davies j m et al_the design of perforated cold-formed steel sections

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Thin-Walled Structures Vol. 29, Nos. 1-4, pp. 141-157, 1997 © 1998 Elsevier Science Ltd. All rights reserved

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The Design of Perforated Cold-Formed Steel Sections Subject to Axial Load and Bending

J. Michael Davies a*, Philip Leach b and Angela Tayloff

"Manchester School of Engineering, University of Manchester, Manchester, UK bDepartment of Civil Engineering, University of Salford, Salford, UK

CLink 51 (Storage Products) Ltd, Halesfield 6, Telford, UK

ABSTRACT

The uprights in a typical pallet rack are typically singly-symmetrical cold- formed sections subject to axial load together with bending about both axes. They usually contain arrays of holes in order to enable beams to be clipped into position at heights that are not pre-determined prior to manufacture. Their slenderness is such that their behaviour may be influenced by the three generic forms of buckling, namely local, distortional and global (lateral torsional). In practice, these members have generally been designed on the basis of expensive test programmes. This paper addresses the problem of how they might be designed analytically. The basis of the investigation is a comprehensive set of test results on upright sections in compression which embraces both stub column tests, in which the load position was varied along the axis of symmetry, and longer columns. The test results were analysed using both finite elements and a version of "Generalized Beam Theory" ( GBT) which incorporated system- atic imperfections. Consideration was also given to the design procedures proposed by the "Federation Europeene de la Manutention" (FEM) and recent research into the influence of perforations on the performance of cold-formed steel sections. It is shown that GBT can be modified to take account of perforations so that the lower bound results give a sufficiently accurate column design curve, which takes account of local, distortional and global buckling, thus making extensive testing unnecessary. © 1998 Elsevier Science Ltd. All rights reserved

*Author to whom correspondence should be addressed.

141

142 J. M. Davies, P. Leach, A. Taylor

1 INTRODUCTION

The design of an upright in a typical pallet rack represents a particularly difficult problem in structural engineering. These members are typically singly-symmetrical cold-formed sections subject to axial load together with bending about both axes. Their slenderness is typical of cold-formed sections so that it is necessary to consider the three generic forms of buckling, namely local, distortional and global (lateral torsional). The problem is made more difficult because pallet racking uprights usually contain arrays of holes in order to enable beams to be clipped into position at heights that are not pre-determined prior to manufacture.

In practice, these members have invariably been designed on the basis of expensive test programmes. This is undesirable, not only on the basis of economics but, more fundamentally, because it makes it very difficult for the manufacturer to consider a range of design parameters in order to optimize the design of his uprights.

This paper addresses the problem of how this type of member might be designed analytically.

1.1 Background to the project

Link 51 (Storage Products) Ltd is the leading manufacturer of storage racking and shelving systems in the UK. The company is part of Wagon Indus- trial Holdings plc with large sales in the UK and exports around the world. The Company, in partnership with the Universities of Manchester and Sal- ford, through an EPSRC- sponsored Teaching Company Scheme, has been given a grant for a project to investigate how the reliance on physical testing in the design and development of the columns of pallet rack structures may be minimized. Typical sections produced by the company, which are the subject of the research described in this paper, are shown in Fig. 1.

Heavy Duty Extra Heavy Duty

) I

Fig. 1. "XL Boltless" uprights.

The design of perforated cold-formed steel sections subject to axial load and bending 143

1.2 Design process for storage racking columns

Unperforated cold-formed section columns with one axis of symmetry and simple boundary conditions can be designed analytically, using well established mathematical equations. However, an established analytical method for the design of perforated columns with intermediate restraints is not yet available and, for this reason, they are currently designed using prototype testing. Perforated columns are widely used in the Storage Racking Industry and are usually designed using the process shown in Fig. 2. However, the prototype testing approach to the evaluation of designs is inefficient and uneconomical. This project aims to replace this process with an analytical method for evaluating designs which will be in harmony with European design standards. ~ Fig. 3 shows the proposed design process.

There has been a significant amount of research concerned with the

(

Prototype

__+ Io -r

Fail

F~tss

Fig. 2. Current design process.


Fall

C~Umi~

ValidatedDesign ]

Fig. 3. Proposed design process.

influence of single holes on the behaviour of thin plates and thin-walled sections. However, there appears to have been comparatively little research into the influence of arrays of large numbers of holes. Rhodes and Schne- idez a have given some preliminary consideration to this problem and some of their results are used later in this paper. More recently, Rhodes and Macdonald 3 have given more detailed consideration to the effect of perforation length, an effect which is not considered in detail here because the perforations in the Link 51 uprights are sufficiently short in length for this effect to be relatively unimportant.

