Stability investigation of local buckling behavior of tubular polygon columns under concentric compression

Thin-Walled Structures 53 (2012) 131–140

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Stability investigation of local buckling behavior of tubular polygon columns under concentric compression Ahmed Godat a,n, Frederic Legeron b, DieuDonne´ Bazonga c ´cole de Technologie Supe´rieure, Montreal (Qc), Canada H3C 1K3 Department of Construction Engineering, Universite´ de Que´bec, E Department of Civil Engineering, Universite´ de Sherbrooke, Sherbrooke, Que´bec, Canada J1K 2R1 c CIMA þ , Laval, Que´bec, Canada H7V 3Z2 a

b

a r t i c l e i n f o

abstract

Article history: Received 4 July 2011 Received in revised form 12 December 2011 Accepted 13 December 2011 Available online 8 February 2012

This paper presents experimental tests conducted to investigate the local buckling behavior of thinwalled tubular polygon steel columns. The experimental program consists of six stub columns with three different cross-sections, octagonal (eight-sided), dodecagonal (twelve-sided) and hexdecagonal (sixteen-sided), tested under concentric compression. For each cross-section, two values of the plate slenderness ratio (plate width-to-thickness ratio) are considered. Accurate measurements of geometrical imperfections are taken prior to the test. The experimental results show that the local buckling mode of failure depends on the type of the cross-section. Moreover, the plate slenderness ratio is the main factor controlling the local buckling capacity. Design equations provided in the ASCE 48-05, the EC3 and Migita and Fukumoto to predict the local buckling capacity of tubular polygons are evaluated against experimental results of 22 polygons tested under concentric compression available in the literature. Based on drawbacks observed in the design equations, the Loov’s equation developed on basis of the ultimate stress concept is adjusted with new fitting parameters to fit for tubular polygon columns. The accuracy of the new equation is evaluated through a comparison with the experimental results. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Thin-walled steel Tubular polygon columns Local buckling Experimental data Slenderness ratio Number of faces Design equations Ultimate stress concept

1. Introduction In recent years, thin-walled steel tubular polygon columns have been a popular form of construction for transmission line structures, antennas and sign structures. This is specifically true in urban environment due to their easier integration into the landscape than lattice tower. Tubular polygon columns are generally fabricated with cold formed folded plates with longitudinal welds. Their cost of manufacturing is higher compared to lattice structures and their use is rarely justified only by cost. This might be attributed to the non-optimal use of steel and expensive connections and welds. To reduce high manufacturing cost, engineers need to optimize the cross-section of tubular polygon columns using accurate design methods. Tubular polygons are usually subjected to high flexure and high axial load. Their cross-sectional shape needs to be optimized for adequate stiffness and strength. For a given section area, A, the inertia is a function of A, and w2, w being the width of the section

n

Corresponding author. E-mail addresses: [email protected] (A. Godat), [email protected] (F. Legeron), [email protected] (D. Bazonga). 0263-8231/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2011.12.013

measured to mid-point of thickness across flats. The flexural strength is related to A and w. Hence, increasing w theoretically results in a stronger and stiffer polygon. However, increasing w while keeping A constant means that the thickness t of the plates is reduced, resulting in a possible local buckling that limits the strength increase. A proper understanding of the local buckling of tubular polygons is therefore the key to maximize the optimization of these structural members. As for other cold formed members, the critical buckling stress at which local bucking occurs can be computed in the elastic state as an assembly of restrained plates. For these plates, the restrains along their longitudinal edges are provided through the other plates forming the polygonal section. Using the elastic analysis, the critical buckling stress (scr) of tubular polygon columns is expressed by the following equation [1]:

scr ¼ kp2 E=12ð1n2 Þðw=tÞ2

ð1Þ

where k, E, n and t are the plate buckling coefficient, modulus of elasticity, Poisson ratio and thickness of the plate, respectively. The plate buckling coefficient (k) is 4 in the case of a simply supported plate. Several studies were carried out to propose the value of the plate buckling coefficient for tubular polygons. It was reported that for tubular polygons under concentric compression, the plate buckling coefficient is greater than 4 for sections with

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A. Godat et al. / Thin-Walled Structures 53 (2012) 131–140

Nomenclature A D E e0,gl e0,loc fe fmax Fu fy k

section area section diameter modulus of elasticity imperfection amplitudes calculated across the plate height imperfection amplitudes calculated across the plate width effective buckling stress experimental stress level ultimate buckling stress tensile yield stress local buckling coefficient

