a given objective functionsatisfying certain conditions.

Currently, the design is made for the most stressedsection that delimits the cross section dimensions andthe amount of steel. Optimisation of the steel area, ina predefined cross section, is similar to computing theminimum rate of steel. In practice, the computation ofdifferent rates with different locations, using designtables, approaches the optimal solution by trial. For thedesigner, the purpose is to establish the expression giv-

ing the maximum resistant bending moment, function ofthe area and location of the reinforcement with therelation A’ s/As.

In the present problem the system of Eqs (5) and (8)are the equilibrium equations of the section. Substitutinginto these expressions the definitions of Fi

cdred, Xi, m1 and

m2, they become perfectly defined in terms of the vari-

ables a,A’

s

As,hf

d,ad

,bbw

,fsyd

fcdand can be written respectively in

the form:

w � g�a,A’

s

As,hf

d,ad

,bbw

,fsyd

fcd� (9)

m � f �a,A’

s

As,hf

d,ad

,bbw

,fsyd

fcd� (10)

In practical terms, the purpose is to maximise the bend-ing moment m, Eq. (10), with the constraint defined byEq. (9).

3.2. Design variables

The variables defining the position of the steel in thesection are used in the optimisation process in thepresent work. This is the ratio of steel areas A’

s /As. Thetotal area of reinforcement is given by the sum A’

s andAs and is considered in the constrain Eq. (9). In non-dimensional terms, it is defined by wt, that is the percen-tile reinforcement given by:

wt �As � A’

s

bdfsyd

fcd

� �1 �A’

s

As�w (11)

This total reinforcement can also be written in functionof all variables, by using Eq. (9):

wt � �1 �A’

s

As� g�a,

A’s

As

,hf

d,ad,bbw

,fsyd

fcd� (12)

The argumentsad

andfsyd

fcd

in Eqs (9) and (10) are not

considered design variables sincead

is usually imposed

by durability and other construction requirements. The

strength ratiofsyd

fcd

is a known value after the choice of

the materials. The geometry parametershf

dand

bbw

can

be used as design variables in the case of shape optimis-ation of the section.

After eliminating the described constant parameters,Eqs (12) and (10) become respectively:

wt � �1 �A’

s

As� g�a,

A’s

As� (13)


m � f �a,A’

s

As� (14)

Choosing m as the objective function and wt as animposed variable, Eq. (13) can be solved in terms ofa, that is:

a � h �A’s

As� (15)

The value a is turned into the objective function, Eq.(14), that becomes:

m � f �h�A’s

As�,

A’s

As� (16)

This equation shows that, in the present work, there isonly one design variable, A’

s /As, but the objective func-tion is defined as a piecewise function, giving as manydifferent expressions as the cases of Fig. 5. As the con-

stant characteristics of the arguments are accepted,ad,

bbw

,hf

dand

fsyd

fcd

, the analytical optimal solution is always

written in terms of those arguments.

3.3. Optimal neutral axis depth

In the present work, the choice of the formula for themathematical optimisation problem does not allow thecalculus of the sensibilities due to the characteristic dis-continuity of the objective function. The optimal solutionis obtained by considering an increment of the designvariable A’ s/As, that is dA’ s/As, giving the following valueof the objective function:

m∗ � f �h�A’s

As�

dA’s

As�,

A’s

As�

dA’s

As� (17)

The function has a stationary value since a maximum ora minimum point exists and Eqs (16) and (17) shouldbe equal when dA’ s/As approaches zero, such as:

f�a,A’

s

As� � lim

dA’s

As→0

f �h�A’s

As

�dA’

s

As�,

A’s

As

�dA’

s

As� (18)

This equation is solved for each of the cases in Fig. 5,giving for high flange depth (cases I, III and VI) theoptimal value of neutral axis depth, a∗ , given by:

a∗ �119198�1 �

ad� (19)

With this optimal value, the minimum flange depthavailable for these cases I, III and VI can be estab-lished by:

hf

d�a∗ �

119198�1 �

ad� (20)

The optimal solution obtained from Eq. (18), for thecases of small flange depth (cases II, V and VIII), is thesame value a∗ of expression (19). The limit value of the

flange depth,hf

d, is obtained by imposing the limits of

these cases, as shown in Fig. 6, and becomes:

hf

d�

1766�1 �

ad� (21)

For the intermediate values of the flange depth, corre-sponding to cases IV and VII, the optimal solution cannot be obtained analytically through Eq. (18) in functionof the parameters. The computation of the optimal neu-tral axis depth using Eq. (18) is possible only when dis-crete values of the different parameters are imposed. Forthis reason an approximate solution for cases IV and VIIis proposed in section 3.6.

