OPTIMUM DESIGN OF REINFORCED CONCRETE RECTANGULAR BEAMS USING SIMULATED ANNEALING

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1 The Iraqi Journal For Mechanical An Material Engineering, Special Issue (B) OPTIMUM DESIGN OF REINFORCED CONCRETE RECTANGULAR BEAMS USING SIMULATED ANNEALING Dept. of Civil Eng g College of Eng g University of Basrah, Basrah, Iraq ABSTRACT: This paper presents the application of Simulate Annealing optimization metho (SA) for solving the problem of the optimum esign of reinforce concrete beams base on the recommenations of American Builing Coe Requirements for structural concrete (ACI ) an the ultimate strength esign metho. Cost of concrete, cost of steel reinforcement an cost of formworks are consiere. The constraints of the problem inclue the concrete beam strength, with-height ratio, minimum with, an eflection constraints. This optimization problem is implemente by constructing a computer program using Matlab. A number of examples are solve using the evelope program an prove that the prouce esign is economical; also it is prove that the evelope program is efficient an versatile. Keywors: optimum, esign, reinforce concrete, beams, simulate annealing الخلاصة : يقدم هذا البحث تطبيق طريقة (مماثلة التلدين) لحل مسا لة التصميم الامثل للعتبات الخرسانية المسلحة طبقا لطريقة المدونة الامريكية وباستخدام طريقة المقاومة القصوى في التصميم. دالة الهدف في المسا لة اخذت بنظر الاعتبار كلفة متر طول من العتبة وتتضمن كلفة الخرسانة وكلفة حديد التسليح وكلفة القالب. محددات التصميم وضعت لتحقيق متطلبات المقاومة ونسبة العرض الى العمق و الحد الادنى للعرض والهطول. كتب برنامج باستخدام Matlab لحل مسالة التصميم الامثل. عدد من المساي ل التطبيقية حلت باستخدام البرنامج المكتوب وبمساعدة حقيبة البرمجيات الجاهزة في برنامج Matlab والتي استنتج منها ان التصميم الناتج من حل المسا لة باستخدام البرنامج تصميم اقتصادي. 201

2 OPTIMUM DESIGN OF REINFORCED CONCRETE RECTANGULAR BEAMS USING SIMULATED ANNEALING INTRODUCTION: Optimum esign of reinforce concrete elements plays an important role in economical esign of reinforce concrete structures. Structural esign requires jugment, intuition an experience, besies the ability to esign structures to be safe, serviceable an economical. The esign coes o not irectly give a esign satisfying all of the above conitions. Thus, a esigner has to execute a number of esign-analyze cycles before converging on the best solution. The intuitive esign experience of an expert esigner can give a goo initial solution, which can reuce the number of esign-analyze cycles. Cost optimum esign of reinforce concrete beams is receiving more an more attention from the researchers. Balagura [2], Brown [3], Friel [4], an Traum [5] have presente algorithms ealing with optimum esign of single reinforce concrete beams an one way slabs. In 1980 Balaguru [6] use Lagrangian multiplier technique to calculate the optimum imensions an the amount of reinforcements for a reinforce rectangular beam. AL-Nassiry [7] use, in 2001, linearization techniques to solve the nonlinear optimization problem of reinforce rectangular beams. In this paper, the optimum esign of reinforce concrete beams, subjecte to strength, with-height ratio, minimum with, an eflection constraints, is sought using Simulate Annealing. As such a computer program using Matlab has been evelope. The problem is formulate base on the requirements of ACI coe an the ultimate strength esign metho. h As t t. b Fig.(1) Singly Reinforce Concrete Beam Cross-Section Details STATEMENT OF THE PROBLEM The optimization problem can be state as: obtain the optimum imensions an the amount of tension reinforcement for the reinforce rectangular beam shown in Fig.(1), given: 202

