Materials and Structures

, Volume 47, Issue 1, pp 367–379

Load bearing capacities of cold formed steel sections subjected to axial load

Authors

    • Faculty of Civil Engineering SuboticaUniversity of Novi Sad
Original Article

DOI: 10.1617/s11527-013-0066-9

Cite this article as:
Bešević, M. Mater Struct (2014) 47: 367. doi:10.1617/s11527-013-0066-9
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Abstract

The paper describes a test program on cold-formed axially compressed steel members. Research has been conducted in order to explain how cold working technology causes considerable enhancement of the material properties in finished element, comparing to the base material. In analysis, we have used results obtained with experimental and numerical methods. “PAK” software, designed for nonlinear finite element analysis of structures, derived results for ultimate bearing capacity with corresponding force–deflection graphs and buckling curves. These results are compared with those obtained in experiments on built-up members. Experiments were conducted on five series each with six specimens with global slenderness values of 50, 70, 90, 110 and 120. Compressed members were analyzed on Amsler Spherical pin support with unique electronically equipment and software. As well as force–deflection curves stress values for several cross sections along the height of the members were also determined. Both groups of results (numerical and experimental) were compared with European buckling curves. These give the value for the reduction factor of the resistance of the column as a function of the reference slenderness for different kinds of cross-sections (referred to different values of the imperfection factor).

Keywords

Axially compressed cold formed membersExperimental analysisFinite element methodBuckling curvesCarbon steelStainless highstrength steel

1 Introduction

The use of cold-formed steel structures has increased rapidly in recent times due to significant improvements in production technology of thin highstrength and stainless steel elements. We have to point out that now nominal yield limit of steel sheet is in the range of 250–550 MPa, while the most commonly used thickness is less than 1.00 mm. Cold rolled steel sections have some distinct structural stability problems, that haven’t observed in hot rolled one. [18] have presented detailed study of buckling and ultimate strength behavior for a series of cold-formed steel members. They took into the consideration axially compressed members with cross-sectional thickness in the range of d = 0.8–1.0 mm. This research shows that columns made of cold formed profiles are very sensitive to the geometric imperfection, and emhasize the importance of this factor into design procedures. Investigation made by [9] shows that similar conclusion can be made for stainless steel. In order to simplify designing process and avoid frequent testing materials and elements design rules and codes were defined. That way calculation replaced testing [19]. So far, there are three sources of International Code Families: USA, Europe and Australia. Each consisting of design codes in connection with product standards and testing results. International design codes for cold formed stainless steel structural members are: Specification for the Design of Cold-Formed Stainless Steel Structural Members [2], the Aust/NZS for Cold-Formed Stainless Steel Structures [3] and the Euro code 3—Design of Steel Structures [11]. Design codes are mainly based on the investigations of pin-ended columns but in practice there is some degree of deviation comparing to test specimens. Gao et al. [13] studied the load-carrying capacity of thin-walled box-section stub columns fabricated by high strength steel 18Mn2CrMoBA. They carried out uniaxial compression experiments of specimens with different geometrical dimensions and obtained results compared with predicted values by the AISI Code [1] Comparison shows that test results for load-carrying capacity of the high strength thin-walled box-section stub columns are much larger than those predicted by codes. Therefore suggested design method is too conservative for the high strength thin-walled box-section stub columns. [20] investigated strength and behavior of cold-formed high strength stainless steel columns. Test strengths were compared with the design strengths predicted using different codes: American codes [2], Australian/New Zealand rules [3], European specifications for cold formed stainless steel [11]. This investigation demonstrates that strength values predicted by codes are too conservative for the cold-formed high strength stainless steel columns. Ellobody [10] in its own investigations used same international design rules for comparison. Study has been performed to investigate the effect of cross-section geometries on the strength and behavior of cold-formed high strength stainless steel stiffened and unstiffened slender hollow section columns. The results obtained from the finite element software ABAQUS have been compared with those predicted by design rules for cold-formed stainless steel structures. Comparison indicates that strengths calculated with design code are generally conservative for the cold-formed stainless steel unstiffened slender square and rectangular hollow section columns, but slightly unconservative for the stiffened slender square and rectangular hollow section columns. Results of the parametric study shows that high strength stainless steel columns with stiffened slender hollow section have a considerable increase in the column strength comparing to the one with unstiffened slender hollow section.

