GENETICALLY OPTIMIZED ARTIFICIAL NEURAL NETWORK BASED OPTIMUM DESIGN OF SINGLY AND DOUBLY REINFORCED CONCRETE BEAMS

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1 ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 7, NO. 6 (2006) PAGES GENETICALLY OPTIMIZED ARTIFICIAL NEURAL NETWORK BASED OPTIMUM DESIGN OF SINGLY AND DOUBLY REINFORCED CONCRETE BEAMS B. Saini a, V.K. Sehgal a and M.L. Gambhir b a Department of Civil Engineering, National Institute of Technology, Kurukshetra, Haryana , India b Thapar Institute of Engineering and Technology, Patiala, Punjab, India ABSTRACT Optimum design of singly and doubly reinforced beams with uniformly distributed and concentrated load has been done by incorporating actual self weight of beam, parabolic stress block, moment-equilibrium and serviceability constraints besides other constraints. Also, this design expertise has been incorporated into a genetically optimized artificial neural network based on steepest descent, adaptive and resilient back-propagation learning techniques. The initial solution for the optimization procedure has been obtained using limit state design as per IS: Keywords: reinforced concrete, cost optimization, beam design, artificial neural network, genetic algorithms, hybrid systems Notation: The following symbols are used in this paper: a st = area of tension reinforcement; a sc = area of compression reinforcement; b = width of beam in mm; C c = cost of concrete per unit volume in Rs./m 3 ; C f = cost of form work per unit area in Rs./m 2 ; C s = cost of steel per unit volume in Rs./m 3 ; d = effective depth of beam in mm; d ' = effective cover to reinforcement in mm; Err_g = Error goal of ANN; F(x) = squared error function of ANN; f ck = characteristic compressive cube strength of concrete in MPa; f y = characteristic yield strength of steel in MPa; δ all = allowable deflection in mm; -address of the corresponding author: babitasaini6@rediffmail.com

2 604 B. Saini, V. K. Sehgal and M. L. Gambhir δ tot = sum of short and long term deflection in mm; l = span in m; l r = Learning rate of ANN; M c = moment capacity of beam in kn-m; M r = dead load moment to maximum moment ratio; M u = bending moment due to given loading and self weight in kn-m; nh1 = No. of neurons in first hidden layer; nh2 = No. of neurons in second hidden layer; p = percentage of tension or compression reinforcement; p t = percentage of tension reinforcement; p c = percentage of compression reinforcement; r = d to b ratio; w = uniformly distributed load (UDL) in kn/m; W = concentrated load (CL) in kn; x a = balanced position of neutral axis in mm; x u = actual position of neutral axis in mm; and Z = cost per unit length of beam in Rs. / m; β = chromosome number. 1. INTRODUCTION Optimum design of Reinforced Cement Concrete (RCC) elements play an important role in economic design of RCC structures. Structural design requires judgment, intuition and experience, besides the ability to design structures to be safe, serviceable and economical. The design codes do not directly give a design satisfying all of the above conditions. Thus, a designer has to execute a number of design-analyze cycles before converging on the best solution. The intuitive design experience of an expert designer can give a good initial solution, which can reduce the number of design-analyze cycles. In this work, the intuitive optimal design expertise has been incorporated into an artificial neural network (ANN) which gives optimal design, satisfying all of the above criteria in one step. The optimization involves choosing of the design variables in such a way that the cost of the beam is the minimum, subject to the satisfaction of behavioral and geometrical constraints as per recommended method of design codes. Doubly reinforced beams (DRB) are required to be designed when the depth of the beam is restricted by architectural considerations and the beam has to take moment greater than limiting moment of resistance of the corresponding singly reinforced beam (SRB). The compression steel increases ductility and reduces long term deflections significantly. Some structure optimization work deals with minimization of weight of the structure [1-4], whereas most of the researchers have worked on cost optimization of the structure [5-16]. Though, weight of a structure may be proportional to its cost, minimization of the cost should be the actual objective in economic design of RCC structure elements. Most of the researchers have used ultimate load method for design of beams [4-6,11,12,14,15], whereas a few have used limit state method [7,16-18]. While ultimate load method provides realistic assessment of safety, it does not guarantee the satisfactory serviceability at service loads. On the other hand, the limit state method aims for

