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Finite element study of steel single angle beam–columns
Engineering Structures 32 (2010) 2087–2095
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Finite element study of steel single angle beam–columns Yi Liu ∗ , Linbo Hui Department of Civil and Resource Engineering, Dalhousie University, Halifax, NS, Canada
article
abstract
info
Article history: Received 12 February 2010 Received in revised form 3 March 2010 Accepted 4 March 2010 Available online 7 April 2010
A finite element study was conducted to investigate the response of steel single angles subjected to eccentric compression and the numerical results were used to examine the effectiveness of the beam–column interaction approach suggested in the current AISC design standard. Numerical results demonstrated inconsistencies and overall conservatism inherent in the design equations and the possible sources of this conservatism were also identified and discussed. Results also indicated the existence of a critical eccentricity associated with each slenderness ratio in the case of eccentric compression causing major axis bending. Within this eccentricity, any reduction in the ultimate capacity due to eccentricity was found to be marginal. A new interaction equation for combined compression and principal axis bending was proposed. The proposed equation achieved the target reliability index with an improved test-toprediction capacity ratio over a wide range of angle geometric parameters when compared with AISC design equations. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Numerical modeling Single angle Eccentric compression Flexure AISC Specification
1. Introduction
for
Due to their ease for installation and simple aesthetic appeal, steel single angles are extensively used in a variety of structures such as steel joists and trusses and braced structures where they are usually either welded or bolted to other structural members by one leg. This connection detail results in eccentric compressive loading with respect to both major and minor principal axes of the angle section. Combined with their asymmetric characteristics, angles subjected to axial compression and bi-axial bending often exhibit torsional or flexural torsional behavior if the bending of the angle is not restrained with respect to certain axis. The design and analysis of such angles is more complex than their simple shape may suggest. Currently, the design of the eccentrically loaded single angles is governed by the Specification for Structural Steel Buildings [1] in the United States, hereafter referred to as AISC Specification 2005 and the Canadian standard ‘‘Limit States Design of Steel Structures’’ [2] in Canada. Both standards adopt the approach of treating eccentrically loaded angles as beam–columns and evaluating their capacity using interaction between axial load and bending. The AISC Specification 2005 suggests that the interaction of flexure and axial compression applicable to specific locations on a cross-section of a single angle shall be limited according to: for
∗
Pr
φc Pn
≥ 0.2,
Pr
φc Pn
+
8 9
Mrx
φb Mnx
+
Mry
φb Mny
≤ 1.0
Corresponding author. Tel.: +1 902 494 1509; fax: +1 902 494 3108. E-mail address: [email protected] (Y. Liu).
0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2010.03.009
(1a)
Pr
φc Pn
< 0.2,
Pr 2φc Pn
+
Mrx
φb Mnx
+
Mry
φb Mny
≤ 1.0
(1b)
where Pr and Pn are the required compressive strength and the nominal compressive strength, Mr and Mn are the required flexural strength and the nominal flexural strength, φc and φb are resistance factors for compression and flexure, and x and y refer to the major and minor principal axes. In Canada, a similar interaction approach is proposed but with axial force and moment interaction factors different from the AISC Specification 2005. Additionally, this standard does not provide explicit provisions for determining the nominal flexural strength for mono-symmetric or asymmetric angles, but rather, it directs users to the SSRC Guide [3] for design methods. Some research has been conducted by others to evaluate the performance of AISC Specification 2005. Based on a comparison study of 71 test results, Adluri and Madugula [4,5] pointed out that these interaction equations are highly conservative. Temple and Sakla [6] reported experimental capacities two to three times the values suggested by AISC Specification 2005. Sakla [7] evaluated the efficacy of these equations for both equal- and unequalleg angles. He concluded that AISC Specification 2005 provides a better prediction of the ultimate capacity of equal-leg single angles than it does for unequal-leg single angles. It underestimates all unequal-leg test results and overestimates capacities of some equal-leg angles. Earls [8] suggested that AISC Specification 2005 for single angle beam design is, in most cases, unnecessarily conservative while in some other situations overestimates angle capacities. Based on a finite element study, he proposed equations for predicting the flexural strength with respect to major principal
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Notations b/t E Fy K KL/ry L M Mn Mr My P Pexp PFEM Pn Pr Py Rm Rn
Angle leg width-to-thickness ratio Modulus of elasticity of angle steel Yield stress of steel Effective length factor = 1.0 Slenderness ratio of angle Length of angle Applied moment Nominal flexural strength Required flexural strength Yield moment about the axis of bending Applied compressive load Experimental compressive capacity Finite element results of compressive load capacity Nominal compressive strength Required compressive strength Yield capacity of angle cross-section Mean resistance obtained from experimental results Averaged nominal resistance obtained from design equations Coefficient of variation of Rm /Rn Eccentricity with respect to major principal axis Critical eccentricity with respect to the major principal axis Eccentricity with respect to minor principal axis Initial imperfection along the length of angle Amplitude of the imperfection at the mid-height of angle q
Vr ex e xo ey
w wo λ β φb φb φc
Slenderness parameter =
KL ry
Fy
π 2E
Reliability index Resistance factor Resistance factors for flexure Resistance factors for compression Distance between shear centre and centroid of angle cross-section
xo
axis of single equal-leg angles as follows [9]: For 8 ≤
For
b t
b t
≤ 14,
> 14,
Mn My
Mn My
=−
b/t
10 000
= −0.00184
Lb rz
− 50 + 1.5
(2a)
7 16 22.4 − 50 + rz b/t b/t
Lb
(2b) where Mn is the ultimate flexural strength, My is the yielding moment, Lb is the unbraced length of beams, rz is the radius of gyration with respect to minor axis, b is the leg width and t is the leg thickness. 2. Scope of the current research Although the previous research indicated a significant conservatism inherent in AISC Specification 2005 for the design of single angles, the angle geometric parameters and loading eccentricities used in those evaluations were limited. To gain a better understanding of the singe angle behavior and have a comprehensive evaluation of the AISC Specification 2005, a numerical study using finite element modeling was carried out to study the effects of several influential parameters on the load carrying capacity of single equal-leg angles eccentrically loaded with respect to either the major or minor principal axis. The finite element model was verified using experimental results available in literature for single
angles subjected to compression with simple support conditions. The verified finite element model was then employed to generate numerical results to further examine the effectiveness of the AISC Specification 2005 interaction equations over a wide range of parameters. The paper was concluded with the proposal of a new interaction equation which provides improved estimations of angle capacity when compared with AISC Specification 2005 results. 3. Finite element model Finite element modeling (FEM) was carried out using the commercial program ANSYS 10.0 [10]. Angle specimens were meshed with general-purpose 4-node Shell 181 elements as specified in ANSYS. Nonlinearities relating to both geometric and material characteristics were considered. An iterative approach using the arc-length method was used in the nonlinear load–displacement analysis and the stress–strain relationship for steel was assumed to be elastic–perfectly plastic. Considering the symmetry of the member and loading, only half of the specimen was modeled and simple support conditions were implemented. Fig. 1 depicts a characteristic mesh for half an angle specimen where x and y axes intersect at the centroid of the cross-section and represent the major and minor principal axes, respectively. Also presented in the figure is the cross-section of the angle with key notations used throughout this paper. The mesh size of 25 × 6.0 mm was selected based on a convergence study performed for typical angles with various slenderness ratios. A fictitious plate attached to the end of the angle was used to enable the modeling of the eccentric point load applied outside the legs of the angle. This plate was made as a rigid region to avoid possible bending and consequent interference with the behavior of the angle elements. The end plate was meshed using 6.0 × 6.0 mm elements in the middle plane with the nodes on two adjacent edges contacting the angle legs coinciding with those of the angle legs. A similar technique used for load application by Adluri et al. [11] was proven to deliver satisfactory results. To simulate the simple end conditions, the centroid of the cross-section at the support was restrained from lateral movement but allowed to rotate freely in both x and y directions. Initial imperfections were considered by assuming the initial geometry in the form of a half sine-wave applied over the entire length of the specimen about its minor principal axis. The amplitude of the imperfection at the mid-height of the angle, wo was taken as L/1500, which is in line with experimental measurements conducted by Adluri and Madugula [5] and Hui [12] on single angles with various cross-sections. They showed that L/1500 is a reasonable assumption for maximum amplitude of the imperfection at mid-height of angles. Previous research [13,6] revealed that while residual stresses had a marked influence on the capacity of concentrically loaded steel angles, marginal effect of about 5% or less on the capacity was observed for eccentrically loaded ones and the maximum residual stress level in steel angles did not generally exceed 25% of the yield stress. In this research, the residual stress distribution pattern as suggested in ECCS [14] was adopted and is shown in Fig. 2. These residual stresses were first transformed into a series of discrete uniform stresses and were then applied to each element of the model as initial stresses during the first sub-step of the first load step of the analysis. 4. Verification of the finite element model The finite element model was verified with test results obtained from the experimental portion of this research program [12] as well as those available in the reported literature. It is noted that the focus of this research was to study single angles as beam–columns with bending not restrained about any particular axis. Hence, only
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Table 1 Comparison of experimental and analytical results [12]. No.
