EngOpt 2008 - International Conference on Engineering Optimization Rio de Janeiro, Brazil, 01 - 05 June 2008.
Design Optimization of EOT Crane Bridge 1
2
3
Rehan H Zuberi , Dr. Long Kai , Prof. Zuo Zhengxing
(1) Doctoral Student, School of Mechanical and Vehicular Engineering, Beijing Institute of Technology (BIT), China,
[email protected] (2) Post Doctoral Student, School of Mechanical and Vehicular Engineering, Beijing Institute of Technology (BIT), China,
[email protected] (3) Prof. & Dean, School of Mechanical and Vehicular Engineering, Beijing Institute of Technology (BIT), China,
[email protected] 1. Abstract
Electric Overhead Travelling (EOT) Crane is one of the essential industrial equipment for material handling job. In recent years little attention has been paid to the optimal design of Heavy Electric Overhead Travelling Crane Bridges. The motive might be, but not limited to the availability of prevailing FEM, DIN, ISO, CMAA, BS, Chinese and now CEN standards for the design of cranes. Most of the crane manufacturers have standardized the single dimensioned box section for multiple spans and duties of Crane Bridges for manufacturing simplicity. Owing to the recent upward trend in the price of structural materials not to mention the demanding dimensions of heavy duty crane box girders, utilization of modern design optimization tools is inevitable. This paper demonstrates design optimization of EOT crane thin walled welded box girder subjected to rolling loads. A simple and innovative procedure has been introduced to use Generalized Reduced Gradient (GRG2) nonlinear optimization code for optimization of various parameters of the welded box section bridge and then comparing the results with the FE simulation. Local buckling of web and compression flange has also been taken in account while performing GRG2 optimization. Later vigorous random and 1st order Design Optimization is performed to establish design space and convergent solutions utilizing commercially available FEA software. Maximum allowable bending stress, allowable shear stress and deflection are constrained to recommended limits of design norms, keeping the volume of the bridge as objective function. The size and thicknesses of plates, stiffeners and reinforcement are included as design parameters (DV’s). Optimal girder so designed is efficient in respect of design technique and verified as cost-effective. 2. Keywords: Design Optimization, EOT Crane Bridge, Optimal Box Girder, Rolling wheel load 3. Introduction
The escalating price of structural material and energy is a global problem consequently optimal consumption of the both can not be considered redundant. Overhead crane, which is a synonym for material handling in the industrial environment, utilizes structural steel for its girder and energy (mostly electrical) for its operation. Light girder for overhead cranes not only save material cost but also trim down energy expenditure because of subsequent employment of low powered drive units. The general procedure for design of EOT crane girders is accomplished through guidance stipulated in the prevailing codes and standards. Thus optimal design in such case is not the one which just exhibit stress criteria offered in structural design methods but the one which follows the limits restrained by the aforementioned codes and safety rules. Shape optimization of closed box type section was studied by Gibczynska et al [1]. Regarding optimal design of simple symmetrical welded box beam Farkas J & Jarmai K [2] incorporated bending stress, shear stress and buckling constraints while kept cost, mass and deflection as objective function. Megson, T H.G. Hallak [3], parametrically and numerically analysed load bearing diaphragms girder at single support point. Narayanan [4], in his two consecutive papers examined strength capacity of webs with cut-outs and rectangular holes and emphasized on prediction of stress in such cases. Recently little literature is found which chiefly reviews crane box beam optimization except for Tadeusz Niezgodzin´skia, Tomasz Kubiakb [6] who considered buckling problem of web sheets in box girders of overhead cranes due to welding of the backing strips.