3 GENETICALLY OPTIMIZED ARTIFICIAL NEURAL NETWORK a comprehensive and rational solution to the design problem, by considering safety at ultimate loads and serviceability at working loads and hence is a better design method. In satisfaction of constraints, [5,7,13,16] have given designs satisfying only moment capacity constraint while [6,11,12,14,15] included self weight of structure in their analysis and [19] have given designs satisfying equivalent allowable deflection corresponding to factored loads which is a better approach. The constraint satisfaction procedure using a structural design method or optimization method requires execution of a number of design and analysis cycles. If an ANN is made to learn the optimized design, it can give designs much faster than that obtained by optimizer. Some researchers have used ANN, with architecture of ANN obtained by trial and error method. The optimized design in such cases was based upon ultimate load method by satisfying moment capacity behavior constraint and an error of 5 % has been reported by [15]. In this paper, an ANN, whose architecture and user-defined parameters, viz. learning rate and error goal have been optimized by genetic algorithm for the problem concerned, is made to learn optimum design for a) simply supported singly reinforced beam (SS-SRB) with uniformly distributed load (UDL). b) simply supported singly reinforced beam with concentrated load (CL) at the center. c) simply supported doubly reinforced beam (SS-DRB) with uniformly distributed load. d) simply supported doubly reinforced beam with concentrated load at the center. An initial solution for each case is obtained using the limit state method, by including self weight of the beam and considering parabolic stress block. The limit state design and the optimization is performed, subject to satisfaction of moment capacity, actual deflection and durability behavioral constraints, besides other geometrical constraints as recommended in IS: [20]. 2. PROBLEM FORMULATION AND OPTIMALITY CRITERIA The objective function, Z to be minimized consists of concrete, steel and form work cost of the beam, where for SRB and ' ' Z = C [b(d + d ) (p bd)] + C p bd + C [b + 2.0(d d )], (1) c t s t f + ' ' Z = C [b(d + d ) (a + a )] + C (a + a ) + C [b + 2.0(d d )], (2) c st sc s st sc f + for DRB, subject to satisfaction of the following constraints:

4 606 B. Saini, V. K. Sehgal and M. L. Gambhir Geometrical constraints: 1. Ductility constraint: x u x a 2. Minimum flexural strength constraint: p 0.85/f y 3. Maximum limit on steel: p 0.04b(d+d ) 4. Depth to width ratio constraint: r = d/b; 1.5 r 4.0 Behavioral constraints: 1. Durability constraint: nominal cover 40 mm 2. Moment-equilibrium constraint: M u M c 3. Deflection constraint (serviceability constraint): δ tot δ all In this work, optimal design of SRB and DRB has been done for different material combinations of M20, M25, M30 grades of concrete and Fe250, Fe415, Fe500 grades of steel. The cost of materials for different grades and form work are given in Table 1. The optimal design has been done using optimization toolbox of MATLAB by obtaining an initial solution from the limit state design method as per IS: by including the self weight of the beam and incorporating moment-equilibrium and serviceability constraints besides other constraints. Table 1. Cost of materials Concrete grade C c (Rs/m 3 ) Steel grade C s (Rs/kg) M Fe C f = 90 Rs./ m 2 M Fe M Fe NEURAL NETWORK LEARNING TECHNIQUES Design of singly reinforced beam has been done by [14,15] using supervised neural learning techniques, employing Steepest Descent Back Propagation (SDBP). This neural algorithm has a drawback of slow convergence rate. The SDBP takes consecutive steps in the direction of negative gradient of the performance surface. The performance function of the feed forward neural network (FFNN) is a squared error function F(x). The surface of this performance function contains long ravines that are characterized by sharp curvature across the ravine and a gently sloping floor. There has been considerable research on the methods to accelerate the convergence of the SDBP algorithm. Adaptive learning (ALBP) and

