Optimal design of a hoist structure frame
P.E. Uys a,*, K. Jarmai a,b, J. Farkas b
a 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 Snymans 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
Applied Mathematical Modelling 27 (2003) 963–982
www.elsevier.com/locate/apm
Nomenclature
Ai cross-sectional area of the beam (mm2)
As surface area of the frame to be painted (mm2)
aw weld size (mm)
bi profile width (mm)
gk constraints
Cw welding technology constant
E elasticity modulus (MPa)
F load on the frame (N)
fy yield stress (MPa)
hi profile height (mm)
H height of the frame (mm)
HA, HD1 horizontal reaction force (N)
Ix, Iy second moments of inertia (mm4)
Kxi, Kyi effective length factors
kp painting cost factor (R/m2)
km manufacturing cost factor (R/kg)
kw welding cost factor (R/m3)
L frame width (mm)
Lw weld length (mm)
MI moments about points I ¼ A, B, C, D
Ni axial forces (N)
V volume of structure (mm3)
VA1;D1 vertical reaction force (N)
Wxi elastic section modulus (mm3)
Greeks
cM1 safety factor
cMf fatigue safety factor
vi flexural buckling factor
vLT lateral-torsional buckling factor
j number of structural parts
q material density (kg/m3)
hw difficulty factor for complexity of structure
DrNi 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
964 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
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.
Variables
ti width of beam profile (mm)
hi height of beam profile (mm)
Terminology
SHS square hollow section
RHS rectangular hollow section
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982 965
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
ME ¼ FL
4 MB; ð1Þ
1,5 m
2 m 1 m 1.2 m
Fig. 1. Frame for a hoisting device on a regular farm truck (bakkie).
966 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
where MB is the moment at the horizontal end points of the horizontal beam given by
MB ¼ FL
4ðk þ 2Þ and k ¼ Ix2H
Ix1L : 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 ; ð4Þ
and the horizontal reaction due to the horizontal force equals
F
A
B E C
M H N
D F
F F
F
(a) (b)
2
2 2
2
L
N
(d)
M
0.1F
(c)
Fig. 2. Forces and moments on the rear frame.
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982 967
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
Ixi ¼ ðhi tiÞ
3
ti
6
”
þ
ti
2
ðbi tiÞðhi tiÞ
2
#
1
0:86 4ti
bi þ hi 2ti
; ð7Þ
and
Iyi ¼ ðbi tiÞ
3
ti
6
”
þ
ti
2
ðbi tiÞ
2
ðhi tiÞ
#
1
0:86 4ti
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]
Ai ¼ 2tiðbi þ hi 2tiÞ 1
0:43 4ti
bi þ hi 2ti
; ð10Þ
b
y
x x b
y
t t
y
x x
y
h
b
Fig. 3. Dimensions of cross-sectional profiles.
968 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
and the surface area of the rear frame is given by
As ¼ 4ðhi þ biÞH þ 2Lðhi þ biÞ: 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
v2: minA2fy1
þ
kM2ME
Wx2fy1
6 1; ð12Þ
where fy1 ¼ fy
cM1 is the yield stress and cM1 ¼ 1:1 is a safety factor.
Here vi min ¼ 1
/i þ ð/2
i k2
i maxÞ
0:5 is the flexural buckling factor, with
/i ¼ 0:5b1 þ 0:34ðki max 0:2Þ þ k2
i maxc and ki max ¼ maxðkxi; kyiÞ; i ¼ 1; 2;
ksub ¼ KsubL
rsubkE
; rsub ¼ Isub
Ai
0:5
; kE ¼ p
E
fy
0:5
; sub ¼ x2; y2
and Kx2 ¼ Ky2 ¼ 0:5 is the effective length factors.