Figure 1. Schematic of an EOT Top Mounted EOT Crane
However in recent past thin walled cold formed steel beams have drawn interest of various researchers in design optimization discipline. G.J. Hancock’s [7] study on beam-columns is very useful for the buckling and strength investigation in thin walled structures. Tuan Tran, Long-Yuan Li [8] carried out global design optimization of the cold formed lipped channel cross-section beams subjected to uniformly distributed loading analytically using Trust Region Method. Local, lateral and distortion buckling had
also been addressed in this paper. For the optimization of cold formed thin walled I-beam and open sections K. Magnuki et al [9, 10, 11] large contributions can not be left un-stated. At Last but not the least Seok Heo et al [12, 13] distortion theory in thin-walled closed beam section design has been very helpful for the present research work for overhead box girder crane. The novel approach behind this paper is to introduce simultaneous computational optimization and FEA design optimization tools in the area of overhead crane bridge design which is a highly complex problem in the sense of its dynamic nature of operation [see figure 1]. This paper suggests a method to determine optimized design parameters of torsion box girder for the EOT crane bridge against minimal weight objective, utilizing Reduced Generalized Gradient Method 2 (GRG2) which is available solver in MS Excel spreadsheet program [14, 15]. To ensure the validity of computational method, Random and First Order Design optimization of the ® same single plate box beam is performed in Numerical optimizer utilizing a comprehensive APDL (ANSYS Parametric Design Language). An overhead crane can be a single girder configuration as well as a twin girder type for which the box girder is primarily subjected to two (or more) concentrated vertical wheel loads as can be seen in figure 2. The problem has been kept confined to one beam of twin girder crane for the reason that the two are identical in almost all aspects. During its life of operation an EOT crane is subjected to rolling load in addition to various other live loads. In our method the design structural loads are computed by the spreadsheet automatically keeping in view guidelines laid down in Crane Manufacturing Association of America (CMAA) [21] and FEM [22] Crane design Standards. Combined bending and shear stresses, maximum deflection and Local buckling safety factors referenced in CMAA-70 standard, are employed as the constraints. Following a detailed research, design parameters have been restricted against practical limits found in codes and handbooks related to this field. The optimum box girder mass for various span and capacities are calculated with the inclusion of the effect of uniform end tapering, diaphragm thickness and spacing. The design variables are Web, Top and bottom flange sizes and thicknesses along with Diaphragm spacing and thickness (see figure 2). Apart from the span of the girder whole model has been parameterized in computational as well as Numerical model [19]. The full capability of solver as well as Numerical simulator is put to test and the two methods have established their effectiveness for design of an optimal Girder.
P
P
L
Figure 2. Girder schematic with section details 4. Optimization Problem Statement:
As illustrated in figure2 we here take an account of a box girder of span L with depth ‘h’ at the middle portion and ‘he’ at the ends while web heights at middle and end is ‘h0’ and ‘h0e’. The breadth of the top flange is ‘ b1’ and that of bottom flange is ‘b2’ , whereas thicknesses of top flange, bottom flange, main web, auxiliary web, diaphragms are tf1, tf2, tw1, tw2 and td respectively. Seeing that the beam section is not constant along the span we choose volume as objective function instead of area as done in normal cases. The approximate objective function ‘V’ for optimal volume is depicted as below:
[
]
[
V = ( L − 2Le) b1t f1 + b2 t f2 + h0 (t w1 + t w2 ) + Le h0e (t w1 + t w2 ) + b1t f1 +
+ b2 t f2
1 2
(h0 − h0e )(t w1 + t w2 ) ]
(1)
(h0 − h0e )2 + Le 2 + N d t d b0 h0
whereas; L N d = S d In equation 1 all of the independent variables are design variables and N d is number of the vertical stiffeners or diaphragms calculated from the stiffener spacing, S d which is in fact a design variable too. The formula for volume of stiffener proposed here is kept simple here but it is quite complicated due to taper in the girder at the end. In the actual spreadsheet this fact has been regarded at length. Sensitivity studies show that the rate of change of the aforementioned Design DV’s has direct bearing upon the structural response. The restrictions on bending stresses, Combined Stresses, Critical buckling stresses and deflection applied by the CMAA code are imposed as constraints to the optimization problem which are specified as under.