1.3 European design procedures for the uprights in pallet racks

The current state of the art is enshrined in the Federation Europeene de la Manutention (FEM) design recommendations ~ which, in turn, have been


written to be in harmony with Part 1.3 of Eurocode 3. 4 In general, the uprights of pallet racks are subject to axial load and bending about both axes. This aspect of the design is treated by a simple interaction equation in which the three effects are separated. The axial load term in this interaction equation requires that a "column curve" is established which relates the axial load capacity to the slenderness. It is the derivation of this column curve which requires the extensive testing that this paper is intended to reduce or eventually eliminate.

It is well known that, when singly-symmetrical columns are tested, the axial load capacity is sensitive to the load position. This is particularly so with relatively short "stub columns". The load position giving the highest capacity is not easy to determine because it is influenced by local buckling and, in the case of racking uprights, by the influence of the arrays of perforations. However, both of the above design codes allow the design to be based on the maximum axial load that can be achieved and the load position can be adjusted along the axis of symmetry in order to find the optimum. It follows that the stub column behaviour is a rather important aspect of the design which must be considered carefully.

Having established the optimum load position and the stub column capacity, the design procedure then requires that the remainder of the column curve is established by testing a range of column lengths. However, uprights in pallet racks are assembled into "upright frames" which provide the stability of the pallet rack system in the cross-aisle direction. These frames include bracing members which both connect and stabilize the uprights at regular intervals. In order to obtain realistic results from a testing programme, the FEM code I requires that the upright is tested as part of a frame. This, of course, adds both expense and complexity to the testing procedure.

Similarly, it is necessary that the corresponding analytical procedure can take account of intermittent restraints in the cross-aisle direction. This rules out most of the classical methods of analysis but Generalized Beam Theory (GBT) proves to offer particularly appropriate facilities because the individ- ual displacement modes can be separated and restrained individually.

In common with all relatively slender members which fail in modes involving buckling, the uprights of pallet racks are sensitive to imperfections. This means that there is a scatter of test results and the European design procedures provide a relatively complex method of dealing with these statistically in order to obtain a reliable column curve for design. Similarly, imperfections are included in the analyses described in this paper. Finding an appropriate methodology for including imperfections in GBT proved to be a particularly interesting part of the project.


1.4 Scope of the work

Two methods of analysis have been used in this study to predict the load bearing capacity of a number of stub columns subject to eccentric axial loading. The first method used a non-linear finite element package (ANSYS) to model the members. The second method extended the use of GBT in order to allow for the influence of perforations and yielding of the section.

The GBT analysis was then extended to consider longer column lengths taking into account the influence of intermittent restraints in restraining certain of the displacement modes at intervals along the member.

The analytical studies were supported by a comprehensive test programme and the results of each analysis method were compared with the results of this large test series.

2 FINITE ELEMENT ANALYSIS

Stub columns, 300 mm long, were analysed using the finite element analysis software package ANSYS 5.0a. The generated model was designed to be an approximation to the test setup used in the stub column testing program.

Table 1 summarizes the properties of the elements 5 that were used in the analysis.

Fig. 4(a) illustrates the mesh used for the analysis of a typical heavy- duty column and Fig. 4(b) and (c) give further details.

TABLE 1 Properties of the Finite Elements Used in the Analysis

Element name SHELL43 SOLID45 CONTA C49

Position of element

Description

Number of nodes Degrees of freedom

Upright Loading plate Contact between upright and plates

Plastic shell element 3-D structural solid 3-D point-to-surface element contact element

4 8 5 x, y, and z x, y, and z x, y, and z

translational and t r ans la t iona l translational rotational d i sp l acem en t s displacements,

displacements temperature.


(b) Detail around perforation

(a) General arrangement (c) Detail of upright

Fig. 4. Finite element mesh used for stub-column analysis.

2.1 Boundary conditions for finite element analysis

The boundary conditions proved to be particularly troublesome and a great deal of attention was given to this aspect of the modelling. Heavy loading plates were incorporated in the model and contact elements between the upright and these loading plates simulated the frictional contact that occurs in practice. The loading plates were fixed in space while allowing the following degrees of freedom:

• vertical translational movement of the bottom plate only • rotational movement of both plates

The loading was applied to the bottom plate using the controlled displacement method. This simulated the action of the loading jack. The displace-


ment was applied at one of the nodes on the axis of symmetry along the underside of the bottom plate. Different nodes could be chosen in order to allow different loading positions to be analysed.

