- Email: [email protected]
An application of geometric programming to structural design
Compuwrr & Srrurfrrrrs Primed in Great Brirain.
Vol. 22. No. 6. pp. 965-971.
1986
004s7949M
53.00 f .MJ
0 1986Pergamon Press Ltd.
AN APPLICATION OF GEOMETRIC PROGRAMMING TO STRUCTURAL DESIGN Faculty
N. C. DAS
of Engineering,
(Received
GUPTA, H. PAUL and CHIEW HUI Yu National University of Singapore, Kent Ridge, 30 May 1984: in revised form
Singapore
OS1 I
I I March 1985)
Abstract-This paper presents an application of generalised geometric prog~mming to an optimal design of a modular floor system, which consists of reinforced solid concrete and voided slab units supported on steel beams. A function representing the cost of the floor system in terms of design variables, length, width and thickness of components, and other engineering and cost parameters is minimized subject to various constraints depending on stresses and deflections. A dualabased algorithm has been used for solving the design problem. Analyses are performed to determine the changes in the optimal values of the design variables with respect to the changes in imposed loads on the floor system.
I. INTRODUCTION
In structural engineering, the design is first governed by the choice of the structural system. Based on the functional requirements, the general layout is then defined. The structure is then analysed to suit certain performance criteria, and the acceptable designs must satisfy such performance criteria. The design also aims at finding the least cost of the structure. The methodology adopted to achieve the specific goal is optimization. One of the optim~tion techniques which has been used in some structural design problems is geometric programming (GP), The basic theory and formal proofs for this optimization technique can be found in [I] and [2]. A GP problem is an optimization with objective functions and constraints, which can be expressed in the form of signomials. Many structural design equations as stipulated by the engineering codes of practice are expressed in the form of signomials. Hence, the computational algorithms to solve GP problems are well suited to such structural optimization problems. To solve these optimization problems, Templeman[3] and Mo~s[41 used the standard GP of D&in ef af.[5], incorporating the iterative technique of Avriel and Williams161.Ramamurthy[7] used the GP model for the optimal design of a prestressed concrete slab and a light gage structural element. Templeman applied the GP model to the optimal design of bridge structures made of steel sections and steel deck plates. This paper presents the formulation of a GP model for the optimal design of a modular floor system consisting of solid and voided precast slabs supported on steel I-sections. Figure 1 shows plan and sectional views of such a floor system. The design problem presented in this paper is formulated based on the geometry, imposed loads, stresses and deflection criteria permitted by the relevant British Codes of Practice.
2. THEORY
The following is a summary of the most general formulation of the GP problem. A complete discussion and derivation of these results may be found in f2] and [Bj. A general ~lynomiaf, y,(x), is defined as
n=I
1-I
m = 0, 1, . - - , .+I,
(I)
where umt = -c t is a Signum function, and the coefficients c,~ are all positive. The primal problem is to minimize Y = Yo
(2)
subject to the constraints Ym G (Jm tE
2 f),
m=
t ,...,
M,
(3)
and .&I >
0,
ff=
1t * . * * N,
(4)
For the associated dual problem, consider a set of T variables o satisfying a normality condition 001 war = u (= +-I),
(3
t-1
N orthogonality conditions
s
*5,
CJmIamrn
QJmr =
T-nonnegativity conditions %Ir * 0,
m=O,I...., M,I=
965
I,..
0,
(6)
966
I
I
I
I SECTtON
I
I
A-A
f
;;rl.-._.-s ---!
L
1. SOLtO
As
SLAB
MlOED
SECTlON 5-a
SECTION
SLAB
EQUIYALEKT VOIDED
B-B
SECTION
SLAB B-8
i’ II
DETAIL
-I---
A
Fig. t . Plan and sectional view of modular floor system.
and M linear inequality
constraints
T,,,. From these variables, plus where T = peso the c,,,~. o,,,* and o, the dual function can be formed as follows:
and, in addition,
Then for every point x0 where v is locally minimum there exists a set of dual variables o0 satisfying eqns (348) such that
d(fJ.l”)= ?;(_r~)* d(o) In this function,
oU is defined
to be equal to + 1
(iI)
The dual function is stationary at I_&’with respect to all nonnegative w,,,~;in particular, with the global minimum .r*. if it exists, the corresponding dual var-
An application iables
w*
are such that
+
d(o”) = +*).
c,
(9) + (2”:; df)
(12)
x L, ps, c, Once the dual variables w are known, the corresponding values of the primal variables .V are found from the following relations: N Co,
n
1, . . . . T”,,
(13)
t = I, . . . , T,,, m = I, . . . , M.