∑ σ b =
∑ M t Z t
+
∑ M l Z l
≤ σ all
(2)
2/9
τ
= Q x S x I xxe t w1 + M tor 2 An t w1 ≤ τ all t =
σ
2 σ x
2 + σ y2 − σ xσ y + 3τ xy ≤ σ all
σ
2
δ max
(3) (4)
+ 3τ 2 ≤ σ all
(5)
≤ δ all
(6)
σ 1 K σ comb
≤ DFB
(7)
The variables are defined as follows:
∑ M t = Summation of computed moments due to loads along transverse direction ∑ M l = Summation of computed moments due to loads along lateral direction b = Summation of computed flexural stress due to transverse and lateral loads Z t = Critical Section modulus of designed box girder along transever se direction σ
Z l = Critical Section modulus of designed box girder along lateral direction Q x = vertical shear force when crab is positioned on girder to produce maximum reaction at support S x = Moment of area on certain point of section concerned about centroid I xxe = Moment of Inertia box girder at end support M tor = Torsional moment at weakest section of box Girder An = Nominal area of weakest section of box girder
= Summation of computed shear stress at bridge crane highest shear and weakest position on bridge σ = Computed combined stress where state of normal orthogonal stresses σ or σ and shear stress τ xy exists t x y
τ
+ 3τ 2 = Computed special combined stress where state of only tensile or compressive stresses along with shear stress exists δ max = Computed total vertical deflection due to total bending moments σ
2
δ all =
Maximum allowable vertical deflection
all = allowable stress from CMAA 70 Crane standard for the particular load case
σ
The primary objective of this paper is to optimize the crane box bridge in view of CMAA 70 crane design standard therefore the allowable stresses and deflections in the inequalities 2 to 7 are used to constrain the model. At present design norms like CMAA depict few confines to the parameters of the box girder which will be our DV’s limitations. These restraints may be explained as below:
• • • • • • • •
Ratio L/h should not exceed 25 [21] Ratio L/b should not exceed 65 [21] Ratio h/t w1 should not exceed 240 [22] Ratio b/t f1 should not exceed 60 [22] S d should not exceed depth h of the section [21] he should be 0.35h to 0.4h [25] Le should be 1/8th to 1/6 th of L [25] t dia is kept between 2 to 6 mm [25]
5. Loads and stresses and stability of EOT Crane Bridge:
The optimization problem of box girder crane bridge is not the straight forward problem for the reason that the girder has to be optimized against several load cases simultaneously. The bridge of crane is always subjected to dead load, inertial loads, skewing and concentrated live loads etc. The inertial loads calculated according to CMAA rules which require pre-selection of all the dynamic data like duty, speeds, load spectrum etc. Optimization of Crane Bridge therefore requires the comprehensive knowledge of crane design. With several years of crane design experience it has been established that the spans and capacities of cranes have much variation from case to case situations. During EOT crane operation the girder is also subjected to fatigue load. However for the optimization against fatigue due to fluctuation of stress has not been considered in this paper. The most important task is to calculate wheel load P which is compute from lifted load, crab or trolley dead load and vertical inertial loads. Here we consider two wheels per girder of span L and these are ‘a’ distance apart (see figure 2). The said wheel load is collectively referred to as rolling load which causes maximum bending moment in the beam under one of the wheel of the crab when center of gravity of twin loads and wheel are equidistant from opposite supports. To calculate the already defined maximum bending moment M t due to moving or rolling load we use following formula which is available in AISC manual [23]:
3/9
M t =
P ⎛ a ⎞ ⎜ L − ⎟ 2 L ⎝ 2 ⎠
2
(8)
As mentioned above there are lateral and skewing loads due to the motion of the crane which also cause compression and tension laterally. Due to inertial action of trolley the girder top flange is subjected to direct compressive stress which adds to the flexural stresses already induced due to transverse moment. The top flange is under compressive stress as well as local bending stress. The bottom flange is also essentially to be designed against tensile stress. In addition to stress constraints the CMAA 70 design code limits the vertical deformation of the girder. Also positive camber in the girder is produced by fabrication. The webs and top flange are also designed against local buckling conditions. Buckling largely depends upon the ratio of plate thickness to the width of the plate and spacing of diaphragm or stiffeners. The web is subjected to local buckling because the distribution of flexural stress changes sign over its entire depth. About half of upper region of the web is under compression which can originate half wave length local buckling in the girder. Similarly compression flange is also susceptible to buckling due to compressive stress. The box girder is under lateral compression also, consequently non uniform compression is induced in the flange i.e. compression is higher at one edge and lower at the other. For both cases the web and top flange are checked against corresponding critical buckling stress which is a multiple of Euler’s buckling stress [21,24] and buckling coefficient K σ [21,24]. Finally Critical stress is calculated and compared with empirically obtained buckling design buckling factor [21,24]. Being a torsional box girder it is eccentrically loaded causing torsion therefore maximum shear constraint is also taken in consideration as described in constraint inequality 5. However for box girder torsion is not a big problem due to the presence of internal diaphragms which resist distortional effects. 6. Design Optimization Methodology