N r R t w weff w/t Zi,1, Z1,j

ey ey lp l

n scr

number of faces radius of gyration bending radius of plates plate thickness width of plate effective width of plate width-to-thickness ratio imperfection measurements in the I or j section yield strain ultimate strain plate buckling parameter equivalent slenderness ratio Poisson ratio critical buckling stress

odd number of faces and 4 for even number of faces [2,3]. However, starting from the octagonal-section and more number of faces, it was shown theoretically that the plate buckling coefficient can be approximately taken as 4 for practical applications [2]. Bulson [4] obtained experimentally that up to 18 faces, the critical buckling stress of tubular polygon columns can be computed as an assembly of plates with k¼4. In addition, he observed that for 18 faces and more, the buckling mode resembles that of circular tubes and should be computed as so. Based on numerical simulations, Teng et al. [5] proposed higher values for the plate buckling coefficient (k) for tubular polygons under bending. In bending, due to stress variation along the column length and through the cross section, plates in tension and plates under a compression are not sufficient to initiate the buckling and provide an additional restraint compared to the concentric compression. They estimated that the plate buckling coefficient in pure bending is 5.3, 5.1 and 4.9 for square, hexagonal and octagonal sections, respectively. The previous studies are related to the elastic buckling capacity. However, in practice, most tubular polygons fail in the elasto-plastic state. This state of failure was confirmed by Aoki et al. [6], who tested 15 stub polygon columns with 4 to 8 faces. They observed that the strength of the specimens is related to the plate slenderness parameter:

The authors believe that proper understanding of tubular polygon columns under concentric compression will definitely facilitate accurate analysis of these members in the presence of eccentricity. From revision of the past literature, it is observed that there is no agreement among industry documents for the design of tubular polygons under concentric compression. The origin of such differences may be the lack of experimental data. Actually, there is no experimental data on tubular polygons with more than 8 faces tested in concentric compression except for Bulson data [4] on very thin plates outside the scope of normal use for tubular polygons. The objective of this article is (i) to enrich the experimental database of tubular polygon columns under concentric compression; (ii) to evaluate the influence of the number of faces as well as the plate slenderness ratio on the strength of tubular polygon columns. The study concentrated on a number of faces between 8 and 16 faces, and in the elasto-plastic range of local buckling where there is a lack of data and significant differences are found among the design guidelines. In addition, an equation is proposed on the basis of the equivalent stress concept to account for the local buckling capacity of thinwalled tubular polygon steel columns.

lp ¼ ðf y =scr Þ1=2

The experimental program involved six thin-walled tubular polygon stub steel columns with three different cross-sections, octagonal, dodecagonal and hexdecagonal, tested under concentric compression. All specimens were selected with a low plate slenderness ratio (width-to-thickness w/t) in order to avoid the global buckling failure and focus on the local elasto-plastic buckling.

ð2Þ

where lp is the buckling parameter; fy is the tensile yield strength of the steel plate. Their study did not comment on the impact of the number of faces on the buckling capacity under concentric compression. Recent research carried out by Cannon and LeMaster [7] produced different design equations in the elasto-plastic buckling state for the octagonal, dodecagonal and hexdecagonal cross-sections of polygon members in flexure. These equations were slightly modified and adopted by the ASCE 48-05 code [8], which is exclusively developed for the tubular pylons of transmission lines. In this code, for very thin plates, the buckling stress converges toward the elastic buckling of k¼4. Other design guidelines, such as the CENELEC [9], the Eurocode 3 parts 1–1 [10], 1–3 [11] and 1–5 [12] and other codes and standards [13–15], provide equations that can be used to design tubular polygon columns. In these documents, the effective width concept is used with slight variations. The effective width concept, developed by Von Ka´rma´n et al. [16] and Winter et al. [17], implies that after buckling, a plate can have a strain hardening behavior and can reach the full yielding on an ‘‘effective’’ width. Therefore, the capacity predicted by these documents is significantly greater for a slender plate than what is predicted by Eq. (1) using k¼4.

2. Experimental program

2.1. Fabrication of specimens and material tests The buckling behavior of tubular members is considerably affected by the method of fabrication. Hence, fabrication techniques employed were typical of those used in structural fabrication shops. The test specimens were fabricated by welding two halfsection pieces made of folded plates with a nominal height of 780 mm. The steel grade used in this study was A-36. Fig. 1 shows the specimens when delivered for testing. For the tested specimens, a diaphragm was welded at both extremities to mitigate end effects. Details of the specimens are shown in Fig. 2; the weld connections in the cross-section are shown with a solid triangular mark. The dimensions of the tested specimens are summarized in Table 1. In the table, D, t, w and A are, respectively, the overall