(14)
xy”
=
w(),cr~“,
I
=
%7
of geometric programming
n=I 1 and
From eqn (13) it can be seen that u will have the same sign as y”. Since there will always be more terms than variables x, N equations can be found which are solvable for the N primals. in addition, the solution of these equations is not difficult since they are linear in log I,,.
L,
( > + 1 .
7
Cc, Cf, C;, C,. C,, C, are unit costs of concrete (S/m’). formwork for solid slabs ($/m’), formwork for voided slabs ($/m’), reinforcement (%/kg), steel beam ($/kg) and erection ($[ W,/l361’,“, where one unit weighs WS kg), respectively. It may be noted that C, includes the cost of material, fabrication and erection of the steel beam. s is the spacing of steel beams (ml; L and h. the width (m) and overall depth (mm) of the slab, respectively; L, and L,., length (m) and breadth (m) of the floor system. respectively; T, I, b and d, flange thickness (mm), web thickness (mm), flange width (mm) and depth of web (mm) of the steel beam, respectively; p,, and pr densities (kg/m’) of steel and concrete, respectively. B, b,, and f are the spacing of ribs (mm), width of rib (mm) and flange thickness (mm) of an equivalent voided section, respectively; A, and A are the area of the reinforcement or prestressing steel (mm’) and the cross-sectional area of the voided section (mm’), respectively.
3. OBJECTIVE FLXCTION
The objective function of the GP model incorporates the cost of concrete, formwork, reinforcing bars, steel beams and erection of precast slabs. The total variable costs are obtained as follows: The following expression gives the overall total variable cost of the floor system using precast solid slabs. go = cost of concrete of reinforcement of steel beam
+ cost of formwork + cost of erection
+ C,(A,L,)
(y)
+ (2 “;;
dr) L, Ps, C, ($
+ cost + cost
p,-, + C, (9) + I) .
(15)
Similarly the overall total variable cost of the floor system using precast voided slabs is worked out as follows: go = IO-’ C,. AB-’
L,L,,
4. CONSTRAINTS
The constraints of the geometric programming model are obtained from the following considerations: allowable bending and shear stresses of concrete and reinforcement steel; allowable bending, shear, web crippling and web crushing stresses of steel beams; allowable deflection of the slab and steel beam; minimum slab thickness and minimum reinforcement to satisfy cover and temperature stress requirements; the minimum requirements of the ratios of flange width to flange thickness and web depth to web thickness of the steel beam; and the upper limit of the ratio of depth to flange width of the steel beam. The formulation of the constraints is carried out in accordance with the BS codes of Practice CPI 10[9] for reinforced concrete slabs and BS449[ IO] for steel beams. 4.1 Bending stress requirement of slubs
At the ultimate limit state the rectangular parabolic stress block is similar in shape to the stressstrain curve for concrete, having a maximum stress of 0.45 fc,, at the ultimate strain of 0.0035. The following expression, which satisfies the ultimate bending moment at the mid-span, is obtained for the solid slab: ? x$+1.6”
f )
+ IO-’ C, A,,B-’ L,L,. p.,,
s 0.87
f?.A., 4 (
k2 0.87 A,rf>
k, x lo3
)
7 (17)
N. C. D&s Gum.-\et al.
968
f3-l) where f<,,, _f? and k, are characteristic concrete (O.jW~.~) G l~f(2~ f v’?n* strength (N/mm’), characteristic steel strength (NI 5 &J < L mm’) and mean concrete stress (N/mm’), respec(2St Z (lJs EIYGi’ tively. k2 is the ratio of the depth of the centroid of the stress block to the depth of the neutral axis. ri< where W, the distributed load carried by the steel is the effective sfab depth (mm), and w is the disbeam (N/ml for the solid slab floor system and the tributed imposed load (kN/m’). voided slab system is as follows: Similarly, the bending stress constraint for
4.2 Deflection requirement ofdabs
The ratio of span to effective depth is required to be satisfied per CPI 10 stipulation. For both solid and voided slabs, the deflection constraint generates the expression s/d= s 20kb x k,. x k,,
471:)
+ 9.81 pst
Since the flange and web of the steet beam are assumed to be unstiffened. the BS449 requirements lead to the constraints
(191 O.Sb - 03
where k, is the modi~cation factor for compression reinforcement (Table 11 of CPI IO), kb is the modification factor for the type of beam (Tables 8 and 9 of CPllO), and k, is the modification factor for tension reinforcement (Table 10 of CPE10). 4.3 Shear stress requirement of stab The effective depth of both solid and voided slabs is determined by considering the allowable shear stress. The follo~ng constraint will guarantee the minimum thickness of the slab: V bci,
s
16T.
d SG8%.