3.4. Ductility limitation

The position of neutral axis affects the deformation ofthe steel, that can be in the elastic or in the plasticdomain. In the design of reinforced concrete, it isimportant to guarantee a certain ductility. The conditionfor the plasticity of the lower steel As, defined by a limitvalue of a, termed ap, depends on the designed yieldstrain �syd, that varies for each class of steel. This limitfor the plastic zone of the steel ap is the following:

ap �7

7 � 2000esyd

. (22)

The plastic limits ap Eq. (22), defined for three classesof steel, namely S235, S400 and S500, are representedin Fig. 7. The optimal value a∗ Eq. (19), is also rep-resented in Fig. 7. Since the condition of applicabilityof a∗ is the yielding of steel As, as imposed in cases I–VIII, the corresponding equation is only valid when itsatisfies the following inequality:

a∗�ap (23)

Fig. 6. Limit zone forhf

din case VIII.


Fig. 7. Optimal neutral axis depth: a∗ or ap.

As a result, for a given a/d, the optimum value for ais either a∗ or ap whichever the smaller value might be.

3.5. Optimal solution and design

The optimal solution of the design variable (A’s /As)∗

is obtained by solving Eq. (15) in terms of A’ s/As andsubstituting in it a∗ or ap, that is:

�A’s

As�∗

� h�1(a∗ or ap) (24)

For a small bending moment there is no reinforcementin the compression zone, meaning that A’ s = 0. The limitvalue of the bending moment can be termed as economicmoment me and is obtained by replacing a∗ or ap andA’ s /As =0 into Eq. (14), which becomes:

me � f(a∗ or ap) (25)

This value is relevant for the practical design ofreinforced concrete sections as well as for the optimalarea of reinforcement w∗

t . The optimal area of reinforce-ment is obtained by replacing the values a∗ and (A’

s /As)∗ in Eq. (13), that is:

w∗t � �1 � �A’

s

As�∗� g�a∗ or ap,�A’

s

As�∗� (26)

If only the area As is considered, the optimum percentilelower reinforcement w∗, Eq. (9), becomes:

w∗ � g�a∗ or ap,�A’s

As�∗� (27)

3.6. Approximate solution in cases IV and VII

For the cases IV and VII the optimal solution is onlyobtained if values for hf /d and bw/b are prescribed, pre-venting a generally analytical solution. In practicalterms, it may be relevant to have design expressionssince they have small errors. For this reason, the

approximate value considered in cases IV and VII is a∗,Eq. (19), satisfying the following condition:

aapr �119198�1 �

ad��ap (28)

In order to determine the error introduced by theapproximate neutral axis depth aapr in the optimal designof the section, the computation of optimal neutral axisdepth a∗ was reached, according to section 3.3, for arepresentative number of prescribed values of hf/d andbw/b and a/d. Replacing these optimal a∗ successivelyin expressions (14) and (13), the corresponding percen-tile of reinforcement wreal was found. The values of thepercentile of reinforcement wapro, obtained when aapr isconsidered in expressions (14) and (13), were also com-puted. The error introduced in the optimal design wasevaluated by the following expression:

wapro�wreal

wreal� 100 (29)

Table 3 gives the values of this definition of error forb /bw = 2, 3, 4, 5, 6, 7, 8, 9, 10 , with a /d = 0.05 andhf/d varying from 0.28 to 0.37.

Table 4 exemplifies the errors that can be obtained bythe variation of parameter a/d. In Table 4, the values ofexpression (23) are plotted for a /d = 0.04, 0.05, 0.06,0.07, 0.08, 0.09, 0.1, with b /bw = 4 and hf/d varyingfrom 0.27 to 0.41. The blank cells in Tables 3 and 4correspond to the optimal value given by ap.

Other calculations, similar to the ones related abovewere made, namely fixing the parameter b/bw and com-puting the error with variable hf/d and a/d, as made inTable 4. The results are missed out but they are alwaysless than 0.14%. This is the reason why expression (22)was considered a good approximation to the optimal sol-ution.