5 GENETICALLY OPTIMIZED ARTIFICIAL NEURAL NETWORK resilient back-propagation (RPROPBP) techniques can be used to accelerate the convergence of the SDBP algorithm. 3.1 Adaptive learning backpropagation method (ALBP) The learning rate (α) is varied according to whether or not an iteration decreases the performance index (the total error for all patterns). If an weight update results in reduced total error, α is multiplied by a factor φ > 1 for the next iteration. If a step produces a network with a total error more than a few (typically, 1-5) percent above the previous value, all changes to the weights are rejected, α is multiplied by a factor β < 1, momentum (γ) is set equal to zero, and the step is repeated. When a successful step is then taken, α is reset to its original value. The rationale behind this maneuver is that as long as the topography of the terrain is relatively uniform and the descent is in a relatively smooth line, the memory implicit in γ will aid convergence. If, however, a step results in a degradation of the performance of the system, then clearly the topography of the terrain demands a change in the direction of the optimization, and prior experience (incorporated in the term in γ) will be more misleading than beneficial. Hence, γ is set to 0 so that memory from previous steps is lost. Only after the network takes a step that reduces the total error will γ again assume a non-zero value. As learning rate varies according to the terrain of performance surface, it is known as Adaptive Learning rate Backpropagation method (ALBP) [21]. 3.2 Resilient backpropagation learning (RPROPBP) RPROPBP stands for 'resilient propagation' and is an efficient new learning scheme that performs a direct adaptation of the weight step based on local gradient information [22]. In a crucial difference to previously developed adaptation techniques, the effort of adaptation is not blurred by gradient behavior whatsoever. To achieve this, an individual update value is introduced for each weight, which solely determines the size of the weight update. This adaptive update value evolves during the learning process based on its local sight on the error function F, according to the following learning rule: ( t) = + η η * *,, if, if + where 0 < η < 1 < η * > 0 ( t ) * < 0 else (3) In other words, the adaptation rule works as follows: Every time the partial derivative of the corresponding weight w changes its sign, which indicates that the last update was too big and the algorithm has jumped over a local minimum, the update value is decreased by the factorη. If the derivative retains its sign, the update value is slightly increased in

6 608 B. Saini, V. K. Sehgal and M. L. Gambhir order to accelerate convergence in shallow regions. Once the update value for each weight is adapted, the weight update itself follows a very simple rule: if the derivative is positive (increasing error), the weight is decreased by its update value, if the derivative is negative, the update value is added: w = ( ( ( + t 1), t) t), if, if ( ( t) t) > 0 < 0 else (4) w = w + w ( t+ 1) ( t ) (5) However, there is one exception: If the partial derivative changes sign, i.e. the previous step was too large and the minimum was missed, the previous weight update is reverted: w = w, if * < 0 (6) Due to that 'backtracking' weight step, the derivative is supposed to change its sign once again in the following step. In order to avoid a double punishment of the update value, there should be no adaptation of the update value in the succeeding step. In practice this can be (t 1) done by setting = 0 in the adaptation rule above. The update values and the weights are changed every time the whole pattern set has been presented once to the network (learning by epoch) RPROPBP algorithm The following pseudo code fragment shows the kernel of the RPROPBP adaptation and learning process. The minimum (maximum) operator will deliver the minimum (maximum) of two numbers; the sign operator returns +1, if the argument is positive, -1, if the argument is negative and 0 otherwise. For all weights and biases {if * > 0 then}

7 GENETICALLY OPTIMIZED ARTIFICIAL NEURAL NETWORK w w (t) (t+ 1) = min (t) = w } else if w } ( t+ 1) (t 1) + ( * η, ) F sign = = 0 (t) = max = w + w * (t) (t) max * (t) < 0 then { ( * η, ) w min (7) (8) else if * = 0 then { w w } ( t+ 1) F sign = = w + w * (9) At the beginning, all update values are set to an initial value 0. For 0 directly determines the size of the first weight step, it is preferably chosen in proportion to the size of the initial weights. The values of different parameters are taken as: 0 = 0.07, η = 0.5, + η =1.2, max = 50, min = The main reason for the success of this algorithm lies in the concept of 'direct adaptation' of the size of weight update. In contrast to all other algorithms, only the sign of the partial derivative is used to perform both learning and adaptation. This leads to a transparent and yet powerful adaptation process, that can be straight forward and very efficiently computed with respect to both time and storage consumption. Another often discussed aspect of common gradient descent is that the size of the derivative decreases exponentially with the distance between the weight and the output layer, due to the limiting influence of the slope