(28, 119)
The moment of inertia. 1, of the steel beam is an artificial variable used for sim~ti~~ation of intqualities[7, 8, 111, and the value of I as expressed in terms of sectional dimensions becomes I = 0.5 bTdd’ + bdT2 i
0.5 b’f” “r 0.083 rd3. (301
5. AN EXAMPLE
c
55vc,
(20)
where V is the maximum ultimate shear force at the support, v, is the ultimate shear stress obtained by regression from the values given in Table 5 of CPlM, and & has the vaiue shown in Table 14 of CPf 10.
The various Dimensions of the steei beam are constrained by the atlowable bending stress, allowable shear stress, allowable buckling stress of the web, allowable bearing stress at the web fillet, and the maximum mid-span deflection, the values of which are governed by BS449 stipulation. The above five constraints lead to the following expressions: O.l25~~~~O.~d + 2’) < pbt
(21) (22)
(23)
In order to illustrate the application of the method, an example of a modular floor system using both s&d and voided precast reinforced concrete slabs is presented in this section. 5. I The deGgn probim
A II-m-by-46-m modular floor system is to be designed to carry a range of imposed load of 3 kK m* to I5 kN/m:. The design specifications are given in Table 1. Based on the speci~~ations in Table I, the rrlp evant design data are estimated. The assumed costs of various items are C,. = $88.30 per m”,
Cf = 513.50perm’.
C, = $0.77 per kg.
C, = SO.55per kg.
C; = $6.00 per m”, = $( W$136ji)‘~‘, c, where Ws = weight of each slab in kg. The objective function and the constraints are obtained by suhstituting the values based on design spe~i~~ation and cost factors and are summa~zed below:
An application of geometric programming
%9
Table I. Design speci!ications codes
Design Concrete
strength
Reinforcement Cover
:
CPllO for
:
f cu - 25 N/mm2
:
fy - 410 Nfmm2
:
25 mm
:
100 mm (minimum)
:
Grade 43
:
3 kN/m2 to
strength
to reinforcement
Thickness Steel
of slab
beam
Imposed
load
g, = 49.44h + 13.61L-‘h
+ 4.11&-
+ 13.61s-‘h
+ 3.18A,
+ 6806 + 8.23bTs-’
g0
= 44.63AB-’
+ 13.61fL-’
+ 13.61s-‘AB-’
+ 6806 + 8.23bTs-’
+ 0.02 A,d,-’
subject to
22.49sd;’
f 0.19AA,d;’ - O.OlA:d;’
O.lZhsA;’
+ 5.77 OS A;’
45.83WdI-’
-
+ 91.67WZT
-
d 1, O.OlAs’A; Id,- ’ + 0.56 wb ?A;
- l.l8A;‘d,
1.79A:dp2
0.02bd TWI-‘r-’
+ 17.83A,B-‘d;’
s 1,
29.86sd,’
=G 1,
+ 0.02 bT2WZ-‘t-’ 141TW-’
G 1,
- 0.71T2W-’
+ 34.78TrW-’
- 0.18&W-’
+ 17.39dtW-’
a 1,
+ 59.76TrW-’ S
a 1,
O.O12dt-’ c 1. Table 2. Optimum
thickness
mm
unit unit
Flange
thickness
Flange
width
100
(L), (s),
reinforcement
Web thickness
It may be noted that the seven constraints for voided slabs on account of steel beams remain the same as those for solid slabs.
GP Method
(h),
Width of slab
Area of
Id,- ’
d 1,
design variables for solid slab system (0 = 7.2 kN/m2)
Variables
Span of slab
0.07Asbo - Id,- ’ + 3.48 obsb,
s 1,
5.2 Solution of the design problem and discussion There are several dual-based algorithms which can be used for solution of the design problem described in Section 5.1. The SIGNOPT, developed
1,
O.O3bT- ’ - 0.311T_’ C 1,
Slab
‘d; ’
s 1,
+ 75.13A,B-‘d;’
- 3OOA,B-‘d,-’
+ 6938rW-’
- 0.7ldTW-’
3.12 x 10’ Wsl-'
+ 4.lldts-’
+ 0.18bT + 0.09 dr
s 1,
+ O.O7A,d;’
+ 6072hB-’
+ 3176AA,B-’ + 1.22A’.5B-‘,5s0.5L0.’