Specimen ID#
ex (mm)
ey (mm)
Fy (MPa)
Pexp (kN)
PFEM (kN)
Pexp /PFEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
E900-1 E900-2 E900-3 E900-4 E900-5 E900-6 E900-7 E900-8 E900-9 E900-10 E1200-1a E1200-1b E1200-2 E1200-3a E1200-3b E1200-4 E1200-5 E1500-1a E1500-1b E1500-1c E1500-2 E1500-3 E1500-4 E1500-5 E1500-6 E1500-7 E1500-7 E1500-7
0.0 4.2 10.0 16.8 29.4 50.4 0.0 0.0 0.0 0.0 4.2 4.2 10.0 16.8 16.8 29.4 50.4 0.0 0.0 0.0 8.4 16.8 29.4 50.4 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 8.4 16.8 29.4 50.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8.4 16.8 29.4 50.4
330 330 330 330 330 330 348 348 348 348 330 330 330 348 330 330 348 348 348 348 348 348 348 348 348 348 348 348
117.0 110.9 105.3 84.5 59.0 46.1 65.6 46.4 34.4 23.3 68.1 69.7 66.7 66.4 71.3 57.0 42.5 45.3 41.1 42.8 42.9 41.3 40.1 37.0 35.2 27.8 23.4 18.0
105.5 105.1 104.9 77.6 56.1 39.7 59.5 44.4 31.9 22.7 66.5 66.1 66.0 65.3 65.2 52.9 38.3 43.8 43.9 43.8 42.9 41.1 40.9 36.0 32.4 26.5 21.5 16.3 Avg.
1.11 1.06 1.00 1.09 1.05 1.16 1.11 1.10 1.05 1.08 1.03 1.02 1.05 1.01 1.02 1.09 1.08 1.11 1.03 0.94 0.98 1.00 1.01 0.98 1.03 1.09 1.05 1.09 1.05
C.O.V.
4.7%
Table 2 Comparison of experimental and analytical results for single equal-leg angles. Source
No. of specs
Slenderness ratio
Angle size
Fy (MPa)
Loading condition
Wakabayashi and Nonaka [15] Adluri and Madugula [5]
10 26
20–150 68–188
303
Concentrically loaded
335–411
Concentrically loaded
330, 348 294–323
Hui [12] Wakabayashi and Nonaka [15] Ishida [16]
4 40
95, 155 20–150
90 × 90 × 7 64 × 64 × 9.5 76 × 76 × 4.8 76 × 76 × 12.7 102 × 102 × 7.9 127 × 127 × 9.5 51 × 51 × 6.4 90 × 90 × 7
7
20–100
75 × 75 × 6
58.8–63.9
Mueller and Erzurumlu [17]
14
60, 110, 192
75 × 75 × 6
349–422
Hui [12]
24
95, 125, 155
51 × 51 × 6.4
330, 348
53.3, 100
75 × 75 × 9 75 × 75 × 12 350 × 350 × 35
295–433
Madugula et al. [18]
6
results from experimental research with set-up consistent with this loading conditions were considered. Although considerable experimental results have been reported on the capacity of single angles attached to structural tee sections which were in turn subjected to axial compression, they were not included since the bending of the angle was restrained about the geometrical axis of the angle. The test results used in the verification process included results of 36 concentrically loaded angles [15,5], 89 eccentrically loaded angles [15–17,12], and 6 angles under flexure only [18]. All angles included in the verification study were tested as simply supported on both ends and were not retrained to bend with respect to any axis. The simple support was formed by welding the end of the angle to a rigid plate, which was in turn supported through a ball bearing arrangement similar to that suggested by Galambos [3].
Avg. of Pexp /PFEM
C.O.V. (%)
1.17
9.3
Concentrically loaded Eccentrically loaded causing major, minor and bi-axial bending Eccentrically loaded causing minor axis bending Eccentrically loaded causing bi-axial bending Eccentrically loaded causing major and minor bending
1.05 1.02
6.7 8.4
0.96
3.0
1.00
10.4
1.05
4.0
Pure bending about minor principal axis
0.98
1.7
A summary comparison of analytical and experimental ultimate loads is presented in Table 1 for results obtained from the experimental portion of this research and in Table 2 for results from other research reported in the available literature. It is noted that test results covered a wide range of angle cross-sections, slendernesses, yield stresses and various loading conditions. Overall, numerical results compared well with experimental responses. The ratios of experimental and numerical ultimate load, Pexp /PFEM , for all specimens varied from 0.96 to 1.17 with COVs ranging from 1.7% to 10.4%. In some cases, the initial imperfections and residual stresses for tested angles were not reported in the literature and assumptions as described previously were then made, which is believed to attribute to relatively high Pexp /PFEM ratios in these cases. Typical comparisons of numerical and experimental results [11] are illustrated in Fig. 3(a) for load vs. mid-height deflection curves and in Fig. 3(b) for
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a y
80
x 60 Load (kN)
z
1
40
2
20
-20
-15
Leg 1-Test Leg 2-Test Leg 1-FEM Leg 2-FEM
0 -10 -5 0 5 10 Mid-height deflection (mm)
15
20
b
Fig. 1. Typical finite element mesh and cross-section details of the angle.
Experimental
FEM
Fig. 3. Comparison of analytical and experimental results: (a) load vs. deformation response; (b) failure modes.