8 610 B. Saini, V. K. Sehgal and M. L. Gambhir of the sigmoid activation function. Consequently, weights far away from the output layer are less modified and do learn much slower. Using RPROPBP, the size of the weight step is only dependent on the sequence of signs, not on the magnitude of the derivative. For that reason, learning is spread equally all over the entire network; weights near the input layer have the equal chance to grow and learn as weights near the output layer. The parameter used for comparing the performance of learning techniques is the mean absolute percentage error (MAPE). Thus, the quality of the output of ANN is compared on the basis of MAPE. MAPE N i = 1 X i O *100 i X i N = (10) where X i is the actual value, O i is the output given by ANN and N is the total number of values predicted. 4. NEURAL NETWORK IMPLEMENTATION Four ANNs, two for preliminary optimal design of SS SRB with UDL and CL (SRB-NN) respectively and two for SS DRB with UDL and CL (DRB-NN) respectively are synthesized. The number of inputs and outputs for SRB-NN has been fixed as 5 and 2, respectively, as per the requirement of design problems. Thus the input of both SRB-NN consists of 5 inputs viz. load, span, d to b ratio, f ck and f y, and the output are optimum tensile steel reinforcement percentage (p t ) and optimum depth of beam (d). A set of 4846 and 4573 optimum design examples were used for training and a set of 476 and 457 unseen examples were used for testing of trained ANN for SS SRB with UDL and SS SRB with CL, respectively. The number of input and output for DRB-NN network has been fixed as 6 and 2, respectively, as per the requirement of design problem. Thus the input of both DRB-NN consists of 6 inputs, viz. load, span, d to b ratio, f ck and f y, and depth of beam. The output of both ANN are cover to reinforcement ( d ' ) and optimum tensile steel reinforcement (p t ). The optimum compression steel reinforcement is derived from input and output variable values of ANN. A set of 3843 and 3751 optimum design examples were generated for training and a set of 342 and 261 unseen examples were used for testing of trained ANN for SS DRB with UDL and SS DRB with CL, respectively. Four layered feed forward neural networks (FFNN) consisting of two hidden layers has been simulated using MATLAB 7.0 developed by [23] for learning of the optimal design examples. To avoid the trapping of algorithm in local minima, the number of neurons in first and second hidden layer and user defined parameters of ANN viz. learning rate and error goal have been optimized for minimum learning error using genetic algorithm with optimization technique given by [24]. The population size was taken as 100 and number of generations as 40. The plot of training error vs. β is shown in Figure 1 for 250 epochs of learning for each chromosome of the population. The optimized parameters of ANN (OPT-ANN) variables (best fit chromosome) for minimum training error are shown in Table 2. In order to increase the learning speed, the weights of the ANN are initialized according to the method given by [25]. In this, first

9 GENETICALLY OPTIMIZED ARTIFICIAL NEURAL NETWORK the weights are initialized according to a uniform random distribution between 1 and 1 and then the magnitude of weight vectors is adjusted in a manner such that each hidden node is linear over only a small interval. Each ANN has been trained for epochs with training data on an Intel P-IV, 3.0 GHz based PC using various neural learning techniques. Parameter Table 2. Optimum parameters of ANN SS-SRB with UDL SS-SRB with CL SS-DRB with UDL SS-DRB with CL nh nh lr Err_g Training error (%) β Figure 1. Training error vs. β for optimal design of ANN using genetic algorithm 5. ARTIFICIAL NEURAL NETWORK RESULTS For SRB, optimized values of only two variables viz. percentage of steel and depth of beam have been predicted by respective OPT-ANN. From five input variables, two predicted variables, and using other problem constants like unit cost etc., the values of dependent variables viz. optimal cost per unit length, total deflection and moment capacity of beam are derived. For DRB, optimized values of only two variables viz. cover to reinforcement and

10 612 B. Saini, V. K. Sehgal and M. L. Gambhir percentage of steel has been predicted by ANN. From six input variables, two predicted variables, and using other problem constants like unit cost etc., the values of dependent variables, viz. compression reinforcement percentage, optimal cost per unit length, total deflection, bending moment and moment capacity of beam are derived. Hence, these design results will have exact correlation with the two predicted variables. If these results are also predicted by ANN as is done by [14,15], then they may not have exact relation with predicted variables. Tables 3 and 4 give the average recall error of OPT-ANN for all the unseen examples of SRB and DRB respectively. Table 5 compares the training time taken by different ANN learning techniques to learn the optimal design. From Tables 3 and 4 it is evident that the recall error was the maximum using SDBP learning in all the four cases. The minimum recall error in case of SS-SRB with UDL and CL for 476 and 457 unseen examples was 2.01 and 0.51 percent using the ALBP and RPROPBP learning technique respectively. From Table 4 it is evident that for SS-DRB, the recall error was the minimum as 0.52 and 1.15 by using RPROPBP learning for 342 and 261 unseen examples of SS-DRB with UDL and CL respectively. The time taken by optimizer and ANN to produce a result was and 0.06 sec respectively. Thus, ANN gave optimum design nine times faster than that given by optimizer. The output of ANN for some unseen examples is shown in Tables 6, 7, 8 and 9 for SS SRB with UDL, SS SRB with CL, SS DRB with UDL and SS DRB with CL, respectively. From these Tables it is inferred that the quality of design given by ANN is as good as that of optimizer. Table 3. Recall MAPE of ANN for SS-SRB with UDL and SS SRB with CL using different neural learning techniques. SS SRB with UDL SS SRB with CL Learning algorithm MAPE in percentage of steel MAPE in depth of beam Average MAPE MAPE in percentage of steel MAPE in depth of beam Average MAPE SDBP ALBP RPROPBP ILLUSTRATIVE EXAMPLES Comparison of results of design problem for SRB and DRB has been illustrated using conventional limit state design and proposed optimization algorithm. Whereas, comparison of ANN techniques with proposed optimization algorithm (Optimizer) have been done for five problems each for SRB and DRB with UDL and CL in Table 6-9 respectively.