+ 561 us2 A;‘d;’
11.77hA;‘d;‘S’
6909W-‘r
15 kNfm2
- 6072fB-’
+ 0.18bT + 0.09dz
subject to
70.51dW-’
and BS4449 for beams
For voided slabs minimize
For solid slabs minimize
+ 1.22s”2h3nL
slabs
m m (A,), m
of beam (T) mm
of
beam (b),
of beam ct.),
Depth of beam (d),
mm
mm mm
Conventional method
100
1.37
1.38
1.93
1.92
299
12.7 197 6.7 567
293
13.2 208.7 9.6 501.9
Table
3. Optimum
design variables
for voided
thickness
(h),
mm
Flange thickness voided section
of (t),
Spacing
(B),
of void
Width of slab Span of slab Area of
unit unit
the equivalent mm mm CL), m
(s),
reinforcement
Flange
width
Flange
thickness
Web thickness
Conventional method
GP Method
Variables
Slab
slab system IO = 7.2 kN.m’)
m (A,),
mm
beam (b),
of beam (t),
Depth of beam cd),
100
50
50
1500
1500
0.998
0.998
1.825
1.769 373
392 12.3
of beam CT), mm of
100
17.7
196
mm
192
6.3
mm
10.6
562
mm
428
45 degrees of difficulties. The model uses an average of 110 seconds of CPU time on an IBLM3033 and an average of 14 iterations for the given range of imposed loads. The design problem was also solved by a conventional method which uses the various stipulations of relevant codes of practice and the nearest equivalent rolled steel sections for the supporting beams. Tables 2 and 3 compare the design parameters
by Templemar(31 and used in this problem, is one such algorithm. It uses simplex linear programming to find feasible starting points and FletcharReeves’s conjugate gradient method to determine the approximate location of the optimum. The final optima are obtained by using the quadratic interpolation technique. The validity of the computational procedure has been verified on certain test problems given in [ 1I]. This GP model has I I independent variables and
-2 vi
2.:
1.9
I 1.5
IMPOSED Fig. 2. Optimum
values of design parameters
IS
11
7 LOAD, for various
0
I kN/m
imposed
1
loads.
971
An application of geometric programming Table 4. Optimum cost components for solid slab system (w = 7.2 kN/m’) Component Concrete Reinforcement Forwork Erection Steel
of
slab
beams Total
cost
X Total
$
4944
13.8
951
23.7
8505
2.6
1984
5.5
19534
54.4 100.00
35,918 I
obtained by the GP method with those obtained by the conventional method for solid slabs and voided slabs, respectively. Imposed loading for both the slabs is 7.2 kN/m’. Table 4 shows the optimum costs of components. The total optimum cost of the floor system is $35,918, while the total estimated cost of the same floor system designed by the conventional method is $39,453. Savings in total cost, if an optimum GP design is apblied, is about 9.8%. Both types of floor systems have been analysed by the GP model for imposed loads ranging from 3 to 15 kN/m’. The variation of the optimum values of a few design parameters is plotted against the imposed loads in Fig. 2, and their nonlinear relationship is clearly indicated. CONCLUSIONS It has been shown in this paper how geometric programming can be applied to optimal design of a modular floor system. From the final solutions obtained it can be seen that this computer algorithm of the GP problem is quite reliable for the class of problems it can solve. The GP model can incorporate nonlinear cost functions and code constraints. For the number of variables and degrees of difficulty encountered in this problem, the amount of CPU time required in an IBM 3033 computer, or an equivalent, is considered to be reasonable. The use of the GP model shows a cost saving of about 10% over the conventional design model. Savings in design time and cost of materials will be
significant if the model is used in computer-aided design of a large number of such floor systems with varying specifications. Aclknowle~gemenrs-The authors acknowledge with thanks the assistance of MIS Seah Harry and Lim Hin Soon, seniors in civil engineering, in this study.
REFERENCES G. S. Beveridge and R. S. Schecter. Optimization: Theory andPractice. McGraw-Hill, New York (1970). 2. D. J. Wilde and C. S. Beightler, Foundation of Optimization. Prentice-Hall, Englewood Cliffs, N.J. 1.
(1%7).
3. A. B. Templeman, Geometric programming with examples of the outimum design of floor and roof SYSten&. Int. Symb. on Cornpurer Aided Structural besign, University of Warwick, July (1972). 4. A. J. Morris, Structural optimization by geometric programming. Int. J. Solids Struct. 8.847-864 (1972). 5. R. J. Dufftn, E. L. Peterson and C. Zener, Geomerrir Programming, Theory and Applications. Wiley. New York (1%7). 6. M. Avriel and A. C. Williams, Complementary geometric programming. J. Appl. Muth. 19, 126- 141 (1970).
7. S. Ramamurthy, Structural optimization using geometric programming, pp. 63-96. Ph.D. Thesis, Cornell University (1977). 8. D. T. Phillips, A. Ravindran and J. Solberg, Operations Research Principles and Practice, pp. 552-561. Wiley, New York. 9. British Standards Institution. Code of Practicefor the Structural Use of Concrete. CPI IO.~February (1976). 10. British Standards Institution. The Use of Strucrural Steel in Buildings. BS 449. Part 2. (lW9j. 11. C. S. Beightler and D. T. Phillips, Applied Geometric Programming. Wiley, New York (1976).