Fig. 2. Residual stress distribution used in the model.
failure modes. The figure shows that the finite element model is capable of predicting both the ultimate load capacity and behavior of single angles. Fig. 4 shows the normalized ultimate load versus slenderness parameter, λ, for eccentrically loaded angles under bi-axial bending and concentrically loaded angles, respectively. In general, reasonably good agreement is obtained between the numerical and experimental results. It is reasonable to consider that the finite element model is accurate to provide the ultimate capacity for single angles with various configurations and subjected to various loading conditions.
single angle behavior and capacity. Parameters included in the numerical simulations were the slenderness parameter, λ, applied eccentricity, ex or ey , and angle leg width-to-thickness ratio, b/t. The test population included nine slenderness parameters (λ = 0.27, 0.53, 0.80, 1.06, 1.33, 1.60, 1.86, 2.13, and 2.39), six eccentricities (0, xo /4, xo /2, xo , 3xo /2, 2xo ) where xo is the distance from the shear centre to the centroid of the crosssection measured in the major-axis direction and five leg widthto-thickness ratios (b/t = 5.3, 6.4, 8.0, 10.7, and 16.0). Majority of the study was carried out for an L51 × 51 × 6.4 eccentrically loaded with respect to either principal axis. For study of b/t ratios, cross-sections in L51 × 51 series with varying angle leg thickness were used. The dimensions of angle sections were taken as the nominal values according to CSA S16-01 [2]. An elastic–perfectly plastic material model was assumed for steel with a yield stress of 350 MPa, an elastic modulus of 200 000 MPa, and a Poisson’s ratio of 0.3. 6. Evaluation of AISC 2005 Specification
5. Parametric study
6.1. Angles eccentrically loaded with respect to major principal axis
Subsequent to the verification, the finite element model was used to study the effects of several influential parameters on the
The load–moment interaction (Eq. (1)) as suggested in the AISC Specification 2005 was determined and compared with numerical
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a
2091
a
b
b
Fig. 6. Normalized flexural strength vs. slenderness parameter λ curves.
6.1.1. Axial load capacity (M
Fig. 4. Comparison of analytical and experimental results: (a) bi-axial bending; (b) concentric compression.
Fig. 5. Comparison of normalized load–moment interaction diagram of analytical and AISC Specification 2005 results (major axis bending).
results for five representative slenderness values. Fig. 5 shows the normalized interaction diagram where the load was normalized by the yield strength of the cross-section, Py , and the moment was normalized by the yield moment, My .
(P = 0)
= 0) and pure moment capacity
For pure axial capacities (M = 0), the numerical and AISC results agreed well with each other except in the case of λ = 0.80. Residual stresses and initial imperfections tend to have a larger effect on angles with an intermediate slenderness. It is therefore believed that the discrepancy noted above was due to the difference between the assumed values used in the numerical model and the actual values. For pure moment capacities (P = 0), both numerical and AISC moment capacity decreased with an increase in the slenderness, but the numerical values were consistently greater than the AISC values for all investigated slendernesses. A typical comparison of numerical and AISC flexural capacities for major-axis bending for b/t = 8.0 is illustrated in Fig. 6(a) where normalized flexural strength vs. slenderness parameter λ curves are plotted. Also shown in the figure are the curves obtained from Eq. (2) proposed by Earls [9]. Numerical results and Eq. (2) results were consistently agreeable over the entire investigated range of λ with an average variation less than about 3%. It also shows that the reduction in the flexural capacity as a result of an increase in slenderness was not as significant as indicated by AISC values. Comparing with numerical results, AISC increasingly underestimated the flexural capacity over the region of λ > 1.0 with a maximum variation of 20%. For the case of low slenderness (λ < 0.5) where angles may reach plasticity, AISC proposes 1.5My as the upper limit of the flexural strength whereas both numerical and Earls’ equations suggest that the angle is able to attain higher plastic moment. The underestimation of flexural strength about major axis is believed
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(a) ex < exo .
(c) ex > exo .
(b) ex = exo .
Fig. 7. Numerical von Mises stress distribution on the cross-section.
to be one contributor to the conservatism observed in AISC Specification 2005 load–moment interaction. Fig. 6(b) shows the numerical flexural capacity in comparison with values obtained from Eq. (2) for other b/t ratios. It is noted that Earls suggested the leg slenderness ratio b/t not less than 8 for the application of Eq. (2). The comparison, however, shows that even for the values of b/t = 5.3 and 6.4, the angle flexural capacities have a reasonably good agreement with those obtained from Eq. (2). 6.1.2. Axial load and moment interaction For combined axial load and moment cases, numerical results, in general, provided higher estimates of capacity in comparison with AISC results. The numerical load–moment interaction demonstrated two distinctive regions where, in one region the axial load remained almost constant as the moment increased, and in the other region the axial load decreased as the moment increased. The first region became increasingly larger as the slenderness ratio increased. It appears that for a given slenderness, there existed a critical eccentricity below which the change of the ultimate load with an increase in eccentricity was marginal to almost unnoticeable. This indicates that an eccentrically loaded angle can reach its concentric compressive capacity as long as the applied eccentricity was within the critical eccentricity. Defined as exo in this paper, the critical eccentricity assumed an increasingly greater value with increased slenderness. Similar findings noted by Schafer [19] suggested that the presence of moment does not necessarily result in a reduction in the axial capacity of a singly symmetrical member as the interaction equation would suggest. In some cases, it actually benefits the axial behavior of the member. As shown in Fig. 5, the underestimation by AISC Specification 2005 was most pronounced in the region of critical eccentricities. For example, for λ = 1.33, 1.86, and 2.39, AISC values in the region of critical eccentricities can be as low as 70%, 64% and 57% of the numerical results, respectively. On the other hand, there were instances where AISC Specification 2005 overestimated angle compressive capacity. For example, in the region of M > 0.8My for λ = 0.80, the AISC overestimated the capacity by a maximum amount of 20%. Fig. 7 demonstrates the numerical von Mises stress distribution on the cross-section at mid-height of a specimen when the applied eccentricity was less than, equal to, and greater than the critical eccentricity. Noting that the red gradation indicates maximum stress, the figure shows that as the applied eccentricity varied from less to greater than the critical eccentricity, the location where maximum stress occurred shifted from the heel to the toes of the angle. As the applied eccentricity equaled exo , the maximum stress occurred simultaneously at both the heel and the toe. Numerical results also showed that angles subjected to eccentric compression resulting in major axis bending failed in torsional–flexural buckling
Table 3 Critical eccentricity ratio for various cross-sections. b/t
5.3 6.4 8.0 10.7 16.0
Critical eccentricity ratio, mo = exo /xo
λ = 0.27
0.80
1.07
1.33
1.86
2.40
0.01 0.01 0.01 0.01 0.01
0.18 0.19 0.20 0.21 0.21
0.42 0.44 0.49 0.51 0.53
0.58 0.61 0.66 0.71 0.77
1.25 1.34 1.48 1.59 1.75
2.45 2.79 3.07 3.32 3.58
mode. The flexural deflection and twist of the cross-section reached maximum at the mid-height of the angle and resulted in re-orientation of the principal axes. To satisfy equilibrium on the deformed shape, additional stresses due to the twist of the crosssection resulted at both the heel and toes of the cross-section. These additional stresses either add to or subtract from the stresses caused by axial and flexural deformations at the heel and toe locations. The critical eccentricity is that at which the combined stresses at both the heel and the toe simultaneously reach a maximum value. The value of this critical eccentricity is dependent on the magnitude of flexural deflection and twist of the angle. 6.1.3. Critical eccentricity The critical eccentricity phenomenon was further studied for various angle leg width-to-thickness ratios, b/t. Five crosssections, L51 × 51 × 3.2, L51 × 51 × 4.8, L51 × 51 × 6.4, L51 × 51 × 7.9, and L51 × 51 × 9.5 with b/t ratio ranging from 5.3 to 16.0 were used in this study. A term was introduced as the critical eccentricity ratio mo , which is defined as the ratio of critical eccentricity, exo and xo . Results of mo for various b/t ratios are listed in Table 3 and mo vs. λ curves are plotted in Fig. 8. It shows that for all b/t ratios, mo and thus exo increased with an increase in the slenderness. However, for a given slenderness, the variation of exo with b/t ratios was more complex. For λ < 1.0, exo remained almost constant for all b/t ratios. For λ > 1.0, the variation in exo for different b/t ratios became more pronounced and an increase in b/t ratio resulted in an increase in exo values. It has been established that the b/t ratio affected the flexural deflection and the twist of the angle failing in elastic flexural torsional buckling [20], which in turn would affect the critical eccentricity values. As the angle became stocky in the length direction and hence λ decreased, it seemed that the b/t ratio had a diminishing effect on the flexural deflection and twist and thus the critical eccentricity values. A lower-bound curve showing relationship of mo and λ is also plotted in the figure. Obtained using curve-fitting technique, this curve is expressed as mo = 0.3λ2.26 . This equation was used in later sections in an effort to develop a general load–moment interaction equation.