11 GENETICALLY OPTIMIZED ARTIFICIAL NEURAL NETWORK Table 4. Recall MAPE of ANN for SS DRB with UDL and SS DRB with CL using different neural learning techniques SS DRB with UDL SS DRB with CL Learning algorithm MAPE in cover to reinforcement MAPE in percentage of steel Average MAPE MAPE in cover to reinforcement MAPE in percentage of steel Average MAPE SDBP ALBP RPROPBP Table 5. Time taken by different techniques Algorithm SS SRB with UDL Training time (sec.) for epochs SS SRB with CL SS DRB with UDL SS DRB with CL SDBP ALBP RPROPBP Design example 1: singly reinforced beam Design a singly reinforced concrete beam having an effective simply supported span of 5.5 m. The beam is required to support live and superimposed loads of 15 kn/m, and 10 kn/m respectively. The materials used are M20 grade concrete and HYSD steel of grade Fe Conventional limit state solution b = mm, d = mm, d ' = 40 mm, p t = , δ tot = mm, δ all = 22 mm, M u = kn-m, M c = kn-m, Cost = Rs /m Solution by proposed technique b = mm, d = mm, ' d = 40 mm, p t = 0.953, tot δ = mm, all δ = 22 mm, M u = kn-m, M c = kn-m, Cost = Rs /m Design given by proposed technique is safe while that given by conventional limit state method fails in moment capacity.

12 614 B. Saini, V. K. Sehgal and M. L. Gambhir Table 6. Results of optimal design for SS SRB with UDL Output Input Predicted variables Derived variables Method p t d M u (kn-m) M c (kn-m) δ all δ tot Z (Rs./ m) w = 35 kn/m, Optimizer l = 4 m, SDBP r = 2.0, ALBP M25, Fe415 RPROPBP w = 65 kn/m, Optimizer l = 6 m, SDBP r = 2.0, ALBP M25, Fe415 RPROPBP w = 45 kn/m, Optimizer l = 5 m, SDBP r = 2.5, ALBP M20, Fe500 RPROPBP w = 60 kn/m, Optimizer l = 8 m, SDBP r = 1.5, ALBP M30, Fe415 RPROPBP w = 50 kn/m, Optimizer l = 6 m, SDBP r = 2.5, ALBP M30, Fe500 RPROPBP

13 GENETICALLY OPTIMIZED ARTIFICIAL NEURAL NETWORK Table 7. Results of optimal design for SS SRB with CL Output Input Predicted variables Derived variables Method p t d M u (kn-m) M c (kn-m) δ all δ tot Z (Rs./ m) W = 200 kn, Optimizer l = 6 m, SDBP r = 1.5, ALBP M25, Fe500 RPROPBP W = 600 kn, Optimizer l = 6 m, SDBP r = 1.5, ALBP M25, Fe415 RPROPBP W = 250 kn, Optimizer l = 7 m, SDBP r = 2.5, ALBP M20, Fe415 RPROPBP W = 500 kn, Optimizer l = 6 m, SDBP r = 1.5, ALBP M30, Fe500 RPROPBP W = 100 kn, Optimizer l = 10 m, SDBP r = 2.0, ALBP M30, Fe415 RPROPBP

