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Optimal design of a hoist structure frame
Applied Mathematical Modelling 27 (2003) 963–982 www.elsevier.com/locate/apm
Optimal design of a hoist structure frame P.E. Uys a
a,*
, K. Jarmai
a,b
, J. Farkas
b
Multidisciplinary Design Optimisation Group (MDOG), Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria 0002, South Africa b Metal Structures Group, Faculty of Mechanical Engineering, University of Miskolc, H-3515 Miskolc Egytemvanoc, Hungary Received 10 July 2001; received in revised form 31 March 2003; accepted 9 June 2003
Abstract In an attempt to find the most cost effective design of a multipurpose hoisting device that can be easily mounted on and removed from a regular farm vehicle, cost optimisation including both material and manufacturing expenditure, is performed on the main frame supporting the device. The optimisation is constrained by local and global buckling and fatigue conditions. Implementation of SnymanÕs gradientbased LFOPC optimisation algorithm to the continuous optimisation problem, results in the economic determination of an unambiguous continuous solution, which is then utilised as the starting point for a neighbourhood search within the discrete set of profiles available, to attain the discrete optimum. This optimum is further investigated for a different steel grade and for the manufacturing and material cost pertaining to different countries. The effect of variations in the formulation of the objective function for optimisation is also investigated. The results indicate that considerable cost benefits can be obtained by optimisation, that costing in different countries do not necessarily result in the same most cost effective design, and that accurate formulation of the objective function, i.e. realistic mathematical modelling, is of utmost importance in obtaining the intended design optimum. 2003 Elsevier Inc. All rights reserved. Keywords: Structural optimisation; Optimal design; Optimisation algorithm; Fatigue; Buckling constraints; Cost calculation
*
Corresponding author. E-mail address: [email protected] (P.E. Uys).
0307-904X/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0307-904X(03)00128-8
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Nomenclature Ai cross-sectional area of the beam (mm2 ) surface area of the frame to be painted (mm2 ) As weld size (mm) aw profile width (mm) bi constraints gk welding technology constant Cw E elasticity modulus (MPa) F load on the frame (N) yield stress (MPa) fy profile height (mm) hi H height of the frame (mm) HA , HD1 horizontal reaction force (N) Ix , Iy second moments of inertia (mm4 ) Kxi , Kyi effective length factors painting cost factor (R/m2 ) kp manufacturing cost factor (R/kg) km welding cost factor (R/m3 ) kw L frame width (mm) weld length (mm) Lw moments about points I ¼ A, B, C, D MI axial forces (N) Ni V volume of structure (mm3 ) VA1;D1 vertical reaction force (N) elastic section modulus (mm3 ) Wxi Greeks cM1 cMf vi vLT j q hw DrNi
safety factor fatigue safety factor flexural buckling factor lateral-torsional buckling factor number of structural parts material density (kg/m3 ) difficulty factor for complexity of structure fatigue stress range for N cycles
Subscripts i ¼ 1 pertaining to vertical beam i ¼ 2 pertaining to horizontal beam k ¼ 1; . . . ; 16 pertaining to constraints w pertaining to weld min minimum max maximum
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Variables width of beam profile (mm) ti height of beam profile (mm) hi Terminology SHS square hollow section RHS rectangular hollow section 1. Introduction Within the farming community in South Africa there exists a real demand for a heavy-duty lightweight hoisting device that can easily be mounted on and removed from the regular farm vehicle. This vehicle is normally a two or four-wheel driven 1-ton light commercial vehicle referred to as a ‘‘bakkie’’. Farmers often tend to make use of scrap iron and commercially available cranes and self-construct devices that meet their demands. This poses a safety threat to users, because stress and strain strength requirements are not verified. On the other hand the economics of farming force farmers to opt for the least expensive option. These contrasting aspects are addressed in this paper in which economic factors are weighed against load and safety requirements. With respect to the mathematical modelling of the structure, rather than reverting to finite element analysis, which may be costly both in terms of setting up the model and computational time, an analytical approach proposed by Jarmai et al. [1] is used. The maximum moments in the different structural components are derived. Criteria for buckling and yielding at the maximum stressed sections of the local as well as the global structure are formulated in terms of load to be supported, and the local and global dimensions of the structure. Fatigue requirements are formulated in the same way. These criteria constitute constraints on the acceptable dimensions of the structure. A further complicating factor is that only a discrete range of structural profiles is available. The economics of the structure is optimised with due consideration to material as well as actual manufacturing (cutting, material preparation, welding, finishing, surface preparation and painting) cost. This approach constitutes a more realistic approach to modelling actual costing compared to costing based only on material costs (or structural mass), which is generally used. The importance of costing the various aspects of manufacturing is underlined by the fact that labour and manufacturing costs vary from country to country. Allowing for the refinement in the costing model can result in one structure being the most economical in one country while another structure will be more economical in another, as is indeed shown in this paper (Section 6.2). This study underlines the importance of the correct formulation of the objective function to be used for optimisation by pointing out that the computed optimum is only as reliable as the mathematical model used in its determination (Section 6.3). For optimisation the LFOPC algorithm of Snyman is used because of its proven robustness and economics in the optimisation of engineering problems. Optimisation is successfully pursued by continuous optimisation subject to maximum and minimum overall bounds on the geometry, followed by a neighbourhood search for the discrete optimum in the vicinity of the indicated continuous optimum.
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2. Formulation of the problem A supporting frame constitutes part of a hoisting device mounted on a 1-ton bakkie, a regular South African farm vehicle, as shown in Fig. 1. The front end of the device is securely fastened to the roller bar mounted on the deck adjacent to the cab. At the rear end, the frame to be optimised is fastened to the deck by bolts and supports a channel bar upon which an electric hoist is mounted. The hoist controls the vertical motion of the lifting cable. The channel rail, along which the hoist runs, extends beyond the back of the bakkie to enable the lifting of a container with mass up to 420 kg containing either liquid fuel or dry mass such as cattle feed. The overhang provided for is 1 m and the length of the bakkie back deck is 2 m implying an effective load of 6300 N at the centre of the horizontal beam. To take account of unsymmetrical load distribution and side forces during the lifting process, a horizontal load 10% the size of the vertical load, is considered for design purposes. The width of the deck is L ¼ 1:2 m and a frame height of H ¼ 1:566 m is required to ensure that the container can be lifted onto the deck and kept upright. If no longitudinal movement of the frame during lift is imperative, longitudinal braces are necessary to secure the frame. A design for the frame constructed from hollow profiles, either square or rectangular, is required which will at minimum cost, have the necessary strength to function appropriately. This constitutes a constrained design optimisation problem. In Fig. 2a the load on the supporting frame is represented by the vertical force F acting at the centre of the cross member mounted on the rear of the bakkie deck [1]. The non-centred loads are accounted for by the a horizontal force 0.1 F , acting sideways on the rear supporting frame, as introduced in Fig. 2c. The moments (MB , MC , and MA ) generated in the rear frame by the applied vertical force and the reaction forces HA and F =2 are also shown in Fig. 2a and b. Fig. 2c and d depict the vertical reactions (VA1 , VD1 ), horizontal reactions, HA1 and HD1 , and axial forces generated by the horizontal force. The maximum moment at the midpoint of the horizontal beam ME , (Fig. 2a) is given by FL MB ; ME ¼ ð1Þ 4
2m
1m
1.2 m
1,5 m
Fig. 1. Frame for a hoisting device on a regular farm truck (bakkie).
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F
B
E
C
F 2
L
F 2
F 2
D
A
N
H
M
F 2
(a)
(b)
M
N
(c)
(d)
0.1F
Fig. 2. Forces and moments on the rear frame.
where MB is the moment at the horizontal end points of the horizontal beam given by MB ¼
FL Ix2 H and k ¼ : 4ðk þ 2Þ Ix1 L
ð2Þ
The moment at A is MA ¼
MB ; 2
ð3Þ
and by symmetry MC ¼ MB and MD ¼ MA . The horizontal reaction HA due to the vertical force, is given by HA ¼
3MA ; H
and the horizontal reaction due to the horizontal force equals
ð4Þ
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HD1 ¼
0:1F ðk þ 1Þ : 2ðk þ 2Þ
ð5Þ
The moment at C due to the horizontal force is M1 ¼
0:1F 3k : 2ð6k þ 1Þ
ð6Þ
In order to apply the buckling and stress constraints to the frame, it is necessary to determine the elastic modulus, second moments of inertia and cross-sectional area of each profile. The second moments of inertia of the vertical ði ¼ 1Þ and horizontal ði ¼ 2Þ profile about the x- and y-axes respectively (see Fig. 3) are defined by " # 3 ðhi ti Þ ti ti 4ti 2 þ ðbi ti Þðhi ti Þ Ixi ¼ 1 0:86 ; ð7Þ 6 2 bi þ hi 2ti and
" Iyi ¼
# ðbi ti Þ3 ti ti 4ti 2 þ ðbi ti Þ ðhi ti Þ 1 0:86 ; 6 2 bi þ hi 2ti
ð8Þ
where bi , hi and ti are the width, height and thickness respectively of the profiles of the vertical ði ¼ 1Þ and horizontal ði ¼ 2Þ beams (Fig. 3) [1] and allowance has been made for the rounding of the corners by a radius r ¼ 2t according to Eurocode 3 [2]. The elastic section modulus is Wxi ¼
2Ixi : hi
ð9Þ
The cross-sectional area of a square or rectangular profile with rounded corners of r ¼ 2t is [2] 4ti ; ð10Þ Ai ¼ 2ti ðbi þ hi 2ti Þ 1 0:43 bi þ hi 2ti
b b y y
h
x
x t
y
b x
x t
y
Fig. 3. Dimensions of cross-sectional profiles.