Y. Liu, L. Hui / Engineering Structures 32 (2010) 2087–2095
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a
b Fig. 8. Critical eccentricity ratio vs. slenderness parameter for various b/t ratios.
6.2. Angles eccentrically loaded with respect to minor principal axis Numerical load–moment interaction curves for combined axial load and minor axis bending of a 51 × 51 × 6.4 angle with various λ values are compared with corresponding curves determined using AISC Specification 2005 in Fig. 9(a) for the case of toes in compression and in 9(b) for the heel in compression. The numerical interaction curves for the minor axis bending case did not show evident critical eccentricity phenomenon as in the case of major axis bending and the ultimate load decreased when the moment and thus the applied eccentricity increased. The general trends and associated load–moment interaction values are nearly the same for cases of toes in compression and heel in compression. The comparison of the numerical and AISC results shows that, while the AISC is generally conservative, the correlation between two curves was reasonably good especially for angles with λ values greater than 0.8. It seems that the discrepancy in flexural capacity between numerical and AISC results may be attributed mainly to the conservatism of AISC Specification 2005, especially for case with λ < 0.8. Fig. 10 further demonstrates the normalized flexural capacities of angles with b/t = 8.00 about minor-axis bending with either toes or heel in compression. It is noted that AISC Specification 2005 specifies the flexural strength for minor-axis bending as a shape factor of 1.5 applied to the yield moment regardless of the slenderness although some research [21] showed that the shape factor for L51 × 51 × 6.4 with respect to minor principal axis can reach 1.80. Numerical results show that this limit is markedly conservative for angles L51 × 51 × 6.4 in practical slenderness range and is reasonable when slenderness exceeds 2.0. Since AISC Specification 2005 limits the flexural capacity to be 1.5My for all the slenderness ratios and the application of eccentricity causing either principal axis bending or geometric axis bending, it is felt that 1.5My represents a general lower bound value of angle flexure capacity which can be expected in practice [22]. 7. Proposal of the new equation The previous sections have demonstrated the inconsistencies and overall conservatism inherent in the interaction equations for single angle design suggested in the current AISC Specification 2005. The possible sources of this conservatism were also identified and discussed. In an effort to provide a better estimate of the capacity of single angle beam–columns, a new interaction equation (Eq. (3)) for combined compression and principal axis bending is
Fig. 9. Comparison of normalized load–moment interaction diagram of analytical and AISC Specification 2005 results (minor axis bending): (a) toes in compression; (b) heel in compression.
proposed based on approximately 2000 finite element model results and curve-fitting technique. While the equation maintained the format of interaction between axial load and moment, two key differences are noted including the incorporation of the critical eccentricity concept and Earls’ flexural capacity equation (Eq. (2)) for the determination of major axis flexural capacity, Mnx . Pr
φ c Pn
+
Pr (ex )m
φb Mnx
+
8 Pr ey 9 φb Mny
= 1.
(3)
The critical eccentricity concept is incorporated in the term (ex )m . Defined as virtual eccentricity, it can be evaluated as (ex )m = n 0 ex − exo
(ex ≤ exo ) (ex > exo ) , where exo
= mo xo and mo is the critical ec-
centricity ratio expressed as mo = 0.3λ2.26 . The determination of Pn and Mny are kept to be the same as in AISC Specification 2005. Fig. 11 shows a comparison of load–moment interaction curves generated using the proposed equation (3) and the finite element model for major axis bending. As can be seen, the interaction curves obtained from Eq. (3) are in a good agreement but on the conservative side compared with finite element curves. Comparing with AISC Specification 2005, the proposed equation improves the load carrying capacity predictions, in particular, around the critical eccentricity region. 7.1. Evaluation of the proposed equation using available experimental database The experimental database used to verify the finite element model was used to evaluate the performance of the proposed equation. Resistance factors for compression and flexure were
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a
Table 4 Summary comparison of test-to-prediction ratios. Source
AISC Specification 2005
Proposed equation (3)
Avg.
C.O.V. (%)
Avg.
C.O.V. (%)
Wakabayashi and Nonaka [15] Ishida [16] Mueller and Erzurumlu [17] Hui [12]
1.18 1.07 1.10 1.23
11.7 7.8 11.9 11.4
1.12 1.07 1.04 1.08
7.7 7.8 11.6 7.4
Total
1.17
14.0
1.09
9.5
Table 5 Reliability index for studied angles.
b
All experimental data
Experimental data (major axis bending only)
Rm /Rn AISC Specification 2005 1.17 Eq. (3) 1.09
Vr (%)
β
Rm /Rn
Vr (%)
β
14.0 9.5
3.4 3.7
1.26 1.07
19.5 7.5
3.1 4.2
This indicates an average increase in the predicted load carrying capacity of 8.0%, and less scattered predicted values by using the proposed equation. It is noted that in the case of comparison with results by Ishida [16], the test-to-prediction ratios and associated COVs were identical obtained using either the proposed equation or AISC Specification 2005. This is attributed to the fact that for angles under eccentric load causing minor axis bending only, equations from both procedures were essentially the same. However, in the case of angles under eccentric loading causing major axis bending, the capacities predicted using AISC Specification 2005 were usually significantly lower than the test results and results from the proposed equation. 7.2. Reliability index, β
Fig. 10. Normalized flexural strength vs. slenderness parameter λ curves for minor axis bending of angles with b/t = 8.00: (a) toes in compression; (b) heel in compression.
The current structural design philosophy adopted by either AISC Specification 2005 or CSA S16 is based on a first-order probabilistic design procedure [23]. The single angle design equations in AISC Specification 2005 were derived from beam–column theory and adjusted using experimental results to achieve an acceptable level of complexity. The load factors and resistance factors are arrived at by using a uniform ‘‘probability of failure’’ level for the full range of basic variables of all member types. The probability of failure, i.e. the level of safety, is defined by reliability index, β . To assess the safety margin of the AISC Specification 2005 procedure and the proposed equation for single angle design, the reliability indices β for both cases were examined using the following equation [23]:
φ=
Fig. 11. Normalized load–moment interaction diagram generated using the proposed equation and numerical results (major axis bending).
taken as 1.0 to facilitate the comparison between the equation and the raw experimental results. The test-to-prediction ratios based on AISC Specification 2005 and proposed equation (3) are summarized in Table 4 for experimental investigations conducted by various researchers. The average test-to-prediction ratios for AISC Specification 2005 and the proposed equation were found to be 1.17 and 1.09 with COVs equal to 14.0% and 9.5%, respectively.
Rm −αβ Vr e Rn
(4)
where φ is the resistance factor, Rm is the mean resistance obtained from experimental results, Rn is the averaged nominal resistance obtained from design equations, α is a constant taken as 0.55 as proposed by Galambos and Ravindra [24], and Vr is the coefficient of variation of Rm /Rn . Using experimental results as listed in Table 4 and resistance factor φ of 0.9, the reliability index β was calculated for (1) all experimental data from different sources, and (2) data concerning eccentric load causing major principal axis bending only. The results are summarized in Table 5. The table shows that the proposed equation provided results closer to the experimental results and improved coefficients of variation. The reliability indices based on either AISC Specification 2005 or proposed equations are greater than 3.0, which is acceptable for limit states design. It is however noted that the AISC Specification 2005 provides lower reliability indices for both cases of data sources than those of proposed equation.