Vol. 22. No. 6. pp. 965-971.
1986
004s7949M
53.00 f .MJ
0 1986Pergamon Press Ltd.
AN APPLICATION OF GEOMETRIC PROGRAMMING TO STRUCTURAL DESIGN Faculty
N. C. DAS
of Engineering,
(Received
GUPTA, H. PAUL and CHIEW HUI Yu National University of Singapore, Kent Ridge, 30 May 1984: in revised form
Singapore
OS1 I
I I March 1985)
Abstract-This paper presents an application of generalised geometric prog~mming to an optimal design of a modular floor system, which consists of reinforced solid concrete and voided slab units supported on steel beams. A function representing the cost of the floor system in terms of design variables, length, width and thickness of components, and other engineering and cost parameters is minimized subject to various constraints depending on stresses and deflections. A dualabased algorithm has been used for solving the design problem. Analyses are performed to determine the changes in the optimal values of the design variables with respect to the changes in imposed loads on the floor system.
I. INTRODUCTION
In structural engineering, the design is first governed by the choice of the structural system. Based on the functional requirements, the general layout is then defined. The structure is then analysed to suit certain performance criteria, and the acceptable designs must satisfy such performance criteria. The design also aims at finding the least cost of the structure. The methodology adopted to achieve the specific goal is optimization. One of the optim~tion techniques which has been used in some structural design problems is geometric programming (GP), The basic theory and formal proofs for this optimization technique can be found in [I] and [2]. A GP problem is an optimization with objective functions and constraints, which can be expressed in the form of signomials. Many structural design equations as stipulated by the engineering codes of practice are expressed in the form of signomials. Hence, the computational algorithms to solve GP problems are well suited to such structural optimization problems. To solve these optimization problems, Templeman[3] and Mo~s[41 used the standard GP of D&in ef af.[5], incorporating the iterative technique of Avriel and Williams161.Ramamurthy[7] used the GP model for the optimal design of a prestressed concrete slab and a light gage structural element. Templeman applied the GP model to the optimal design of bridge structures made of steel sections and steel deck plates. This paper presents the formulation of a GP model for the optimal design of a modular floor system consisting of solid and voided precast slabs supported on steel I-sections. Figure 1 shows plan and sectional views of such a floor system. The design problem presented in this paper is formulated based on the geometry, imposed loads, stresses and deflection criteria permitted by the relevant British Codes of Practice.
2. THEORY
The following is a summary of the most general formulation of the GP problem. A complete discussion and derivation of these results may be found in f2] and [Bj. A general ~lynomiaf, y,(x), is defined as
n=I
1-I
m = 0, 1, . - - , .+I,
(I)
where umt = -c t is a Signum function, and the coefficients c,~ are all positive. The primal problem is to minimize Y = Yo
(2)
subject to the constraints Ym G (Jm tE
2 f),
m=
t ,...,
M,
(3)
and .&I >
0,
ff=
1t * . * * N,
(4)
For the associated dual problem, consider a set of T variables o satisfying a normality condition 001 war = u (= +-I),
(3
t-1
N orthogonality conditions
s
*5,
CJmIamrn
QJmr =
T-nonnegativity conditions %Ir * 0,
m=O,I...., M,I=
965
I,..
0,
(6)
966
I
I
I
I SECTtON
I
I
A-A
f
;;rl.-._.-s ---!
L
1. SOLtO
As
SLAB
MlOED
SECTlON 5-a
SECTION
SLAB
EQUIYALEKT VOIDED
B-B
SECTION
SLAB B-8
i’ II
DETAIL
-I---
A
Fig. t . Plan and sectional view of modular floor system.
and M linear inequality
constraints
T,,,. From these variables, plus where T = peso the c,,,~. o,,,* and o, the dual function can be formed as follows:
and, in addition,
Then for every point x0 where v is locally minimum there exists a set of dual variables o0 satisfying eqns (348) such that
d(fJ.l”)= ?;(_r~)* d(o) In this function,
oU is defined
to be equal to + 1
(iI)
The dual function is stationary at I_&’with respect to all nonnegative w,,,~;in particular, with the global minimum .r*. if it exists, the corresponding dual var-
An application iables
w*
are such that
+
d(o”) = +*).
c,
(9) + (2”:; df)
(12)
x L, ps, c, Once the dual variables w are known, the corresponding values of the primal variables .V are found from the following relations: N Co,
n
1, . . . . T”,,
(13)
t = I, . . . , T,,, m = I, . . . , M.
(14)
xy”
=
w(),cr~“,
I
=
%7
of geometric programming
n=I 1 and
From eqn (13) it can be seen that u will have the same sign as y”. Since there will always be more terms than variables x, N equations can be found which are solvable for the N primals. in addition, the solution of these equations is not difficult since they are linear in log I,,.