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and the surface area of the rear frame is given by As ¼ 4ðhi þ bi ÞH þ 2Lðhi þ bi Þ:
ð11Þ
3. Formulation of the design constraints For the first design iteration of the hoisting device the design constraints are formulated with regard to the rear main supporting frame only. It is assumed that the bases of the column beams are fixed and that the horizontal to vertical joints are rigidly welded. Furthermore it is assumed that longitudinal movement of the frame during lift is prevented by the presence of longitudinal braces. With these assumptions in mind, the following constraints have to be satisfied by the structure: 3.1. Global stress constraint of the horizontal beam The horizontal beam, i ¼ 2, has to comply with the overall stress constraint for bending and axial compression given by Eurocode 3 [2]: HA þ HD1 kM2 ME þ 6 1; v2: min A2 fy1 Wx2 fy1
ð12Þ
f
y where fy1 ¼ cM1 is the yield stress and cM1 ¼ 1:1 is a safety factor. 1 is the flexural buckling factor, with Here vi min ¼ 2 /i þ ð/i k2i max Þ0:5
/i ¼ 0:5b1 þ 0:34ðki max 0:2Þ þ k2i max c and ki max ¼ maxðkxi ; kyi Þ; Ksub L ; ksub ¼ rsub kE
rsub ¼
Isub Ai
0:5 ;
0:5 E kE ¼ p ; fy
i ¼ 1; 2;
sub ¼ x2; y2
and Kx2 ¼ Ky2 ¼ 0:5 is the effective length factors. Furthermore kM2 ¼ 1 þ
1; 2kx2 ðHA þ HD1 Þ ; ðv2 A2 fy Þ
where v2 is calculated the same as v2 min with kx2 . 3.2. Local buckling of the horizontal beam Constraints on local buckling of the horizontal beam require that b2 3t2 6 42e2 ; t2
ð13Þ
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where e2 ¼
235 rmax 2
0:5 and rmax 2 ¼
HA þ HD1 ME þ ; A2 Wx2
to prevent compression of the flange of the beam. With regard to the webs it is necessary that [2] h2 3t2 42e2 6 t2 0:67 þ 0:33w21
if w2 > 1;
ð14aÞ
and h2 3t2 6 62e2 ð1 w2 Þðw2 Þ0:5 t2
if w2 6 1;
ð14bÞ
where ME N2 Wx2 A2 and N2 ¼ HA þ HD1 : w2 ¼ ME N2 þ Wx2 A2 3.3. Global buckling of the column Stress constraints on the global buckling of the column (the stress criteria at point C, Fig. 2) imply that [2] N1 kM1 ðM1 þ MC Þ þ 6 1; Wx1 fy1 v1 min A1 fy1
ð15Þ
where kM1 ¼ 1
0:3kx1 N1 ; vx1 A1 fy1
N1 ¼
F þ VD1 2
and VD1 ¼
2M1 ; L
and the same equations for v1 min , u1 and k1 max as above apply for i ¼ 1, but Kx1 H Ky1 H ; Kx1 ¼ 2:19; ky1 ¼ and Ky1 ¼ 0:5: kx1 ¼ rx1 kE ry1 kE 3.4. Local buckling of the column To prevent local buckling of the vertical columns the same criteria as summarised for constraint 2 apply, but in this case, i ¼ 1 b1 3t1 6 42e1 ; t1 to prevent compression of the flange. With regard to the webs
ð16Þ
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h1 3t1 42e1 6 t1 0:67 þ 0:33w1
if w1 > 1;
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ð17aÞ
and h1 3t1 6 62e1 ð1 w1 Þðw1 Þ0:5 t1
if w1 6 1;
ð17bÞ
with e1 ¼
235 rmax 1
0:5 ;
rmax 1 ¼
N1 M1 þ MC þ ; A1 Wx1
and M1 þ MC N1 Wx1 A1 : w1 ¼ M1 þ MC N1 þ Wx1 A1 3.5. Fatigue stress Because of the cyclic mode of the loading and unloading process it is also necessary to consider the fatigue stress constraint for the horizontal beam at the midpoint (point E) and for the columns at the welded joints (point C). Complying with the requirements of the International Institute of Welding as amended by Hobbacher, Jarmai et al. [1] derived the constraints HA þ HD1 ME DrN 2 þ 6 ; A2 Wx2 cMf
ð18Þ
N1 M1 þ MC DrN1 þ 6 ; A1 Wx1 cMf
ð19Þ
and
N2 N1 where Dr ¼ 231 MPa and Dr ¼ 146 MPa. cMf cMf These values have been derived for 105 cycles, a static safety factor of 1.5 and a fatigue safety factor of 1.25. Clearly the satisfaction of the above stress, buckling and fatigue constraints, Eqs. (12)–(19), depends on the physical dimensions of the profiles (see Fig. 3). These dimensions xi , i ¼ 1; 2; . . . ; n, represented by the vector x ¼ ðx1 ; x2 ; . . . ; xn Þ, may be taken as the design variables.
4. Formulation of the objective function The particular objective function to be minimized here with respect to the design variables x, takes into account material costs Km , painting costs Kp and welding costs Kw , i.e. the cost function f ðxÞ is defined by
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f ðxÞ ¼ KðxÞ ¼ Km ðxÞ þ Kw ðxÞ þ Kp ðxÞ:
ð20Þ
4.1. Material cost function The material cost Km is found by multiplying the mass of the beam by the material cost factor km . The price lists used for rectangular and square tubing, were obtained from Robor Stewardts & Lloyds [3], the distributor of one of the main steel manufacturers, Robor Steel, in South Africa. The average prices of the standard profiles were found to be R 10.80/kg thus km ¼ 10.80 R/kg (R 8 ¼ $1). 4.2. Welding cost function The expression for welding costs, Eq. (21), has been derived by Jarmai and Farkas [4]. ! X 0:5 n Kw ¼ kw Hw ðjqV Þ þ 1:3 Cwi awi Lwi ;
ð21Þ
i
where kw is the welding cost factor in Rands/minute and the other parameters are discussed below. As indicated in the report on the welded tubular frame for a special truck [1], shielded metal arc welding (SMAW) of the tubes and braces is considered. A difficulty factor of hw ¼ 3 is assumed which reflects the complexity of the structure with regard to assembly and welding. The number of members is j ¼ 7, since there are 3 bars, 2 splice plates and 2 base plates to be assembled [1]. For fillet welds made by hand welding, the welding technology constant [1,4] equals Cw ¼ 0:7889 103 and the time of welding, deslagging, changing the electrode etc, (i.e. the second term of Eq. (21)) depends on the welding technology, type of welds, weld size ðaw Þ and weld length Lw , where i refers to the ith element and the value of n is derived from curve fitting calculations for the various welding techniques [4]. In a previous study relating to British Constructional Steel Tables and European manufacturing costs a welding cost factor kw , of $1/min is used by Jarmai et al. [1]. South African industrial statistics indicate that the labour costs for specialized welding is R 50/h (R 8 $1), i.e., R 0.83/min [5]. Welding rods applicable for the welding of thin walled tubes are available at R 18.52/ kg [6]. At a consumption rate of 0.0986 kg/m and a welding rate of 1.6 m/min [7], electrode costs amount to R 1.12/min. Adding the electrode and labour costs imply a welding cost factor of km ¼ 1:95 R/min. This figure does not include overhead costs. 4.3. Painting cost function The painting cost Kp is obtained by multiplying the painting cost factor kp with the surface area. A painting cost factor of kp ¼ R 20:27 106 /mm2 has been determined. This amount includes R 123.18 (without value added tax, VAT) for 5 l of undercoat which covers 6 m2 and R 181.59 for 5 l of car duco enamel topcoat that covers 7 m2 (tax excluded) [8]. Combining the paint application factors of 3 · 106 min/mm2 for ground coat application and 4.15 · 106 (min/mm2 ) given by Jarmai et al. [4], with the labour rate of R 20/h [5] and the paint costs for one layer of undercoat and two layers of topcoat, gives the stated result.
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5. Optimisation methodology The optimisation problem is solved by means of the leap-frog algorithm for constrained optimisation (LFOPC) of Snyman [9–11]. This gradient-based method, requiring no explicit line searches, is a proven robust and reliable method, being relatively insensitive to local inaccuracies and discontinuities in the gradients. As the gradients are to be computed here by relatively rough forward finite difference approximations, the leap-frog method should be ideally suitable for the current problem. The algorithm in general aims to minimize the objective function f ðxÞ, x 2 Rn subject to inequality constraints gi ðxÞ ¼ 0, i ¼ 1; 2; . . . ; m and equality constraints hj ðxÞ ¼ 0, j ¼ 1; 2; . . . ; r. The particular choice of design variables, being the width and wall thickness of the respective profiles of the columns and transverse beams, are as listed in Table 1. The objective cost function is related to these design variables by Eq. (20). This function includes the material costs and the painting costs for the structure as well as the cost of welding the transverse beam to the columns, the cost of welding the braces and the cost of welding the columns to the base, where cost of welding includes preparation, change of electrodes, deslagging and finishing. The constraints gi ðxÞ ¼ 0, i ¼ 1; 2; . . . ; 16, are listed in Table 2. In addition to the stress, buckling and fatigue constraints already discussed and described by Eqs. (12)–(19), upper and
Table 1 Design variables Variable
Description
Symbol
x1 x2 x3 x4
Width of column Wall thickness of column Width of transverse beam Thickness of transverse beam
b1 t1 b2 t2
Table 2 Description of inequality constraints Nature of constraint
Symbol
Minimum width of column, lower bound Maximum width of column, upper bound Minimum thickness of transverse beam, lower bound Maximum thickness of transverse beam, upper bound Overall buckling of the transverse beam, (Eq. (12)) Overall buckling of the column, (Eq. (15)) Local buckling of column flanges, (Eq. (16)) Local buckling of the flanges of the transverse beam, (Eq. (13)) Local buckling of the webs of the transverse beam, (Eqs. (14a),(14b)) Local buckling of column webs, (Eqs. (17a),(17b)) Fatigue constraints on the column, (Eq. (19)) Fatigue constraints on the transverse beam, (Eq. (18)) Minimum width of transverse beam, lower bound Maximum width of transverse beam, upper bound Minimum thickness of column, lower bound Maximum thickness of column, upper bound
g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 g16
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lower bounds on the design variables are also imposed in line with available profile dimensions. No equality constraints are prescribed. Initially the problem is treated as being continuous in the solution space and the associated optimum solution is obtained. An acceptable discrete optimum solution is then sought by inspection of several candidate discrete solutions in the neighbourhood of the continuous optimum. The candidate (available) discrete solutions are obtained from the steel tables of the Southern African Institute of Steel Construction [12] and Robor Cold Form [13]. The candidate discrete solutions are rated by evaluating the corresponding objective function for each candidate and determining to what extent they also satisfy the constraints. Finally, a design constructed from the available profiles with the lowest objective function value, and which also complies with the constraints within reasonable tolerances, is chosen as the final discrete optimum solution. Two profiles are considered: rectangular hollow sections (RHS) and square hollow sections (SHS). With regard to these profiles two possibilities are considered, one where the transverse beams and columns have the same profiles and the other where the column and transverse beam profiles may differ.