4. Examples

The optimal neutral axis depth and the correspondingexpressions of the design, obtained according to section3.5, was found for a general T-section. The expressionsobtained in these computations are developed in Appen-dix B and they are presented in Table 5, with theirrespective limitations.

4.1. Abacus for optimal design

The developed expressions are applied in a T-sectionfor a /d = 0.1 with variable geometric definitions ofb/bw:b/bw equal 1/10, 1/8, 1/6, 1/4 and 1/2. The con-sidered steel classes are S400 or S500. The values of theeconomic reduced bending moment me are plotted in Fig.8. Fig. 8 emphasises the limits of expressions (6a), (7a)


Table 3Error for different values of hf/d and b/bw

b/bwhf/d 2 3 4 5 6 7 8 9 10

0.28 0.000001 0.000003 0.000005 0.000007 0.000009 0.000011 0.000013 0.000015 0.0000170.29 0.000021 0.000055 0.000089 0.000123 0.000156 0.000189 0.000221 0.000225 0.0002800.30 0.000110 0.000278 0.000449 0.000616 0.000775 0.000927 0.001072 0.001210 0.0013410.31 0.000346 0.000865 0.001385 0.001881 0.002347 0.002786 0.003198 0.003585 0.0039500.32 0.000832 0.002059 0.003265 0.004396 0.005445 0.006416 0.007317 0.008155 0.0089370.33 0.001689 0.004133 0.006496 0.008675 0.010668 0.012492 0.014165 0.015707 0.0171330.34 0.003041 0.007374 0.011489 0.015228 0.018604 0.021660 0.024439 0.026977 0.0293090.35 0.005020 0.012056 0.018629 0.024521 0.029782 0.034498 0.038751 0.042609 0.0461290.36 0.007747 0.018428 0.028256 0.036956 0.044645 0.051479 0.057597 — —0.37 0.011325 0.026694 0.040635 0.052836 0.063520 — — — —0.38 0.015834 0.036997 0.055939 0.072348 —0.39 0.021322 0.049409 0.074241 — —0.40 0.027800 0.063917 0.095489 — —0.41 0.035235 0.080418 — — —

Table 4Error for different values of hf/d and a/d

a/dhf/d 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.27 0.000002 0.000000 0.000000 — — — —0.28 0.000056 0.000029 0.000014 0.000010 0.000000 0.000000 -0.0000000.29 0.000335 0.000226 0.000146 0.000089 0.000051 0.000026 0.0000120.30 0.001121 0.000846 0.000624 0.000449 0.000313 0.000211 0.0001350.31 0.002770 0.002230 0.001771 0.001385 0.001065 0.000802 0.0005910.32 0.005680 0.004768 0.003966 0.003265 0.002589 0.002139 0.0016970.33 0.010255 0.008868 0.007618 0.006496 0.005497 0.004612 0.0038350.34 0.016752 0.014923 0.013130 0.011488 0.009995 0.008641 0.0074210.35 0.025870 0.023285 0.020873 0.018629 0.016552 0.014636 0.0128760.36 0.037498 0.034241 0.031160 0.028256 0.025529 0.022975 0.0205940.37 0.051922 0.047990 0.044225 0.040634 0.037220 0.033983 —0.38 0.069199 0.064629 0.060207 0.055940 0.051836 — —0.39 0.089266 0.084143 0.079130 0.074241 0.069488 — —0.40 0.111929 0.106387 0.100899 0.095488 — — —0.41 0.136864 0.131085 0.125290 — — — —

Table 5Resume of equations in Appendix B, for the different cases

Case of Fig. 5 II, V, VII IV, VII I, III, VI

Optimal sol. a∗ a∗ ap a∗

Equations (7a), (7b), (7c) (8a), (8b), (8c) (9a), (9b), (9c) (6a), (6b), (6c)Limit values

0 � hf /d�1766�1 +

ad� 17

66�1 +ad� � hf /d�a∗ 17

66�1 +ad� � hf /d�ap hf /d

1766�1 +

ad�

and (8a), given in Appendix B, depending on the depthof neutral axis that changes with the values of hf /d. Theother involved variables, such as a/d; bw /b and steelclass lead to different curves.

