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) 963â€“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 Â¼ 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 Â¼ 1 pertaining to vertical beam

i Â¼ 2 pertaining to horizontal beam

k Â¼ 1; … ; 16 pertaining to constraints

w pertaining to weld

min minimum

max maximum

964 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963â€“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 â€˜â€˜bakkieâ€™â€™. 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) 963â€“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 Â¼ 1:2 m and a frame height of H Â¼ 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 Â¼ FL

4 MB; Ã°1Ãž

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) 963â€“982

where MB is the moment at the horizontal end points of the horizontal beam given by

MB Â¼ FL

4Ã°k Ã¾ 2Ãž and k Â¼ Ix2H

Ix1L : Ã°2Ãž

The moment at A is

MA Â¼ MB

2 ; Ã°3Ãž

and by symmetry MC Â¼ MB and MD Â¼ MA.

The horizontal reaction HA due to the vertical force, is given by

HA Â¼ 3MA

H ; Ã°4Ãž

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) 963â€“982 967

HD1 Â¼ 0:1F Ã°k Ã¾ 1Ãž

2Ã°k Ã¾ 2Ãž : Ã°5Ãž

The moment at C due to the horizontal force is

M1 Â¼ 0:1F 3k

2Ã°6k Ã¾ 1Ãž

: Ã°6Ãž

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 Ã°i Â¼ 1Ãž and horizontal Ã°i Â¼ 2Ãž profile about the

x- and y-axes respectively (see Fig. 3) are defined by

Ixi Â¼ Ã°hi tiÃž

3

ti

6

”

Ã¾

ti

2

Ã°bi tiÃžÃ°hi tiÃž

2

#

1

0:86 4ti

bi Ã¾ hi 2ti

; Ã°7Ãž

and

Iyi Â¼ Ã°bi tiÃž

3

ti

6

”

Ã¾

ti

2

Ã°bi tiÃž

2

Ã°hi tiÃž

#

1

0:86 4ti

bi Ã¾ hi 2ti

; Ã°8Ãž

where bi, hi and ti are the width, height and thickness respectively of the profiles of the vertical

Ã°i Â¼ 1Ãž and horizontal Ã°i Â¼ 2Ãž beams (Fig. 3) [1] and allowance has been made for the rounding of

the corners by a radius r Â¼ 2t according to Eurocode 3 [2].

The elastic section modulus is

Wxi Â¼ 2Ixi

hi

: Ã°9Ãž

The cross-sectional area of a square or rectangular profile with rounded corners of r Â¼ 2t is [2]

Ai Â¼ 2tiÃ°bi Ã¾ hi 2tiÃž 1

0:43 4ti

bi Ã¾ hi 2ti

; Ã°10Ãž

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) 963â€“982

and the surface area of the rear frame is given by

As Â¼ 4Ã°hi Ã¾ biÃžH Ã¾ 2LÃ°hi Ã¾ biÃž: Ã°11Ãž

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 Â¼ 2, has to comply with the overall stress constraint for bending and

axial compression given by Eurocode 3 [2]:

HA Ã¾ HD1

v2: minA2fy1

Ã¾

kM2ME

Wx2fy1

6 1; Ã°12Ãž

where fy1 Â¼ fy

cM1 is the yield stress and cM1 Â¼ 1:1 is a safety factor.

Here vi min Â¼ 1

/i Ã¾ Ã°/2

i k2

i maxÃž

0:5 is the flexural buckling factor, with

/i Â¼ 0:5b1 Ã¾ 0:34Ã°ki max 0:2Ãž Ã¾ k2

i maxc and ki max Â¼ maxÃ°kxi; kyiÃž; i Â¼ 1; 2;

ksub Â¼ KsubL

rsubkE

; rsub Â¼ Isub

Ai

0:5

; kE Â¼ p

E

fy

0:5

; sub Â¼ x2; y2

and Kx2 Â¼ Ky2 Â¼ 0:5 is the effective length factors.