6. Numerical results 6.1. Results in terms of the optimisation process Table 3 lists the computational results of the optimisation process for a nominal yield stress of fy ¼ 235 MPa. The optimisation was carried out with the LFOPC convergence tolerances set at ex ¼ 105 and eg ¼ 105 and the values of the penalty parameters given by l0 ¼ 102 and l1 ¼ 104 . The maximum prescribed step size was chosen p toffiffiffibe d ¼ 5 which is of the order of the diameter of the region of interest by the relation d ¼ Rmax n, where Rmax is the maximum variable range and n the number of variables. In computing the forward finite difference approximation to the gradients a variable step size of Dxi ¼ 106 was used. For each optimum the components of the corresponding design vector are listed to the right, followed by the associated cost function value. Further information is listed regarding the active and violated constraints. In the final column to the right, the total number of LFOPC algorithm steps required for convergence to the specified accuracy, is given for the continuous solution. The results show that apart from dimensional restrictions, the constraints which are most active is that of cyclic fatigue at the welded joints of the column and global buckling of the transverse beam. Optimising the square profiles tends to require more iterations for convergence than that for rectangular profiles, as is apparent from Table 3 by comparing differing RHS and SHS profiles (this is for the same starting point, x1 ¼ 30, x2 ¼ 3, x3 ¼ 30, x4 ¼ 3). In order to verify that a global optimum had indeed been obtained, the optimisation was performed with different initial designs for the case of differing rectangular sections (with fy ¼ 355 MPa and kp ¼ 11:68 R/min). The same optimum was obtained regardless of the initial values. The importance of the optimisation parameters, ex , eg , l0 , l1 and d was apparent when problems occurred due to violation of constraints. It was experienced that the LFOPC parameter values used and which correspond with the directives given by Snyman [11], ensured convergence in all cases.
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Table 3 Results of optimisation ðf ðxÞ ¼ Km ðxÞ þ Kw ðxÞ þ Kp ðxÞÞ x1
x2
x3
x4
f
Active constraints
35.9 30 30 30 38.1 38.1 38.1
1.6 2 2.5 2 1.6 1.6 1.6
61.4 63.5 63.5 60 63.5 60 60
2 2 2 2 2 2 2
253.1 257.2 283.8 251.2 264.2 258.2 258.2
g3 , g11 , g3
44.1 40 40 50 50
2 2.5 3 2 2
44.1 40 40 50 50
2 2.5 3 2 2
285.5 312.8 364.5 321 321
c3 , c5
None g5 ¼ 0:025 None None None
200
56.9 57.2 50 57.2 57.2 50 50 50 57.2
1.6 1.6 2 1.6 1.6 1.6 1.6 2 1.6
96.4 90 90 90 100 100 100 100 100
2 2 2 3 2 2 2.5 2 2
264.4 257.8 265.1 305.1 269.2 252.22 278.6 276.5 269.2
g3 , g11 , g13
None g11 g11 None None g11 g11 None None
4041
2 2 2.5 3 2 2
69.7 70 63.5 63.5 76.2 70
2 2 2.5 3 2 2
299.3 300.3 329.4 384 325.1 300.3
g3 , g5
Best
69.7 70 63.5 63.5 76.2 70
Best continuous Different RHS
35.9
1.6
61.4
2
253.1
69.7
2
69.7
2
299.3 46.2
38.1
1.6
60
2
258.2
50
2
50
2
321 62.8
Scenario Different RHS Continuous optimum Discrete candidate
Best Equal RHS Continuous Discrete
Best Different SHS Continuous Discrete
Best Equal SHS Continuous Discrete
Worst continuous Equal SHS Difference Best discrete Different RHS Worst discrete Equal RHS Difference
Constraints violated
LFOPC steps 2140
g11 None g11 None None None
468 None g5 None None None
g3 , g5
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Figs. 4 and 5 reflect the convergence histories of the objective function and the design variables for the continuous problem of different RHS profiles for the vertical columns and horizontal beams. This convergence behaviour is typical for the different scenarios. 350 300 250 200
f 150 100 50 0 0
500
1000
1500
2000
2500
step
Fig. 4. Convergence history of objective function for different RHS.
70
Design variables
60 50 40
x1 x3
30 20 10 0 0
500
1000
(a)
1500
2000
2500
Steps
3.5
Design variables
3 2.5 2
x2 x4
1.5 1 0.5 0 0
(b)
200
400
600
800
1000
1200
Step
Fig. 5. Convergence history of design variables for different RHS.
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Table 4 Optimal dimension in mm using UK (British Steel, Tizani [16]) cost data Profiles
Continuous solution 3
Discrete solution
Dimension (mm )
Cost $
Dimension (mm3 )
Equal RHS Different RHS
44.1 · 88.2 · 2 40.45 · 80.9 · 2 47.15 · 94.3 · 2
73.9 71.6
40 · 80 · 3 40 · 80 · 3 40 · 80 · 3
Equal SHS Different SHS
69.7 · 2 59.4 · 59.4 · 2 80.1 · 80.1 · 2
76.8 73.2
70 · 70 · 3 50 · 50 · 2.5 80 · 80 · 3
Cost $ 90.3 90.3 102.1 82
6.2. Results with regard to the physical quantities It is apparent from Table 3 that the best final candidate is rectangular tubing with different dimensions for the transverse beams and columns. The worst case is equal RHS sections. The difference between the worst and the best value is R321 R258 ¼ R63($7.88) (a 20% improvement relative to the worst). It is of interest to compare these values with that of Jarmai et al. [1] for Hungarian conditions based on British Steel sections, which are listed in Table 4. They found the most expensive solution to be similar SHS profiles and the cheapest to be two different SHS profiles, the price difference being $20.10 (20% variation relative to the worst)––which is a similar result to that found for South African conditions by way of percentage. In the current study, for the continuous optima, the cheapest alternative is different RHS profiles and the most expensive equal SHS profiles. The difference between the worst and best solutions is R299:30 R253:10 ¼ R46:20 ($5.78) representing a 15% improvement relative to the worst. Similarly Jarmai et al. [1] found different RHS profiles to be the cheapest alternative for the continuous optima and similar SHS profiles were the most expensive. They obtained the difference in the price extremes in this case to be $5.20 (i.e.7%). The differences between the Hungarian and South African determined optima are of course due to the differences in cost structures and available profiles. Table 5 summarizes the differences between the price and profile structures of the two countries for comparable scenarios. 6.3. Results concerning formulation of the objective function The importance of the various terms of the cost function was also investigated. The results are given in Table 6 for different rectangular hollow sections of the column and beam. The value of the objective function and the related column and beam dimensions are given for four formulations of the objective function, i.e. where the objective function (1) includes only material costs, (2) consists of material and welding costs, (3) is defined only in terms of welding costs, (4) takes account of painting costs only and (5) includes both welding and painting costs but not material costs. Whereas the material costs constitute 63% of the total cost (total objective function value) and the welding costs constitute 28%, the painting costs contribute some 7% to the total cost function. Considering either material cost or welding cost or both in the cost function, result in very much the same optimum, but if the painting cost is considered the optimum design differs considerably. This difference can be ascribed to the fact that painting costs increase as surface area increases and
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Table 5 Comparable Hungarian (British Steel sections) and South African Scenarios (Dimensions in mm) Hungary
South Africa
Different RHS 40.45 · 80.9 · 2; 47.15 · 94.3 · 2 71.60
Different RHS 35.9 · 71.8 · 1.6; 60 · 120 · 2 31.64
Different SHS 50 · 50 · 2.5; 80 · 80 · 3 82.00
Different RHS 38.1 · 76.2 · 1.6; 60 · 120 · 2 32.28
Equal SHS 69.7 · 69.7 · 2 76.80
Equal SHS 69.7 · 69.7 · 2 33.65
Profile Cost in $
Equal SHS 70 · 70 · 3 $102.10
Equal RHS 50 · 100 · 2 $37.41
Cost function Welding cost constant Painting cost constant Material cost constant
$1.00/min $14.40/m2 $1.00/kg
$0.24/min $2.53/m2 $1.35/kg
Available material SHS minimum SHS maximum RHS minimum RHS maximum
20 · 20 · 2 150 · 150 · 4 50 · 25 · 2 100 · 200 · 4
12.7 · 12.7 · 1.6 300 · 300 · 10 12.7 · 25.4 · 1.6 100 · 200 · 10
Continuous optimum Profile Cost in $ Discrete optimum Profile Cost in $ Continuous worst Profile Cost in $ Discrete worst
Table 6 Optimisation results for the continuous problem for different RHS using different cost functions Cost function
x1
x2
x3
x4
Function value
Km þ Kw þ Kp Km Km þ Kw Kw Kw þ Kp Kp
35.9 35.9 36.1 37.1 36.1 26.1
1.6 1.6 1.6 1.6 1.6 4
61.4 61.5 61.1 59.3 61.1 46.9
2 2 2 2 2 4
253.1 158.6 230.4 71.8 94.5 16.8
the best result would be that of minimum surface area. On the other hand material and welding costs are to a large extent related to material thickness. If painting costs are equated to that of the Hungarian option, i.e., R115.20/m2 , the best design is 35.1 · 70.2 · 1.6; 62.1 · 124.2 · 2 mm3 compared to 38.1 · 76.2 · 1.6; 60 · 120 · 2 for painting costs evaluated at R 20–27/m2 .