3 Dr. Alaa, The Iraqi Journal For Mechanical An Material Engineering, Special Issue (B) 1. The ultimate External bening moment (M u ). 2. Cost of concrete per unit volume (c c ). 3. Cost of steel per unit weight (C s ). 4. Cost of formwork per unit area (C f ). 5. Ratio of effective cover thickness of beam (t t ). 6. Compressive strength of concrete (f c ). 7. Yiel strength of steel (fy). 8. Unit weight of concrete (γ c ). 9. Unit weight of steel (γ s ). ASSUMPTIONS The proceure recommene for the optimum esign of reinforce rectangular beam in this paper is base on the following assumptions: 1. Plane sections before bening remain plane after bening. 2. At ultimate capacity, strain an stress are not proportional. 3. Strain in the concrete is proportional to the istance from the neutral axis. 4. Tensile strength of concrete is neglecte in flexural computations. 5. The ultimate concrete strain is The moulus of elasticity of the reinforcing steel is 200,000 MPa. 7. The average compressive stress in the concrete is 0.85 fc'. 8. The average tensile stress in the reinforcement oes not excee f y. 9. Only bening an eflection effects on the critical cross section are consiere. So, the beam has to be checke for shear consierations. FORMULATION OF THE PROBLEM Consiering the cost of unit length of the beam, the problem can be mathematically state as: Minimize, (C)=C c.b.h+c s. (A s ).γ s +C f [b+2h]...(1) subject to the following constraints: 1) Resisting moment of the cross section given external moment. 2) The maximum eflection the allowable eflection The following notations an efinitions are to be use 203

4 OPTIMUM DESIGN OF REINFORCED CONCRETE RECTANGULAR BEAMS USING SIMULATED ANNEALING α=(1+t t ).(2) b β 1 =.(3) Using equations (2) an (3), eq.(1) can be written as: Minimize C=C c α β 2 + C s A s.γ s +C f (2α+β)...(4) Using the ACI coe, the resisting moment capacity of the reinforce concrete section can be written as: 2. f 5 Mu= 2 y φ As. β1. f y As. β1..(5) f c ' where φ is the strength reuction factor (φ=0.9) [7] Self weight moment =M s = 1.4 γ c.β 1..α. τ. l 2.(6) where γ c is the unit weight of reinforce concrete an τ is the multiplying factor epening on the beam support conitions (τ=1/8 for simply supporte beam). Using eqs.(5) an (6), the first constraint (g 1 ) can be written as: 2 1 φ As. β 5. f y As. β1. 2. f y 2 2 M u γ c. α. β1. l..(7) fc ' or f 5 g 1 = ( M u γ c. α. β1. l. ) - 2 y φ As. β f y Asβ1 0 fc' DEFLECTION COMPUTATION: Immeiate eflection shall be compute with the moulus of elasticity E c for concrete as (4700 f c ' ) an with the effective moment of inertia, I e, as follows, but not greater than I g [7]. I e Mcr = I M where M cr g f. I r Y g + 3 Mcr 1 Icr (8) M =, cracking moment M=unfactore external bening moment, 204

5 Dr. Alaa, The Iraqi Journal For Mechanical An Material Engineering, Special Issue (B) Y= istance from centroi of gross section to extreme fiber in tension. I g = moment of inertia of gross concrete section about centroial axis, neglecting reinforcement. I cr = moment of inertia of cracke section transforme to concrete. For normal weight concrete f r, moulus of rapture of concrete is taken as (0.62 f c ' ). For simply supporte rectangular beams subjecte to uniformly istribute loa, the maximum eflection (at mi span) is equal to: 4 5wl =..(9) 384E c. I c where w=uniformly istribute live loa over span length. The allowable eflection is taken as (l/360). The cracking moment can be compute as: M = = cr f r 2 β α (10) DEPTH OF NEUTRAL AXIS As shown in Fig.(2), the neutral axis epth (x) for cracke section can be compute as follows: Taking moment about the neutral axis: b(x 2 /2)=n.As.(-x) (11) β 1 (x 2 /2)=n. As.(-x ) (12) where n=e s /E c E s = steel moulus of elasticity. Solving eq.(12) gives the epth of neutral axis (x). b x n. As Fig. (2) Concrete Cracke Section Details 205