3 GENERALIZED BEAM THEORY

GBT is a method of analysis that can be used for any unbranched open cross section which considers both local and overall buckling together with cross-section distortion. The basis of the method has been extensively described elsewhere 6-8 and will not be repeated here. An important feature is the separation and orthogonalization of the displacement and buckling modes so that these can be considered individually or in selected combinations. This facility has proved to be particularly useful in studies of the buckling of cold-formed sections, notably those susceptible to distortional buckling. Here it has the added advantage of allowing appropriate modes to be restrained by intermediate bracing members as discussed above. Ref- erence 9 illustrates the use of this method to predict the elastic buckling load of a series of steel channels and then uses the interaction formula of EC3 Part 1.3 to calculate the failure load. A similar procedure will be used in the present paper.

The method as presented here therefore accounts for:

• Overall buckling • Local buckling • Buckling involving distortion of the cross section • Yielding of the section • The effects of initial imperfections • The influence of intermediate restraints

3.1 The second-order equation of equilibrium

According to 9, the following equation can be applied to all first- and second- order problems in order to calculate the stress distribution in a section subject to an arbitrary load, or to calculate the bifurcation load of a section when subject to any load (or series of loads).

Et, kCkV ,,,, - GkkDkW + k~BkV + ~ ~ UkX,~(iWJV)" i = l j = l

.q- ijkXv(iWYt J V "1- 2 i W 'j V' ) = kqQ

(1)

where:


*k B

~,kC1

~ C ~kDl

~kD 2

kk D E G P

,J% i W

JV ~qQ

= Bending stiffness in mode k with respect to a load applied in mode k

= First-order longitudinal stiffness in mode k with respect to a load applied in mode k

= Second-order longitudinal stiffness in mode k with respect to a load applied in mode k

= kkC 1 + k~C 2 = First-order torsional stiffness in mode k with respect to a load

applied in mode k = Second-order torsional stiffness in mode k with respect to a

load applied in mode k = 2(1 - u)*kD1 -- v(kkD 1 + kkD2) = Young's modulus = Shear modulus = Poisson's ratio = Second-order terms arising from longitudinal stresses = Second-order terms arising from shear stresses = Stress resultant in mode i = Non-dimensional displacement function for mode j = Applied load in mode k

3.2 Solution of the second-order equation of equilibrium

Before attempting to solve this family of equations by any method, the first step is to calculate the cross section properties according to GBT. m Eqn (1) is then rewritten to account for initial imperfections in each of the "n" orthogonal modes of displacement that are included in GBT and to enable an iterative solution to be more readily implemented.

~t

[EkkCkV ' ' ' ' - G**DkV " + **B*V"] k - I

= ? kqQ _ + okXfW,, Vo + 2 w, jgo) = i - - j = l

(2)

In these equations, the value of kV 0 on the right-hand side is the initial (assumed) imperfection in each mode. In this study, the size of this imperfection was related to the length of the columns for the overall buckling modes (major axis bending, minor axis bending and torsion) as length/1000. For the local cross section distortion modes the size was given a fixed value of 0.1 mm. The shape of all of the initial imperfections was assumed to


take the form of (1 - cosTrx/length). Parametric studies revealed that the solution was not too sensitive to this assumed initial imperfection shape.

This series of "k" equations is then solved by applying a small increment of load kQ (i.e. a combination of axial force [k = 1] and bending moment [k = 3] to the equations and solving for kV. This gives a revised set of values of kVo that can then be substituted into the right-hand sides of the equations to give an improved solution for ~Vo in each mode. Since the equations are coupled by the iJ~X~ and iJ~X~ terms, the equations cannot be solved directly and numerical techniques must be used. In this study a finite difference approach was used with 11 nodes along the length of the section.

If this iterative procedure converges in all modes for this magnitude of the applied loads, the section is stable under these loads and they can then be increased and the governing equation solved again. A method for check- ing the convergence of the solution for all modes has been devised which appears to work in all cases. The method involves calculating the displacements in each of the "m" modes considered in turn. These then become the "initial" displacements. The "m" equations are then solved iteratively until the rate of growth of the displacements in all of the modes is a constant ratio for each mode. The final displacements in each mode can then be calculated by using an infinite geometric series which will converge if all of the ratios are less than unity but will not converge if any of the ratios exceed unity. In this latter case, the section has failed by elastic buckling. In the former case, the stresses in all of the modes can be calculated from the displaced shapes and summed at all nodes of the cross-section. If any stress exceeds the yield stress of the material, the section is deemed to have failed by yielding. If the yield stress is not exceeded, the load can be increased and the solution cycle restarted.