A. Godat et al. / Thin-Walled Structures 53 (2012) 131–140

dimension measured from mid width of flats, thickness, plate width and cross section area of the specimens. Each specimen was labeled by a short form of number of faces, followed by a number

that denotes the specimen number in the series. In order to examine the influence of the plate slenderness ratio on the buckling resistance, two values were proposed for each crosssection. The ratio varied from 56 to 32. As listed in Table 1, this ratio was in the range of the elasto-plastic buckling, relatively high for the first four specimens, while it was low for the other two specimens. Reaching the same width to thickness ratio for hexdecagonal and octagonal section would produce sections or thickness that were not practical for the test. The ASCE 48-05 [8] limit for the slenderness ratio (w/t)limit are provided in Table 1 for sake of comparison. This limit is calculated with the actual tensile yield strength. For w/t greater than (w/t)limit, the elastoplastic local buckling starts according to the ASCE 48-05. The plate slenderness ratio calculated with the actual tensile yield strength is within 0.65–1.08. Flat plate coupons were cut from the material prior to fabrication to determine the elastic modulus and the tensile yield strength. The coupons were prepared in accordance with the ASTM A370-05 standard [18]. Coupons were oriented so that material properties would be measured to what would later be the longitudinal axis of the specimens. For each specimen, tensile tests on four coupons were performed and the average values were taken. In Table 2, the tensile yield stress (fy), the elastic modulus (E) as well as the tensile yield strain (ey) and ultimate strain (eu) are summarized.

Fig. 1. Test specimens when delivered.

3mm welding

DETAIL A

133

2.2. Measurements of geometric imperfections

780

DETAIL A A

Geometric imperfections of the specimens result from the fabrication process specifically due to the thermal gradients that are induced during welding. Knowledge of the magnitude of member imperfections is required because it affects the critical buckling capacity [19]. In this study, the imperfection in the plate surfaces was measured with respect to longitudinal reference lines. The device used to measure the imperfections is shown in Fig. 3. A dial gage with high accuracy was fixed on a firm steel bar to measure the flatness profile of the specimens. After placing the specimen horizontally, the dial gage was moved on the test specimen to measure the profile. The data measured for all the specimens were analyzed using a numerical analysis method proposed by Mennink [20]; a brief description of the method is given herein. For each plate, the amplitudes were calculated at

A

3mm welding

DETAIL B

DETAIL A

w D

ELEVATION

w w

DETAIL B

Table 2 Mechanical properties of tested specimens.

t

BR D/2

BR

DETAIL B

D

Typical dimension of the width of the section

SECTION A - A

Fig. 2. Specimens fabrication details.

Specimen

F y (MPa)

E (GPa)

ey  10  3

eu (%)

OCT-1-A OCT-4-A DODE-1-A DODE-2-A HEXA-1-A HEXA-4-A

279 265 273 305 277 302

200 199 206 218 199 200

1.4 1.3 1.3 1.4 1.4 1.5

26 27 24 25 26 26

Table 1 Geometrical properties of tested specimens. Specimen

OCT-1-A OCT-4-A DODE-1-A DODE-2-A HEXA-1-A HEXA-4-A

No. of faces

8 8 12 12 16 16

D (mm)

247 194 297 297 275 317

t (mm)

1.897 1.367 1.367 1.897 1.519 1.897

w (mm)

95 75 76 75 52 60

A (mm2)

1556 880 1307 1814 1333 1918

w/t

50 55 56 40 34 32

(w/t)limit

41 42 38 36 34 32

kp

0.98 1.05 1.08 0.81 0.67 0.65

Results

Absolute max. imperfection (mm)

Load (kN)

Stress (MPa)

327 198 325 515 317 508

210 225 249 284 238 265

2.2352 2.1590 2.5400 2.0066 1.2700 1.8288

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A. Godat et al. / Thin-Walled Structures 53 (2012) 131–140

Fig. 3. Geometrical imperfection measuring apparatus.

predetermined discrete points, five points across the plate width and thirteen points along the plate height. In this research, the term ‘‘local’’ imperfection is used for the imperfections amplitudes calculated across the plate width (X-axis), while the ‘‘global’’ imperfection is used for those along the plate height (Y-axis). The global imperfection amplitude (e0,gl) and the local imperfection amplitude (e0,loc) are characterized by the deviation amplitude (e0) defined as the offset between discrete points on the plate and a straight line joining both extremities of the plates. The imperfection amplitudes were computed based on the following equations:    z z e0,gl ¼ maxi ¼ 1,m maxj ¼ 1,n zi,j zi,1  i,n i,1 yi,j yi,n    zm,j z1,j xi,j e0,loc ¼ maxj ¼ 1,n maxi ¼ 1,m zi,j z1,j  xm,j

ð3Þ

Fig. 4. Typical test setup and arrangement of LDVTs.

Table 3 Imperfection results. Specimen Absolute max. imperfection (mm)

Absolute average imperfection

Standard deviation

OCT-1-A OCT-4-A DODE-1A DODE-2A HEXA-1-A HEXA-4-A

2.2352 2.1590 2.5400

0.8366 0.8128 0.9071

0.329 0.395 0.365

2.0066

0.9434

0.334

1.2700 1.8288

0.6277 0.9684

0.228 0.341

ð4Þ

where the terms zi,1, zj,i, etc. are the imperfection measurements in the i or j section. In calculating the deviation amplitudes, no distinction was made between inward and outward deviations since absolute values of the deviations were used.