Furthermore

kM2 Â¼ 1 Ã¾

1; 2kx2Ã°HA Ã¾ HD1Ãž

Ã°v2A2fyÃž ;

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; Ã°13Ãž

P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963â€“982 969

where

e2 Â¼ 235

rmax 2 0:5

and rmax 2 Â¼ HA Ã¾ HD1

A2

Ã¾

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 Ã¾ 0:33w21

if w2 > 1; Ã°14aÃž

and

h2 3t2

t2

6 62e2Ã°1 w2ÃžÃ°w2Ãž

0:5 if w2 6 1; Ã°14bÃž

where

w2 Â¼

ME

Wx2

N2

A2

ME

Wx2

Ã¾

N2

A2

and N2 Â¼ HA Ã¾ 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

Ã¾

kM1Ã°M1 Ã¾ MCÃž

Wx1fy1

6 1; Ã°15Ãž

where

kM1 Â¼ 1 0:3kx1N1

vx1A1fy1

; N1 Â¼ F

2 Ã¾ VD1 and VD1 Â¼ 2M1

L ;

and the same equations for v1 min, u1 and k1 max as above apply for i Â¼ 1, but

kx1 Â¼ Kx1H

rx1kE

; Kx1 Â¼ 2:19; ky1 Â¼ Ky1H

ry1kE

and Ky1 Â¼ 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 Â¼ 1

b1 3t1

t1

6 42e1; Ã°16Ãž

to prevent compression of the flange. With regard to the webs

970 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963â€“982

h1 3t1

t1

6

42e1

0:67 Ã¾ 0:33w1

if w1 > 1; Ã°17aÃž

and

h1 3t1

t1

6 62e1Ã°1 w1ÃžÃ°w1Ãž

0:5 if w1 6 1; Ã°17bÃž

with

e1 Â¼ 235

rmax 1 0:5

; rmax 1 Â¼ N1

A1

Ã¾

M1 Ã¾ MC

Wx1

;

and

w1 Â¼

M1 Ã¾ MC

Wx1

N1

A1

M1 Ã¾ MC

Wx1

Ã¾

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 Ã¾ HD1

A2

Ã¾

ME

Wx2

6

DrN2

cMf

; Ã°18Ãž

and

N1

A1

Ã¾

M1 Ã¾ MC

Wx1

6

DrN1

cMf

; Ã°19Ãž

where DrN2

cMf Â¼ 231 MPa and DrN1

cMf Â¼ 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)â€“(19),

depends on the physical dimensions of the profiles (see Fig. 3). These dimensions xi, i Â¼ 1; 2; … ; n,

represented by the vector x Â¼ Ã°x1; x2; … ; xnÃž, 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 Ã°xÃž is defined by

P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963â€“982 971

f Ã°xÃž Â¼ KÃ°xÃž Â¼ KmÃ°xÃž Ã¾ KwÃ°xÃž Ã¾ KpÃ°xÃž: Ã°20Ãž

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 Â¼ 10.80 R/kg

(R 8 Â¼ $1).

4.2. Welding cost function

The expression for welding costs, Eq. (21), has been derived by Jarmai and Farkas [4].

Kw Â¼ kw HwÃ°jqV Ãž

0:5

Ã¾ 1:3

X

i

Cwian

wi

Lwi

!

; Ã°21Ãž

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 Â¼ 3 is assumed

which reflects the complexity of the structure with regard to assembly and welding. The number

of members is j Â¼ 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 Â¼

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 Ã°awÃž 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 Â¼ 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 Â¼ 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 Â· 106 min/mm2 for ground coat application and 4.15 Â· 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) 963â€“982

5. Optimisation methodology

The optimisation problem is solved by means of the leap-frog algorithm for constrained optimisation (LFOPC) of Snyman [9â€“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 Ã°xÃž, x 2 Rn subject to

inequality constraints giÃ°xÃž Â¼ 0, i Â¼ 1; 2; … ; m and equality constraints hjÃ°xÃž Â¼ 0, j Â¼ 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 giÃ°xÃž Â¼ 0, i Â¼ 1; 2; … ; 16, are listed in Table 2. In addition to the stress,

buckling and fatigue constraints already discussed and described by Eqs. (12)â€“(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) 963â€“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 Â¼ 235 MPa.

The optimisation was carried out with the LFOPC convergence tolerances set at ex Â¼ 105 and

eg Â¼ 105 and the values of the penalty parameters given by l0 Â¼ 102 and l1 Â¼ 104. The maximum

prescribed step size was chosen to be d Â¼ 5 which is of the order of the diameter of the region of

interest by the relation d Â¼ 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 Â¼ 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 Â¼ 30, x2 Â¼ 3, x3 Â¼ 30, x4 Â¼ 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 Â¼ 355