One refinement of the method was to allow the initial imperfections to take either a positive or negative value in each mode. For a section with "n" GBT modes, this required 2 n solutions to cover all possible combinations of the imperfections. Each of these 2 n possibilities was solved and the results were summarized as an upper and lower bound to the solutions for that particular assumed set of initial imperfections.

4 TREATMENT OF PERFORATIONS

An empirical method for the treatment of perforations in GBT has been developed. This involves the use of an "effective thickness" whereby perforated plates are treated as unperforated plates with an equivalent thickness which can be shown to result in the same behaviour as the perforated plate. The section properties for the local and distortional buckling modes are calculated using this effective thicknesses but the cross-sectional area is


taken as that of the minimum net cross-section. The gross section properties are used for the global buckling modes.

The equivalent thickness is given by eqn (3) and is based on the ratios of gross and net effective widths. This is then weighted, taking into account the length of the perforations along the axis of the column.

bn-{-(1- LL) b~]t teq ~- [~P- bg (3)

where: t~q Lp

L bo bg

b t

= equivalent thickness of the plate = length of perforation multiplied by the number of perforations

along the length of the plate = length of the plate = net effective width of the plate (eqn (4)) = gross effective width of the plate (eqn (5) and eqn (6)) = actual width of the plate = actual thickness of the plate

The effective widths of the perforated plates were calculated using BS 5950: Part 5 and a method presented by Rhodes and Schneider in a recent studyfl Perforations were considered by using the effective width over the net cross-section area. A constant stress over the effective areas of the plate was assumed. Where perforations intersect this area, the stress is disre- garded as shown in Fig. 5.

The net effective width of a plate is given by:

b n = bg - bp (4)

where: bp -- total width of perforations within the effective area of the plate

- - r / . , 2 -4

Fig. 5. Uni form stress distribution over the effective net section.

152 ]. M. Davies, P. Leach, A. Taylor

K

pcr = 185000K(b) 2

= buckling factor = 4

(5)

If Per < 0.123, then bg = b, otherwise: { /4] b g = b 1+ 1 4 / ~ p e r - 0 . 3 5 (6)

T A B L E 2 H-duty Columns

Loading position Length of column (mm)

300 600 900 1200

1 2 3 2 2 2A 3 2 3 3 2 3 3 3 3 3 2 2 2 2

T A B L E 3 E-duty Columns

Loading position Length of column (mm)

300 600 900 1200

1 2 2 1 0 2A 3 2 3 3 2 3 3 3 3 3 2 2 ( + 1 that 1 0

came out of the fig)


i

i i I

2 2 A

3 I • I I I I j I

(a) H-duty (b) E-duty Fig. 6. Loading positions.

TABLE 4 Distance Between the Loading Position and the Web of the Cross-section (mm)

Loading position

Column duty 1 2A 2 3

H-duty 16.2 25.8 28.2 40.2 E-duty 14.9 24.5 26.9 38.9

65

8 4 85 .28

24.61 THICKNESS 1 2.0 2.64 29.36

~ . . ~ Corner radii

~o.~ ~.,~ ,.o-~ H-duty E-duty

Fig. 7. Dimensions of the upright cross-sections (mm).

5 EXPERIMENTAL INVESTIGATION

Both stub column and longer column tests were carried out on two different grades of storage racking columns from the Company 's "XL Boltless" pallet racking range. The column duties tested were Heavy (H) duty, and Extra heavy (E) duty as shown in Fig. 1. Tests were carried out with different loading positions for different lengths of column. Tables 2 and 3 give the


number of tests carried out. Failure was taken as the maximum load carried by the stub column.

The different loading positions that were used in the tests are shown in Fig. 6 and given in Table 4 (dimensions in ram). The detailed cross-sectional dimensions of the columns are given in Fig. 7.

5.1 Material properties of the steel

The steel used by Link 51 (Storage Products) Ltd is cold-reduced on the coil before cold-forming into the shape of the section. This gives rise to a minimum value for the yield stress of Ormi n = 429 N/mm 2. The average yield and ultimate stresses and the modulus of elasticity for the steel in each column duty tested is given in Table 5.

5.2 Test setup

The stub columns were tested in a compression testing machine. Fig. 8 shows the test layout in which the load was applied using a 100 ton single- acting ram through steel balls and ball plates. This allowed the ends of the column to rotate in any direction. The column was connected to the ball plates with a metal pin at each end which prevented it from moving out of the rig.

The ball plates fixed to each end of the column were in two parts, namely a loading plate and a fixing plate. A pin fixes the column into position on the fixing plate which is bolted to the loading plate as shown in Fig. 9.