Fig. 9(a) is an abacus giving the optimal total area ofreinforcement (A’ s + As) as α function of the appliedreduced bending moment m, and the ratio hf/d, for a /d= 0.1 and b /bw = 10. The distribution of this area of

steel between the upper and lower faces, is obtainedfrom Fig. 9(b)). Figs. 10–13, have the same interpret-ation of Fig. 9, and correspond to the different values ofthe geometric parameters and a/d=0.1. These figures areresumed in Table 6 and cover some current sections per-mitting the direct calculation from the figures oremploying linear interpolation. Design abacuses forother values of a/d are in [12].


Fig. 8. Economic me for a /d = 0.1.

Fig. 9. (a) Increasing in µ with ωt, for b/bw=10 (b) µ variation with A’ s/A s , for b/bw=10.

Fig. 10. (a) Increase in m with w t, for b /bw = 8; (b) m variation with A’ s/A s, for b /bw = 8.

4.2. Numerical results

In this section a comparison is made between thedesign obtained in the present formulation and theapproximated solutions given by CEB [11]. The percen-tile of total reinforcement, the amount of upper steel areaand the errors in percentile are detailed in Table 7.

The percentile of error is calculated by:

Error �Re f.[11] � mod el

mod el� 100

As can be observed in Table 7, the total area ofreinforcement has a maximum difference of 2.05% whencompared with the CEB solution. Generally the differ-ences in the distribution A’ s/A s. is always larger with amaximum of 10.4%. Although the differences for therequired reinforcement do not exceed 10.4% in the stud-ied case, it seems that in repeated structures, such as inprefabrication, this difference can be relevant in econ-


Fig. 11. (a) Increase in m with w t, for b /bw = 6; (b) m variation with A’ s/A s, for b /bw = 6.

Fig. 12. (a) Increase in m with wt, for b /bw = 4; (b) m variation with A’ s/A s, for b /bw = 4.

Fig. 13. (a) Increase in m with wt, for b /bw = 2; (b) m variation with A’ s/A s, for b/bw=2.


Table 6Resume of geometric definitions in Figs. 9–13

S400 S500 b/bw

10 8 6 4 2

wt (9a) (10b) (11a) (12a) (13a)As /A s (9b) (10b) (11b) (12b) (13b)

omical terms. It can be emphasised that L-, I- and T-sections are commonly used in buildings.

Since the solution obtained in this work is an exactone, it can be noted that the errors in the CEB solutionhave consequences in terms of: not satisfying the equi-librium Eqs (5) and (8); deviation of the neutral axis anderror in the rotation of the section at rupture.

5. Conclusion

In the present work a model is developed for theoptimisation of reinforced concrete T-sections, in ulti-mate design, under bending moment. In the model, thenon-linear behaviour of concrete, with parabola–rec-tangle law for compression and no tensile strength, andelasto-plastic behaviour for steel are considered.

In this work, the equation of maximum value of thebending moment considering only single reinforcement,for T-section is obtained. The equations for the optimalarea of reinforcement and corresponding localisation arealso presented. These equations are expressed in termsof the geometry and mechanical characteristics as indi-cated in Appendices A and B. The equations are plottedfor current T-section geometric characteristics in Figs.9–13, for S400 and S500 steel.

The comparison between the results obtained with theoptimal design and current practice methods (CEBsolution) is made. The advantages of the present modelare: the correct solution is economic when compared tocurrent practice solutions; the use of non-linear behav-iour of the materials; the development of a methodologythat can be extended to other sections; the establishmentof the optimal design equations that can be implementedin computer codes.

Acknowledgements

This work was performed under the financial supportof the Portuguese Minister of Science and Technologyby Programa Operacional do Quadro Comunitario deApoio (POCTI) and by FEDER grantPOCTI/EMC/12126/1998, Fase II.

Appendix A

The resulting reduced compressive force Ficdred

in theconcrete and the location Xi of the force Fi

cd, indicatedin Fig. 4, are listed bellow, for the different cases (seeFig. 5):

FIcdred

�1712a2

(a�1)2(3�8a); XI �149a�48a�3

a d

FIIcdred

�1712

1(a�1)2��21

hf

da2�6

hf

da�18�hf

d�2

a � 3�hf

d�2

� 5bw

b �hf

d�3��bw

b�1��8

bw

ba3 � 3

bw

ba2�

XII �17d

48FIIcdred

(a�1)2 ��42a2 � 48hf

da � 12a�8

hf

d

�15�hf

d�2��hf

d�2�1�bw

b � � 4bw

ba3�9

bw

ba4�

FIIIcdred

�6875a�

17300

; XIII �d20

1 � 171a2�22a16a�1

FIVcdred

�17

300(a�1)2 �216a3 � 18a�1

� a2�75bw

b�108� � �bw

b�1�hf

d(525a2�150a)