If crane girders are designed to sustain the loads only the process is simple and straightforward but for the optimal design a long comprehensive iterative procedure is needed which is normally difficult by usual computation efforts. To undertake this complex task an intelligent and automated computational procedure is needed which is outlined here. The design optimization methodology, as viewed in figure 3, is to start the process by feeding crane specifications and previously defined box girder design variables into the pre-designed computational spreadsheet in Ms Excel. The work sheet instantly calculates the required bending and shear stresses, transverse and lateral deflections, critical buckling stresses and buckling design safety factors. Already modelled GRG2 solver for objective, constraints and design parameters limits is allowed to iterate to obtain minimal mass beam as a subsequent step. The transformed initial variables are then inspected for their expediency. In case they are appropriate the solution is qualified as optimal. In certain situations if the solution is impracticable the solver is permitted to generate few more solutions by initialization of different set of variables. The Spread sheet has very elaborate answer and sensitivity reports along with scenario report capability. Consequently the results are stored easily and retrievable for future comparisons and utilization in design and fabrication drawings. Since the solver can solve continuos or integer values for DV’s the values of thicknesses or other dimensions may be needed to be rounded of to the nearest feasible. The other alternative path is the numerical verification. There are two choices either we take the best design variables set from the spread sheet and try the model through FEM. The other mode which has been recommended in figure 3 is to use design optimization simulations to compute optimal volume of the girder. Later the optimized thicknesses are compared with the spread sheet optimizer. Optimized DV’s suitability is left to further judgement against fabrication and practical aspects.
Start
Specify Crane capacity, span, speed and accelerations
Stress & Deformations computation and their summations as per CMAA70 standard in a single spread sheet Stability Calculations
Design variables & loads initialization
Design Optimization in ANSYS
Comparison & Best Design set Chosen GRG2 Optimizatio n
APDL Feasible
Not feasible
DV’s Feasibility check
Figure 3. Crane Optimization Methodology Flowchart
4/9
7. Computational Optimization Results and Discussions
Computational optimal solution has been accomplished through Microsoft Excel spreadsheet in combination with bundled GRG2 Solver. The reason behind the selection of the said solver is that it design optimization of crane is highly nonlinear and GRG solvers and their variants are well suited for such problems [28]. Crane girders of capacity 10, 20, 30, 40 and 50-ton capacities and 22, 25.5, 27.5 and 30 meter span lengths have been optimized in this study. The optimal objective is single girder mass and as previously mentioned the corresponding eleven optimized design variables h, b0, tw1, tw2, b2, tf1, tf2, Sd, tdia, he, Le examined against allowable flexural stress, transverse & lateral flexural deformation and buckling resistance. As seen in graph of figure 4 the depth ‘h’ of the girder rises linearly with the increase in the span for 10 ton crane girder. The values of h in 10 ton capacity remained near DV lower limit which is L/25 but when the tonnage increased it showed some variations. This situation mainly occurs due to compensatory thickness change of the web. The same situation has been observed with web spacing b. There are two sets of design parameters which have been assigned in this problem one type can be categorised as the size variables and the other are thickness variables. The optimizer depending upon the design sensitivity tries to first gradually modify only thickness variables keeping the overall size at the lowest design limits. This is because of the fact that overall size increase may raise mass to maximal level quickly. Keeping in mind this behaviour transformation of overall DV’s with same tonnage and different spans can be viewed in graphs in figure 4. Adjustment of optimal thicknesses of webs and flanges with the span are given in figure 3. Another important aspect of this particular crane beam optimization procedure is the induction of diaphragm spacing ‘S d ’ and thickness ‘t d ’ as design variable to create a stiffened girder. Optimal diaphragm spacing and thicknesses is highly desirable in minimization of mass of crane box girder. The optimization of beam depth along the span has also been considered by including Le and he variables. These two parameters reduce sufficient amount of weight by tapering the beam near the ends of span under limitations imposed by design constraints. Design sensitivity of Le is very hight therefore the graph for all the capacities is supper imposed. 10-ton
20-ton
30-ton
40-ton
50-ton
10-ton
2000.00
20-ton
30-ton
40-ton
50-ton
700.00 600.00
) m m ( b , g n i c a p s b e w
1500.00
) m m ( h , h t p e d
500.00 400.00
1000.00
300.00 200.00
500.00
100.00
0.00
0.00 22
23
24
25
26
27
28
29
30
22
23
24
25
span, L (m)
10-ton
20-ton
26
27
28
29
30
span length, L (m)
30-ton
40-ton
50-ton
10-ton
800.00
20-ton
30-ton
40-ton
50-ton
6000.00 ) m ( e L , h t g n e l
5000.00
) m m ( e h , h t p e d d n e
600.00
4000.00
400.00
3000.00
r e p a T d n E
200.00 0.00
2000.00 1000.00 0.00
22
23
24
25
26
27
span length, L (m)
28
29
30
22
23
24
25
26
27
28
29
30
span length, L (m)
Figure 4: Transformation in optimized overall size parameters of box girder of 10, 20, 30, 40, 50 ton crane with 22, 25.5, 27.5 and 30m spans
The alteration in optimal diaphragm thickness and spacing with various spans can be seen at figure 6. In table1 design parameters of an actually manufactured crane girder and the one optimized by the said computational method have been compared. Almost 38% mass reduction has been achieved by using the computational optimizer.