Liu and Young [17] described test program on axially compressed cold-formed stainless steel square hollow section (SHS). The tests were performed over a range of column lengths, which involved local buckling and overall flexural buckling. Measurements of overall geometric imperfections and material properties of the specimens were carried out. The experimental column strengths were compared with the design strengths predicted by the American [2], Australian/New Zealand [3] and European [11] specifications for cold formed stainless steel structures. Results of comparison were similar to the former one. Among the all other researchers, Jandera et al. [16], Goggins et al. [15] as well as Cruise and Gardner [7] have carried out similar investigations with corresponding results. In experiments on stainless steel hollow sections [14] specially treated one particular feature of stainless steel that differs from carbon steels. That is the form of its basic material stress–strain curve. Although studies of the behavior of stainless steel are less numerous, sufficient exist to demonstrate that the material possesses a rounded stress–strain curve, with no sharp yield point. The stress–strain properties of the corner regions in cold-formed stainless steel cross-sections differ from the properties of the flat regions due to the material’s response to deformation. Corner regions of cold-formed stainless steel sections (SHS and RHS) have 0.2 % proof strengths, commonly between 20 and 100 % higher than proof strengths of flat regions, due to the very pronounced degree of strain hardening that stainless steel have. This is accompanied by a corresponding loss in ductility. Material properties of stainless steel are sensitive to plastic deformation that causes an increase in yield strength by thermal process. Different strain paths formed around cold-formed cross sections during manufacture create unique material strength distributions for sections from different forming routes and influence on residual stress patterns. Strength enhancements induced during the forming routes of cold-formed sections, which are the most common type of stainless steel cross section, offer higher material strength than that currently assumed in design codes. Cruise and Gardner [8] proposed method for predicting the distribution of 0.2 % proof stress around press-braked and cold-rolled stainless steel sections based on an experimental program comprising tensile coupon tests and hardness tests. The magnitude and distribution of residual stresses in cold-formed stainless steel sections has been quantified earlier but it was not treated as a factor who significantly affect on structural behavior. Achieved enhancements in efficiency are significant and highlight the importance and benefit of utilizing the strength increases occurred during process of cold- working production in designing.

2 Carbon steel properties and changes caused by cold working

In case of compressed complex cold formed members there are many parameters that affects its global and local stability. Therefore design of this type is function with numerous variables and we analyzing only those related to structural changes caused by production technology.

The first part of investigations deals with the increase of mechanical characteristics, particulary strain hardening, of cold formed profiles while second deals with determination of compressive yield limit, namely section resistance on compression, for stub column tests. Based on these analyses input mechanical parameters and geometrical imperfections for determination of appropriate buckling curves of axially compressed complex members were defined. Tests were performed on coupons obtained in two ways: from the basic sheet or strip and from the finished profile [5]. Coupons were tested in order to determine: yielding point, tensile strength, elongation percentage, Young’s modulus and ductility by applying Sharpy’s pendulum. Effect of production technology on mechanical characteristics of rolling steel mean values can be observed in the summary diagrams on Fig. 1: yield point (fyl), tensile strength (fu), elongation percentage (A) and Young’s modulus (E). In corners, changes are most pronounced and values for yield point are 560–606 MPa, while on the straight parts going between 302 and 320 MPa (Table 1) [12].
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Fig. 1

Mechanical characteristics of steel

Table 1

Mechanical properties of basic sheet steel—domestic producer

Producer

Tensile tests results

fy (N/mm2)

fu (N/mm2)

A (%)

E (N/mm2)

Eu (J)

Č0451

~EN 10292 H300LAD

280

390

27

193,000

26.67

Č0452

~EN10268 H280LA

279

373

39

Č0453

~EN10149 S315NC

296

420

28

fy yield strength, fu tensile strength, A elongation percentage, E Young’s modulus, Eu total impact energy

During definition of necessary input data we had to determine: geometry of the sample (thickness area, bend radius and static inertia moments), initial deflection, tensile and compression tests and the state of stress distribution. Profile taken for this research is cold formed C profile (Fig. 2). Cold working process causes deviations in the geometry of base section and composite member as a whole. Initial deflection of the sample can be significant if internall stresses are insignificant. Important influence on a bearing capacity of pressed column have deviation from the theoretical axis direction. Initial imperfections cause additional moment of inertia as a result between normal force and eccentricity, that affect on bearing capacity. In case when initial imperfections exceeding over theoretical limits as a results we could have earlier loss of capacity.
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Fig. 2

Cross section for profile C 90 × 45 × 20 × 2.5

Initial deflection (straightness) of samples was measured with leveling instrument Ni 0.02 in two points on perfectly horizontal plane. Precision of leveler was 0.2 mm on 1,000 m. Measurements was performed in a laboratory on a short (3.5 m) basis. Readings were obtained by optical micrometer with precision of 1/1,000 ± 5/100 mm. Along the length of a profile, shorter samples had five measurement points, middle size samples 9, longer samples 13 and longest samples had 17 measurement points both for x and y direction. Compensator insured verticality of α = 0.1″. Applied technique required a micrometer screw connected to the perfectly horizontal plate that enables 5 mm movements in a vertical plane.