14 616 B. Saini, V. K. Sehgal and M. L. Gambhir Table 8. Results of optimal design for SS DRB with UDL Output Input Predicted variables Derived variables Method d' p t p c M u (kn-m) M c (kn-m) δ all δ tot Z (Rs./m) w = 40 kn/m, Optimizer l =5 m, r = 2.0 SDBP M20, Fe415 ALBP d = mm RPROPBP w = 30 kn/m, Optimizer l = 6 m, r = 2.5 SDBP M25, Fe415 ALBP d = mm RPROPBP w = 50 kn/m, Optimizer l = 6 m, r = 1.5 SDBP M30, Fe500 ALBP d = mm RPROPBP w = 70 kn/m, Optimizer l = 4 m, r = 1.5 SDBP M30, Fe415 ALBP d = mm RPROPBP w = 50 kn/m, Optimizer l = 5 m, r = 2.5 SDBP M20, Fe500 ALBP d = mm RPROPBP

15 GENETICALLY OPTIMIZED ARTIFICIAL NEURAL NETWORK Table 9. Results of optimal design for SS DRB with CL Output Input Predicted variables Derived variables Method d' p t p c M u (kn-m) M c (kn-m) δ all (mm ) δ tot Z (Rs./m) W = 350kN, Optimizer l = 4 m, r=1.5 SDBP M20, Fe415 ALBP d = mm RPROPBP W = 200 kn, Optimizer l = 6 m, r = 1.5 SDBP M30, Fe500 ALBP d = mm RPROPBP W = 250 kn, Optimizer l = 4 m, r = 1.5 SDBP M25, Fe415 ALBP Design example 2: doubly reinforced beam Design a simply supported doubly reinforced beam of span 6 m, effective depth 440 mm subjected to the following loads and specifications: Superimposed load = 10 kn/m, Live load = 20 kn/m, using M30 grade concrete and Fe415 grade steel Conventional limit state solution d = 400 mm, b=266 mm, M u = kn-m, d ' = 40 mm, p t = , p c = , M c = kn-m, δ = mm, tot δ = 24 mm, Cost = Rs./m. all Solution by proposed technique ' d = 40 mm, d = 400 mm, b = 266 mm, p t = , p c = , tot δ = mm, all δ = 24 mm, M u = kn-m, M c = kn-m, Cost = Rs./m. Design given by proposed technique is safe in comparison to conventional limit state method which fails in deflection.

16 618 B. Saini, V. K. Sehgal and M. L. Gambhir 7. CONCLUSIONS In this work, the optimal design of SS-SRB, SS-DRB with UDL and CL has been done by taking initial solution from limit state method by including the self weight of the beam and incorporating moment-equilibrium and serviceability constraints, besides other constraints. An optimum ANN was synthesized to learn the optimal design and intuitive design expertise for each of the four problems. To avoid trapping of the ANN in local minima, the structure of ANN and user-defined parameters like learning rate and error goal have been optimized using genetic algorithm. In order to accelerate learning speed and to achieve better accuracy in learning, ALBP and RPROPBP techniques have been used and their performance has been compared with that of SDBP technique. Hence, among the four neural learning techniques, RPROPBP has been adjudged to be the best learning techniques for optimal design of beams as it gave recall error of minimum order. In this work, a best average MAPE of 2.01 and 0.51 percent in case of SS-SRB with UDL and CL respectively and a MAPE of 0.52 and 1.15 has been obtained for unseen design examples in case of SS-DRB with UDL and CL respectively, in comparison to an error of 5 percent as reported in [15]. The main reason of better performance of ANN is attributed to accelerated learning techniques as well as due to optimization of architecture, learning and error goal parameters of ANN using genetic algorithms instead of using trial and error method of finalizing these parameters as is done in [14,15]. Also, it is inferred that the neural networks give optimum design nine times faster than that given by optimizer. The design given by the proposed technique is safe and economical in comparison to the conventional limit state method. REFERENCES 1. Haug, E.J. Minimum weight design of beams with inequality constraints on stress and deflection. Department of Mechanical Engineering, Kansas State University, Kansas, Haug, E.J. and Krimser, P.G. Minimum weight design of beams with inequality constraints on stress and deflection, Journal of Applied Mechanics Transactions of the ASME, (1967) Venkayya, V.B. Design of optimal structures, Computers and Structures, 1(1971) Karihaloo, B.L. Optimal design of multipurpose tie beams, Journal of Optimization Theory and Applications, 27(3), (1979) Friel, L.L., Optimal singly reinforced concrete sections, ACI Journal, No. 11, 71(1974) Balaguru, P.N. Cost optimum design of doubly reinforced concrete beams, Building and Environment, 15(1980) Prakash, A., Agarwala, S.K. and Singh, K.K. Optimum design of reinforced concrete sections, Computers and Structures, 30(4), (1988)

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