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The formulation of the cost function clearly influences what would be considered the best result. This emphasizes the importance of formulating the objective function correctly and of weighting the various criteria against one another in order to obtain the most acceptable result. The results also indicate that considering only material costs for this kind of structure may give a good approximation to the best design.
7. The influence of minimum yield stress value The guaranteed minimum yield stress fy for hollow sections in South Africa is given as 200 MPa in the South African Steel Construction Handbook (1987) [14]. A switch over has been made to steel with a yield strength of 300 MPa [15]. In comparison, a yield stress value of 235 MPa apply for the British steel profiles distributed in Europe and this value was indeed used in the first part of this study. To determine the effect of changing the prescribed value of fy to 300 MPa, the analysis was repeated with the latter value and the results are as listed in Table 7. In Table 8 the optimum results for the two cases are compared. As is apparent from Table 8, there is little difference in the continuous optima (with all the constraints satisfied) for the two different cases, although there is a difference in the overall buckling constraint of both the column ðg6 Þ and transverse ðg5 Þ beams and in the fatigue stress constraint values, g11 and g12 . For similar profiles in the columns and transverse beams, the cost functions and optimum profiles differed more. The difference in the optimum solutions can be ascribed to the fact that the constraint on the overall buckling of the transverse beam ðg5 Þ becomes active in the case where fy ¼ 235 MPa. Even though the dimensions of the optimum profiles for fy ¼ 235 and 300 MPa differ from one another, the differences are small and given the available profiles, it can be seen by comparing Tables 3 and 7, that the discrete solutions for the case fy ¼ 300 MPa also satisfies the constraints in the case of fy ¼ 235 MPa except for the case of equal RHS, where constraint g5 is just violated. The solution is thus not very sensitive to the value of fy but overall buckling of the transverse beam should be given particular attention.
8. Conclusion and recommendations This study shows that significant savings can be realised by seeking an optimised design via mathematical programming. The use of a realistic mathematical model that not only takes into consideration material costs but also manufacturing costs, constitute an additional refinement that may prove to be of considerable importance, particularly in the case of the design of more complex structures requiring sophisticated manufacturing procedures. The particular usage of hollow sections, i.e., rectangular and square tubing, has been considered. Standards for buckling, welding and fatigue constraints for hollow sections have only recently been formulated by the European Committee for Standardization in 1996 [2]. Experience with the application of these criteria has further enhanced the value of the presented study and underlines the applicability of the criteria spelt out in these Standards. In view of the difference in manufacturing, material and painting costs in South Africa and Hungary/Britain, it is recommended that a more complete study be made of the calculation of
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P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
Table 7 Optimisation results for fy ¼ 300 MPa Scenario Different RHS Continuous Discrete
Best Equal RHS Continuous Discrete Best Different SHS Continuous Discrete
Best Equal SHS Continuous Discrete Best Best continuous Different RHS Worst continuous Equal SHS Difference Best discrete Different RHS Worst discrete Equal RHS Difference
x4
f
Active constraints
61.4 60 60 60 60 60
2 2 2 2 2 2
253.1 258.2 286.5 251.2 277.9 258.2
g3 , g11 , g13
2 2.5 2 2.5
43 40 50 40
2 2.5 2 2.5
278.9 312.8 321 312.8
g3 , g11
57 57.2 57.2 50 45 45 45 50 57.2
1.6 1.6 1.6 2 2 2.5 2.5 2.5 1.6
96.1 90 100 100 100 100 90 90 100
2 2 2 2 2 2 2 2 2
264.4 257.8 269.2 276.2 261.7 288.2 276.9 294.8 269.2
g3 , g11 , g13
67.9 70 63.5 70
2 2 2.5 2
67.9 70 63.5 70
2 2 2.5 2
291.8 300 329 300
g3 , g11
35.9
1.6
61.4
2
253.1
g3 , g11 , g13
67.9
2
67.9
2
291.8 38.7
g3 , g11
38.1
1.6
60
2
258.2
None
40
2.5
40
2.5
312.8 54.6
None
x1
x2
35.9 38.1 38.1 30 30 38.1
1.6 1.6 2 2 2.5 1.6
43 40 50 40
x3
Constraints violated
LFOPC steps 2662
None None c11 None None 247 None None None 3219 g11 None None g11 None g11 None None 383 None None None
these values. This will be of prime importance if cost functions are to be extended to include manufacturing costs, and if a comparison with international values is to be made. Although only the rear frame of the framework has been taken into account in this study, the advantages of design optimisation have been illustrated and the indications are that the techniques can in general be extended to the complete and more complex structure.
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Table 8 Effect of yield strength values on optimal results f ¼ 300 MPa
Optimum profile
f ¼ 235 MPa
Optimum profile
Profile Function value g5 g6 g11 g12
Different RHS 253.1 0.5006 0.35 0.4236 0.9493
35.9 · 1.6; 61.4 · 2
Different RHS 253.1 0.3622 0.2193 0.3427 0.9488
35.9 · 1.6; 61.4 · 2
Profile Function value g5 g6 g11 g12
Equal RHS 278.8 0.1735 0.4162 0.4624 0.6429
43.0 · 2
Equal RHS 285.4 0.8323 0.3141 0.7325 0.1814
44.1 · 2
Profile Function value g5 g6 g11 g12
Different SHS 264.6 0.4752 0.3255 0.1146 0.8801
56 · 1.6; 94.2 · 2
Different SHS 264.4 0.3531 0.1882 0.32 0.9292
56.9 · 1.6; 96.4 · 2
Profile Function value g5 g6 g11 g12
Equal SHS 291.8 0.1702 0.399 0.6259 0.5441
67.9 · 2
Equal SHS 299.3 0.2582 0.302 0.7877 0.1806
69.7 · 2
Finally the ease with which optimum constrained solutions were computed in this study confirms the applicability of the LFOPC optimisation algorithm for structural problems where a variety of different physical constraints such as buckling, fatigue and dimensional constraints apply. Acknowledgements The authors wish to acknowledge financial support for this study from the Hungarian and South African Governments via the Hungarian––South African Intergovernmental S&T Cooperation programme for 2000–2002. This work was done within the project ‘‘Optimum design of tubular and framed structures’’ with coordinators Prof J Karoly (Hungary) and Prof JA Snyman (South Africa), OTKA, FKFP. References [1] K. Jarmai, J. Farkas, P. Visser-Uys, Minimum cost design of welded tubular frames for a special truck, IIWDoc.XV-1085-WG9-09-01,XV-1085-01 International Institute of Welding Annual Assembly, Ljubjana, 8–11 July 2001, 12 p.
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[2] European Prestandard ENV 1993-1-3 Eurocode 3: Design of steel structures––Part 1–3: General rules–– Supplementary rules for cold formed thin gauge members and sheeting, 22, 25–26, 69. [3] Fax quotation, Robor Stewardts & Loyds, 2001. [4] K. Jarmai, J. Farkas, Cost calculation and optimisation of welded steel structures, J. Constr. Steel Res. 50 (1999) 115–135. [5] Metal and Engineering Industries Bargaining Council, private communication, April 2001. [6] Afrox, private communication, 2001. [7] The Lincoln Electric Company; The Procedure Handbook of Arc Welding, 13th ed., The Lincoln Electric Company, Cleveland, OH, 1994, pp. 6-2-24. [8] Paint Sales Warehouse, private communication, 2001. [9] J.A. Snyman, A new dynamic method for unconstrained minimization, Appl. Math. Model. 7 (1983) 216–218. [10] J.A. Snyman, An improved version of the original leap-frog dynamic method for the unconstrained minimization LFOP1(b), Appl. Math. Model. 6 (1982) 449–462. [11] J.A. Snyman, The LFOPC leap frog algorithm for constrained optimisation, Comput. Math. Appl. 40 (2000) 1085– 1096. [12] Structural Steel Tables, Seventh ed., The Southern African Institute of Steel Construction, Johannesburg, 1977. [13] Robor Cold Form; Product catalogue, 2001. [14] South African Steel Construction Handbook, The South African Institute of Steel Construction, Johannesburg, 1987, 2.41. [15] South African Steel Construction Handbook, The South African Institute of Steel Construction, Johannesburg, 1999, 1.5. [16] W.M.K. Tizani, G. Davies, A.S. Whitehead, A knowledge based system to support joint fabrication decision making at the design stage––case studies for CHS trusses, in: J. Farkas, K. Jarmai (Eds.), Tubular Structures VII. Balkema, Rotterdam, 1996, pp. 483–489.
