Com/~er~ d Sfrucrures Vol. 30. No. 4, 9~. IOCW-IOI 1. 1988 Printed in Great Britain.

0045-7949/M $3.00 + 0.00 0 1988 Pqmon Press plc

OPTIMUM DESIGN OF REINFORCED CONCRETE

SECTIONS

ANAND PRAKASH, S. K. AGARWALA and K. K. SINGH Civil Engineering Department, University of Roorkee, Roorkee, India

Abstract-In the ideal case, optimization should consider the structure as a whole and take into account its initial cost, maintenance cost and functional benefits. However, in most designs such an approach is too complicated to be of practical use. Hence optimization of individual structural components is commonly adopted. The basis of optimization is minimum weight or minimum cost. The former is better for high rise buildings in which the same component is repeated storey after storey. For low rise buildings the minimum cost is a better criterion for the optimum design of components.

The main factors to be considered are the costs of steel, concrete and shuttering. The problem is considerably simplified by neglecting the latter and treating the cost ratio of steel to concrete as a variable to obtain the optimum designs. The authors have adopted this approach using computers and relatively simple optimization techniques to obtain optimum design results for singly and doubly reinforced beams, T-beams and columns with eccentric loads. In the paper the authors present a brief review of the relevant literature along with the procedures adopted, results and conclusions.

NOTATION

width of section (subscript f denotes flange) cost of unit volume of concrete (subscript r denotes ratio of costs of unit volumes of steel and concrete) effective depth of a beam section (subscript s denotes reinforcement cover) total depth of section characteristic strength of concrete characteristic strength of steel design moment percentage of steel (subscript m for maximum per- centage permitted in an SRB) axial load in column maximum depth of neutral axis p/l00 (subscripts c and t denote compression rein- forcement and additional tensile reinforcement for a DRB)

INTRODUCTION

Ideally, optimization should consider the structure as a whole, accounting for its construction cost, mainte- nance cost and functional benefits. However, it is simpler to optimize on the basis of either the mini- mum weight or minimum cost of individual structural components. In reinforced concrete const~ction the former is limited to high rise construction only and so the latter is adopted here.

Friel [I] and Chow [2] have optimized singly rein- forced and T-sections for flexure using the Lagran- gian multiplier method. Misra [3] has optimized singly and doubly reinforced and T-sections, while Mebrahtu [4] has optimized beam column sections under the supervision of the authors. The paper is based on this and further work. Optimization has been carried out using computer programs based on the Lagrangian method, Simplex method or search techniques. All designs are by the limit state concept of IS: 456 IS] elaborated in SP: 16 [6]. Due to similar- ities in the concrete design codes of different countries

this information and the conclusions reached may be of general use. Limitation of space has resulted in the presentation of only sample results.

Several common grades of concrete (Ml& MZO. M25, M20) and steel (Fe250, Fe415, FeSOO) have been considered (numbers indicate characteristic strength in N/mm2). The effect of the cost of form- work is neglected. The range of the cost ratio of unit volumes of steel and concrete is varied from 50 to 100, as this covers the practical range.

Singly reinforced beam (SRB)

The cost function is:

Z = bdC[l + d,ld + (C, - I)],

subject to the design condition

M = 0.87f,p[l - l.00S~p/f,]bd2.

Optimi~tion has been carried out for p and d/b, which are the main variables.

Doubly reinforced beam (DRB)

The cost function is:

Z=bdC[l+~/d+(C~-I)(p+p~),

subject to the design condition

M = 0.36f,bx,(d - 0.414~3 + 0.87fp,bd(d -d,),

where

x,/d = 0.0035/(0.0055 + 0.87fY/E,).

1009

1010

For any design, d is fixed and so optimization can be design depending on the combination of steel and done only with respect to the d/b ratio. concrete grades adopted.