� �75�450a � 125hf

d��bw

b�1��hf

d�2�XIV �

17 dFIV

cdred6000(a�1)2 �(36a2�12a � 1)2

� 1875�hf

d�4�bw

b�1� � � 5250�bw

b�1��hf

d�2

a2

� �bw

b�1��hf

d�2�1000hf

d(1�6a)�1500a�

� 500bw

ba3�1125

bw

ba4�


Tab

le7

Des

ign

ofa

T-s

ectio

nfo

rm

sd=

0.5,

1.0,

2.0

and

3.0

h f/d

0.05

0.40

b/b w

210

210

a/d

0.05

0.1

0.15

0.05

0.1

0.15

0.05

0.1

0.15

0.05

0.1

0.15

msd

=0.5

wt

Mod

el0.

9103

0.94

730.

9881

0.98

841.

0387

1.09

480.

8171

0.84

050.

8661

0.82

070.

8464

0.87

52R

ef.

[8]

0.91

10.

949

0.99

10.

988

1.03

91.

095

0.81

70.

841

0.86

90.

821

0.84

70.

877

%E

rror

0.07

0.18

0.29

�0.

040.

030.

02�

0.01

0.06

0.33

0.04

0.07

0.21

Mod

el0.

5850

0.58

410.

5846

0.84

730.

8508

0.85

460.

3597

0.35

960.

3608

0.40

790.

4165

0.42

66A

’ s

As

Ref

.[8]

0.59

30.

606

0.61

90.

850

0.85

50.

862

0.36

60.

397

0.39

30.

411

0.42

40.

438

%E

rror

1.37

3.75

5.88

0.32

0.49

0.86

1.75

10.4

08.

920.

761.

802.

67m

sd=

1.0

mod

el1.

9629

2.05

842.

1646

2.04

102.

1498

2.27

131.

8697

1.95

162.

0426

1.87

331.

9575

2.05

16R

ef.

[8]

1.96

42.

060

2.16

72.

041

2.15

02.

272

1.87

01.

952

2.04

51.

874

1.95

92.

054

%E

rror

0.06

0.08

0.11

0.00

0.01

0.03

1.60

2.05

0.12

0.04

0.08

0.12

Mod

el0.

7834

0.78

440.

7862

0.92

300.

9250

0.92

720.

6587

0.66

270.

6678

0.68

880.

6976

0.70

73A

’ s

As

Ref

.[8

]0.

787

0.79

60.

806

0.92

40.

927

0.93

00.

668

0.67

50.

687

0.69

00.

702

0.71

3%

Err

or0.

461.

482.

520.

110.

220.

301.

411.

862.

880.

170.

630.

81m

sd=

2.0

Mod

el4.

0682

4.28

064.

5175

4.14

634.

3720

4.62

423.

9750

4.17

384.

3955

3.97

864.

1798

4.40

46R

ef.

[8]

4.06

94.

282

4.52

04.

146

4.37

24.

625

3.97

54.

175

4.39

73.

979

4.18

14.

407

%E

rror

0.02

0.03

0.06

�0.

010.

000.

020.

000.

030.

030.

010.

030.

05M

odel

0.88

930.

8902

0.89

150.

9614

0.96

240.

9636

0.82

350.

8267

0.83

060.

8403

0.84

600.

8521

A’ s

As

Ref

.[8

]0.

892

0.89

60.

902

0.96

20.

964

0.96

60.

826

0.83

40.

841

0.84

10.

848

0.85

6%

Err

or0.

300.

651.

180.

060.

170.

250.

300.

881.

250.

080.

240.

46m

sd=

3.0

Mod

el6.

1734

6.50

296.

8705

6.25

156.

5942

6.97

716.

0803

6.39

616.

7484

6.08

386.

4020

6.75

75R

ef.

[8]

6.17

56.

504

6.87

36.

252

6.59

46.

977

6.08

06.

397

6.75

16.

084

6.40

36.

759

%E

rror

0.03

0.02

0.04

0.01

0.00

0.00

0.00

0.01

0.04

0.00

0.02

0.02

Mod

el0.

9256

0.92

630.