L,
( > + 1 .
7
Cc, Cf, C;, C,. C,, C, are unit costs of concrete (S/m’). formwork for solid slabs ($/m’), formwork for voided slabs ($/m’), reinforcement (%/kg), steel beam ($/kg) and erection ($[ W,/l361’,“, where one unit weighs WS kg), respectively. It may be noted that C, includes the cost of material, fabrication and erection of the steel beam. s is the spacing of steel beams (ml; L and h. the width (m) and overall depth (mm) of the slab, respectively; L, and L,., length (m) and breadth (m) of the floor system. respectively; T, I, b and d, flange thickness (mm), web thickness (mm), flange width (mm) and depth of web (mm) of the steel beam, respectively; p,, and pr densities (kg/m’) of steel and concrete, respectively. B, b,, and f are the spacing of ribs (mm), width of rib (mm) and flange thickness (mm) of an equivalent voided section, respectively; A, and A are the area of the reinforcement or prestressing steel (mm’) and the cross-sectional area of the voided section (mm’), respectively.
3. OBJECTIVE FLXCTION
The objective function of the GP model incorporates the cost of concrete, formwork, reinforcing bars, steel beams and erection of precast slabs. The total variable costs are obtained as follows: The following expression gives the overall total variable cost of the floor system using precast solid slabs. go = cost of concrete of reinforcement of steel beam
+ cost of formwork + cost of erection
+ C,(A,L,)
(y)
+ (2 “;;
dr) L, Ps, C, ($
+ cost + cost
p,-, + C, (9) + I) .
(15)
Similarly the overall total variable cost of the floor system using precast voided slabs is worked out as follows: go = IO-’ C,. AB-’
L,L,,
4. CONSTRAINTS
The constraints of the geometric programming model are obtained from the following considerations: allowable bending and shear stresses of concrete and reinforcement steel; allowable bending, shear, web crippling and web crushing stresses of steel beams; allowable deflection of the slab and steel beam; minimum slab thickness and minimum reinforcement to satisfy cover and temperature stress requirements; the minimum requirements of the ratios of flange width to flange thickness and web depth to web thickness of the steel beam; and the upper limit of the ratio of depth to flange width of the steel beam. The formulation of the constraints is carried out in accordance with the BS codes of Practice CPI 10[9] for reinforced concrete slabs and BS449[ IO] for steel beams. 4.1 Bending stress requirement of slubs
At the ultimate limit state the rectangular parabolic stress block is similar in shape to the stressstrain curve for concrete, having a maximum stress of 0.45 fc,, at the ultimate strain of 0.0035. The following expression, which satisfies the ultimate bending moment at the mid-span, is obtained for the solid slab: ? x$+1.6”
f )
+ IO-’ C, A,,B-’ L,L,. p.,,
s 0.87
f?.A., 4 (
k2 0.87 A,rf>
k, x lo3
)
7 (17)
N. C. D&s Gum.-\et al.
968
f3-l) where f<,,, _f? and k, are characteristic concrete (O.jW~.~) G l~f(2~ f v’?n* strength (N/mm’), characteristic steel strength (NI 5 &J < L mm’) and mean concrete stress (N/mm’), respec(2St Z (lJs EIYGi’ tively. k2 is the ratio of the depth of the centroid of the stress block to the depth of the neutral axis. ri< where W, the distributed load carried by the steel is the effective sfab depth (mm), and w is the disbeam (N/ml for the solid slab floor system and the tributed imposed load (kN/m’). voided slab system is as follows: Similarly, the bending stress constraint for
4.2 Deflection requirement ofdabs
The ratio of span to effective depth is required to be satisfied per CPI 10 stipulation. For both solid and voided slabs, the deflection constraint generates the expression s/d= s 20kb x k,. x k,,
471:)
+ 9.81 pst
Since the flange and web of the steet beam are assumed to be unstiffened. the BS449 requirements lead to the constraints
(191 O.Sb - 03
where k, is the modi~cation factor for compression reinforcement (Table 11 of CPI IO), kb is the modification factor for the type of beam (Tables 8 and 9 of CPllO), and k, is the modification factor for tension reinforcement (Table 10 of CPE10). 4.3 Shear stress requirement of stab The effective depth of both solid and voided slabs is determined by considering the allowable shear stress. The follo~ng constraint will guarantee the minimum thickness of the slab: V bci,
s
16T.
d SG8%.