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8. Conclusions
References
A finite element study was conducted to investigate the effects of several parameters on the load carrying capacity of equalleg single angles eccentrically loaded resulting in either major or minor principal axis bending. The performance of design equations suggested in AISC Specification 2005 and proposed in this paper was evaluated using both experimental and numerical results. The essential conclusions are listed as follows. For eccentric compression resulting in major axis bending, numerical results suggest that a critical eccentricity seemed to exist within which when the eccentricity was applied, the equal-leg single angles attained almost their concentric axial capacity. Values of the critical eccentricity were associated with slenderness and leg width-to-thickness ratio of the angle. A lower bound expression was proposed for the calculation of the critical eccentricity. For eccentric compression resulting in minor axis bending, the effects of all studied parameters on angle ultimate capacities showed negligible variations for cases of either the angle toes or the heel in compression. In contrast with the major axis bending, the critical eccentricity was not evident and the ultimate capacity consistently decreased with an increase in the eccentricity. Compared with the experimental and numerical results, AISC Specification 2005, in general, markedly underestimated the ultimate load capacity of single angles under eccentric loading causing major axis bending, especially in the region of critical eccentricity. Results also identified a few instances where AISC overestimated ultimate capacities of angles in the case of major axis bending. For eccentric loading causing minor axis bending, AISC Specification 2005 was shown to be markedly conservative for angles with a low λ value (λ = 0.27) but provide lower-bound and reasonably accurate predictions for angles with relatively large slenderness. A new interaction equation incorporating the concept of critical eccentricity and Earls’ flexural capacity equations was proposed and the comparison with the AISC results indicated that the proposed equation provided improved capacity estimates and acceptable reliability index for equal-leg angles with a wide range of parameters.
[1] AISC specification. Specification for structural steel buildings. Chicago (IL): American Institute of Steel Construction; 2005. [2] CAN/CSA S16-01. Limit states design of steel structures. Mississauga (Ontario, Canada): Canadian Standards Association; 2001. [3] Galambos TV. Guide to stability design criteria for metal structures. In: Structural stability research council. 5th ed. New York: John Wiley & Sons, Inc.; 1998. [4] Adluri SMR, Madugula MKS. Eccentrically loaded single angle struts. Eng J AISC 1992;29(2):59–66. [5] Adluri SMR, Madugula MKS. Flexural buckling of steel angles: experimental investigation. J Struct Eng, ASCE 1996;122(3):309–17. [6] Temple MC, Sakla SSS. Single-angle compression members welded by one leg to a gusset plate, II, a parametric study and design equation. Canad J Civ Eng 1998;25(3):585–94. [7] Sakla SSS. Performance of the AISC LRFD specification in predicting the capacity of eccentrically loaded single-angle struts. Eng J AISC 2005;42(40):239–46. [8] Earls CJ. Design of single angles bent about the major principal axis. Eng J AISC 2003;40(3):159–66. [9] Earls CJ. Proposed equal leg single angle flexural design provisions for consideration in the development of future AISC specification editions. Eng J AISC 2003;40(3):167–75. [10] ANSYS 10.0. Release 10.0 documentation for ANSYS: guide to the ANSYS documentation. ANSYS. Inc. and ANSYS Europe, Ltd.; 2006. [11] Adluri SMR, Madugula MKS, Monforton GR. Finite element failure analysis of schifflerized angles. Comput Struct 1991;41(5):1087–93. [12] Hui LB. Beam–column behaviour of steel single angles. MASc. thesis. Halifax (NS, Canada): Department of Civil and Resource Engineering. Dalhousie University; 2007. [13] Usami T, Fukumoto Y. Compressive strength and design of bracing members with angle or tee sections. Proceedings of Japanese society of civil engineering, vol. 201. 1972. p. 31–50. [14] ECCS. Recommendations for angles in lattice towers. Technical Committee 8. Publication No. 39. Brussels (Belgium): European Convention for Constructional Steelwork; 1985. [15] Wakabayashi M, Nonaka T. On the buckling strength of angles in transmission towers. Bull Disast Prev Res Inst 1965;15(91):1–18. (Part 2). [16] Ishida A. Experimental study on column carrying capacity of ‘SHY steel’ angles. Yawata Technical Report 265. Tokyo (Japan): Yawata Iron and Steel Co. Ltd. 1968. [17] Mueller WH, Erzurumlu H. Behaviour and strength of angles in compression: an experimental investigation. Oregon (USA): Research Report of CivilStructural Engineering. Division of Engineering and Applied Science. Portland State University; 1983. [18] Madugula MKS, Kojima T, Kajita Y, Ohama M. Minor axis bending strength of angle beams. In: Proceedings of the international conference on structural stability and design. 1995. p. 73–8. [19] Schafer BW. Designing cold-formed steel using the direct strength method. In: 18th international specialty conference on cold-formed steel structures. 2006. [20] Dabrowski R. Thin-walled members under biaxial eccentric compression. Der Stahlbau, vol. 30. Germany: Wihelm Ernst and Sohn; 1961. p. 360–5. [21] Dong YC, Madugula MK. Elastic and plastic section moduli of steel angles abut principal axes. Eng J AISC 2004;41(1):7–14. [22] Earls CJ, Galambos TV. Practical compactness and bracing provisions for the design of single angle beams. Eng J AISC 1998;35(1):19–25. [23] Bjorhovde R, Galambos TV, Ravindra MK. LRFD criteria for steel beam–columns. J Struct Div ASCE, (New York) 1978;104(ST9):1371–87. [24] Ravindra MK, Galambos TV. Load and resistance factor design for steel. J Struct Div ASCE, (New York) 1978;104(ST9):1337–53.
Acknowledgements The authors wish to recognize the contribution of financial assistance by the Steel Structures Educational Foundation (SSEF) as well as in kind assistance from Marid Industries Ltd. for the donation of specimens.
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Finite element study of steel single angle beam–columns Yi Liu ∗ , Linbo Hui Department of Civil and Resource Engineering, Dalhousie University, Halifax, NS, Canada
article
abstract
info
Article history: Received 12 February 2010 Received in revised form 3 March 2010 Accepted 4 March 2010 Available online 7 April 2010
A finite element study was conducted to investigate the response of steel single angles subjected to eccentric compression and the numerical results were used to examine the effectiveness of the beam–column interaction approach suggested in the current AISC design standard. Numerical results demonstrated inconsistencies and overall conservatism inherent in the design equations and the possible sources of this conservatism were also identified and discussed. Results also indicated the existence of a critical eccentricity associated with each slenderness ratio in the case of eccentric compression causing major axis bending. Within this eccentricity, any reduction in the ultimate capacity due to eccentricity was found to be marginal. A new interaction equation for combined compression and principal axis bending was proposed. The proposed equation achieved the target reliability index with an improved test-toprediction capacity ratio over a wide range of angle geometric parameters when compared with AISC design equations. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Numerical modeling Single angle Eccentric compression Flexure AISC Specification
1. Introduction
for
Due to their ease for installation and simple aesthetic appeal, steel single angles are extensively used in a variety of structures such as steel joists and trusses and braced structures where they are usually either welded or bolted to other structural members by one leg. This connection detail results in eccentric compressive loading with respect to both major and minor principal axes of the angle section. Combined with their asymmetric characteristics, angles subjected to axial compression and bi-axial bending often exhibit torsional or flexural torsional behavior if the bending of the angle is not restrained with respect to certain axis. The design and analysis of such angles is more complex than their simple shape may suggest. Currently, the design of the eccentrically loaded single angles is governed by the Specification for Structural Steel Buildings [1] in the United States, hereafter referred to as AISC Specification 2005 and the Canadian standard ‘‘Limit States Design of Steel Structures’’ [2] in Canada. Both standards adopt the approach of treating eccentrically loaded angles as beam–columns and evaluating their capacity using interaction between axial load and bending. The AISC Specification 2005 suggests that the interaction of flexure and axial compression applicable to specific locations on a cross-section of a single angle shall be limited according to: for
∗
Pr
φc Pn
≥ 0.2,
Pr
φc Pn
+
8 9
Mrx
φb Mnx
+
Mry
φb Mny
≤ 1.0
Corresponding author. Tel.: +1 902 494 1509; fax: +1 902 494 3108. E-mail address: [email protected] (Y. Liu).