2.1 Residual stresses

Residual stresses have important role in calculation and design of structural steel members.

They are caused by uneven cooling of cross section after hot rolling and production procedures such as cold forming, pressing, welding, torch cutting, etc. In cold formed profiles they are usually caused by the effects of cold bending during forming or pressing [4, 6]. Due to production differences residual stresses in cold formed profiles can be significantly different from those in hot rolled. When compared to stresses from working loads, effects of residual stresses can be harmful or beneficial depending on their magnitude, orientation and distribution. The general influence of residual stresses on structural members is to cause premature yielding, leading to loss of stiffness and a reduction in load-carrying capacity. Therefore the significance of these effects should be considered in predicting structural behavior of members. Member residual stresses were defined experimentally and their distribution through the section is given on Fig. 3. With an increase of member’s axial compression plastic deformation of the material first appears in areas where residual stresses are compressive (negative), i.e. on the inner surface of the column. Influence of the residual stresses is taken into account through correction in the initial yield stresses of the compression and tension zones.
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Fig. 3

Measured residual stresses

Investigation also included residual stress distribution along the wall thickness of cold formed stainless profiles with box cross section [16].

3 Experimental testing

Axially compressed member is a member with compression force applied along its centroidal axis. Theoretically geometrically perfect, straight axially compressed member would not have any lateral deflection for loads less then critical, but such member does not exist. In practice lateral deflection occurs from the very beginning of the load application process, due to the bending caused by the initail curvature and eccentricity of the force. Material imperfection,residual stresses and the variable yield point accross the cross section cause emerging of additional effects on lateral defection for loads above the proportional limit. Stress distribution in relation to the main section axes affect on distribution of yield zones. These effects, replaced by the equivalent geometric imprefection of the element, can significantly reduce its load bearing limit. Main issue is to make sure that joints will not influence on the stresses in the mid section of the member. Axially compressed composite members were tested with metod defined by the European convention for obtaining the stress–strain curve with constant increase in force of 10 N/mm2 per minute until the failure. The center of a cross section is the reference point for positioning of a specimen.

The EC-3 recommendations for experimental investigation give the following minimum of pretesting, measurements and testing equipment for testing of load and stability of axially compressed members:
  1. 1.

    Mechanical characteristics of the material obtained by tension tests and by stub column testing (completed for specimens U1–U6, five series each).

     
  2. 2.

    Pretesting of specimens covered cross-section characteristics, dimensions of the web, flanges, width, diameter of corner curves, i.e. exact features (area, momentum/radius of inertia/gyration). These measures were taken nine times in each of five sections along the specimen length.

     
  3. 3.

    Initial deflection i.e. straightness of the member for both axes in five sections along the specimen length.

     
  4. 4.

    Testing of residual stresses in a cross section of a member

     
  5. 5.

    Centric positioning of a specimen on the bearings

     
  6. 6.

    Besides axial compression capacity is necessary to determine: global failure limits, diagram force–deflection for the midsection, diagram force-dilatation for the midsection.

     

Testing of axial compression capacity was performed on five series each with six specimens, with global slenderness values: 50, 70, 90, 110 and 120. Lengths of built up members had the following values 122.11, 170.95, 219.79, 268.64 and 293.06 cm. The base principle of the method is a continuous dynamic loading with strain increment in the critical area in the value of 1 % per minute, until the loading process reach a critical point of force and the complete stabilization.