5/9
8. Numerical Optimization
First order numerical optimization of the full stiffened box girder is performed with the help of parametric design language. The box girder of 22, 25.5, 27.5 and 30 m span and 10 ton loading capacity are parametrically constructed in numerical design FE optimizer. The box beam is loaded with static concentrated transverse, lateral, skewing and inertial loads simultaneously at critical moment point as discussed above. The values of Elastic Modulus and Allowable stress is taken as E = 205000 MPa, and σall = 165 MPa as stipulated in the CMAA Standard. A great amount of time has been spent over preparation of model of girder with APDL (ANSYS parametric design language). A step by step procedure of numerical optimization can be envisaged through the flow chart shown in figure 7. After several attempts a conclusive parametric model was achieved which rendered numerical optimization possible.
10-ton
20-ton
30-ton
40-ton
50-ton
10-ton
12.00 ) m m ( 1 w t , 1 b e w s s e n k c i h t
40-ton
50-ton
10.00
8.00 6.00 4.00 2.00 0.00
8.00 6.00 4.00 2.00 0.00
22
23
24
25
26
27
28
29
30
22
23
24
span length, L (m)
10-ton
20-ton
30-ton
25
26
27
28
29
30
span length, L (m)
40-ton
50-ton
10-ton
20.00
20-ton
30-ton
40-ton
50-ton
14.00 ) m m ( 2 f t , e g n a l f
12.00
15.00
10.00
10.00
p o t T
30-ton
12.00 ) m m ( 2 w t , 2 b e w s s e n k c i k t
10.00
) m m ( 1 f t , s s e n k c i h t e g n a l f
20-ton
m o t t o b
5.00 0.00
8.00 6.00 4.00 2.00 0.00
22
23
24
25
26
27
span length, L (m)
28
29
30
22
23
24
25
26
27
28
29
30
span length, L (m)
Figure 5: Variation of optimal thickness of different tonnage capacity girders with changing span
After model generation and initial FE simulations values of maximum deformation and equivalent stress are stored as parameters to be constrained. Keeping the volume as objective function the FEA simulator after several iterations optimizes the design parameters and structure and various thicknesses of shell elements of FE model change dynamically along with iterative steps. The stress and deflection situations are also clearly observed for the comprehensively optimized model. The stiffener spacing and thickness are also adjusted during numerical optimization. Table 2 shows optimized variable sets obtained for 10 ton and 22m, 25.5 m, 27.5m and 30m spans. The optimised volumes obtained from optimizer exhibit very low mass due to the reason that the girder gives optimal result against stress and deflection. As mentioned in flow chart the stability check is achieved by computational optimizer. 9. Conclusions and Recommendations
Optimization of crane box bridge is a complex nonlinear problem for which a simple computational procedure has been suggested. The worksheet with solver can take more than thousand variables and very small to large scale nonlinear optimization can be performed. The results seem to be encouraging and the optimized girders are lighter than the cranes manufactured and supplied in the prevailing market. Additionally the designed spreadsheet does not cross any limit and therefore robust design can be approached. Computational work sheet as well as numerical optimizer calculates optimum parameters in millimetres and most of the time thicknesses are not in whole number. Such problem can be solved by rounding up to a nearest whole number or introducing the integer constraint in the solver which could bring about fractional rise in the weight of the crane. In crane design environment this methodology might help to save precious design time, without incurring high computational cost. The Numerical optimizer has been assessed for simple unstiffened box beam to girders with stiffener and variety of load cases and it has been