Furthermore
kM2 ¼ 1 þ
1; 2kx2ðHA þ HD1Þ
ðv2A2fyÞ ;
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
t2
6 42e2; ð13Þ
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982 969
where
e2 ¼ 235
rmax 2 0:5
and rmax 2 ¼ HA þ HD1
A2
þ
ME
Wx2
;
to prevent compression of the flange of the beam. With regard to the webs it is necessary that [2]
h2 3t2
t2
6
42e2
0:67 þ 0:33w21
if w2 > 1; ð14aÞ
and
h2 3t2
t2
6 62e2ð1 w2Þðw2Þ
0:5 if w2 6 1; ð14bÞ
where
w2 ¼
ME
Wx2
N2
A2
ME
Wx2
þ
N2
A2
and N2 ¼ HA þ HD1:
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
v1 minA1fy1
þ
kM1ðM1 þ MCÞ
Wx1fy1
6 1; ð15Þ
where
kM1 ¼ 1 0:3kx1N1
vx1A1fy1
; N1 ¼ F
2 þ VD1 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 ¼ Kx1H
rx1kE
; Kx1 ¼ 2:19; ky1 ¼ Ky1H
ry1kE
and Ky1 ¼ 0:5:
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
t1
6 42e1; ð16Þ
to prevent compression of the flange. With regard to the webs
970 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
h1 3t1
t1
6
42e1
0:67 þ 0:33w1
if w1 > 1; ð17aÞ
and
h1 3t1
t1
6 62e1ð1 w1Þðw1Þ
0:5 if w1 6 1; ð17bÞ
with
e1 ¼ 235
rmax 1 0:5
; rmax 1 ¼ N1
A1
þ
M1 þ MC
Wx1
;
and
w1 ¼
M1 þ MC
Wx1
N1
A1
M1 þ MC
Wx1
þ
N1
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
A2
þ
ME
Wx2
6
DrN2
cMf
; ð18Þ
and
N1
A1
þ
M1 þ MC
Wx1
6
DrN1
cMf
; ð19Þ
where DrN2
cMf ¼ 231 MPa and DrN1
cMf ¼ 146 MPa.
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; … ; xnÞ, 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
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982 971
f ðxÞ ¼ KðxÞ ¼ KmðxÞ þ KwðxÞ þ KpðxÞ: 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].
Kw ¼ kw HwðjqV Þ
0:5
þ 1:3
X
i
Cwian
wi
Lwi
!
; ð21Þ
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 ðawÞ 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.
972 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
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 giðxÞ ¼ 0, i ¼ 1; 2; … ; m and equality constraints hjðxÞ 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 giðxÞ 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 Width of column b1
x2 Wall thickness of column t1
x3 Width of transverse beam b2
x4 Thickness of transverse beam t2
Table 2
Description of inequality constraints
Nature of constraint Symbol
Minimum width of column, lower bound g1
Maximum width of column, upper bound g2
Minimum thickness of transverse beam, lower bound g3
Maximum thickness of transverse beam, upper bound g4
Overall buckling of the transverse beam, (Eq. (12)) g5
Overall buckling of the column, (Eq. (15)) g6
Local buckling of column flanges, (Eq. (16)) g7
Local buckling of the flanges of the transverse beam, (Eq. (13)) g8
Local buckling of the webs of the transverse beam, (Eqs. (14a),(14b)) g9
Local buckling of column webs, (Eqs. (17a),(17b)) g10
Fatigue constraints on the column, (Eq. (19)) g11
Fatigue constraints on the transverse beam, (Eq. (18)) g12
Minimum width of transverse beam, lower bound g13
Maximum width of transverse beam, upper bound g14
Minimum thickness of column, lower bound g15
Maximum thickness of column, upper bound g16
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982 973
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 to be d ¼ 5 which is of the order of the diameter of the region of
interest by the relation d ¼ Rmax
ffiffiffi
n p , 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.