0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2010.03.009
(1a)
Pr
φc Pn
< 0.2,
Pr 2φc Pn
+
Mrx
φb Mnx
+
Mry
φb Mny
≤ 1.0
(1b)
where Pr and Pn are the required compressive strength and the nominal compressive strength, Mr and Mn are the required flexural strength and the nominal flexural strength, φc and φb are resistance factors for compression and flexure, and x and y refer to the major and minor principal axes. In Canada, a similar interaction approach is proposed but with axial force and moment interaction factors different from the AISC Specification 2005. Additionally, this standard does not provide explicit provisions for determining the nominal flexural strength for mono-symmetric or asymmetric angles, but rather, it directs users to the SSRC Guide [3] for design methods. Some research has been conducted by others to evaluate the performance of AISC Specification 2005. Based on a comparison study of 71 test results, Adluri and Madugula [4,5] pointed out that these interaction equations are highly conservative. Temple and Sakla [6] reported experimental capacities two to three times the values suggested by AISC Specification 2005. Sakla [7] evaluated the efficacy of these equations for both equal- and unequalleg angles. He concluded that AISC Specification 2005 provides a better prediction of the ultimate capacity of equal-leg single angles than it does for unequal-leg single angles. It underestimates all unequal-leg test results and overestimates capacities of some equal-leg angles. Earls [8] suggested that AISC Specification 2005 for single angle beam design is, in most cases, unnecessarily conservative while in some other situations overestimates angle capacities. Based on a finite element study, he proposed equations for predicting the flexural strength with respect to major principal
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Notations b/t E Fy K KL/ry L M Mn Mr My P Pexp PFEM Pn Pr Py Rm Rn
Angle leg width-to-thickness ratio Modulus of elasticity of angle steel Yield stress of steel Effective length factor = 1.0 Slenderness ratio of angle Length of angle Applied moment Nominal flexural strength Required flexural strength Yield moment about the axis of bending Applied compressive load Experimental compressive capacity Finite element results of compressive load capacity Nominal compressive strength Required compressive strength Yield capacity of angle cross-section Mean resistance obtained from experimental results Averaged nominal resistance obtained from design equations Coefficient of variation of Rm /Rn Eccentricity with respect to major principal axis Critical eccentricity with respect to the major principal axis Eccentricity with respect to minor principal axis Initial imperfection along the length of angle Amplitude of the imperfection at the mid-height of angle q
Vr ex e xo ey
w wo λ β φb φb φc
Slenderness parameter =
KL ry
Fy
π 2E
Reliability index Resistance factor Resistance factors for flexure Resistance factors for compression Distance between shear centre and centroid of angle cross-section
xo
axis of single equal-leg angles as follows [9]: For 8 ≤
For
b t
b t
≤ 14,
> 14,
Mn My
Mn My
=−
b/t
10 000
= −0.00184
Lb rz
− 50 + 1.5
(2a)
7 16 22.4 − 50 + rz b/t b/t
Lb
(2b) where Mn is the ultimate flexural strength, My is the yielding moment, Lb is the unbraced length of beams, rz is the radius of gyration with respect to minor axis, b is the leg width and t is the leg thickness. 2. Scope of the current research Although the previous research indicated a significant conservatism inherent in AISC Specification 2005 for the design of single angles, the angle geometric parameters and loading eccentricities used in those evaluations were limited. To gain a better understanding of the singe angle behavior and have a comprehensive evaluation of the AISC Specification 2005, a numerical study using finite element modeling was carried out to study the effects of several influential parameters on the load carrying capacity of single equal-leg angles eccentrically loaded with respect to either the major or minor principal axis. The finite element model was verified using experimental results available in literature for single
angles subjected to compression with simple support conditions. The verified finite element model was then employed to generate numerical results to further examine the effectiveness of the AISC Specification 2005 interaction equations over a wide range of parameters. The paper was concluded with the proposal of a new interaction equation which provides improved estimations of angle capacity when compared with AISC Specification 2005 results. 3. Finite element model Finite element modeling (FEM) was carried out using the commercial program ANSYS 10.0 [10]. Angle specimens were meshed with general-purpose 4-node Shell 181 elements as specified in ANSYS. Nonlinearities relating to both geometric and material characteristics were considered. An iterative approach using the arc-length method was used in the nonlinear load–displacement analysis and the stress–strain relationship for steel was assumed to be elastic–perfectly plastic. Considering the symmetry of the member and loading, only half of the specimen was modeled and simple support conditions were implemented. Fig. 1 depicts a characteristic mesh for half an angle specimen where x and y axes intersect at the centroid of the cross-section and represent the major and minor principal axes, respectively. Also presented in the figure is the cross-section of the angle with key notations used throughout this paper. The mesh size of 25 × 6.0 mm was selected based on a convergence study performed for typical angles with various slenderness ratios. A fictitious plate attached to the end of the angle was used to enable the modeling of the eccentric point load applied outside the legs of the angle. This plate was made as a rigid region to avoid possible bending and consequent interference with the behavior of the angle elements. The end plate was meshed using 6.0 × 6.0 mm elements in the middle plane with the nodes on two adjacent edges contacting the angle legs coinciding with those of the angle legs. A similar technique used for load application by Adluri et al. [11] was proven to deliver satisfactory results. To simulate the simple end conditions, the centroid of the cross-section at the support was restrained from lateral movement but allowed to rotate freely in both x and y directions. Initial imperfections were considered by assuming the initial geometry in the form of a half sine-wave applied over the entire length of the specimen about its minor principal axis. The amplitude of the imperfection at the mid-height of the angle, wo was taken as L/1500, which is in line with experimental measurements conducted by Adluri and Madugula [5] and Hui [12] on single angles with various cross-sections. They showed that L/1500 is a reasonable assumption for maximum amplitude of the imperfection at mid-height of angles. Previous research [13,6] revealed that while residual stresses had a marked influence on the capacity of concentrically loaded steel angles, marginal effect of about 5% or less on the capacity was observed for eccentrically loaded ones and the maximum residual stress level in steel angles did not generally exceed 25% of the yield stress. In this research, the residual stress distribution pattern as suggested in ECCS [14] was adopted and is shown in Fig. 2. These residual stresses were first transformed into a series of discrete uniform stresses and were then applied to each element of the model as initial stresses during the first sub-step of the first load step of the analysis. 4. Verification of the finite element model The finite element model was verified with test results obtained from the experimental portion of this research program [12] as well as those available in the reported literature. It is noted that the focus of this research was to study single angles as beam–columns with bending not restrained about any particular axis. Hence, only
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Table 1 Comparison of experimental and analytical results [12]. No.