It should be noted that, although the FEM design code I recommends that uprights should be tested as part of a frame including the influence of bracing members, and the design procedure includes for this effect, for this initial study the uprights were tested in isolation.

6 EXPERIMENTAL AND THEORETICAL RESULTS

The theoretical results are compared with the stub column test results in Graphs 1 to 4 and with the tests over the full range of column lengths in

TABLE 5 Results From Tensile Tests

Column du~ Average yield stress Average ultimate Average Young's (2%)~(N/mm e) stress~.(N/mm 2) modulus E (kN/mm e)

H 473 512 195 E 451 483 193


I ~ LOAD CELL & HOUSING

~ LOAD BALL

FIXING PIN

"~"~"-'~'~- TEST SAMPLE

' L FIXING PLATE j ' ,

I (~-~) y BALL PLATES I v .-.-

I .,I ~__~ LOAD APPLIED

Fig. 8. Test setup.

Load ball Loading plate

Perforation in flange Stub column

Fig. 9. Fixing details.


Graphs 5 and 6. The GBT results include the upper and lower bounds when all imperfection combinations are considered as described in section 3.2.

t - " G ~ T L ~ E v ~ I

I I I I

tamd ~ l l ~ l ~ ot ~

~ I ~ 1

: ...... . . . . . . . . . L ? 7 ~ L

i ~ i r

Gr.ph 2

]

,~ ~ a . y ~ ~ ,

Tea Imllll

I T I I

G r a t ~ h 3

600 mm E x ~ Duty Sn~t

- - G~' I " Upp l r L i J~ - - ~ l ' r Lowu LJ i t x T u t ~

o i i |

G - r a p h 4

]

J

Ax2tlly Loaded, Exlm t-t~ry ~ Slxut

: - - G ~ T U R ~ I . i m l - - ~ ! I " l , o ~ s L i m l

i i i x x

- - ~ I T U~pa L i ~ t - G~T l~,nr kit~

Tinl~lJ qm~m ~0

I 1 . . . . ' : . . . . .

Graph 5 GrJph 6

Evidently, the GBT approach provides a satisfactory theoretical solution to an extremely complex problem and the lower bound results offer a con- sistently safe design approach.


7 CONCLUSIONS

Two theoretical methods for predicting the failure load of perforated light gauge steel columns have been compared with test results. GBT gives better results than Finite Element Analysis in less time. GBT easily considers geometric imperfections and gives safe results for the whole range of column lengths. The higher test results for the H-duty column (Graphs 1 and 3) could be due to post-elastic effects. Explaining the elevated test results for higher slenderness ratios of the H-duty column is more difficult. The effects of perforations could be less pronounced at longer lengths.

REFERENCES

1. Recommendations for the Design of Steel Static Pallet Racking and Shelving. Federation Europeenne de la Manutention, Section X, McLaren Building, 35 Dale End, Birmingham, January 1996.

2. Rhodes, J. and Schneider, F. D., The compressional behaviour of perforated elements. Proc. Twelfth International Speciality Conference on Cold-Formed Steel Structures, St Louis, MO, 1994.

3. Rhodes, J. and Macdonald, M., The effects of perforation length on the behaviour of perforated elements in compression. Proc. Thirteenth International Speciality Conference on Cold-Formed Steel Structures, St Louis, MO, 1996.

4. CEN ENV 1993-1-3, Eurocode 3: Design of Steel Structures--Part 1.3: Gen- eral rules--Supplementary Rules for Cold Formed Thin Gauge Members and Sheeting, Ref: CEN/TC250/SC3, 1996-04--24.

5. ANSYS User's Manual for Revision 5: Volume 3: Elements. Swanson Analysis Systems Inc., Houston, PA, 1994.

6. Davies, J. M. and Leach, P., First-order Generalized Beam Theory. Journal of Constructional Steel Research, 1994, 31, 187-220.

7. Davies, J. M., Leach, P. and Heinz, D., Second-order Generalized Beam Theory. Journal of Constructional Steel Research, 1994, 31, 221-241.

8. Davies, J. M. and Leach, P., Some applications of Generalized Beam Theory. Proc. Eleventh International Speciality Conference on Cold-Formed Steel Structures, St Louis, MO, 1992, pp. 479-501.

9. Leach, P. and Davies, J. M., An experimental verification of the Generalized Beam theory applied to interactive buckling problems. Thin-Walled Structures, 1996, 25, 61-79.

10. Leach, P., The calculation of modal cross-section properties for use in the Generalized Beam Theory. Thin-Walled Structures, 1994, 19, 61-79.

davies j m et al_the design of perforated cold-formed steel sections

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