Optimal design of a hoist structure frame P.E. Uys a
a,*
, K. Jarmai
a,b
, J. Farkas
b
Multidisciplinary Design Optimisation Group (MDOG), Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria 0002, South Africa b Metal Structures Group, Faculty of Mechanical Engineering, University of Miskolc, H-3515 Miskolc Egytemvanoc, Hungary Received 10 July 2001; received in revised form 31 March 2003; accepted 9 June 2003
Abstract In an attempt to find the most cost effective design of a multipurpose hoisting device that can be easily mounted on and removed from a regular farm vehicle, cost optimisation including both material and manufacturing expenditure, is performed on the main frame supporting the device. The optimisation is constrained by local and global buckling and fatigue conditions. Implementation of SnymanÕs gradientbased LFOPC optimisation algorithm to the continuous optimisation problem, results in the economic determination of an unambiguous continuous solution, which is then utilised as the starting point for a neighbourhood search within the discrete set of profiles available, to attain the discrete optimum. This optimum is further investigated for a different steel grade and for the manufacturing and material cost pertaining to different countries. The effect of variations in the formulation of the objective function for optimisation is also investigated. The results indicate that considerable cost benefits can be obtained by optimisation, that costing in different countries do not necessarily result in the same most cost effective design, and that accurate formulation of the objective function, i.e. realistic mathematical modelling, is of utmost importance in obtaining the intended design optimum. 2003 Elsevier Inc. All rights reserved. Keywords: Structural optimisation; Optimal design; Optimisation algorithm; Fatigue; Buckling constraints; Cost calculation
*
Corresponding author. E-mail address: [email protected] (P.E. Uys).
0307-904X/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0307-904X(03)00128-8
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Nomenclature Ai cross-sectional area of the beam (mm2 ) surface area of the frame to be painted (mm2 ) As weld size (mm) aw profile width (mm) bi constraints gk welding technology constant Cw E elasticity modulus (MPa) F load on the frame (N) yield stress (MPa) fy profile height (mm) hi H height of the frame (mm) HA , HD1 horizontal reaction force (N) Ix , Iy second moments of inertia (mm4 ) Kxi , Kyi effective length factors painting cost factor (R/m2 ) kp manufacturing cost factor (R/kg) km welding cost factor (R/m3 ) kw L frame width (mm) weld length (mm) Lw moments about points I ¼ A, B, C, D MI axial forces (N) Ni V volume of structure (mm3 ) VA1;D1 vertical reaction force (N) elastic section modulus (mm3 ) Wxi Greeks cM1 cMf vi vLT j q hw DrNi
safety factor fatigue safety factor flexural buckling factor lateral-torsional buckling factor number of structural parts material density (kg/m3 ) difficulty factor for complexity of structure fatigue stress range for N cycles
Subscripts i ¼ 1 pertaining to vertical beam i ¼ 2 pertaining to horizontal beam k ¼ 1; . . . ; 16 pertaining to constraints w pertaining to weld min minimum max maximum
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965
Variables width of beam profile (mm) ti height of beam profile (mm) hi Terminology SHS square hollow section RHS rectangular hollow section 1. Introduction Within the farming community in South Africa there exists a real demand for a heavy-duty lightweight hoisting device that can easily be mounted on and removed from the regular farm vehicle. This vehicle is normally a two or four-wheel driven 1-ton light commercial vehicle referred to as a ‘‘bakkie’’. Farmers often tend to make use of scrap iron and commercially available cranes and self-construct devices that meet their demands. This poses a safety threat to users, because stress and strain strength requirements are not verified. On the other hand the economics of farming force farmers to opt for the least expensive option. These contrasting aspects are addressed in this paper in which economic factors are weighed against load and safety requirements. With respect to the mathematical modelling of the structure, rather than reverting to finite element analysis, which may be costly both in terms of setting up the model and computational time, an analytical approach proposed by Jarmai et al. [1] is used. The maximum moments in the different structural components are derived. Criteria for buckling and yielding at the maximum stressed sections of the local as well as the global structure are formulated in terms of load to be supported, and the local and global dimensions of the structure. Fatigue requirements are formulated in the same way. These criteria constitute constraints on the acceptable dimensions of the structure. A further complicating factor is that only a discrete range of structural profiles is available. The economics of the structure is optimised with due consideration to material as well as actual manufacturing (cutting, material preparation, welding, finishing, surface preparation and painting) cost. This approach constitutes a more realistic approach to modelling actual costing compared to costing based only on material costs (or structural mass), which is generally used. The importance of costing the various aspects of manufacturing is underlined by the fact that labour and manufacturing costs vary from country to country. Allowing for the refinement in the costing model can result in one structure being the most economical in one country while another structure will be more economical in another, as is indeed shown in this paper (Section 6.2). This study underlines the importance of the correct formulation of the objective function to be used for optimisation by pointing out that the computed optimum is only as reliable as the mathematical model used in its determination (Section 6.3). For optimisation the LFOPC algorithm of Snyman is used because of its proven robustness and economics in the optimisation of engineering problems. Optimisation is successfully pursued by continuous optimisation subject to maximum and minimum overall bounds on the geometry, followed by a neighbourhood search for the discrete optimum in the vicinity of the indicated continuous optimum.
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2. Formulation of the problem A supporting frame constitutes part of a hoisting device mounted on a 1-ton bakkie, a regular South African farm vehicle, as shown in Fig. 1. The front end of the device is securely fastened to the roller bar mounted on the deck adjacent to the cab. At the rear end, the frame to be optimised is fastened to the deck by bolts and supports a channel bar upon which an electric hoist is mounted. The hoist controls the vertical motion of the lifting cable. The channel rail, along which the hoist runs, extends beyond the back of the bakkie to enable the lifting of a container with mass up to 420 kg containing either liquid fuel or dry mass such as cattle feed. The overhang provided for is 1 m and the length of the bakkie back deck is 2 m implying an effective load of 6300 N at the centre of the horizontal beam. To take account of unsymmetrical load distribution and side forces during the lifting process, a horizontal load 10% the size of the vertical load, is considered for design purposes. The width of the deck is L ¼ 1:2 m and a frame height of H ¼ 1:566 m is required to ensure that the container can be lifted onto the deck and kept upright. If no longitudinal movement of the frame during lift is imperative, longitudinal braces are necessary to secure the frame. A design for the frame constructed from hollow profiles, either square or rectangular, is required which will at minimum cost, have the necessary strength to function appropriately. This constitutes a constrained design optimisation problem. In Fig. 2a the load on the supporting frame is represented by the vertical force F acting at the centre of the cross member mounted on the rear of the bakkie deck [1]. The non-centred loads are accounted for by the a horizontal force 0.1 F , acting sideways on the rear supporting frame, as introduced in Fig. 2c. The moments (MB , MC , and MA ) generated in the rear frame by the applied vertical force and the reaction forces HA and F =2 are also shown in Fig. 2a and b. Fig. 2c and d depict the vertical reactions (VA1 , VD1 ), horizontal reactions, HA1 and HD1 , and axial forces generated by the horizontal force. The maximum moment at the midpoint of the horizontal beam ME , (Fig. 2a) is given by FL MB ; ME ¼ ð1Þ 4
2m
1m
1.2 m
1,5 m
Fig. 1. Frame for a hoisting device on a regular farm truck (bakkie).
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
967
F
B
E
C
F 2
L
F 2
F 2
D
A
N
H
M
F 2
(a)
(b)
M
N
(c)
(d)
0.1F
Fig. 2. Forces and moments on the rear frame.
where MB is the moment at the horizontal end points of the horizontal beam given by MB ¼
FL Ix2 H and k ¼ : 4ðk þ 2Þ Ix1 L
ð2Þ
The moment at A is MA ¼
MB ; 2
ð3Þ
and by symmetry MC ¼ MB and MD ¼ MA . The horizontal reaction HA due to the vertical force, is given by HA ¼
3MA ; H
and the horizontal reaction due to the horizontal force equals
ð4Þ
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HD1 ¼
0:1F ðk þ 1Þ : 2ðk þ 2Þ
ð5Þ
The moment at C due to the horizontal force is M1 ¼
0:1F 3k : 2ð6k þ 1Þ
ð6Þ
In order to apply the buckling and stress constraints to the frame, it is necessary to determine the elastic modulus, second moments of inertia and cross-sectional area of each profile. The second moments of inertia of the vertical ði ¼ 1Þ and horizontal ði ¼ 2Þ profile about the x- and y-axes respectively (see Fig. 3) are defined by " # 3 ðhi ti Þ ti ti 4ti 2 þ ðbi ti Þðhi ti Þ Ixi ¼ 1 0:86 ; ð7Þ 6 2 bi þ hi 2ti and
" Iyi ¼
# ðbi ti Þ3 ti ti 4ti 2 þ ðbi ti Þ ðhi ti Þ 1 0:86 ; 6 2 bi þ hi 2ti
ð8Þ
where bi , hi and ti are the width, height and thickness respectively of the profiles of the vertical ði ¼ 1Þ and horizontal ði ¼ 2Þ beams (Fig. 3) [1] and allowance has been made for the rounding of the corners by a radius r ¼ 2t according to Eurocode 3 [2]. The elastic section modulus is Wxi ¼
2Ixi : hi
ð9Þ
The cross-sectional area of a square or rectangular profile with rounded corners of r ¼ 2t is [2] 4ti ; ð10Þ Ai ¼ 2ti ðbi þ hi 2ti Þ 1 0:43 bi þ hi 2ti
b b y y
h
x
x t
y
b x
x t
y
Fig. 3. Dimensions of cross-sectional profiles.