T-beam and floors

The cost function for a single T-section is

2 = bf dC[w(l + k/d) + (I - w)t + pw(C, - l)],

(3) For a single T-beam, Fig. 4 shows that the variation of optimum depth with moment is parabolic. However for optimization of the T-beam floor, mere optimi~tion of a single T-beam is not sufficient and it woutd be necessary to adopt the program prepared for the T-beam floor. A

and different design conditions are adopted depend- ing on whether the neutral axis lies within or outside the flange. After making some simplifying assump- tions the Lagrangian method can be applied to obtain optimum depth and reinforcement formulae. How- ever, for optimizing the T-beam floor the cost of steel in the slab, slab thickness, number of T-beams etc. have to be considered. A suitable cost function was adopted and a computer program prepared to opti- mize the complete T-beam floor. Similarly for two- way slabs (simply supported edges) another program has been prepared to obtain optimum thickness and steel percentage in both directions using the yield line theory.

d/b Fig. I. Cost variation with d/b for SRB.

Column section

Circular and rectangular columns have been opti- mized by the simplex method for uniaxial moment by treating only the steel percentage and b/D as vari- ables.

a \

d/b=3,Cr=90

7

H

$6 bl

>

._ ii

CONCLUSIONS 0 0.2 0.4 0.6 0.8 1.0 p/ Pm

(1) For the SRB, cost continually reduces the d/b ratio (Fig. 1) but it is not practicable to use d/B > 4 in design. The optimum steel content is independent of d/b, but depends on C, and grades of steel and concrete (Fig. 2). For some cases the steel percentage should be pm but for a combination of low grade steei with high grade concrete its value is much lower (Table 1).

Fig. 2. Cost variation with reinf for SRB.

(2) From Fig. 3 it is obvious that for the DRB cost varies moderately with d/b. It increases with d/b for a combination of low grade steel with high grade concrete and high C, value while it reduces with increasing values of d/b for high grade steel with low grade concrete. Thus, either the minimum or maxi- mum practical value of d/b results in an optimum

~~~~

d/b 3

Fig. 3. Cost variation with d/b for DRB.

Steel

Fe250

Fe415

Table 1. Optimum steel percentage for SRB

c, Concrete 50 60 70 80

Ml5 1.32 1.32 1.32 1.27 M20 1.76 1.70 1.56 1.45 M25 2.09 1.89 1.72 1.68 M30 2.27 2.04 1.85 1.69

M 15, M20 p = pm where pm = 19.82~~~~ M25 1.19 1.19 1.19 1.19 M30 I .43 1.43 1.43 1.39

90 100

1.19 1.12 1.35 1.26 1.47 1.37 1.57 1.48

1.19 1.13 1.30 1.22

Fe500 All grades

p = pm where p, = 18.87j&

Reinforced concrete sections 1011

0 to 20 30 40 50 60 d (cm)

Fig. 4. Optimum depth variation with moment for T-beam

0123456

M/P

Fig. 5. Cost variation with column shape.

7’ cr=a5, Fe 415, M 20

0 I I I I J 0 I 2 M/P3 4 5

Fig. 6. Cost variation with b/D.

cr-85, b/D 20.9 Fe415. M20

Fig. 7. Optimum depth contours for column.

comparison of two-way slabs and T-beams showed that for a span of up to 6m, with normal residential building loads, the slab is more economical while for heavier loads or longer spans the T-beam floor is economical.

(4) For columns with uniaxial moment, Fig. 5 shows the comparative cost for circular and rectangu- lar columns. Figure 6 indicates that there is large economy with reduction of b/D ratio. Optimum D value of a column can be determined from Fig. 7. It was further seen that the optimum steel percentage lies in the range of l-2 times the minimum permitted by the code, the higher value being for larger M/p and lower C, values.

4.

5.

6.

REFERENCES

L. Friel, Optimum singly reinforced concrete sections. ACI, No. 74, p. 556. T. Chou, Optimum reinforced concrete T-beam sections. J. Struct. Div., ASCE 103, 1605-1617 (1977). A. K. Misra, Optimum design of R.C: sections as per IS : 456- 1978. M.E. Thesis. Deoartment of Civil Entineer- ing, University of Roorkee (1981). A. Mebrahtu, Optimal design of beam column R.C.C. sections. M.E. Thesis, Civil Engineering Department, University of Roorkee (1985). 18:456-1978, Code of Practice for Plain and Reinforced Concrete. Indian Standards Institution, New Delhi (1979). SP: 16, Design Aids for Reinforced Concrete, to IS: 456-1978. Indian Standards Institution, New Delhi (1980).