2.3. Test setup and instrumentation The test setup used and the instrumentation are shown in Fig. 4. The hydraulic testing machine was used to apply the concentric forces. The machine has a capacity of 5500 kN. The load was applied over a large diameter steel plate to ensure a uniform compression loading on the perimeter of the tested specimens. The base of the tubular polygon stub columns was seated on steel plates. Therefore, specimens were simply supported along the reaction surfaces but effectively restrained against radial movement by friction. The load was applied by a displacement control method with high rate up to 50% of the estimated failure load then decreasing the rate as the expected failure load was approached. For a better visual indication of the failure progress, the specimens were white-painted and a black grid was drawn on them. For each specimen, six LVDTs (Linear Variable Differential Transformers) were attached with equal space adjacent to the welded plate at different locations along the height of the specimens. This allowed the direct measurements of the shortening in the displacement.

3. Experimental results The results presented in the following sections are the initial geometrical imperfections, critical buckling, and maximum loads as well as failure modes. As there was no noticeable difference in the test between the buckling load and the maximum load, the term buckling load is used as an indication to the maximum load experienced during the test. Results are also provided in terms of axial stress–strain relationships. Special emphasis is given on the influence of the plate slenderness ratio and number of faces on the buckling load capacity.

3.1. Initial geometrical imperfection Table 3 shows the absolute values of maximum imperfection, average imperfection and the standard deviation of imperfection for each specimen. It is seen that the maximum imperfection value obtained is around 2 mm for all specimens. Similar results of imperfection, except specimen HEXA-1-A, are obtained for all specimens. The lower maximum imperfection value is measured in specimen HEXA-1-A. Also, this specimen shows the lowest average as well as standard deviation among the tested specimens. The values measured on other faces are generally within 1 mm. The highest imperfection values with their corresponding surfaces are shown in Fig. 5. Generally, it can be stated that the initial imperfection shape has one half wave over the width and one or

A. Godat et al. / Thin-Walled Structures 53 (2012) 131–140

135

within the practical limits. The maximum initial imperfection measured is 2.54 mm. This level of imperfection is within the quality class A or excellent as proposed by the EC3-1-6 for almost all the tubular polygon sections. According to the EC3-1-6, Specimen OCT-4-A is the only specimen whose level of imperfection is slightly above class A and would qualify for Class B-high quality. The initial imperfection values are generally within 1/30 to 1/42 of the width of the plate. For the surfaces having the highest imperfection values, it is usually within 1/100 to 1/200 of the width, which is acceptable in practice. Plat e leng th

(mm)

3.2. Compression test

Imperfection (mm)

The results of the compression test are discussed in terms of critical buckling stress based on the influence of number of faces and the influence of plate slenderness ratio, and the failure modes.

m)

Imperfection (mm)

Plate le ngth (m

Imperfection (mm)

Plate leng th (mm)

3.2.1. Critical buckling stress Table 1 shows the critical buckling load as well as the critical buckling stress. The average axial stress–strain relation for the tested specimens are plotted in Fig. 6. The average stress is the applied force divided by the cross-section area and the average strain is the relative displacement between the top and the bottom of the specimen divided by the length of the specimen. It is understood that the average stress and strain do not capture directly the complex stress and strain distribution at the bulging locations and after buckling, where out of plan deflection occurs together with bending moment and horizontal membrane stress. However, the average stress and stress provide interesting information on the overall column behavior specifically in terms of ultimate capacity, the pre-peak column stiffness and post-peak ductile behavior. As shown in the figure, all specimens, except for specimen HEXA-4-A, exhibited similar ascending and descending behavior. During the initial loading stage, nonlinearity is observed. Following this, linear elastic behavior is obtained up to the maximum buckling stress. Beyond this point, there is a gradual reduction in the axial stress up to the failure. It is important to mention that the strain corresponds to the maximum axial stress and is almost similar for all specimens, in spite of initial geometrical imperfections of the plates. For specimen HEXA-4-A, significant deviation of the slope of the ascending branch from the other specimens is observed. This deviation is attributed to the test setup. The highest buckling stress is obtained in specimen DODE-2-A (284 MPa), while the lowest value is measured in specimen OCT-1-A (210 MPa). 350

OCT-4-A OCT-4-A DODE-1-A DODE-2-A HEXA-1-A HEXA-4-A

Plate leng

th (mm)

Fig. 5. Typical example of geometrical imperfection measurements for the face having the highest imperfection value.

two waves over the length of the plate. The maximum initial imperfection occurs close to the ends of the polygonal section. For tubular polygon columns, few limitations for the initial out of roundness imperfection are provided in available design guidelines. The values measured for the maximum initial imperfections are compared with the initial out of roundness imperfection limitations proposed by the EC3 1-6 [21] for cylinders. Although these limits are applicable for cylinders, it is of interest to evaluate these limits with tubular polygon columns. The values of imperfection measured are

Axial stress (MPa)

300 250 200 150 100 50 0

0

0.005

0.01

0.015

0.02

Axial strain (mm/mm) Fig. 6. Axial stress–strain relationships of tested specimens.