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
969
and the surface area of the rear frame is given by As ¼ 4ðhi þ bi ÞH þ 2Lðhi þ bi Þ:
ð11Þ
3. Formulation of the design constraints For the first design iteration of the hoisting device the design constraints are formulated with regard to the rear main supporting frame only. It is assumed that the bases of the column beams are fixed and that the horizontal to vertical joints are rigidly welded. Furthermore it is assumed that longitudinal movement of the frame during lift is prevented by the presence of longitudinal braces. With these assumptions in mind, the following constraints have to be satisfied by the structure: 3.1. Global stress constraint of the horizontal beam The horizontal beam, i ¼ 2, has to comply with the overall stress constraint for bending and axial compression given by Eurocode 3 [2]: HA þ HD1 kM2 ME þ 6 1; v2: min A2 fy1 Wx2 fy1
ð12Þ
f
y where fy1 ¼ cM1 is the yield stress and cM1 ¼ 1:1 is a safety factor. 1 is the flexural buckling factor, with Here vi min ¼ 2 /i þ ð/i k2i max Þ0:5
/i ¼ 0:5b1 þ 0:34ðki max 0:2Þ þ k2i max c and ki max ¼ maxðkxi ; kyi Þ; Ksub L ; ksub ¼ rsub kE
rsub ¼
Isub Ai
0:5 ;
0:5 E kE ¼ p ; fy
i ¼ 1; 2;
sub ¼ x2; y2
and Kx2 ¼ Ky2 ¼ 0:5 is the effective length factors. Furthermore kM2 ¼ 1 þ
1; 2kx2 ðHA þ HD1 Þ ; ðv2 A2 fy Þ
where v2 is calculated the same as v2 min with kx2 . 3.2. Local buckling of the horizontal beam Constraints on local buckling of the horizontal beam require that b2 3t2 6 42e2 ; t2
ð13Þ
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where e2 ¼
235 rmax 2
0:5 and rmax 2 ¼
HA þ HD1 ME þ ; A2 Wx2
to prevent compression of the flange of the beam. With regard to the webs it is necessary that [2] h2 3t2 42e2 6 t2 0:67 þ 0:33w21
if w2 > 1;
ð14aÞ
and h2 3t2 6 62e2 ð1 w2 Þðw2 Þ0:5 t2
if w2 6 1;
ð14bÞ
where ME N2 Wx2 A2 and N2 ¼ HA þ HD1 : w2 ¼ ME N2 þ Wx2 A2 3.3. Global buckling of the column Stress constraints on the global buckling of the column (the stress criteria at point C, Fig. 2) imply that [2] N1 kM1 ðM1 þ MC Þ þ 6 1; Wx1 fy1 v1 min A1 fy1
ð15Þ
where kM1 ¼ 1
0:3kx1 N1 ; vx1 A1 fy1
N1 ¼
F þ VD1 2
and VD1 ¼
2M1 ; L
and the same equations for v1 min , u1 and k1 max as above apply for i ¼ 1, but Kx1 H Ky1 H ; Kx1 ¼ 2:19; ky1 ¼ and Ky1 ¼ 0:5: kx1 ¼ rx1 kE ry1 kE 3.4. Local buckling of the column To prevent local buckling of the vertical columns the same criteria as summarised for constraint 2 apply, but in this case, i ¼ 1 b1 3t1 6 42e1 ; t1 to prevent compression of the flange. With regard to the webs
ð16Þ
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
h1 3t1 42e1 6 t1 0:67 þ 0:33w1
if w1 > 1;
971
ð17aÞ
and h1 3t1 6 62e1 ð1 w1 Þðw1 Þ0:5 t1
if w1 6 1;
ð17bÞ
with e1 ¼
235 rmax 1
0:5 ;
rmax 1 ¼
N1 M1 þ MC þ ; A1 Wx1
and M1 þ MC N1 Wx1 A1 : w1 ¼ M1 þ MC N1 þ Wx1 A1 3.5. Fatigue stress Because of the cyclic mode of the loading and unloading process it is also necessary to consider the fatigue stress constraint for the horizontal beam at the midpoint (point E) and for the columns at the welded joints (point C). Complying with the requirements of the International Institute of Welding as amended by Hobbacher, Jarmai et al. [1] derived the constraints HA þ HD1 ME DrN 2 þ 6 ; A2 Wx2 cMf
ð18Þ
N1 M1 þ MC DrN1 þ 6 ; A1 Wx1 cMf
ð19Þ
and
N2 N1 where Dr ¼ 231 MPa and Dr ¼ 146 MPa. cMf cMf These values have been derived for 105 cycles, a static safety factor of 1.5 and a fatigue safety factor of 1.25. Clearly the satisfaction of the above stress, buckling and fatigue constraints, Eqs. (12)–(19), depends on the physical dimensions of the profiles (see Fig. 3). These dimensions xi , i ¼ 1; 2; . . . ; n, represented by the vector x ¼ ðx1 ; x2 ; . . . ; xn Þ, may be taken as the design variables.
4. Formulation of the objective function The particular objective function to be minimized here with respect to the design variables x, takes into account material costs Km , painting costs Kp and welding costs Kw , i.e. the cost function f ðxÞ is defined by
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f ðxÞ ¼ KðxÞ ¼ Km ðxÞ þ Kw ðxÞ þ Kp ðxÞ:
ð20Þ
4.1. Material cost function The material cost Km is found by multiplying the mass of the beam by the material cost factor km . The price lists used for rectangular and square tubing, were obtained from Robor Stewardts & Lloyds [3], the distributor of one of the main steel manufacturers, Robor Steel, in South Africa. The average prices of the standard profiles were found to be R 10.80/kg thus km ¼ 10.80 R/kg (R 8 ¼ $1). 4.2. Welding cost function The expression for welding costs, Eq. (21), has been derived by Jarmai and Farkas [4]. ! X 0:5 n Kw ¼ kw Hw ðjqV Þ þ 1:3 Cwi awi Lwi ;
ð21Þ
i
where kw is the welding cost factor in Rands/minute and the other parameters are discussed below. As indicated in the report on the welded tubular frame for a special truck [1], shielded metal arc welding (SMAW) of the tubes and braces is considered. A difficulty factor of hw ¼ 3 is assumed which reflects the complexity of the structure with regard to assembly and welding. The number of members is j ¼ 7, since there are 3 bars, 2 splice plates and 2 base plates to be assembled [1]. For fillet welds made by hand welding, the welding technology constant [1,4] equals Cw ¼ 0:7889 103 and the time of welding, deslagging, changing the electrode etc, (i.e. the second term of Eq. (21)) depends on the welding technology, type of welds, weld size ðaw Þ and weld length Lw , where i refers to the ith element and the value of n is derived from curve fitting calculations for the various welding techniques [4]. In a previous study relating to British Constructional Steel Tables and European manufacturing costs a welding cost factor kw , of $1/min is used by Jarmai et al. [1]. South African industrial statistics indicate that the labour costs for specialized welding is R 50/h (R 8 $1), i.e., R 0.83/min [5]. Welding rods applicable for the welding of thin walled tubes are available at R 18.52/ kg [6]. At a consumption rate of 0.0986 kg/m and a welding rate of 1.6 m/min [7], electrode costs amount to R 1.12/min. Adding the electrode and labour costs imply a welding cost factor of km ¼ 1:95 R/min. This figure does not include overhead costs. 4.3. Painting cost function The painting cost Kp is obtained by multiplying the painting cost factor kp with the surface area. A painting cost factor of kp ¼ R 20:27 106 /mm2 has been determined. This amount includes R 123.18 (without value added tax, VAT) for 5 l of undercoat which covers 6 m2 and R 181.59 for 5 l of car duco enamel topcoat that covers 7 m2 (tax excluded) [8]. Combining the paint application factors of 3 · 106 min/mm2 for ground coat application and 4.15 · 106 (min/mm2 ) given by Jarmai et al. [4], with the labour rate of R 20/h [5] and the paint costs for one layer of undercoat and two layers of topcoat, gives the stated result.
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5. Optimisation methodology The optimisation problem is solved by means of the leap-frog algorithm for constrained optimisation (LFOPC) of Snyman [9–11]. This gradient-based method, requiring no explicit line searches, is a proven robust and reliable method, being relatively insensitive to local inaccuracies and discontinuities in the gradients. As the gradients are to be computed here by relatively rough forward finite difference approximations, the leap-frog method should be ideally suitable for the current problem. The algorithm in general aims to minimize the objective function f ðxÞ, x 2 Rn subject to inequality constraints gi ðxÞ ¼ 0, i ¼ 1; 2; . . . ; m and equality constraints hj ðxÞ ¼ 0, j ¼ 1; 2; . . . ; r. The particular choice of design variables, being the width and wall thickness of the respective profiles of the columns and transverse beams, are as listed in Table 1. The objective cost function is related to these design variables by Eq. (20). This function includes the material costs and the painting costs for the structure as well as the cost of welding the transverse beam to the columns, the cost of welding the braces and the cost of welding the columns to the base, where cost of welding includes preparation, change of electrodes, deslagging and finishing. The constraints gi ðxÞ ¼ 0, i ¼ 1; 2; . . . ; 16, are listed in Table 2. In addition to the stress, buckling and fatigue constraints already discussed and described by Eqs. (12)–(19), upper and
Table 1 Design variables Variable
Description
Symbol
x1 x2 x3 x4
Width of column Wall thickness of column Width of transverse beam Thickness of transverse beam
b1 t1 b2 t2
Table 2 Description of inequality constraints Nature of constraint
Symbol
Minimum width of column, lower bound Maximum width of column, upper bound Minimum thickness of transverse beam, lower bound Maximum thickness of transverse beam, upper bound Overall buckling of the transverse beam, (Eq. (12)) Overall buckling of the column, (Eq. (15)) Local buckling of column flanges, (Eq. (16)) Local buckling of the flanges of the transverse beam, (Eq. (13)) Local buckling of the webs of the transverse beam, (Eqs. (14a),(14b)) Local buckling of column webs, (Eqs. (17a),(17b)) Fatigue constraints on the column, (Eq. (19)) Fatigue constraints on the transverse beam, (Eq. (18)) Minimum width of transverse beam, lower bound Maximum width of transverse beam, upper bound Minimum thickness of column, lower bound Maximum thickness of column, upper bound
g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 g16
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lower bounds on the design variables are also imposed in line with available profile dimensions. No equality constraints are prescribed. Initially the problem is treated as being continuous in the solution space and the associated optimum solution is obtained. An acceptable discrete optimum solution is then sought by inspection of several candidate discrete solutions in the neighbourhood of the continuous optimum. The candidate (available) discrete solutions are obtained from the steel tables of the Southern African Institute of Steel Construction [12] and Robor Cold Form [13]. The candidate discrete solutions are rated by evaluating the corresponding objective function for each candidate and determining to what extent they also satisfy the constraints. Finally, a design constructed from the available profiles with the lowest objective function value, and which also complies with the constraints within reasonable tolerances, is chosen as the final discrete optimum solution. Two profiles are considered: rectangular hollow sections (RHS) and square hollow sections (SHS). With regard to these profiles two possibilities are considered, one where the transverse beams and columns have the same profiles and the other where the column and transverse beam profiles may differ.