6 OPTIMUM DESIGN OF REINFORCED CONCRETE RECTANGULAR BEAMS USING SIMULATED ANNEALING MOMENT OF INERTIA OF CRACKED SECTION The moment of inertia of cracke section with respect to the neutral axis can be compute as: I cr =β 1 (x 3 /12)+n.A s (-x) 2 (13) DEFLECTION CONSTRAINTS After etermination of neutral axis epth using eq.(12) an the cracke section moment of inertia using eq. (13), the eflection constraint can be formulate as: 4 5wl 384E. I or c e l..(14) wl g 2 = 384E. I c e l SIDE CONSTRAINTS In orer to get a reasonable esign of reinforce concrete beam, the esign variables of the problem (, As) must satisfy the following sie constraints: 1) As min As As max 2) min max SIMULATED ANNEALING Simulate Annealing (SA) is an artificial intelligence base optimization technique that resembles the cooling process of molten metal through annealing. At high temperature, atoms in molten metal can move freely with respect to each another, but as the temperature is reuce, the movement of atoms gets restricte. The atoms start to get orere an finally form the crystals having the minimum possible energy. However, the formation of the crystal mostly epens on the cooling rate. If the temperature is reuce at a very fast rate, the crystalline state may not be achieve at all; instea, the system may en up in a polycrystalline state, which may have a higher energy state than the crystalline state. Therefore, in orer to achieve the absolute minimum energy state, the temperature nees to be reuce at a slow rate. Controlling a temperature-like parameter introuce with the concept of the Boltzmann probability 206

7 Dr. Alaa, The Iraqi Journal For Mechanical An Material Engineering, Special Issue (B) istribution simulates the cooling phenomenon. Accoring to the Boltzmann probability istribution, a system in thermal equilibrium at a temperature T has its energy istribute probabilistically accoring to P(E)=exp(-E/KT), where K is the Boltzmann constant. Simulate annealing is a point-by-point metho. The algorithm begins with an initial point an a high temperature (T). A secon point is create at ranom in the vicinity of the initial point an the ifference in function values ( E) at these two points is calculate. If the secon point has a smaller function value, the point is accepte; otherwise the point is accepte with a probability exp (- E/T). This completes the one iteration of simulate annealing proceure. In the next generation, another point is create at ranom in neighborhoo of current point an Metropolis algorithm is use to accept or reject the point. The algorithm is terminate when a sufficiently small temperature is obtaine or a small enough change in function values is foun. SIMULATED ANNEALING ALGORITHM 1- Choose ranom initial values for the esign variables ( i,as i ), select the initial system temperature, an specify the cooling i.e. annealing scheule (T=kT) where k is system parameter. 2- Evaluate C( i, As i ) using eq.(1). 3- Perturb (i, As i ) to obtain a neighboring esign vector ( i+1, As i+1 ). 4- Evaluate C( i+1, As i+1 ). 5- If C( i+1, As i+1 )< C( i, As i ), then, ( i+1, As i+1 ) is the new current solution. 6- If C( i+1, As i+1 )> C( i, As i ), then accept ( i+1, As i+1 ) as the new current solution with a probability e (- E/T) where Ε=C( i+1, As i+1 )-C( i, As i ). 7- The current solution is checke against the constraints an it is aopte as the new current solution when it is feasible, i.e. when it satisfies the constraints. On the other han, the current solution is iscare when it oes not satisfy the constraints. 8- Reuce the system temperature (T) accoring to the cooling scheule (T=k.T). 9- The algorithm stops when T is a small percentage of the initial temperature. APPLICATION EXAMPLE In this work, optimum esign of reinforce concrete beam has been one. A number of examples were run to evaluate the valiity of the evelope formulation an computer program. 207