0.025

A. Godat et al. / Thin-Walled Structures 53 (2012) 131–140

1.4

1.4

1.2

1.2

1

1

Fcr / Fy

Fcr / Fy

136

0.8 OCT-1-A OCT-4-A DODE-1-A DODE-2-A HEXA-1-A HEXA-4-A

0.6 0.4 0.2 0

0

4

0.8 OCT-1-A OCT-4-A DODE-1-A DODE-2-A HEXA-1-A HEXA-4-A

0.6 0.4 0.2 0

8

12

16

20

Number of sides

0

10

20

30

40

50

60

Slenderness ratio (w/t)

Fig. 7. Influence of number of faces and slenderness ratio on critical buckling capacity.

3.2.2. Influence of number of faces It has been stated by some researchers that as the number of faces increased over eight, there is no increase reported in the local buckling capacity [2,5]. Other researchers [6] reported that the increase in the local buckling capacity is related to the increase in the number of faces. The effect of number of faces on the buckling resistance of tubular polygon columns is verified in this study. Specimens OCT-4-A and DODE-1-A have almost identical plate width-to-thickness ratio, while the cross-section is different. The cross-section for specimen OCT-4-A is octagonal, while it is dodecagonal for specimen DODE-1-A. The critical buckling stress measured for specimen OCT-4-A is 223 MPa. For specimen DODE-1-A, the critical buckling stress is 216 MPa, which is 3% lower than specimen OCT-4-A. In order to draw a suitable conclusion for the influence of this parameter, it is favorable to be nondimensional. A relation between the critical buckling stress versus the number of faces is presented in Fig. 7(a). From the figure, it appears that the critical buckling stress has a tendency to increase with the increase of number of faces. However, this increase is insignificant when the polygonal members are compared at constant plate slenderness ratio. This observation agrees with what was predicted by Avent and Robinson [2]. 3.2.3. Influence of plate slenderness ratio The plate slenderness ratio (plate width-to-thickness ratio) is undoubtedly one of the important parameters affecting the critical buckling stress of tubular polygon columns. In this study, different values of the plate slenderness ratio are used in order to verify the influence of such parameter. A relation between the critical buckling stress versus the plate slenderness ratio is shown in Fig. 7(b). The critical buckling stress obtained from the test is divided by the tensile yield stress to obtain nondimensional values. Fig. 7(b) indicates that the critical buckling stress tends to increase with the decrease of the slenderness ratio (w/t) regardless of the number of faces. This result is in accordance with previous findings and design codes. 3.2.4. Modes of failure Representative failure modes for the tested specimens are shown in Fig. 8. All specimens failed as a result of the local buckling at an axial compressive stress level below the tensile yield stress of the specimen. The local buckling responsible for the failure occurs at few centimeters close to one end of the specimen. This type of failure is typical to that obtained for thin wall section columns experimentally [22] and numerically [23]. The failure occurs due to restrains provided by stiff plates at the ends, and also due to potential accidental loading eccentricities. For the

octagonal cross-section specimens, the local buckling occurs in a symmetric mode. Each plate element buckles in an opposite direction to the adjacent plate in a typical polygonal cross section with an even number of faces. When the number of faces increase, the inwards and outwards failure mode occurred in the octagonal cross-section specimens and is not observed in the hexdecagonal section. The hexdecagonal and one of the two dodecagonal crosssection specimens buckle in a diamond shape close to the cylinder buckling mode (bulging buckling). This indicates that with number of faces starting from twelve, the behavior of the tubular polygon columns become close to the circular tubes. This is in contradiction with the results of Bulson [4] who showed that the bulging mode appeared for tubular polygons with more than 18 faces. However, Bulson tests [4] were for very high w/t (above 100 for 16 faces and below). The bulging mode can be evaluated from an equivalent cylinder having the same perimeter. In this case, assuming a bending radius (R) of plates of 4t as in the following: R ¼ ðnw þ 9ptÞ=2p

ð5Þ

where N is the number of faces. For all specimens, no indication of local yielding is observed in the tested specimens prior to local buckling. This reinforces the assumption that linear behavior is in effect until failure. Table 4 compares between the experimental buckling stress results and the theoretical cylinder buckling strength results; the latter is calculated as 0.3Et/R [24]. A comparison is made between the stress levels obtained experimentally and the cylinder buckling strength to verify the assumption of Bulson [4]. It can be observed that the difference between the ‘stress level’ in Table 1 and the ‘cylinder buckling strength’ is very high for the first two specimens (OCT-1-A and OCT-4-A), while it decreases with the increase of number of faces. Based on large discrepancies observed between the experimental stress results and the cylinder buckling strength, an appropriate design equation to better evaluate the buckling stress of tubular polygon columns is required.