MPa and kp Â¼ 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) 963â€“982

Table 3

Results of optimisation Ã°f Ã°xÃž Â¼ KmÃ°xÃž Ã¾ KwÃ°xÃž Ã¾ KpÃ°xÃžÃž

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 Â¼ 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) 963â€“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) 963â€“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 Â¼ 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)â€“â€“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 Â¼ 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 Â· 88.2 Â· 2 73.9 40 Â· 80 Â· 3 90.3

Different RHS 40.45 Â· 80.9 Â· 2 71.6 40 Â· 80 Â· 3 90.3

47.15 Â· 94.3 Â· 2 40 Â· 80 Â· 3

Equal SHS 69.7 Â· 2 76.8 70 Â· 70 Â· 3 102.1

Different SHS 59.4 Â· 59.4 Â· 2 73.2 50 Â· 50 Â· 2.5 82

80.1 Â· 80.1 Â· 2 80 Â· 80 Â· 3

P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963â€“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 Â· 70.2 Â· 1.6; 62.1 Â· 124.2 Â· 2 mm3 compared to 38.1 Â· 76.2 Â· 1.6; 60 Â· 120 Â· 2 for painting

costs evaluated at R 20â€“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 Â· 80.9 Â· 2; 47.15 Â· 94.3 Â· 2 35.9 Â· 71.8 Â· 1.6; 60 Â· 120 Â· 2

Cost in $ 71.60 31.64

Discrete optimum

Different SHS Different RHS

Profile 50 Â· 50 Â· 2.5; 80 Â· 80 Â· 3 38.1 Â· 76.2 Â· 1.6; 60 Â· 120 Â· 2

Cost in $ 82.00 32.28

Continuous worst

Equal SHS Equal SHS

Profile 69.7 Â· 69.7 Â· 2 69.7 Â· 69.7 Â· 2

Cost in $ 76.80 33.65

Discrete worst

Equal SHS Equal RHS

Profile 70 Â· 70 Â· 3 50 Â· 100 Â· 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 Â· 20 Â· 2 12.7 Â· 12.7 Â· 1.6

SHS maximum 150 Â· 150 Â· 4 300 Â· 300 Â· 10

RHS minimum 50 Â· 25 Â· 2 12.7 Â· 25.4 Â· 1.6

RHS maximum 100 Â· 200 Â· 4 100 Â· 200 Â· 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 Ã¾ Kw Ã¾ Kp 35.9 1.6 61.4 2 253.1

Km 35.9 1.6 61.5 2 158.6

Km Ã¾ Kw 36.1 1.6 61.1 2 230.4

Kw 37.1 1.6 59.3 2 71.8

Kw Ã¾ 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) 963â€“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 Ã°g6Ãž and transverse Ã°g5Ãž 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 Ã°g5Ãž becomes active in the case where fy Â¼ 235 MPa.

Even though the dimensions of the optimum profiles for fy Â¼ 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 Â¼ 300 MPa also satisfies the constraints

in the case of fy Â¼ 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) 963â€“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 Â¼ 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) 963â€“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 Hungarianâ€“â€“South African Intergovernmental S&T Cooperation programme for 2000â€“2002. This work was done within the project â€˜â€˜Optimum design of

tubular and framed structuresâ€™â€™ 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, 8â€“11 July

2001, 12 p.

Table 8

Effect of yield strength values on optimal results

f Â¼ 300 MPa Optimum profile f Â¼ 235 MPa Optimum profile

Profile Different RHS 35.9 Â· 1.6; 61.4 Â· 2 Different RHS 35.9 Â· 1.6; 61.4 Â· 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 Â· 2 Equal RHS 44.1 Â· 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 Â· 1.6; 94.2 Â· 2 Different SHS 56.9 Â· 1.6; 96.4 Â· 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 Â· 2 Equal SHS 69.7 Â· 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) 963â€“982 981

[2] European Prestandard ENV 1993-1-3 Eurocode 3: Design of steel structuresâ€“â€“Part 1â€“3: General rulesâ€“â€“

Supplementary rules for cold formed thin gauge members and sheeting, 22, 25â€“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)

115â€“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) 216â€“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) 449â€“462.

[11] J.A. Snyman, The LFOPC leap frog algorithm for constrained optimisation, Comput. Math. Appl. 40 (2000) 1085â€“

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 stageâ€“â€“case studies for CHS trusses, in: J. Farkas, K. Jarmai (Eds.), Tubular Structures VII.

Balkema, Rotterdam, 1996, pp. 483â€“489.

982 P.E. Uys et al. / Appl. Math. Modelling 27 (2003) 963â€“982

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