Specimen ID#
ex (mm)
ey (mm)
Fy (MPa)
Pexp (kN)
PFEM (kN)
Pexp /PFEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
E900-1 E900-2 E900-3 E900-4 E900-5 E900-6 E900-7 E900-8 E900-9 E900-10 E1200-1a E1200-1b E1200-2 E1200-3a E1200-3b E1200-4 E1200-5 E1500-1a E1500-1b E1500-1c E1500-2 E1500-3 E1500-4 E1500-5 E1500-6 E1500-7 E1500-7 E1500-7
0.0 4.2 10.0 16.8 29.4 50.4 0.0 0.0 0.0 0.0 4.2 4.2 10.0 16.8 16.8 29.4 50.4 0.0 0.0 0.0 8.4 16.8 29.4 50.4 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 8.4 16.8 29.4 50.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8.4 16.8 29.4 50.4
330 330 330 330 330 330 348 348 348 348 330 330 330 348 330 330 348 348 348 348 348 348 348 348 348 348 348 348
117.0 110.9 105.3 84.5 59.0 46.1 65.6 46.4 34.4 23.3 68.1 69.7 66.7 66.4 71.3 57.0 42.5 45.3 41.1 42.8 42.9 41.3 40.1 37.0 35.2 27.8 23.4 18.0
105.5 105.1 104.9 77.6 56.1 39.7 59.5 44.4 31.9 22.7 66.5 66.1 66.0 65.3 65.2 52.9 38.3 43.8 43.9 43.8 42.9 41.1 40.9 36.0 32.4 26.5 21.5 16.3 Avg.
1.11 1.06 1.00 1.09 1.05 1.16 1.11 1.10 1.05 1.08 1.03 1.02 1.05 1.01 1.02 1.09 1.08 1.11 1.03 0.94 0.98 1.00 1.01 0.98 1.03 1.09 1.05 1.09 1.05
C.O.V.
4.7%
Table 2 Comparison of experimental and analytical results for single equal-leg angles. Source
No. of specs
Slenderness ratio
Angle size
Fy (MPa)
Loading condition
Wakabayashi and Nonaka [15] Adluri and Madugula [5]
10 26
20–150 68–188
303
Concentrically loaded
335–411
Concentrically loaded
330, 348 294–323
Hui [12] Wakabayashi and Nonaka [15] Ishida [16]
4 40
95, 155 20–150
90 × 90 × 7 64 × 64 × 9.5 76 × 76 × 4.8 76 × 76 × 12.7 102 × 102 × 7.9 127 × 127 × 9.5 51 × 51 × 6.4 90 × 90 × 7
7
20–100
75 × 75 × 6
58.8–63.9
Mueller and Erzurumlu [17]
14
60, 110, 192
75 × 75 × 6
349–422
Hui [12]
24
95, 125, 155
51 × 51 × 6.4
330, 348
53.3, 100
75 × 75 × 9 75 × 75 × 12 350 × 350 × 35
295–433
Madugula et al. [18]
6
results from experimental research with set-up consistent with this loading conditions were considered. Although considerable experimental results have been reported on the capacity of single angles attached to structural tee sections which were in turn subjected to axial compression, they were not included since the bending of the angle was restrained about the geometrical axis of the angle. The test results used in the verification process included results of 36 concentrically loaded angles [15,5], 89 eccentrically loaded angles [15–17,12], and 6 angles under flexure only [18]. All angles included in the verification study were tested as simply supported on both ends and were not retrained to bend with respect to any axis. The simple support was formed by welding the end of the angle to a rigid plate, which was in turn supported through a ball bearing arrangement similar to that suggested by Galambos [3].
Avg. of Pexp /PFEM
C.O.V. (%)
1.17
9.3
Concentrically loaded Eccentrically loaded causing major, minor and bi-axial bending Eccentrically loaded causing minor axis bending Eccentrically loaded causing bi-axial bending Eccentrically loaded causing major and minor bending
1.05 1.02
6.7 8.4
0.96
3.0
1.00
10.4
1.05
4.0
Pure bending about minor principal axis
0.98
1.7
A summary comparison of analytical and experimental ultimate loads is presented in Table 1 for results obtained from the experimental portion of this research and in Table 2 for results from other research reported in the available literature. It is noted that test results covered a wide range of angle cross-sections, slendernesses, yield stresses and various loading conditions. Overall, numerical results compared well with experimental responses. The ratios of experimental and numerical ultimate load, Pexp /PFEM , for all specimens varied from 0.96 to 1.17 with COVs ranging from 1.7% to 10.4%. In some cases, the initial imperfections and residual stresses for tested angles were not reported in the literature and assumptions as described previously were then made, which is believed to attribute to relatively high Pexp /PFEM ratios in these cases. Typical comparisons of numerical and experimental results [11] are illustrated in Fig. 3(a) for load vs. mid-height deflection curves and in Fig. 3(b) for
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a y
80
x 60 Load (kN)
z
1
40
2
20
-20
-15
Leg 1-Test Leg 2-Test Leg 1-FEM Leg 2-FEM
0 -10 -5 0 5 10 Mid-height deflection (mm)
15
20
b
Fig. 1. Typical finite element mesh and cross-section details of the angle.
Experimental
FEM
Fig. 3. Comparison of analytical and experimental results: (a) load vs. deformation response; (b) failure modes.
Fig. 2. Residual stress distribution used in the model.
failure modes. The figure shows that the finite element model is capable of predicting both the ultimate load capacity and behavior of single angles. Fig. 4 shows the normalized ultimate load versus slenderness parameter, λ, for eccentrically loaded angles under bi-axial bending and concentrically loaded angles, respectively. In general, reasonably good agreement is obtained between the numerical and experimental results. It is reasonable to consider that the finite element model is accurate to provide the ultimate capacity for single angles with various configurations and subjected to various loading conditions.
single angle behavior and capacity. Parameters included in the numerical simulations were the slenderness parameter, λ, applied eccentricity, ex or ey , and angle leg width-to-thickness ratio, b/t. The test population included nine slenderness parameters (λ = 0.27, 0.53, 0.80, 1.06, 1.33, 1.60, 1.86, 2.13, and 2.39), six eccentricities (0, xo /4, xo /2, xo , 3xo /2, 2xo ) where xo is the distance from the shear centre to the centroid of the crosssection measured in the major-axis direction and five leg widthto-thickness ratios (b/t = 5.3, 6.4, 8.0, 10.7, and 16.0). Majority of the study was carried out for an L51 × 51 × 6.4 eccentrically loaded with respect to either principal axis. For study of b/t ratios, cross-sections in L51 × 51 series with varying angle leg thickness were used. The dimensions of angle sections were taken as the nominal values according to CSA S16-01 [2]. An elastic–perfectly plastic material model was assumed for steel with a yield stress of 350 MPa, an elastic modulus of 200 000 MPa, and a Poisson’s ratio of 0.3. 6. Evaluation of AISC 2005 Specification
5. Parametric study
6.1. Angles eccentrically loaded with respect to major principal axis
Subsequent to the verification, the finite element model was used to study the effects of several influential parameters on the
The load–moment interaction (Eq. (1)) as suggested in the AISC Specification 2005 was determined and compared with numerical
Y. Liu, L. Hui / Engineering Structures 32 (2010) 2087–2095
a
2091
a
b
b
Fig. 6. Normalized flexural strength vs. slenderness parameter λ curves.
6.1.1. Axial load capacity (M
Fig. 4. Comparison of analytical and experimental results: (a) bi-axial bending; (b) concentric compression.
Fig. 5. Comparison of normalized load–moment interaction diagram of analytical and AISC Specification 2005 results (major axis bending).
results for five representative slenderness values. Fig. 5 shows the normalized interaction diagram where the load was normalized by the yield strength of the cross-section, Py , and the moment was normalized by the yield moment, My .