4. Comparison of test results with design equations 4.1. Summary of design equations Following the previous discussion on the behavior and failure modes of tubular polygon columns, it is of interest to see how the measured local buckling capacity compares with predictions from available design guidelines. Design equations proposed to compute the critical buckling capacity for tubular polygon columns in the elastic buckling might not be sufficient to obtain accurate results. There are some equations provided in design codes based on test results that predict the maximum stress

A. Godat et al. / Thin-Walled Structures 53 (2012) 131–140

137

Fig. 8. Local buckling failure shape of tested specimens.

Table 4 Comparison between experimental stresses results and theoretical cylinder stresses. Specimen

Cylinder buckling stress (MPa)

Plate buckling stress (MPa)

Buckling mode

OCT-1-A OCT-4-A DODE-1-A DODE-2-A HEXA-1-A HEXA-4A

443 406 272 377 329 355

288 240 234 463 617 723

Plate buckling Plate buckling Plate buckling Cylinder buckling Cylinder buckling Cylinder buckling

supported by a section, such as the EC3 part 1–5 [12] and the ASCE 48-05 [8]. The loading capacity in these codes is related to the buckling stress, but there might be some strain hardening capacity available after buckling that may be included. The EC3 part 1–5 provides such post buckling capacity through the effective width concept. In this concept, it is assumed that part of the section reaches yielding and is therefore called the effective section.

The design equation used in the EC3 part 1–5 [12] is also adopted by other codes to cold formed members [13–15]. The effective width is given by wef f ¼ w wef f ¼ rw

if lp r 0:673 if lp 4 0:673

r ¼ ð10:22=lp Þ=lp

ð6Þ ð7Þ ð8Þ

The ASCE relies on the effective stress concept supported by the total cross section. Its equations depend on the number of faces; they are summarized in Table 5. In the table, the notation used in the EC3 is kept for homogeneity. The parameter  qffiffiffiffiffi  w f y =t used in the ASCE 48-05 is converted into 850lp that is obtained from Eq. (1). The factor 850 accounts for the modulus of elasticity of steel (E¼200,000 MPa), which is consistent with the buckling stress provided by the ASCE. A value of E¼210,000 MPa is used in the EC3. There are few other analytical studies performed to propose a design equation for tubular polygon columns. The most significant is the contribution by Aoki et al. [6]. They based their

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A. Godat et al. / Thin-Walled Structures 53 (2012) 131–140

Table 5 ASCE design equations for local buckling capacity of tubular polygon columns. Number of faces

Equation limit

fa/fy design equation

4 to 8

lp o 0.801 0.801 r lp r 1.082 lp 41.082

¼ 1.0 ¼ 1.42(1.0–0.370lp) ¼ 1/l2p

12

lp o 0.740 0.740 r lp r 1.153 lp 41.153

¼ 1.0 ¼ 1.45 (1.0 0.418lp) ¼ 1/l2p

16

lp o 0.662 0.662 r lp r1.269 lp 41.269

¼ 1.0 ¼ 1.45(1.0  0.444lp) ¼ 1/l2p

approach on an effective stress proposed from their experimental tests on polygons with 4 to 8 faces. With additional numerical results, the design equation by Migita and Fukumoto [25] is modified to 0:75

f e =f y ¼ 0:74=lp

o 1:0

Fig. 9. Validation of design equations versus experimental results.

ð9Þ

4.2. Evaluation of design equations and discussion To investigate the accuracy of the design equations, the EC3, ASCE and Migita and Fukumoto design equations are compared to experimental results of 22 polygons with 5 to 16 faces tested under concentric compression by Aoki et al. [6], Harraq [26] and the present study. The data by Cannon and Lemaster [7] are not included because they were tested in flexure. All the experimental tests were stub columns failed due to local buckling. In addition, the global buckling was disregarded in the prediction since the slenderness ratio KL/r in all specimens was lower than 10 corresponding to an equivalent global slenderness parameter l of less than 0.1. Fig. 9 and Table 6 show the experimental results compared to the strength curves for EC3, ASCE (for 8, 12 and 16faces) and Eq. (9). In Table 6, the specimens’ designations correspond to those used in the original references. In Fig. 9 and Table 6, it is observed that the ASCE 48-05 tends to overestimate the experimental results for plate slenderness parameter (lp) lower than 1.5. For plate slenderness parameter close to 1.8 (corresponds to plate width-to-thickness ratio (w/t) of 100), the ASCE seems to be closer to the experimental values. The EC3 overestimates greatly the capacity for high width-to-thickness ratios. On the whole range of plate slenderness parameter, the average theoretical-to-experimental ratio is 0.95 for ASCE 4805 and 0.92 for EC3 with standard deviation of 0.10 and 0.14, respectively. Migita and Fukumoto design equation is unconservative in some cases and conservative in others. It is important to mention that, apparently, Harraq [26] is the only researcher who tested polygonal columns with large plate slenderness ratio (w/t). His four tests showed consistent results and therefore cannot be ignored. Harraq reported in his work a very small difference between the buckling load and the collapse load, which is in disagreement with the effective width concept considered in the EC3 code. It should be mentioned here that most of the development of the effective width approach [16] is not based on regular sections under concentric compression. Indeed, it is uncertain if a section with a local buckling involving all plates of a cross-section can actually provide a collapse load greatly above the buckling load. Based on Harraq data, it is questionable if the EC3 is fully adapted to regular polygons under concentric compression. The equation proposed by Migita and Fukumoto [25] seems appropriate for plate slenderness parameters (lp) up to 1.5, but it does not provide good results beyond

