OPTIMAL COST DESIGN OF RIGID RAFT FOUNDATION

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1 The Tenth Eat Aia-Pacific Conference on Structural Engineering and Contruction Augut 3-5, 2006, Bangkok, Thailand OPTIMAL COST DESIGN OF RIGID RAFT FOUNDATION P. K. BASUDHAR 1, A. DAS 2, S. K. DAS 2, A. DEY 3, K. DEB 3 and S. DE 3 ABSTRACT: Deign of a raft foundation involve proportioning and tructural deign, in which proportioning part mainly depend upon geotechnical apect, i.e. earing capacity and ettlement of the underlying oil. Optimum cot deign of uch a foundation need everal trial deign. In thi paper, an optimum cot deign of rigid raft foundation ha een done conidering oth geotechnical and tructural deign apect. The cot of the raft foundation ha een conidered a the ojective function and the deign requirement have een impoed a ide and ehavioral contraint. The prolem i formulated a a imple mathematical programming prolem. Many of the contraint eing non-linear, the prolem are one of the non-linear programming prolem. Amongt claical optimization technique, equential uncontrained minimization technique uing the penalty function method ha een ued. In thi formulation, the contraint are appended to the ojective function to form a compoite urface through a penalty parameter and uncontrained minimization of the ame i carried out uing Powell conjugate direction earch method for decreaing the equence of the aove mentioned penalty parameter. The aove cheme ha een found to e efficient in determining the optimum cot analyi of the rigid raft foundation. Amongt the non-traditional technique, genetic algorithm for a contrained optimization prolem uing a penalty parameter ha een ued. Parametric tudie have een made to tudy the effect of oil parameter on the optimum analyi. It ha een found that for aturated fine grained oil, thickne of the raft foundation i not affected y the undrained hear trength. For coheionle oil, optimum deign i inenitive to the variation in φ value etween Modulu of elaticity of oil ha a great effect on optimum deign of raft. The optimum cot i more enitive to the ettlement criterion than the criterion of earing capacity. KEYWORDS: Optimal Cot Deign, Raft Foundation, Sequential Uncontrained Minimization Technique, Powell Conjugate Direction Method 1. INTRODUCTION Optimization technique ha een ued uccefully in variou tructural and geotechnical prolem. Mot of the optimization technique have een applied to the lope taility prolem. Suarao et al. [1] developed a method for an optimal dimenioning of footing ujected to axial loading and uniaxial moment. Bavikatti [2] ha propoed a method of optimal deign of iolated column footing uing equential linear programming. They concluded that one can achieve 8 10% economy uing thi technique. But they have not conidered the ettlement criteria in the deign. Madanmohan [3] preented methodology for ettlement controlled optimization deign of rectangular footing with generalized loading uing equential uncontrained minimization technique. Mondal [4] ued the aove technique for optimum cot deign of hallow circular footing ujected to generalized loading. Pandian et al. [5] have found out optimum cot deign of comined footing conidering earing 1 Profeor, Department of Civil Engineering, I. I. T. Kanpur, India 2 Former Pot Graduate Student, Department of Civil Engineering, I. I. T. Kanpur, India 3 Pot Graduate Student, Department of Civil Engineering, I. I. T. Kanpur, India

2 capacity and ettlement criteria. But they have conidered it a a ingle variale prolem, in which the width of the footing wa the deign variale. Sequential uncontrained minimization technique ha een ued in the preent tudy. A the ucce of the numerical cheme i oriented, it i neceary to have an in-depth tudy and a critical appraial of the ame to arrive at a meaningful concluion regarding their efficiency and effectivene. Parametric tudie have alo een conducted to ring out the effect of the oil ehavior on optimum cot. 2. STATEMENT OF THE PROBLEM Adequate earing capacity and retricted ettlement are the two primary and mot important criteria for deign of raft foundation. Figure 1 how the rectangular raft with concrete column. The prolem i to find the optimal cot deign of the rigid raft foundation for a given ite condition, poition of the column and loading condition. The deign of the raft of done y trip method of analyi a per Bowel [6] with linear elatic oil. 3. ANALYSIS 3.1. ASSUMPTIONS In order to develop the generalized procedure for the analyi, it wa aumed that the hape of the raft and the column were rectangular or quare, the pacing of the column were ame along the X- and Y- direction repectively. The oil wa aumed to e a linearly elatic material DESIGN ARIABLES The deign variale for the analyi were edge ditance along the x- and Y- direction, d 1 and d 2 repectively; thickne of the raft foundation, d 3 ; and the diameter of the reinforcing ar, d OBJECTIE FUNCTION The total cot of the raft foundation i conidered a the ojective function for the analyi to e carried out. F = C = R exc exc + C exc conc + R + C conc teel conc + C + R teel teel + R (1) where, F = Total cot of the raft, C exec, Cconc, Cteel & C = Cot of excavation, concrete, teel, and ing of the remaining excavated earth volume, R exec, Rconc, Rteel & R = Rate of repective item & exec, conc, teel & = olume of excavation, concrete, teel and ing repectively which depend upon the deign variale DESIGN CONSTRAINTS The prolem involve contraint retricting ome parameter of the deign to lie within pecified limit. The deign contraint may e claified a tructural contraint and geotechnical contraint. Structural contraint like minimum percentage of reinforcement, pacing of reinforcement, development length etc. are impoed a per IS 456:2000 [7]. Geotechnical contraint are impoed a (i) ettlement of raft hould e within permiile limit (ii) impoed tre hould e le than afe earing capacity of oil and (iii) the trip conidered for analyi of raft hould e rigid. For the trip to