3.1 Testing

The study was conducted in the laboratory of the Institute for Materials, Serbia on the press capacity of 5,000 kN using testing methods for complex centric pressed rods, with a constant force increment of 10 N/mm2 per minute to fracture. Aligning is carried out in relation to the focus section. Testing was conducted on spherical bearing, Amsler manufacturer, that allows unrestricted movement in x and y direction. Electronic deflectometer Hottenger, type 50, connected to the UPM 60 device, was used to measure the application of force. For measurement of deflection along the x axis were used five deflectometers, placed along the length of samples. For perpendicular direction three deflectometers were placed, two of them were by supports and one in the middle. Out of six specimens in the same series two were used to measure dilatations and the test force. Rosette strain gages were positioned in the middle of element, in the number of 12 or 18 of them (Fig. 4).
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Fig. 4

Test specimen with measuring tapes and disposition of rosette strain gages

Centering of the specimens is done with particular reference to anlignment of the end cross section and center of the bearing. Specimens were specially processed in order to have end cross sections perpendicular to their longitudinal axis. Their surfaces were processed by a face milling cutter.Centering have more influence in case of shorter specimens and lower values of slenderness. However, imperfections in centric positioning always influence on the performance and on the load bearing capacity of a specimen. Taking into consideration the initial curvature of a specimen and low values of residual stresses in some cases this influence can become dominant. Buckling of the elements starts from the begining of load application. Data obtained by three types of experimental investigation gave the basis for calculations of the buckling curves—global failure limit, stub column test and elongation of the basic steel material.

3.2 Experimental results for axially compressed members with different global slenderness values

Dilatation testing was conducted wrapping up measuring tapes around the middle section of the column and connecting measuring tapes with a multichannel measuring device UPM-60. Derived results were recorded with corresponding software and automatically transformed into force-dilatation diagrams. Measured dilatations have even values on same side of profile but on other/outher side values change sign under the boundary influences, what is presented on the diagram Fig. 5. Same diagram shows that deformations on pressed part of the element are higher becouse of inner stresses caused by a technological process (cold rolling, pulling through a system of rollers) [4].
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Fig. 5

Relation of force and strain for U61 specimen

From buckling tests on specimens with measurement tapes also were derived data presented on digrams of critical test force and maximum deflection (Fig. 6).
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Fig. 6

Graph of measured deflections for maximum forces (P) obtained for U61 specimen

Results obtained in experimental investigation of axial load bearing capacity for cold formed composite members (2C90 × 45 × 20 × 2.5) were compared to the European buckling curves (A–D) on the basis of stub column tests (St) and elongation of the basic material (fy). The variations from the buckling curves A–D are marked specially with given percentages (Table 2). In deflection test, when factor of time is included, we have a reduction in the fall of the critical force and increase of the deflection.
Table 2

Results of experimental investigation on axial load capacity of the complex element 2C90 × 45 × 20 × 2.5, and deviations from European deflection curves based on tests on short columns

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σE yield point (results for 2C90 × 45 × 20 × 2.5), σT yield point (results from tests on short columns)

From analysis of the experimental results yielded buckling curves according to values from Euro code 3 (buckling curves A–D) concluded:
  • for moderate global slenderness values (λ = 70, 90) members with complex cross sections have to be designed according to the buckling curve C.

  • members with higher global slenderness values (λ = 110, 120) have to be designed according to the buckling curve D because loss of stability is observed prior the yielding stress—failure due to excessive deflection

  • for lower slenderness values additional research has to be conducted in order to precisely determine the appropriate design-buckling curve. From this research, the design according to buckling curve D is safer.

4 Numerical analysis

Numerucal analysis of compressed member made from cold formed steel implies problem that involves deformation and a stability. Numerical analysis was based on Finite Element Model (FEM) and PAK-software, commonly used for nonlinear static and dynamic analysis of structures. In this analysis was used a general beam finite element with deformable cross section and universal geometry. This general element can be used for linear and nonlinear analysis of construction. Proces start from the assuption that structure have one axis (longitudinal) along which the structure have constant geometry and materialisation (Fig. 7a). In cross section plane we can define shape and material of element (Fig. 7b). For better coordination characteristic points were assigned on reference axis of the beam, mathced with longitudinal axis. Each of these replacing elements could have complex structure and contain isoparametric subelements (Fig. 7c). Since beam element comprises subelements (isoparatmeric 3D, shell and beam) it can be regarded as superelement. Cross sections of each subelement can be noticed within representative cross section (Fig. 8). Segments are defined by nodes within representative cross section positioned according to coordinate system for the main beam. They have standard degree of freedom for isoparametric elements 3D, shell and beam,such as three translations and three rotations, also defined according to main beam coordinate system.
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Fig. 7

Complex structure modeling with beam superelement: a longitudinal axes. b Cross section. c Subelements of the beam superelement. d Segments within representative cross section