6/9
established that computation time may increase to a moderate level. The APDL created for girder needs only parameter, loads and spans initialization. Later no further assistance is required for the complete model construction in FE software. The optimization is to be triggered of by another batch command which inputs design constraints and design variable limits to FE optimizer. The computational worksheet has not been considered for fatigue and weld design optimization for the plate girder which has been intentionally left for the sake of simplicity. These cases can be incorporated in the spread sheet in future. 10-ton
20-ton
30-ton
40-ton
50-ton ) m m ( a i d t , s s e n k c i h t
2000.00
) m m ( d S , g n i c a p s
1500.00 1000.00
m a r h p a i d
m g a r h p a i d
500.00 0.00 22
23
24
25
26
27
28
29
30
10-ton
20-ton
24
25
30-ton
40-ton
50-ton
4.00 3.00 2.00 1.00 0.00 22
23
26
27
28
29
30
span length, L (m)
span length, L (m)
Figure 6: Optimal diaphragm spacing with thickness variations
Table 1: Actual Crane and Optimized crane Parameters
Actual Crane Depth h web spacing inner Web1 Thickness Web2 Thickness Overhang (outstand) Top flange thickness Bottom flange thickness diaphragm spacing diaphragm thickness End section Depth Taper length at ends Girder Mass
h b0 tw1 tw2 b2 tf1 tf2
1638 715 12 12 126.9195 15 12
mm mm mm mm mm mm mm
Optimal Crane 1724.71 550.00 10.09 10.09 110.00 9.17 9.17
Sd tdia he Le
500 5 450 2000 15332
mm mm mm mm Kg
689.89 2.00 603.65 4583.33 9537.00
mm mm mm mm mm mm mm mm mm mm mm kg
Figure 7: Numerical Design Optimization process
7/9
Table 2: Feasible Design sets achieved from Numerical Optimizer simulations for 10 ton crane
Span
Equivalent Stress, SMAX
[SV]
Maximum Displacement, MXDSP, [SV] Lateral Displacement DISPX , [SV]
Nominal Height HN
Web spacing BN
22m
25.5m
27.5m
30 m
*SET 5*
*SET 5*
*SET 6*
*SET 6*
[FEASIBLE]
[FEASIBLE]
[FEASIBLE]
[FEASIBLE]
104.66
109.41
142.8
132.56
39.298
45.313
54.142
59.484
8.003
6.9973
7.4706
8.6066
[DV]
[DV]
880
1085.7
1100
1205
338.46
401.74
450.33
461.54
98.685
78.462
84.615
97.789
5.00
5.00
5.00
5.00
5.0767
5.0405
5.0462
5.0326
5.1954
5.00
5.00
5.00
5.00
5.1359
5.1567
5.1846
394.64
483.76
520.82
420
2809.3
3256
4494.9
4068.1
2
2
2.7942
2
790
724.08
1963.5
824.36
VOLUME [OBJ]
3.28E+08
4.90E+08
5.33E+08
6.10E+08
GIRDER MASS
2581.70
3847.78
4188.52
4797.82
Overhang Under load, B2
[DV]
Main Web Thickness, TW1
[DV]
Auxiliary Web Thickness, TW2 Top Flange Thickness TF1
[DV]
Bottom Flange Thickness TF2 End Height HE
[DV] [DV]
[DV]
End Taper Length LE
[DV]
Diaphragm Thickness, TDIA Diaphragms Spacing, SD
[DV]
[DV]
10. References (1) Gibczynska, Teresa. Sularz, Stanislaw. Optimization of shape of the box girders. Archiwum Budowy Maszyn. v 34 n 2 1987 p 153-171. (2) Farkas, J. Jarmai, K. Multiobjective optimal design of welded box beams Microcomputers in Civil Engineering. v 10 n 4 Jul1995. p 249-255. (3) Megson, T H G. Hallak, G. Optimum design of load-bearing box girder diaphragms having a central support Thin-Walled Structures. v 22 n 3 1995. p 203-215. (4) Narayanan, R. and Der Avanessian, N.G.V. 1983. Strength of Webs Containing Cut-Outs, IABSE Proceedings P-64/83. (5) Narayanan, R. and