9273

0.97

420.

9749

0.95

570.

8810

0.88

340.

8863

0.89

260.

8967

0.90

10A

’ s

As

Ref

.[8]

0.92

70.

931

0.93

40.

974

0.97

60.

977

0.88

30.

888

0.89

40.

894

0.89

80.

903

%E

rror

0.15

0.51

0.72

�0.

020.

112.

230.

230.

520.

870.

160.

140.

22


FVcdred

�1720

hf

d�

6875

bw

ba�

1720

hf

dbw

b�

17300

bw

b

XV �d20

150�bw

b�1��hf

d�2

�171a2 � 22a�1

15�bw

b�1�hf

d�16

bw

ba �

bw

b

FVIcdred

�289420a; XVI

cdred�

99238a d

FVIIcdred

�17

6720a2�245bw

ba3�147

hf

da2�bw

b�1�

�441�hf

d�2

a�bw

b�1� � 343�hf

d�3�bw

b�1� � 27a3�

XVIII �3d28

1029bw

ba4�686�hf

d�2

a2�bw

b�1��2744�hf

d�3

a�bw

b�1� � 2401�hf

d�4�bw

b�1� � 27a4

245bw

ba3�147

hf

da2�bw

b�1��441�hf

d�2

a�bw

b�1� � 343�hf

d�3�bw

b�1� � 27a3

FVIIIcdred

�1720

hf

d�

289420

bw

ba�

1720

hf

dbw

b

XVIII �3d14

�49�hf

d�2

� 49bw

b �hf

d�2

�33bw

ba2

�21hf

d�17

bw

ba � 21

bw

bhf

d

Appendix B

The optimal design of the sections referred to in sec-tion 3.5, with the limits referred in Table 5, can be rep-resented by the following expressions:

me �4913

47520�1 �ad��3�

ad�. (6a)

�A’s

As�∗

(6b)

�

14739 � 9826ad

�4913�ad�2

�47520m

�4913 � 9826ad

� 14739�ad�2

�47520m

,

w∗ �

�4913 � 9826ad

� 14739�ad�2

�47520m

47520(ad

�1)(6c)

me�1720

hf

d�1�bw

b � �4913

15840bw

b�

491323760

bw

bad

(7a)

�1740�hf

d�2�1�bw

b ��491347520

bw

b �ad�2

�A’ s

As�∗

�

40392hf

d�1�bw

b ��20196�hf

d�2�1�bw

b � � 14739bw

b� 9826

bw

bad

�4913bw

b �ad�2

�47520m

�20196�hf

d�2�1�bw

b ��4913bw

b� 9826

bw

bad

� 14739bw

b �ad�2

� 40392hf

dad�1�

bw

b ��47520m(7b)

w∗ �1

47520�ad

�1� ��40392hf

dad�20196�hf

d�2��1

�bw

b ��4913bw

b �1�2ad� � 14739

bw

b �ad�2

(7c)

�47520m�me�

�1

284359680�1 �ad�2��105746256

hf

d�1�bw

b

�bw

b �ad�2

� �ad�2��77276577

bw

b�32013108

ad

� � ( 351895104ad

� 1034985600 )�hf

d�3�1

�bw

b �� 512317872�hf

d�4�1�bw

b � � 751689�ad�4

��10921599 �422096400�hf

d�2ad�1�

bw

b � (8a)

� 52873128�hf

d�2�ad�2�1�

bw

b ��30509730�ad�2

� 211492512hf

dad�1�

bw

b ��474969528�hf

d�2�1

�bw

b � � 28647703�ad�4bw

b� 145886622�a

d�2bw

b

�� 203182028bw

bad

� 8666532�ad�3�bw

b�1��

( A’ s /As)∗ � KA /KB, with KA � 8666532ad�bw

b

�1� � 284359680m�1 � �ad�2�

�105746256adhf

d�1 � �ad�2

�bw

b�

bw

b �ad�2� �

� �422096400ad

� 52873128

� 1034985600adhf

d� 351895104

hf

d


� 512317872�hf

d�2

� 474969528�ad�2�.�hf

d�2�bw

b

�1� � 28647703bw

b�10921599�a

d�4

� 751689

� �211492512�bw

b�1�hf

d�30509730��a

d�2

�

� 568719360 mad

� 203182028bw

b �ad�3

� 77276577bw

b �ad�4

� 145886622bw

b �ad�2

(8b)