6. Numerical results 6.1. Results in terms of the optimisation process Table 3 lists the computational results of the optimisation process for a nominal yield stress of fy ¼ 235 MPa. The optimisation was carried out with the LFOPC convergence tolerances set at ex ¼ 105 and eg ¼ 105 and the values of the penalty parameters given by l0 ¼ 102 and l1 ¼ 104 . The maximum prescribed step size was chosen p toffiffiffibe d ¼ 5 which is of the order of the diameter of the region of interest by the relation d ¼ Rmax n, where Rmax is the maximum variable range and n the number of variables. In computing the forward finite difference approximation to the gradients a variable step size of Dxi ¼ 106 was used. For each optimum the components of the corresponding design vector are listed to the right, followed by the associated cost function value. Further information is listed regarding the active and violated constraints. In the final column to the right, the total number of LFOPC algorithm steps required for convergence to the specified accuracy, is given for the continuous solution. The results show that apart from dimensional restrictions, the constraints which are most active is that of cyclic fatigue at the welded joints of the column and global buckling of the transverse beam. Optimising the square profiles tends to require more iterations for convergence than that for rectangular profiles, as is apparent from Table 3 by comparing differing RHS and SHS profiles (this is for the same starting point, x1 ¼ 30, x2 ¼ 3, x3 ¼ 30, x4 ¼ 3). In order to verify that a global optimum had indeed been obtained, the optimisation was performed with different initial designs for the case of differing rectangular sections (with fy ¼ 355 MPa and kp ¼ 11:68 R/min). The same optimum was obtained regardless of the initial values. The importance of the optimisation parameters, ex , eg , l0 , l1 and d was apparent when problems occurred due to violation of constraints. It was experienced that the LFOPC parameter values used and which correspond with the directives given by Snyman [11], ensured convergence in all cases.
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Table 3 Results of optimisation ðf ðxÞ ¼ Km ðxÞ þ Kw ðxÞ þ Kp ðxÞÞ x1
x2
x3
x4
f
Active constraints
35.9 30 30 30 38.1 38.1 38.1
1.6 2 2.5 2 1.6 1.6 1.6
61.4 63.5 63.5 60 63.5 60 60
2 2 2 2 2 2 2
253.1 257.2 283.8 251.2 264.2 258.2 258.2
g3 , g11 , g3
44.1 40 40 50 50
2 2.5 3 2 2
44.1 40 40 50 50
2 2.5 3 2 2
285.5 312.8 364.5 321 321
c3 , c5
None g5 ¼ 0:025 None None None
200
56.9 57.2 50 57.2 57.2 50 50 50 57.2
1.6 1.6 2 1.6 1.6 1.6 1.6 2 1.6
96.4 90 90 90 100 100 100 100 100
2 2 2 3 2 2 2.5 2 2
264.4 257.8 265.1 305.1 269.2 252.22 278.6 276.5 269.2
g3 , g11 , g13
None g11 g11 None None g11 g11 None None
4041
2 2 2.5 3 2 2
69.7 70 63.5 63.5 76.2 70
2 2 2.5 3 2 2
299.3 300.3 329.4 384 325.1 300.3
g3 , g5
Best
69.7 70 63.5 63.5 76.2 70
Best continuous Different RHS
35.9
1.6
61.4
2
253.1
69.7
2
69.7
2
299.3 46.2
38.1
1.6
60
2
258.2
50
2
50
2
321 62.8
Scenario Different RHS Continuous optimum Discrete candidate
Best Equal RHS Continuous Discrete
Best Different SHS Continuous Discrete
Best Equal SHS Continuous Discrete
Worst continuous Equal SHS Difference Best discrete Different RHS Worst discrete Equal RHS Difference
Constraints violated
LFOPC steps 2140
g11 None g11 None None None
468 None g5 None None None
g3 , g5
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Figs. 4 and 5 reflect the convergence histories of the objective function and the design variables for the continuous problem of different RHS profiles for the vertical columns and horizontal beams. This convergence behaviour is typical for the different scenarios. 350 300 250 200
f 150 100 50 0 0
500
1000
1500
2000
2500
step
Fig. 4. Convergence history of objective function for different RHS.
70
Design variables
60 50 40
x1 x3
30 20 10 0 0
500
1000
(a)
1500
2000
2500
Steps
3.5
Design variables
3 2.5 2
x2 x4
1.5 1 0.5 0 0
(b)
200
400
600
800
1000
1200
Step
Fig. 5. Convergence history of design variables for different RHS.
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Table 4 Optimal dimension in mm using UK (British Steel, Tizani [16]) cost data Profiles
Continuous solution 3
Discrete solution
Dimension (mm )
Cost $
Dimension (mm3 )
Equal RHS Different RHS
44.1 · 88.2 · 2 40.45 · 80.9 · 2 47.15 · 94.3 · 2
73.9 71.6
40 · 80 · 3 40 · 80 · 3 40 · 80 · 3
Equal SHS Different SHS
69.7 · 2 59.4 · 59.4 · 2 80.1 · 80.1 · 2
76.8 73.2
70 · 70 · 3 50 · 50 · 2.5 80 · 80 · 3
Cost $ 90.3 90.3 102.1 82
6.2. Results with regard to the physical quantities It is apparent from Table 3 that the best final candidate is rectangular tubing with different dimensions for the transverse beams and columns. The worst case is equal RHS sections. The difference between the worst and the best value is R321 R258 ¼ R63($7.88) (a 20% improvement relative to the worst). It is of interest to compare these values with that of Jarmai et al. [1] for Hungarian conditions based on British Steel sections, which are listed in Table 4. They found the most expensive solution to be similar SHS profiles and the cheapest to be two different SHS profiles, the price difference being $20.10 (20% variation relative to the worst)––which is a similar result to that found for South African conditions by way of percentage. In the current study, for the continuous optima, the cheapest alternative is different RHS profiles and the most expensive equal SHS profiles. The difference between the worst and best solutions is R299:30 R253:10 ¼ R46:20 ($5.78) representing a 15% improvement relative to the worst. Similarly Jarmai et al. [1] found different RHS profiles to be the cheapest alternative for the continuous optima and similar SHS profiles were the most expensive. They obtained the difference in the price extremes in this case to be $5.20 (i.e.7%). The differences between the Hungarian and South African determined optima are of course due to the differences in cost structures and available profiles. Table 5 summarizes the differences between the price and profile structures of the two countries for comparable scenarios. 6.3. Results concerning formulation of the objective function The importance of the various terms of the cost function was also investigated. The results are given in Table 6 for different rectangular hollow sections of the column and beam. The value of the objective function and the related column and beam dimensions are given for four formulations of the objective function, i.e. where the objective function (1) includes only material costs, (2) consists of material and welding costs, (3) is defined only in terms of welding costs, (4) takes account of painting costs only and (5) includes both welding and painting costs but not material costs. Whereas the material costs constitute 63% of the total cost (total objective function value) and the welding costs constitute 28%, the painting costs contribute some 7% to the total cost function. Considering either material cost or welding cost or both in the cost function, result in very much the same optimum, but if the painting cost is considered the optimum design differs considerably. This difference can be ascribed to the fact that painting costs increase as surface area increases and
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Table 5 Comparable Hungarian (British Steel sections) and South African Scenarios (Dimensions in mm) Hungary
South Africa
Different RHS 40.45 · 80.9 · 2; 47.15 · 94.3 · 2 71.60
Different RHS 35.9 · 71.8 · 1.6; 60 · 120 · 2 31.64
Different SHS 50 · 50 · 2.5; 80 · 80 · 3 82.00
Different RHS 38.1 · 76.2 · 1.6; 60 · 120 · 2 32.28
Equal SHS 69.7 · 69.7 · 2 76.80
Equal SHS 69.7 · 69.7 · 2 33.65
Profile Cost in $
Equal SHS 70 · 70 · 3 $102.10
Equal RHS 50 · 100 · 2 $37.41
Cost function Welding cost constant Painting cost constant Material cost constant
$1.00/min $14.40/m2 $1.00/kg
$0.24/min $2.53/m2 $1.35/kg
Available material SHS minimum SHS maximum RHS minimum RHS maximum
20 · 20 · 2 150 · 150 · 4 50 · 25 · 2 100 · 200 · 4
12.7 · 12.7 · 1.6 300 · 300 · 10 12.7 · 25.4 · 1.6 100 · 200 · 10
Continuous optimum Profile Cost in $ Discrete optimum Profile Cost in $ Continuous worst Profile Cost in $ Discrete worst
Table 6 Optimisation results for the continuous problem for different RHS using different cost functions Cost function
x1
x2
x3
x4
Function value
Km þ Kw þ Kp Km Km þ Kw Kw Kw þ Kp Kp
35.9 35.9 36.1 37.1 36.1 26.1
1.6 1.6 1.6 1.6 1.6 4
61.4 61.5 61.1 59.3 61.1 46.9
2 2 2 2 2 4
253.1 158.6 230.4 71.8 94.5 16.8
the best result would be that of minimum surface area. On the other hand material and welding costs are to a large extent related to material thickness. If painting costs are equated to that of the Hungarian option, i.e., R115.20/m2 , the best design is 35.1 · 70.2 · 1.6; 62.1 · 124.2 · 2 mm3 compared to 38.1 · 76.2 · 1.6; 60 · 120 · 2 for painting costs evaluated at R 20–27/m2 .