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Fig. 8

Types of subelements

Cross section is symmetrical, deformation (buckling) is assumed only in one plane so only one half of the cross section is modeled. Since deformation (buckling) is symmetric relative to the middle of the member’s length, calculations are performed only for one-half of the member’s length.One-half of the column’s cross section is modeled with twenty-six 2D segments and length of individual elements in this model is constant along the columns axis. Table 3 shows sample member lengths for numerical simulation, number of elements along the columns length and total number of elements. Figure 9 shows a numerical model of the member of the sample U21 and a detail with cross section and element layers along the length.
Table 3

Properties and number of elements

Member name

Length (mm)

Mumber of elements along the column’s length

Total number of elements

U21

1,221.2

20

26*20 = 520

U33

1,709.7

28

26*28 = 728

U43

2,198.9

36

26*36 = 936

U56

2,686.7

44

26*44 = 1,144

U61

2,932.0

48

26*48 = 1,248

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Fig. 9

Cross section (1/2 of composite member 2 × C90 × 45 × 20 × 2.5) modeled with 2D segments

4.1 Results from numerical simulation

Numerical simulation with beam superelement has considerable advantage, regarding to fulfillment of boundary conditions, comparing to numerical models of beams and columns with shell element. Due to the nature of the force/deflection dependence, in case when force reaches maximum and than fall back, for calculation is used axial load on the top of column. Initial imperfections were defined according to measured values on the individual samples tested in case of ultimate load state. Maximum imperfection values were varied within the numerical simulation in order to estimate its effect on the force value of ultimate load state. Initial member imperfections, in other words deviation from the straight line—longitudinal axis were defined as sin function. Figure 10 gives the numerical model shown in a state of plastic deformations over the one-half of the profile.
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Fig. 10

Maximal stresses of the numerically modeled sample-member

4.2 Comparisons with other results

Results from numerical simulation were compared experimental results for axially compressed built-up member (Table 4). Obtained relations between them are highlighted on following force–deflection diagrams (Fig. 11).
Table 4

Result comparison between experimentally obtained data (PE, σE) and numerical (FEM) simulation of the axial load capacity of complex member 2C90 × 45 × 20 × 2.5 (PN, σN)

Sample

A (mm 2)

λ0

PE (kN)

σE (kN/cm2)

PN (kN)

σN (kN/cm2)

σN/σE (kN/cm 2)

%

\( \overline{\lambda } \)

κ

U21

1,021.62

51.67

263.18

25.76

281.55

27.56

1.07

6.98

0.56

0.85

U33

1,020.35

72.28

221.73

21.73

221.76

21.73

1.00

0.02

0.78

0.67

U43

982.09

92.21

175.07

17.83

162.44

16.54

0.93

–7.22

0.99

0.51

U56

946.07

113.35

115.45

12.20

117.48

12.42

1.02

1.76

1.22

0.38

U61

989.55

123.29

104.71

10.58

102.32

10.34

0.98

–2.28

1.33

0.32

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Fig. 11

Force/deflection diagrams obtained from experiments and FEM for sampless U43 and U61 (eksperimental/numerical value)

Comparing buckling curves of composite members obtained experimentally and numerically, we can see that both groups of averaged values have very high level of conformity (Fig. 12). Same diagram consists Europian buckling curves for another comparision of numerical results from which we can derive almoust the same conclusions as with experimental results:
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Fig. 12

Buckling curves of complex member obtained experimentally and numerically

  • For moderate global slenderness values (λ = 70, 90)—buckling curve C

  • For higher slenderness values (λ = 110, 120) (loss of stability appears before plastic deformations)—buckling curve D.

  • For lower global slenderness values, additional research must be performed in order to precisely define their buckling curve (it is suggested, for safety reasons, to make calculations associated with the buckling curve D).

5 Conclusion

Bearing capacities of cold formed steel section have important role in structural design. Presented investigation considered cold formed steel sections, under axial load, in order to reveal role of cold working technology in material enhancement of finished element compared to base material for stainless, high-grade and carbon steel elements. Research included experimental and numerical method in obtaining valid results for selected parameters (initial imperfections, bearing capacity…). Both results groups were compared to each other and competent Euro code as well, in order to obtain better analysis. From those comparisons we can conclude that this type of structural element have to be design according to the buckling curve “C”, for moderate global slenderness values (λ = 70 and 90), or according to buckling curve “D” for higher slenderness values (λ = 110 and 120) as well as for slenderness values λ = 50. One more conclusion is derived during this investigation, behavior of axially compressed elements made from high grade and stainless steel is similar to behavior of those from carbon steel.

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