�32013108�ad�3

, KB � � 52873128�bw

b

�1�hf

d��ad�2

(hf

d� 2)�2� � 284359680m�1

� �ad�2��32013108

ad77276577

bw

b� � 216�bw

b

�1�hf

d�1954150ad

hf

d� 2198933

hf

d� 979132

ad

� 2371842�hf

d�3��203182028bw

bad��15552�bw

b

�1��hf

d�3�226227ad

� 66550� � 8666532�bw

b

�1��ad�3

� 28647703bw

b �ad�4

� 10921599 �

� 751689�ad�4

�30509730�ad�2

�145886622bw

b �ad�2

� 568719360adm

w∗

��1

284359680�1 �ad�2�a

d�1��1034985600

ad

� 351895104��hf

d�3�1�bw

b �� 8666532ad�1

�bw

b � � 28647703bw

b� 284359680 m�1

� �ad�2�� 512317872�hf

d�4�1�bw

b ��10921599�a

d�4

� 751689 �

� 422096400�hf

d�2ad�1�

bw

b � � (52873128 (8c)

�474969528�ad�2

)�hf

d�2�1�bw

b �� 32013108�ad�3

��105746256�ad�3hf

d�1�bw

b �� 211492512�a

d�2hf

d�1�bw

b ��30509730�ad�2

� 568719360mad

�� 105746256hf

dad�1�

bw

b ��203182028�a

d�3bw

b� 77276577

bw

b �ad�4

�145886622bw

b �ad�2�

me��17

3840(7 � 2000esyd)2��3773bw

b

�216000esyd�675�196 � 104bw

besyd � �bw

b

�1��4116hf

d� � 7203�hf

d�4

�17836�hf

d�3

�1088 � 109�hf

d�3

e3syd�672 � 107�hf

d�3

e2syd

� 2856 � 106�hf

d�2

e2syd�64 � 1012�hf

d�3

e4syd (9a)

� 48 � 1012�hf

d�4

e4syd � 336 � 106�hf

d�e2syd

� 8232 � 103�hf

d�4

esyd � 10290�hf

d�2

�18032

� 103�hf

d�3

esyd � 9408 � 103�hf

d�2

esyd � 2352

� 103�hf

d�esyd � 672 � 109�hf

d�4

e3syd � 288

� 109�hf

d�2

e3syd � 3528 � 106�hf

d�4

e2syd��A

´

s

As�∗

�

69972hf

d�bw

b�1��64141

bw

b� 188160m � 303212�hf

d�3�1�bw

b ��122451�hf

d�4�1�bw

b �52479

bw

b� 188160m � 139944�hf

d�3�1�bw

b ��122451�hf

d�4�1�bw

b � � 1377

(9b)

�

�11475�3672000esyd�18496 � 109�hf

d�3�bw

b�1�e3syd�306544000�hf

d�3�bw

b�1�esyd

�3264 � 109�hf

d�3�bw

b�1�e3syd�119952000�hf

d�3�bw

b�1�esyd�34272 � 106�hf

d�3�bw

b�1�e2syd

�11424 � 107�hf

d�3�bw

b�1�e2syd � 159936000�hf

d�2�bw

b�1�esyd � 48552 � 106�hf

d�2�bw

b�1�e2syd � 39984000

hf

d�bw

b�1�esyd

�19992000�hf

d�2�bw

b�1�esyd�2856∗106�hf

d�2�bw

b�1�e2syd � 11424 � 109�hf

d�4�bw

b�1�3

syde� 8161012�hf

d�4�bw

b�1�e4syd


�

�1088 � 1012�hf

d�3�bw

b�1�e4syd � �hf

d�4�bw

b�1�(11424 � 109e3syd � 816 � 1012e4syd) � 4896 � 109�hf

d�2�bw

b�1�e3syd

�hf

d�4�bw

b�1�(59976 � 106e2syd � 139944000esyd)�34986�hf

d�2�bw

b�1� � 1536 � 107e2sydm � 10752000esydm

5712x106�hf

d��bw

b�1�e2syd � 59976 � 106�hf

d�4�bw

b�1�e2syd � 139944000�hf

d�4�bw

b�1�esyd � 1536 � 107e2sydm

�12852ad

�ad�hf

d�3�bw

b�1�(�163268�15232 � 109e3syd�79968 � 106e2syd�1088 � 1012 � 106e4syd�186592000esyd)