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The formulation of the cost function clearly influences what would be considered the best result. This emphasizes the importance of formulating the objective function correctly and of weighting the various criteria against one another in order to obtain the most acceptable result. The results also indicate that considering only material costs for this kind of structure may give a good approximation to the best design.
7. The influence of minimum yield stress value The guaranteed minimum yield stress fy for hollow sections in South Africa is given as 200 MPa in the South African Steel Construction Handbook (1987) [14]. A switch over has been made to steel with a yield strength of 300 MPa [15]. In comparison, a yield stress value of 235 MPa apply for the British steel profiles distributed in Europe and this value was indeed used in the first part of this study. To determine the effect of changing the prescribed value of fy to 300 MPa, the analysis was repeated with the latter value and the results are as listed in Table 7. In Table 8 the optimum results for the two cases are compared. As is apparent from Table 8, there is little difference in the continuous optima (with all the constraints satisfied) for the two different cases, although there is a difference in the overall buckling constraint of both the column ðg6 Þ and transverse ðg5 Þ beams and in the fatigue stress constraint values, g11 and g12 . For similar profiles in the columns and transverse beams, the cost functions and optimum profiles differed more. The difference in the optimum solutions can be ascribed to the fact that the constraint on the overall buckling of the transverse beam ðg5 Þ becomes active in the case where fy ¼ 235 MPa. Even though the dimensions of the optimum profiles for fy ¼ 235 and 300 MPa differ from one another, the differences are small and given the available profiles, it can be seen by comparing Tables 3 and 7, that the discrete solutions for the case fy ¼ 300 MPa also satisfies the constraints in the case of fy ¼ 235 MPa except for the case of equal RHS, where constraint g5 is just violated. The solution is thus not very sensitive to the value of fy but overall buckling of the transverse beam should be given particular attention.
8. Conclusion and recommendations This study shows that significant savings can be realised by seeking an optimised design via mathematical programming. The use of a realistic mathematical model that not only takes into consideration material costs but also manufacturing costs, constitute an additional refinement that may prove to be of considerable importance, particularly in the case of the design of more complex structures requiring sophisticated manufacturing procedures. The particular usage of hollow sections, i.e., rectangular and square tubing, has been considered. Standards for buckling, welding and fatigue constraints for hollow sections have only recently been formulated by the European Committee for Standardization in 1996 [2]. Experience with the application of these criteria has further enhanced the value of the presented study and underlines the applicability of the criteria spelt out in these Standards. In view of the difference in manufacturing, material and painting costs in South Africa and Hungary/Britain, it is recommended that a more complete study be made of the calculation of
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Table 7 Optimisation results for fy ¼ 300 MPa Scenario Different RHS Continuous Discrete
Best Equal RHS Continuous Discrete Best Different SHS Continuous Discrete
Best Equal SHS Continuous Discrete Best Best continuous Different RHS Worst continuous Equal SHS Difference Best discrete Different RHS Worst discrete Equal RHS Difference
x4
f
Active constraints
61.4 60 60 60 60 60
2 2 2 2 2 2
253.1 258.2 286.5 251.2 277.9 258.2
g3 , g11 , g13
2 2.5 2 2.5
43 40 50 40
2 2.5 2 2.5
278.9 312.8 321 312.8
g3 , g11
57 57.2 57.2 50 45 45 45 50 57.2
1.6 1.6 1.6 2 2 2.5 2.5 2.5 1.6
96.1 90 100 100 100 100 90 90 100
2 2 2 2 2 2 2 2 2
264.4 257.8 269.2 276.2 261.7 288.2 276.9 294.8 269.2
g3 , g11 , g13
67.9 70 63.5 70
2 2 2.5 2
67.9 70 63.5 70
2 2 2.5 2
291.8 300 329 300
g3 , g11
35.9
1.6
61.4
2
253.1
g3 , g11 , g13
67.9
2
67.9
2
291.8 38.7
g3 , g11
38.1
1.6
60
2
258.2
None
40
2.5
40
2.5
312.8 54.6
None
x1
x2
35.9 38.1 38.1 30 30 38.1
1.6 1.6 2 2 2.5 1.6
43 40 50 40
x3
Constraints violated
LFOPC steps 2662
None None c11 None None 247 None None None 3219 g11 None None g11 None g11 None None 383 None None None
these values. This will be of prime importance if cost functions are to be extended to include manufacturing costs, and if a comparison with international values is to be made. Although only the rear frame of the framework has been taken into account in this study, the advantages of design optimisation have been illustrated and the indications are that the techniques can in general be extended to the complete and more complex structure.
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Table 8 Effect of yield strength values on optimal results f ¼ 300 MPa
Optimum profile
f ¼ 235 MPa
Optimum profile
Profile Function value g5 g6 g11 g12
Different RHS 253.1 0.5006 0.35 0.4236 0.9493
35.9 · 1.6; 61.4 · 2
Different RHS 253.1 0.3622 0.2193 0.3427 0.9488
35.9 · 1.6; 61.4 · 2
Profile Function value g5 g6 g11 g12
Equal RHS 278.8 0.1735 0.4162 0.4624 0.6429
43.0 · 2
Equal RHS 285.4 0.8323 0.3141 0.7325 0.1814
44.1 · 2
Profile Function value g5 g6 g11 g12
Different SHS 264.6 0.4752 0.3255 0.1146 0.8801
56 · 1.6; 94.2 · 2
Different SHS 264.4 0.3531 0.1882 0.32 0.9292
56.9 · 1.6; 96.4 · 2
Profile Function value g5 g6 g11 g12
Equal SHS 291.8 0.1702 0.399 0.6259 0.5441
67.9 · 2
Equal SHS 299.3 0.2582 0.302 0.7877 0.1806
69.7 · 2
Finally the ease with which optimum constrained solutions were computed in this study confirms the applicability of the LFOPC optimisation algorithm for structural problems where a variety of different physical constraints such as buckling, fatigue and dimensional constraints apply. Acknowledgements The authors wish to acknowledge financial support for this study from the Hungarian and South African Governments via the Hungarian––South African Intergovernmental S&T Cooperation programme for 2000–2002. This work was done within the project ‘‘Optimum design of tubular and framed structures’’ with coordinators Prof J Karoly (Hungary) and Prof JA Snyman (South Africa), OTKA, FKFP. References [1] K. Jarmai, J. Farkas, P. Visser-Uys, Minimum cost design of welded tubular frames for a special truck, IIWDoc.XV-1085-WG9-09-01,XV-1085-01 International Institute of Welding Annual Assembly, Ljubjana, 8–11 July 2001, 12 p.
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[2] European Prestandard ENV 1993-1-3 Eurocode 3: Design of steel structures––Part 1–3: General rules–– Supplementary rules for cold formed thin gauge members and sheeting, 22, 25–26, 69. [3] Fax quotation, Robor Stewardts & Loyds, 2001. [4] K. Jarmai, J. Farkas, Cost calculation and optimisation of welded steel structures, J. Constr. Steel Res. 50 (1999) 115–135. [5] Metal and Engineering Industries Bargaining Council, private communication, April 2001. [6] Afrox, private communication, 2001. [7] The Lincoln Electric Company; The Procedure Handbook of Arc Welding, 13th ed., The Lincoln Electric Company, Cleveland, OH, 1994, pp. 6-2-24. [8] Paint Sales Warehouse, private communication, 2001. [9] J.A. Snyman, A new dynamic method for unconstrained minimization, Appl. Math. Model. 7 (1983) 216–218. [10] J.A. Snyman, An improved version of the original leap-frog dynamic method for the unconstrained minimization LFOP1(b), Appl. Math. Model. 6 (1982) 449–462. [11] J.A. Snyman, The LFOPC leap frog algorithm for constrained optimisation, Comput. Math. Appl. 40 (2000) 1085– 1096. [12] Structural Steel Tables, Seventh ed., The Southern African Institute of Steel Construction, Johannesburg, 1977. [13] Robor Cold Form; Product catalogue, 2001. [14] South African Steel Construction Handbook, The South African Institute of Steel Construction, Johannesburg, 1987, 2.41. [15] South African Steel Construction Handbook, The South African Institute of Steel Construction, Johannesburg, 1999, 1.5. [16] W.M.K. Tizani, G. Davies, A.S. Whitehead, A knowledge based system to support joint fabrication decision making at the design stage––case studies for CHS trusses, in: J. Farkas, K. Jarmai (Eds.), Tubular Structures VII. Balkema, Rotterdam, 1996, pp. 483–489.