3 e conidered a rigid, the following condition hould e atified: S λ π / 4, where, 4 K B E B E λ = 4 = invere of the characteritic length of eam, K = = modulu 4 2 EI EI 1 µ of ugrade reaction of the oil, B = width of the trip, E = elaticity modulu of 3 concrete ( = 5700 f ck ), I = moment of inertia of the ection of the trip ( = 1/12 d ), d = depth of the eam, E = elaticity modulu of the oil, and µ = Poion ratio of the oil MATHEMATICAL PROGRAMMING PROBLEM To find out the optimum cot of the raft foundation, the cot function i to e minimized. The prolem i tated a an optimization prolem a follow: Find the deciion vector D uch that m Cot = F( Dm ) i the minimum of F (D) ujected to (2) g j ( D) 0; j = 1,2,, m where m i the total numer of inequality contraint, and g j (D) i the inequality contraint function MINIMIZATION PROCEDURE The prolem i converted into an uncontrained minimization prolem with the help of extended penalty function method [8]. The equential uncontrained minimization of the developed compoite function i done y Powell conjugate direction method for the multidimenional earch and quadratic interpolation function for the one-dimenional earch. The advantage of thi method i that even if the deign vector i in an infeaile region, the algorithm i capale of finding the feaile optimum olution. For variou engineering prolem, mot of the time, it i difficult to find the initial feaile point. In uch cae, the developed technique i expected to e helpful in finding a feaile olution from initial infeaile point. 4. RESULTS AND DISCUSSIONS A rigid raft with applied load a hown in Figure 1 i analyzed in the preent tudy. The parametric tudy i done for the aove prolem. The tre and ettlement calculation have een carried out y dicretizing the plan area of the raft into everal mall element and taking the effect of each element at each corner of the raft foundation. The influence of the initial deign vector on the optimal olution ha een tudied and found that the optimal vector i identical tarting with varying initial deign vector (Tale 1). So, the optimal olution can e conidered a the gloal olution to the prolem. It wa oerved that the effect of undrained hear trength ( S u ) i appreciale for the value le than 30 kpa. To tudy it effect on the geometry of the raft, a parameter area ratio i defined A1 A0 a A r = 100%, where, A 0 =ae of the upertructure and A 1 =area of the raft ae. It i A0 found that the change in the area ratio i ignificant when trength value fall elow 30 kpa. A the earing capacity decreae, the area of the raft increae, hence increae in the cot and area ratio. When Su exceed 30 kpa, the earing capacity doe not control the deign. But it i intereting to note that he over the trength variation of 20 kpa to 40 kpa, the depth of the raft remain almot contant. From Figure 2, it can e deduced that cot of teel predominantly influence the total cot over a wide