174930�hf

d�2�bw

b�1��33320000�bw

b �esyd � 107520000esydm

�ad�hf

d�2�bw

b�1�( � 51408 � 106e2syd � 4896 � 109e3syd � 179928000esyd � 209916)�3672000

adesyd�33320000

ad�bw

b �esyd

�1

� 5712 � 106ad�hf

d��bw

b�1�e2syd�116620

bw

bad

�ad�hf

d��bw

b�1�(39984000esyd � 69972)

w∗ � �1

3840(2000esyd � 7)�1�ad�(2000esyd�7)

�52479bw

b� 188160m

� 139944�hf

d�3�1�bw

b � � �1�bw

b ��122451�hf

d�4

� 34986�hf

d�2

� 3264

� 109�hf

d�3

e3syd�59976 � 106�hf

d�4

e2syd� � 1377 � �1�bw

b ��hf

d�3

(34272

� 106e2syd � 119952000esyd) � �1�bw

b ��hf

d�2

(2856 � 106e2syd

� 119992000esyd) � �1�bw

b ��hf

d�4

(�816x1012e4syd�139944000esyd�11424

� 109e3syd)�12852ad

� 163268ad�hf

d�3�1�bw

b ��1088 � 1012ad�hf

d�3

e4syd�bw

b

�1� � 39984000ad�hf

d�esyd�bw

b�1� � 186592000

ad�hf

d�3

esyd�1�bw

b � (9c)

� 5712 � 106hf

dade2syd�bw

b�1� � 4896 � 109

ad�bw

b�1�e3syd�hf

d�2

�15232

� 109ad�bw

b�1�e3syd�hf

d�3

� 79968 � 106ad�hf

d�3

e2syd�1�bw

b ��33320000

adesyd

bw

b� 179928000

ad�bw

b�1�esyd�hf

d�2

�3672000adesyd

�51408 � 106ad�hf

d�2

e2syd � 209916ad�bw

b�1��hf

d�2

� 69972adhf

d�bw

b�1�

�116620adbw

b� 51408 � 106

ad�bw

b ��hf

d�2

e2syd � 1536 � 107e3sydm

� 107520000esydm]

References

[1] Yen JYR. Optimized direct design of reinforced concrete col-umns with uniaxial loads. ACI Structural Journal 1990;May-June:247–51.

[2] Kangasundram S, Karihaloo BL. Minimum cost design ofreinforced concrete structures. Structural Optimization1990;2:173–84.

[3] Adamu A, Karihaloo BL. Minimum cost reinforced concretebeams using continuum-type optimality criteria structures. Struc-tural Optimization 1994;7:91–102.

[4] Adamu A, Karihaloo BL. Minimum cost design of R.C. T-beamsusing DCOC. 2nd International Conference on ComputationalStructures Technology, Athens, Greece, 1994;151–160.

[5] Al-Salloum YA, Siddiqi GH. Cost-optimum minimum design ofreinforced concrete beams. ACI Structural Journal 1994;Nov-Dec:647–55.

[6] Eurocode 2, Design of concrete structures, CEN; 1991.[7] American Concrete Institute, Building code requirements for

structural concrete, ACI 318-95, Detroit; 1995.[8] Rodriguez JA, Ochoa JDA. Biaxial interaction diagrams for short

RC columns of any cross section. Journal Structural EngineeringASCE 1999;125(6):672–83.

[9] Fafitis A. Interaction surfaces for of reinforced concrete sectionsin biaxial bending. Journal Structural Engineering ASCE2001;127(7):840–6.

[10] Bonet JL, Miguel PF, Romero ML, Fernandez MAA. A modifiedalgorithm for reinforced concrete cross sections integration. In:Proceedings of VI Int. Conf. Computational Structures Tech-nology. Scotland: Civil-Comp Press; 2002.

[11] CEB/FIP, Manual on bending and compression, Committee Euro-International du Beton Construction Press, bulletin no.141; 1982.

[12] Ferreira CC, Melao Barros MHF, Barros AFM. Optimizacao daArmadura de Seccoes de Betao Armado em T sob FlexaoSimples, VII Congresso de Mecanica Aplicada e Computacional,Abril 2003, Univ. Evora, Portugal.

optimal design of reinforced concrete t-sections in … design of reinforced concrete t-sections in...

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