974 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
Table 3
Results of optimisation ðf ðxÞ ¼ KmðxÞ þ KwðxÞ þ KpðxÞÞ
Scenario x1 x2 x3 x4 f Active constraints
Constraints
violated
LFOPC
steps
Different RHS
Continuous optimum 35.9 1.6 61.4 2 253.1 g3, g11, g3 2140
Discrete candidate 30 2 63.5 2 257.2 g11
30 2.5 63.5 2 283.8 None
30 2 60 2 251.2 g11
38.1 1.6 63.5 2 264.2 None
38.1 1.6 60 2 258.2 None
Best 38.1 1.6 60 2 258.2 None
Equal RHS
Continuous 44.1 2 44.1 2 285.5 c3, c5 None 200
Discrete 40 2.5 40 2.5 312.8 g5 ¼ 0:025
40 3 40 3 364.5 None
50 2 50 2 321 None
Best 50 2 50 2 321 None
Different SHS
Continuous 56.9 1.6 96.4 2 264.4 g3, g11, g13 None 4041
Discrete 57.2 1.6 90 2 257.8 g11
50 2 90 2 265.1 g11
57.2 1.6 90 3 305.1 None
57.2 1.6 100 2 269.2 None
50 1.6 100 2 252.22 g11
50 1.6 100 2.5 278.6 g11
50 2 100 2 276.5 None
Best 57.2 1.6 100 2 269.2 None
Equal SHS
Continuous 69.7 2 69.7 2 299.3 g3, g5 468
Discrete 70 2 70 2 300.3 None
63.5 2.5 63.5 2.5 329.4 g5
63.5 3 63.5 3 384 None
76.2 2 76.2 2 325.1 None
Best 70 2 70 2 300.3 None
Best continuous
Different RHS 35.9 1.6 61.4 2 253.1
Worst continuous
Equal SHS 69.7 2 69.7 2 299.3 g3, g5
Difference 46.2
Best discrete
Different RHS 38.1 1.6 60 2 258.2
Worst discrete
Equal RHS 50 2 50 2 321
Difference 62.8
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982 975
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.
0
50
100
150
200
250
300
350
0 500 1000 1500 2000 2500
step
f
Fig. 4. Convergence history of objective function for different RHS.
0
(a)
(b)
10
20
30
40
50
60
70
0 500 1000 1500 2000 2500
Steps
Design variables
x1
x3
0
0.5
1
1.5
2
2.5
3
3.5
0 200 400 600 800 1000 1200
Step
Design variables
x2
x4
Fig. 5. Convergence history of design variables for different RHS.
976 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
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
Table 4
Optimal dimension in mm using UK (British Steel, Tizani [16]) cost data
Profiles Continuous solution Discrete solution
Dimension (mm3) Cost $ Dimension (mm3) Cost $
Equal RHS 44.1 · 88.2 · 2 73.9 40 · 80 · 3 90.3
Different RHS 40.45 · 80.9 · 2 71.6 40 · 80 · 3 90.3
47.15 · 94.3 · 2 40 · 80 · 3
Equal SHS 69.7 · 2 76.8 70 · 70 · 3 102.1
Different SHS 59.4 · 59.4 · 2 73.2 50 · 50 · 2.5 82
80.1 · 80.1 · 2 80 · 80 · 3
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982 977
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.
Table 5
Comparable Hungarian (British Steel sections) and South African Scenarios (Dimensions in mm)
Hungary South Africa
Continuous optimum
Different RHS Different RHS
Profile 40.45 · 80.9 · 2; 47.15 · 94.3 · 2 35.9 · 71.8 · 1.6; 60 · 120 · 2
Cost in $ 71.60 31.64
Discrete optimum
Different SHS Different RHS
Profile 50 · 50 · 2.5; 80 · 80 · 3 38.1 · 76.2 · 1.6; 60 · 120 · 2
Cost in $ 82.00 32.28
Continuous worst
Equal SHS Equal SHS
Profile 69.7 · 69.7 · 2 69.7 · 69.7 · 2
Cost in $ 76.80 33.65
Discrete worst
Equal SHS Equal RHS
Profile 70 · 70 · 3 50 · 100 · 2
Cost in $ $102.10 $37.41
Cost function
Welding cost constant $1.00/min $0.24/min
Painting cost constant $14.40/m2 $2.53/m2
Material cost constant $1.00/kg $1.35/kg
Available material
SHS minimum 20 · 20 · 2 12.7 · 12.7 · 1.6
SHS maximum 150 · 150 · 4 300 · 300 · 10
RHS minimum 50 · 25 · 2 12.7 · 25.4 · 1.6
RHS maximum 100 · 200 · 4 100 · 200 · 10
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 35.9 1.6 61.4 2 253.1
Km 35.9 1.6 61.5 2 158.6
Km þ Kw 36.1 1.6 61.1 2 230.4
Kw 37.1 1.6 59.3 2 71.8
Kw þ Kp 36.1 1.6 61.1 2 94.5
Kp 26.1 4 46.9 4 16.8
978 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
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 ðg6Þ and transverse ðg5Þ 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 ðg5Þ 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
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982 979
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.