(28, 119)
The moment of inertia. 1, of the steel beam is an artificial variable used for sim~ti~~ation of intqualities[7, 8, 111, and the value of I as expressed in terms of sectional dimensions becomes I = 0.5 bTdd’ + bdT2 i
0.5 b’f” “r 0.083 rd3. (301
5. AN EXAMPLE
c
55vc,
(20)
where V is the maximum ultimate shear force at the support, v, is the ultimate shear stress obtained by regression from the values given in Table 5 of CPlM, and & has the vaiue shown in Table 14 of CPf 10.
The various Dimensions of the steei beam are constrained by the atlowable bending stress, allowable shear stress, allowable buckling stress of the web, allowable bearing stress at the web fillet, and the maximum mid-span deflection, the values of which are governed by BS449 stipulation. The above five constraints lead to the following expressions: O.l25~~~~O.~d + 2’) < pbt
(21) (22)
(23)
In order to illustrate the application of the method, an example of a modular floor system using both s&d and voided precast reinforced concrete slabs is presented in this section. 5. I The deGgn probim
A II-m-by-46-m modular floor system is to be designed to carry a range of imposed load of 3 kK m* to I5 kN/m:. The design specifications are given in Table 1. Based on the speci~~ations in Table I, the rrlp evant design data are estimated. The assumed costs of various items are C,. = $88.30 per m”,
Cf = 513.50perm’.
C, = $0.77 per kg.
C, = SO.55per kg.
C; = $6.00 per m”, = $( W$136ji)‘~‘, c, where Ws = weight of each slab in kg. The objective function and the constraints are obtained by suhstituting the values based on design spe~i~~ation and cost factors and are summa~zed below:
An application of geometric programming
%9
Table I. Design speci!ications codes
Design Concrete
strength
Reinforcement Cover
:
CPllO for
:
f cu - 25 N/mm2
:
fy - 410 Nfmm2
:
25 mm
:
100 mm (minimum)
:
Grade 43
:
3 kN/m2 to
strength
to reinforcement
Thickness Steel
of slab
beam
Imposed
load
g, = 49.44h + 13.61L-‘h
+ 4.11&-
+ 13.61s-‘h
+ 3.18A,
+ 6806 + 8.23bTs-’
g0
= 44.63AB-’
+ 13.61fL-’
+ 13.61s-‘AB-’
+ 6806 + 8.23bTs-’
+ 0.02 A,d,-’
subject to
22.49sd;’
f 0.19AA,d;’ - O.OlA:d;’
O.lZhsA;’
+ 5.77 OS A;’
45.83WdI-’
-
+ 91.67WZT
-
d 1, O.OlAs’A; Id,- ’ + 0.56 wb ?A;
- l.l8A;‘d,
1.79A:dp2
0.02bd TWI-‘r-’
+ 17.83A,B-‘d;’
s 1,
29.86sd,’
=G 1,
+ 0.02 bT2WZ-‘t-’ 141TW-’
G 1,
- 0.71T2W-’
+ 34.78TrW-’
- 0.18&W-’
+ 17.39dtW-’
a 1,
+ 59.76TrW-’ S
a 1,
O.O12dt-’ c 1. Table 2. Optimum
thickness
mm
unit unit
Flange
thickness
Flange
width
100
(L), (s),
reinforcement
Web thickness
It may be noted that the seven constraints for voided slabs on account of steel beams remain the same as those for solid slabs.
GP Method
(h),
Width of slab
Area of
Id,- ’
d 1,
design variables for solid slab system (0 = 7.2 kN/m2)
Variables
Span of slab
0.07Asbo - Id,- ’ + 3.48 obsb,
s 1,
5.2 Solution of the design problem and discussion There are several dual-based algorithms which can be used for solution of the design problem described in Section 5.1. The SIGNOPT, developed
1,
O.O3bT- ’ - 0.311T_’ C 1,
Slab
‘d; ’
s 1,
+ 75.13A,B-‘d;’
- 3OOA,B-‘d,-’
+ 6938rW-’
- 0.7ldTW-’
3.12 x 10’ Wsl-'
+ 4.lldts-’
+ 0.18bT + 0.09 dr
s 1,
+ O.O7A,d;’
+ 6072hB-’
+ 3176AA,B-’ + 1.22A’.5B-‘,5s0.5L0.’