this slenderness parameter. It is not a significant improvement as compared to EC3 or ASCE 48-05. Most of the stub columns considered herein failed in the elasto-plastic state. In this state, it is difficult to observe a significant impact of the number of faces from the experimental data. The only exception is the data of the present study for the 16 faces specimens. Clearly, the two 16-face polygons are out of the range compared to other data. This can be attributed to the mode of failure observed for these specimens. This study reported a mode of failure similar to a cylinder failure mode, which can significantly reduce the buckling stress calculated as demonstrated before. The ASCE design equations predict high maximum stresses for the 16 faces specimens compared to 8 and 12 faces. Therefore, the equations provided in the ASCE for different number of faces are not accurate to predict the local buckling capacity in uniform compression similar to those obtained experimentally. Hence, the ASCE introduces an effect of number of faces in the design equations that does not seem to be obtained in concentric compression. Note that the ASCE code is based on tests in flexure. The EC3 as well as Migita and Fukumoto design equation may be unsafe for high buckling parameters. 4.3. Design equation proposed 4.3.1. Ultimate stress equation Based on the previous discussion of the discrepancy observed between the design equations and the experimental results, Loov’s design equation [27] is modified to predict the capacity of tubular polygon columns under concentric compression. The new equation proposed is established on the basis of the ultimate stress (Fu) to account for the local buckling capacity of thin-walled tubular polygon columns. The equation should converge towards the elastic buckling for high buckling parameters and should be similar for all number of faces as demonstrated by the tests results. The equation is expressed as follows: 2n

F u ¼ f y ð1 þ lp Þ1=n

ð10Þ

The plate buckling coefficient is taken as 4.0 for the calculation of lp. Regression analysis is used to obtain an appropriate value for the ‘n’ parameter. It is found that the critical buckling capacity is not very sensitive when the ‘n’ parameter varies between 1.5 and 3.0. The best-fit can be obtained from the experimental results for a value of ‘n’ taken as 2.0.

A. Godat et al. / Thin-Walled Structures 53 (2012) 131–140

139

Table 6 Comparison of prediction with design equations to experimental data. Spec.

Number of faces

b (mm)

w (mm)

t (mm)

(w/t)

fy (MPa)

fmax

lp

fmax/fy Exp.

EC3

ASCE

Proposed equation

fmax/fy Theo.

Exp./ theo.

fmax/fy Theo.

Exp./ theo.

fmax/fy Theo.

Exp./ theo.