8 OPTIMUM DESIGN OF REINFORCED CONCRETE RECTANGULAR BEAMS USING SIMULATED ANNEALING A reinforce concrete rectangular simply supporte beam with the following values is consiere: C c = (I.D/m 3 ) C s = (I.D/ton) C f = (I.D/m 2 ) γ c =24 (kn/m 3 ) γ s =7850 (kg/m 3 ) t t =t c =0.1 f c =21MPa f y =414 MPa β=0.25 l=9m min =230mm max =800mm w=10kn/m E s = MPa E c =4700 f c ' =26120MPa Initial temperature (T=100) Cooling parameter (k=0.95). The optimization problem can be state as follows: Fin the values of the esign variables ( an As) which minimize the cost function (C) uner the constraints g 1, g 2 an the sie constraints state above. The solution of this problem by the simulate annealing metho is shown in Table (1). TABLE (1) RESULTS OF R.C.B. BY SIMULATED ANNEALING METHOD solution Initial Final (mm) As(mm2) C(I.D)

9 Dr. Alaa, The Iraqi Journal For Mechanical An Material Engineering, Special Issue (B) Table (2) shows the optimum values of the esign variables for various span length. TABLE (2) OPTIMUM VALUES OF DESIGN VARIABLES FOR VARIOUS SPAN LENGTH l (m) (mm) As(mm 2 ) C(I.D) It is to be note that the results of the effective epth (), obtaine in the two tables above, are larger than the minimum thickness values of the ACI-coe, which is for simply supporte beam is l/16. In orer to investigate the effect of the unit costs of the concrete an steel, the cost function can be written in the following form: C/Cs=C c/ C s α β 2 + A s.γ s +C f /C s (2α+β) Table (3) shows that the increase of the ratio Cc/Cs leas to increase the compression steel area (As) an ecrease the effective epth of beam (). TABLE (3) EFFECT OF INCREASING THE CONCRETE UNIT COST (CC) ON THE OPTIMUM DESIGN Cc/Cs (mm) As(mm 2 ) C(I.D)

10 OPTIMUM DESIGN OF REINFORCED CONCRETE RECTANGULAR BEAMS USING SIMULATED ANNEALING CONCLUSIONS 1. The simulate annealing optimization metho has been presente. The evelope proceure, wherein the optimal cost of a reinforce concrete beam has been formulate as a nonlinear programming problem, has been foun to be quiet efficient in isolating the optimal solution. 2. The increase of the ratio Cc/Cs leas to increase the steel area (As) an ecrease the effective epth of beam (). REFERENCES 1. Balaguru P., Optimum esign of singly reinforce rectangular sections. J. Civ. Eng. Design 2, (1980). 2. Brown H. R., Minimum cost selection of one-way slab thickness Proc. Paper J. Struct. Div. Am. Soc. Civ. Engrs. 101 (ST12), (1975). 3. Friel L.L., Optimum singly reinforce concrete sections. Am. Concr. Inst. 71, (1974). 4. Traum E., Economical esign of reinforce concrete slabs using ultimate strength theory. J. Am. Concr, Inst. 60, (1963). 5. Balaguru P., Cost optimum esign of oubly reinforce concrete beams. 6. Al-Nassiry J.. The applications of Linearization techniques in solving nonlinear optimization problems. M.Sc. thesis, university of Basrah, Iraq, American Concrete Institute, Builing coe requirements for structural concrete ACI American Concrete Institute, Farmington Hills, Michigan,

Analytical Model for Estimating Long-Term Deflections of Two-Way Reinforced Concrete Slabs *

Analytical Model for Estimating Long-Term Deflections of Two-Way Reinforced Concrete Slabs * Dr. Bayan Salim Al-Nu'man Civil Engineering Department, College of Engineering Al-Mustansiriya University, Baghdad,