4 range of Su value. So to ave the cot, one can deign the raft a flexile with low thickne. However, in uch a cae, the effect of differential ettlement at the different column point, have to e taken into account on the overall ehavior of the tructure. For purely coheionle oil with a wide range of φ value, the variation of µ from 0.34 to doe not play any ignificant role on the cot a well a on the optimal dimenion of the raft (Figure 3). In thi cae, the ettlement i alway le than the permiile value. But for E value elow 25 MPa, the effect of E on the raft i greater (Figure 4). At thi tage deign i trictly controlled y ettlement, and conequently the plan area of the raft increae, reulting in a high area ratio. But the depth of the raft did not change a depth required from hear and ending thickne i much le than geometric contraint ( d m). From Figure 5, it can e een that for the given et of oil parameter, C c /( 1+ e0 ) doe not have any ignificant effect on the cot, till it reache a limiting value of ettlement up to which the deign i not governed y ettlement criteria. But a C c /( 1+ e0 ) increae, the rate of ettlement alo increae and when it reache the limiting value (50 mm) the deign i fully controlled y ettlement. The effect of permiile ettlement on the optimum cot i preented in Tale 2. It can e een that for 5% increae in given loading, there i great increae in cot when permiile limit i 25 mm. At the ame loading condition, if the ettlement limit i increaed to 50 mm and 75 mm, the optimal cot i reduced. Similar ehavior i oerved when permiile limit i increaed from 50 mm to 75 mm and the raft can take 280% of the initial loading. In all thee cae, the thickne of the raft i 0.75 m with only increae in area ratio. In all the cae, the high cot can e attriuted to the ettlement limit and correponding increaed area ratio. 5. CONCLUSIONS The developed procedure, wherein the optimal cot deign of a rigid raft foundation ha een formulated a a nonlinear programming prolem, ha een found to e quite efficient in iolating the optimal olution. For the tudie carried out here, it i found that the variation in the cot i due to the variation in area ratio a the thickne of the raft i fixed due to geometric contraint. For aturated fine grained oil under undrained condition, the undrained hear trength play a great role on the optimal dimenion of the raft, when the value of Su fall elow a limiting value. However, aove that value, the effect of Su i negligile. For purely frictional oil over the range of 20 φ 30, there i no effect of φ on the optimum cot. Modulu of Elaticity E ha a greater effect on the optimum deign of raft. For c φ oil, when E value fall elow the limiting value and the µ =0.499, the olution cannot e determined, and the cot of raft turn out to e very high. It i alo found that high cot can e attriuted to ettlement limit. A the ettlement limit increae the optimum cot decreae. Tale 1 Effect of initial variale on the reult Φ = 0; E = 40 MPa; µ = 0.45; Cc/(1+e 0 ) = 0.05; γ = 20 kn/m 3 ; and GWT at 0.0 m depth Initial ariale Final ariale d 1 (m) d 2 (m) d 3 (m) d 4 (m) d 1 (m) d 2 (m) d 3 (m) d 4 (m) Cot (R.)

5 Tale 2 Effect of ettlement contraint on optimum cot Soil Condition: Φ = 15º; E = 50 MPa; µ = 0.499; Cc/(1+e 0 ) = 0.07; γ = 20 kn/m 3 and c = 10 kpa Loading Smax = 25 mm Smax = 50 mm Smax = 75 mm Smax = 100 mm 1.05 R A r = 6.53 d 3 = 0.75 R A r = d 3 = 0.75 R R No Solution R Ar = No Solution No Solution R A r = R A r = d 3 = No Solution No Solution No Solution R R A r = R A r = Column No P (kn) Mx (knm) My (knm) Figure 1 Plan, Elevation and Loading of the Raft Foundation

6 Figure 2 ariation of Cot with S u Figure 3 ariation of Cot with Poion Ratio Figure 4 ariation of Cot with Elaticity Modulu Figure 5 ariation of Cot with C c 1+ e ) 6. REFERENCES /( 0 1. Suarao, K. S., Rao, N. N. and Ranganatham, B.., Plan dimenioning of footing ujected to uniaxial moment, Indian Geotechnical Journal, ol. 2(1), 1972, pp Bavikatti, S. S., Optimum deign of iolated column footing uing equential linear programming, International Conference on Computer Application in Civil Engineering, Roorkee, India, MadanMohan, R., Settlement controlled optimum deign of hallow footing, M. Tech. Thei, Department of Civil Engineering, IIT Kanpur, India, Mondal, A. S., Optimum deign of hallow circular footing, M. Tech. Thei, Department of Civil Engineering, IIT Kanpur, India, Pandian, N. S., Sridharan, A. and Sathidevi, U., Economical conideration of deign of comined footing, Indian Geotechnical Journal, ol. 24(4), 1994, pp Bowle, J. E., Foundation Analyi and Deign. McGraw Hill Inc., New York. 7. IS-456:2000, Code of Practice for Plain and Reinforced Concrete, Bureau of Indian Standard, New Delhi, Kavlie, D. and Moe, J., Automated deign of frame tructure, Journal of the Structural Diviion, ASCE, ol. 97, ST1, 1971, pp