Aoki et al. [6] PEN-24-A PEN-24-1-A HEX20-A HEX25-A HEX30-A HEP17-A OCT-15 OCT15-b OCT-15w OCT20-A OCT25-A OCT30

5 5 6 6 6 7 8 8 8 8 8 8

237 238 198 246 295 167 147 148 150 196 264 296

237 238 198 246 295 167 147 148 150 196 264 296

4.53 4.44 4.49 4.51 4.54 4.50 4.50 4.50 4.50 4.49 4.52 4.51

52.3 53.6 44.2 54.5 65.1 37.1 32.7 33.0 33.3 43.6 58.5 65.7

289 289 289 289 289 289 289 289 289 289 289 289

1.01 1.03 0.85 1.05 1.25 0.72 0.63 0.64 0.64 0.84 1.13 1.27

213.0 207.0 251.0 206.0 177.0 271.0 278.0 274.0 275.0 244.0 206.0 173.0

0.737 0.716 0.869 0.713 0.612 0.938 0.962 0.948 0.952 0.844 0.713 0.599

0.777 0.767 0.872 0.755 0.660 0.968 1.000 1.000 1.000 0.881 0.696 0.656

0.948 0.934 0.996 0.944 0.929 0.968 0.962 0.948 0.952 0.958 1.024 0.912

0.891 0.877 0.972 0.867 0.635 1.000 1.000 1.000 1.000 0.978 0.787 0.624

0.828 0.817 0.893 0.822 0.964 0.938 0.962 0.948 0.952 0.863 0.906 0.959

0.702 0.683 0.809 0.671 0.536 0.890 0.930 0.927 0.924 0.817 0.618 0.529

1.050 1.048 1.073 1.063 1.142 1.054 1.035 1.022 1.030 1.034 1.153 1.131

Harraq [26] OCTRCC-1 OCTRCC-2 OCTIRR-1 OCTIRR-2

8 8 8 8

95 95 95 95

92 92 92 92

0.94 0.94 0.94 0.94

98.1 98.1 98.1 98.1

243 243 243 243

1.78 1.78 1.78 1.78

91.3 79.9 78.7 80.9

0.375 0.329 0.324 0.333

0.506 0.506 0.506 0.506

0.741 0.650 0.639 0.657

0.314 0.314 0.314 0.314

1.195 1.048 1.031 1.060

0.300 0.300 0.300 0.300

1.253 1.098 1.081 1.111

8 8 12 12 16 16

95 75 76 75 52 60

95 75 76 75 52 60

1.90 1.37 1.37 1.90 1.52 1.90

50.2 55.0 55.8 39.5 34.2 31.5

279 265 273 305 277 302

0.98 1.06 1.08 0.78 0.67 0.64

210.2 224.9 248.6 283.9 237.9 264.8

0.753 0.849 0.911 0.931 0.859 0.877

0.804 0.766 0.751 0.927 1.000 1.000

0.937 1.107 1.213 1.004 0.859 0.877

0.903 0.865 0.798 0.979 0.997 1.000

0.835 0.981 1.141 0.951 0.862 0.877

0.718 0.668 0.654 0.856 0.912 0.924

1.049 1.271 1.392 1.088 0.942 0.949

Present study OCT-1-A OCT-4-A DODE-1-A DODE-2-A HEXA-1-A HEXA-4-A Mean Standard deviation

4.3.2. Validation of design equation proposed Fig. 9 and Table 6 compare the results of the equation proposed with the experimental results for the database of the 22 polygons. It is obvious that the design equation is able to give very good predictions for the various test data. The results are within the scatter of the local buckling points; this gives credibility of the equation to predict the critical buckling load with suitable accuracy. The average of the experimental-to-theoretical ratio is 1.09 with a standard deviation of 0.10, which is very satisfactory. If deemed appropriate, a better safety can be obtained through partial security factor. It is important to comment on the two predictions out of the curve of the equation proposed. They are obtained for the two 16face polygons tested in this study. As reported before, the buckling mode of these two specimens was diamond buckling mode instead of inward–outward failure mode. Using a buckling stress of 0.3Et/r (r is the radius of gyration), instead of the theoretical plate buckling stress, the predictions are improved for the specimens DODE-2-A, HEXA-1-A and HEXA-4-A. This gives experimental to predicted ratios of 1.20, 1.12 and 1.15, respectively. The average experimental to prediction ratio is 1.11 with a standard deviation of 0.09. The cylinder buckling mode is well predicted in this case.

5. Conclusion The local buckling capacity of six thin-walled tubular polygon stub columns with different forms of cross sections was investigated. Octagonal (eight-sided), dodecagonal (twelve-sided) and hexdecagonal (sixteen-sided) sections were considered for various values of plate slenderness ratio (plate width-to-thickness ratio). The experimental program was conducted to increase the database

0.916 0.140

0.947 0.101

1.094 0.103

of thin-walled tubular polygon columns associated with a better understanding of the critical buckling stress in the elasto-plastic buckling range. The parameters investigated in this study were the plate slenderness ratio and the number of faces. Based on the results obtained in this study, it can be concluded that thin-walled tubular polygon columns having the same cross-sectional area and smaller plate slenderness ratio may become advantageous with regard to the local buckling capacity. In addition, the number of faces was not an important factor in the local buckling strength when the plate width to thickness ratio is kept constant. In this study, Loov’s equation is modified with new fitting parameters to account for the critical buckling capacity of thinwalled tubular polygon columns with different forms of cross sections. The equation proposed was established on the basis of the ultimate stress concept instead of the equivalent width concept. The design equation was verified with the experimental results of tubular polygon columns available in the literature. Overall, very good accuracy was obtained between the new equation and the experimental results regardless of number of cross-section faces. The new equation is useful for practical design. It is important to emphasize that the number of data on the topic is still very limited and more data are necessary to provide an accurate design equation. Also, the applicability of this work to tubular polygon columns under bending stress is to be evaluated.

Acknowledgment This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC), Hydro-Que´bec and

140

A. Godat et al. / Thin-Walled Structures 53 (2012) 131–140

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