Table 7
Optimisation results for fy ¼ 300 MPa
Scenario x1 x2 x3 x4 f Active
constraints
Constraints
violated
LFOPC
steps
Different RHS
Continuous 35.9 1.6 61.4 2 253.1 g3, g11, g13 2662
Discrete 38.1 1.6 60 2 258.2 None
38.1 2 60 2 286.5 None
30 2 60 2 251.2 c11
30 2.5 60 2 277.9 None
Best 38.1 1.6 60 2 258.2 None
Equal RHS
Continuous 43 2 43 2 278.9 g3, g11 247
Discrete 40 2.5 40 2.5 312.8 None
50 2 50 2 321 None
Best 40 2.5 40 2.5 312.8 None
Different SHS
Continuous 57 1.6 96.1 2 264.4 g3, g11, g13 3219
Discrete 57.2 1.6 90 2 257.8 g11
57.2 1.6 100 2 269.2 None
50 2 100 2 276.2 None
45 2 100 2 261.7 g11
45 2.5 100 2 288.2 None
45 2.5 90 2 276.9 g11
50 2.5 90 2 294.8 None
Best 57.2 1.6 100 2 269.2 None
Equal SHS
Continuous 67.9 2 67.9 2 291.8 g3, g11 383
Discrete 70 2 70 2 300 None
63.5 2.5 63.5 2.5 329 None
Best 70 2 70 2 300 None
Best continuous
Different RHS 35.9 1.6 61.4 2 253.1 g3, g11, g13
Worst continuous
Equal SHS 67.9 2 67.9 2 291.8 g3, g11
Difference 38.7
Best discrete
Different RHS 38.1 1.6 60 2 258.2 None
Worst discrete
Equal RHS 40 2.5 40 2.5 312.8 None
Difference 54.6
980 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
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.
Table 8
Effect of yield strength values on optimal results
f ¼ 300 MPa Optimum profile f ¼ 235 MPa Optimum profile
Profile Different RHS 35.9 · 1.6; 61.4 · 2 Different RHS 35.9 · 1.6; 61.4 · 2
Function value 253.1 253.1
g5 0.5006 0.3622
g6 0.35 0.2193
g11 0.4236 0.3427
g12 0.9493 0.9488
Profile Equal RHS 43.0 · 2 Equal RHS 44.1 · 2
Function value 278.8 285.4
g5 0.1735 0.8323
g6 0.4162 0.3141
g11 0.4624 0.7325
g12 0.6429 0.1814
Profile Different SHS 56 · 1.6; 94.2 · 2 Different SHS 56.9 · 1.6; 96.4 · 2
Function value 264.6 264.4
g5 0.4752 0.3531
g6 0.3255 0.1882
g11 0.1146 0.32
g12 0.8801 0.9292
Profile Equal SHS 67.9 · 2 Equal SHS 69.7 · 2
Function value 291.8 299.3
g5 0.1702 0.2582
g6 0.399 0.302
g11 0.6259 0.7877
g12 0.5441 0.1806
P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982 981
[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.
982 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963–982
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