+ 561 us2 A;‘d;’
11.77hA;‘d;‘S’
6909W-‘r
15 kNfm2
- 6072fB-’
+ 0.18bT + 0.09dz
subject to
70.51dW-’
and BS4449 for beams
For voided slabs minimize
For solid slabs minimize
+ 1.22s”2h3nL
slabs
m m (A,), m
of beam (T) mm
of
beam (b),
of beam ct.),
Depth of beam (d),
mm
mm mm
Conventional method
100
1.37
1.38
1.93
1.92
299
12.7 197 6.7 567
293
13.2 208.7 9.6 501.9
Table
3. Optimum
design variables
for voided
thickness
(h),
mm
Flange thickness voided section
of (t),
Spacing
(B),
of void
Width of slab Span of slab Area of
unit unit
the equivalent mm mm CL), m
(s),
reinforcement
Flange
width
Flange
thickness
Web thickness
Conventional method
GP Method
Variables
Slab
slab system IO = 7.2 kN.m’)
m (A,),
mm
beam (b),
of beam (t),
Depth of beam cd),
100
50
50
1500
1500
0.998
0.998
1.825
1.769 373
392 12.3
of beam CT), mm of
100
17.7
196
mm
192
6.3
mm
10.6
562
mm
428
45 degrees of difficulties. The model uses an average of 110 seconds of CPU time on an IBLM3033 and an average of 14 iterations for the given range of imposed loads. The design problem was also solved by a conventional method which uses the various stipulations of relevant codes of practice and the nearest equivalent rolled steel sections for the supporting beams. Tables 2 and 3 compare the design parameters
by Templemar(31 and used in this problem, is one such algorithm. It uses simplex linear programming to find feasible starting points and FletcharReeves’s conjugate gradient method to determine the approximate location of the optimum. The final optima are obtained by using the quadratic interpolation technique. The validity of the computational procedure has been verified on certain test problems given in [ 1I]. This GP model has I I independent variables and
-2 vi
2.:
1.9
I 1.5
IMPOSED Fig. 2. Optimum
values of design parameters
IS
11
7 LOAD, for various
0
I kN/m
imposed
1
loads.
971
An application of geometric programming Table 4. Optimum cost components for solid slab system (w = 7.2 kN/m’) Component Concrete Reinforcement Forwork Erection Steel
of
slab
beams Total
cost
X Total
$
4944
13.8
951
23.7
8505
2.6
1984
5.5
19534
54.4 100.00
35,918 I
obtained by the GP method with those obtained by the conventional method for solid slabs and voided slabs, respectively. Imposed loading for both the slabs is 7.2 kN/m’. Table 4 shows the optimum costs of components. The total optimum cost of the floor system is $35,918, while the total estimated cost of the same floor system designed by the conventional method is $39,453. Savings in total cost, if an optimum GP design is apblied, is about 9.8%. Both types of floor systems have been analysed by the GP model for imposed loads ranging from 3 to 15 kN/m’. The variation of the optimum values of a few design parameters is plotted against the imposed loads in Fig. 2, and their nonlinear relationship is clearly indicated. CONCLUSIONS It has been shown in this paper how geometric programming can be applied to optimal design of a modular floor system. From the final solutions obtained it can be seen that this computer algorithm of the GP problem is quite reliable for the class of problems it can solve. The GP model can incorporate nonlinear cost functions and code constraints. For the number of variables and degrees of difficulty encountered in this problem, the amount of CPU time required in an IBM 3033 computer, or an equivalent, is considered to be reasonable. The use of the GP model shows a cost saving of about 10% over the conventional design model. Savings in design time and cost of materials will be
significant if the model is used in computer-aided design of a large number of such floor systems with varying specifications. Aclknowle~gemenrs-The authors acknowledge with thanks the assistance of MIS Seah Harry and Lim Hin Soon, seniors in civil engineering, in this study.
REFERENCES G. S. Beveridge and R. S. Schecter. Optimization: Theory andPractice. McGraw-Hill, New York (1970). 2. D. J. Wilde and C. S. Beightler, Foundation of Optimization. Prentice-Hall, Englewood Cliffs, N.J. 1.
(1%7).
3. A. B. Templeman, Geometric programming with examples of the outimum design of floor and roof SYSten&. Int. Symb. on Cornpurer Aided Structural besign, University of Warwick, July (1972). 4. A. J. Morris, Structural optimization by geometric programming. Int. J. Solids Struct. 8.847-864 (1972). 5. R. J. Dufftn, E. L. Peterson and C. Zener, Geomerrir Programming, Theory and Applications. Wiley. New York (1%7). 6. M. Avriel and A. C. Williams, Complementary geometric programming. J. Appl. Muth. 19, 126- 141 (1970).
7. S. Ramamurthy, Structural optimization using geometric programming, pp. 63-96. Ph.D. Thesis, Cornell University (1977). 8. D. T. Phillips, A. Ravindran and J. Solberg, Operations Research Principles and Practice, pp. 552-561. Wiley, New York. 9. British Standards Institution. Code of Practicefor the Structural Use of Concrete. CPI IO.~February (1976). 10. British Standards Institution. The Use of Strucrural Steel in Buildings. BS 449. Part 2. (lW9j. 11. C. S. Beightler and D. T. Phillips, Applied Geometric Programming. Wiley, New York (1976).