Der Avanessian, N.G.V. 1983. Equilibrium Solution for Predicting the Strength of Webs with Rectangular Holes, Proc. ICE, Part 2. (6) Tadeusz Niezgodzin´ski, Tomasz Kubiakb The problem of stability of web sheets in box-girders of overhead cranes.Poland. Thin-Walled Structures 43 [2005] 1913–192 (7) G.J. Hancock, K.J.R. Rasmussen, Recent research on thin-walled beam-columns. Australia .Thin-Walled Structures 32 [1998] 3–18 (8) Tuan Tran, Long-yuan Li.Global optimization of cold-formed steel channel sections .U.K Thin-Walled Structures 44 [2006] 399–406 K. (9) K Magnucki, M. Rodak, J.Lewin ′ skia. Optimization of mono- and anti-symmetrical I-sections of cold-formed thin-walled beams. Poland. Thin-Walled Structures 44 [2006] 832–836 (10) K. Magnucki, M. Mac ′ kiewicz, J. Lewinski. Optimal design of a mono-symmetrical open cross section of a cold-formed beam with cosinusoidally corrugated flanges. Poland. Thin-Walled Structures 44 [2006] 554–562 (11) K. Magnucki. Optimization of open cross section of the thin walled beam with flat web and circular flange .Poland. ThinWalled Structures 40 [2002] 297–310 (12) Seok Heo , Jin Hong Kim , Yoon Young Kim, Significance of distortion in thin-walled closed beam section design International Journal of Solids and Structures 40 [2003] 633 –648 (13) Jeppe Jonsson. Distortional theory of thin –walled beams. Thin-Walled Structures 33 [1999] 269–303 (14) Lasdon, L. S.; Waren, A. D.; Jain, A.; and Ratner,M. 1978, “Design and testing of a generalized reduced gradient code for nonlinear programming, ACM Transactions on Mathematical Software,Vol. 4, No. 1, pp. 34–49. (15) Fylstra D, Lasdon L S, Watson J and Waren A D 1998 Design and use of the Microsoft Excel Solver Interfaces 28 29–55 (16) N. Silvestre. Generalised beam theory to analyse the buckling behaviour of circular cylindrical shells and tubes.Portugal.2007. Thin-Walled Structures (17) P. Vinot , S. Cogan, J. Piranda. Shape optimization of thin-walled beam-like structures. France. Thin-Walled Structures 39 [2001] 611–630 (18) B.P. Gotluru , B.W. Schafer , T. Peko¨z. Torsion in thin-walled cold-formed steel beams. USA. Thin-Walled Structures 37 [2000] 127–145
8/9
(19) M. Mrzyglod and A. Osyczka ,Parallel Computing for Design Optimization with Computationally Expensive Functions using Evolutionary Algorithms, Proceedings of the Eighth International Conference on the Application of Artificial Intelligence to Civil, Structural and Environmental Engineering, Civil-Comp Press, Stirling, Scotland. (20) S.M.B Alfonso, J.Sienz and F.Belblidia. Structural optimization strategies for simple and integrally stiffened plates and shells.2004.International Journal for computer –Aided engineering and Software. (21) CMAA Specification # 70, Revised 1994.The Crane Manufacturers Association of America, Inc. (22) H.G.Greiner, Crane Handbook.,Whiting Corporation, Illinois (23) AISC [American Institute of Steel Construction] Manual. (24) Federation Europeenne de la Mautention [FEM].3rd Edition 1998.Rules for the design of Hoisting Appliances. (25) Lifting Appliances design handbook. Chinese edition. (26) Gill, P. E., E. Murray and M. H. Wright, Practical Optimization, Academic Press, New York [1981]. (27) R.C Hibbeler. Mechanics of Materials. 1997, 3rd edition. (28) A.A. HAGGAG , France, A variant of the generalized reduced gradient algorithm for non-linear programming and its applications, Techniques d'Avant Garde S.A., 75116 Paris
9/9