(P = 0)
= 0) and pure moment capacity
For pure axial capacities (M = 0), the numerical and AISC results agreed well with each other except in the case of λ = 0.80. Residual stresses and initial imperfections tend to have a larger effect on angles with an intermediate slenderness. It is therefore believed that the discrepancy noted above was due to the difference between the assumed values used in the numerical model and the actual values. For pure moment capacities (P = 0), both numerical and AISC moment capacity decreased with an increase in the slenderness, but the numerical values were consistently greater than the AISC values for all investigated slendernesses. A typical comparison of numerical and AISC flexural capacities for major-axis bending for b/t = 8.0 is illustrated in Fig. 6(a) where normalized flexural strength vs. slenderness parameter λ curves are plotted. Also shown in the figure are the curves obtained from Eq. (2) proposed by Earls [9]. Numerical results and Eq. (2) results were consistently agreeable over the entire investigated range of λ with an average variation less than about 3%. It also shows that the reduction in the flexural capacity as a result of an increase in slenderness was not as significant as indicated by AISC values. Comparing with numerical results, AISC increasingly underestimated the flexural capacity over the region of λ > 1.0 with a maximum variation of 20%. For the case of low slenderness (λ < 0.5) where angles may reach plasticity, AISC proposes 1.5My as the upper limit of the flexural strength whereas both numerical and Earls’ equations suggest that the angle is able to attain higher plastic moment. The underestimation of flexural strength about major axis is believed
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(a) ex < exo .
(c) ex > exo .
(b) ex = exo .
Fig. 7. Numerical von Mises stress distribution on the cross-section.
to be one contributor to the conservatism observed in AISC Specification 2005 load–moment interaction. Fig. 6(b) shows the numerical flexural capacity in comparison with values obtained from Eq. (2) for other b/t ratios. It is noted that Earls suggested the leg slenderness ratio b/t not less than 8 for the application of Eq. (2). The comparison, however, shows that even for the values of b/t = 5.3 and 6.4, the angle flexural capacities have a reasonably good agreement with those obtained from Eq. (2). 6.1.2. Axial load and moment interaction For combined axial load and moment cases, numerical results, in general, provided higher estimates of capacity in comparison with AISC results. The numerical load–moment interaction demonstrated two distinctive regions where, in one region the axial load remained almost constant as the moment increased, and in the other region the axial load decreased as the moment increased. The first region became increasingly larger as the slenderness ratio increased. It appears that for a given slenderness, there existed a critical eccentricity below which the change of the ultimate load with an increase in eccentricity was marginal to almost unnoticeable. This indicates that an eccentrically loaded angle can reach its concentric compressive capacity as long as the applied eccentricity was within the critical eccentricity. Defined as exo in this paper, the critical eccentricity assumed an increasingly greater value with increased slenderness. Similar findings noted by Schafer [19] suggested that the presence of moment does not necessarily result in a reduction in the axial capacity of a singly symmetrical member as the interaction equation would suggest. In some cases, it actually benefits the axial behavior of the member. As shown in Fig. 5, the underestimation by AISC Specification 2005 was most pronounced in the region of critical eccentricities. For example, for λ = 1.33, 1.86, and 2.39, AISC values in the region of critical eccentricities can be as low as 70%, 64% and 57% of the numerical results, respectively. On the other hand, there were instances where AISC Specification 2005 overestimated angle compressive capacity. For example, in the region of M > 0.8My for λ = 0.80, the AISC overestimated the capacity by a maximum amount of 20%. Fig. 7 demonstrates the numerical von Mises stress distribution on the cross-section at mid-height of a specimen when the applied eccentricity was less than, equal to, and greater than the critical eccentricity. Noting that the red gradation indicates maximum stress, the figure shows that as the applied eccentricity varied from less to greater than the critical eccentricity, the location where maximum stress occurred shifted from the heel to the toes of the angle. As the applied eccentricity equaled exo , the maximum stress occurred simultaneously at both the heel and the toe. Numerical results also showed that angles subjected to eccentric compression resulting in major axis bending failed in torsional–flexural buckling
Table 3 Critical eccentricity ratio for various cross-sections. b/t
5.3 6.4 8.0 10.7 16.0
Critical eccentricity ratio, mo = exo /xo
λ = 0.27
0.80
1.07
1.33
1.86
2.40
0.01 0.01 0.01 0.01 0.01
0.18 0.19 0.20 0.21 0.21
0.42 0.44 0.49 0.51 0.53
0.58 0.61 0.66 0.71 0.77
1.25 1.34 1.48 1.59 1.75
2.45 2.79 3.07 3.32 3.58
mode. The flexural deflection and twist of the cross-section reached maximum at the mid-height of the angle and resulted in re-orientation of the principal axes. To satisfy equilibrium on the deformed shape, additional stresses due to the twist of the crosssection resulted at both the heel and toes of the cross-section. These additional stresses either add to or subtract from the stresses caused by axial and flexural deformations at the heel and toe locations. The critical eccentricity is that at which the combined stresses at both the heel and the toe simultaneously reach a maximum value. The value of this critical eccentricity is dependent on the magnitude of flexural deflection and twist of the angle. 6.1.3. Critical eccentricity The critical eccentricity phenomenon was further studied for various angle leg width-to-thickness ratios, b/t. Five crosssections, L51 × 51 × 3.2, L51 × 51 × 4.8, L51 × 51 × 6.4, L51 × 51 × 7.9, and L51 × 51 × 9.5 with b/t ratio ranging from 5.3 to 16.0 were used in this study. A term was introduced as the critical eccentricity ratio mo , which is defined as the ratio of critical eccentricity, exo and xo . Results of mo for various b/t ratios are listed in Table 3 and mo vs. λ curves are plotted in Fig. 8. It shows that for all b/t ratios, mo and thus exo increased with an increase in the slenderness. However, for a given slenderness, the variation of exo with b/t ratios was more complex. For λ < 1.0, exo remained almost constant for all b/t ratios. For λ > 1.0, the variation in exo for different b/t ratios became more pronounced and an increase in b/t ratio resulted in an increase in exo values. It has been established that the b/t ratio affected the flexural deflection and the twist of the angle failing in elastic flexural torsional buckling [20], which in turn would affect the critical eccentricity values. As the angle became stocky in the length direction and hence λ decreased, it seemed that the b/t ratio had a diminishing effect on the flexural deflection and twist and thus the critical eccentricity values. A lower-bound curve showing relationship of mo and λ is also plotted in the figure. Obtained using curve-fitting technique, this curve is expressed as mo = 0.3λ2.26 . This equation was used in later sections in an effort to develop a general load–moment interaction equation.
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a
b Fig. 8. Critical eccentricity ratio vs. slenderness parameter for various b/t ratios.
6.2. Angles eccentrically loaded with respect to minor principal axis Numerical load–moment interaction curves for combined axial load and minor axis bending of a 51 × 51 × 6.4 angle with various λ values are compared with corresponding curves determined using AISC Specification 2005 in Fig. 9(a) for the case of toes in compression and in 9(b) for the heel in compression. The numerical interaction curves for the minor axis bending case did not show evident critical eccentricity phenomenon as in the case of major axis bending and the ultimate load decreased when the moment and thus the applied eccentricity increased. The general trends and associated load–moment interaction values are nearly the same for cases of toes in compression and heel in compression. The comparison of the numerical and AISC results shows that, while the AISC is generally conservative, the correlation between two curves was reasonably good especially for angles with λ values greater than 0.8. It seems that the discrepancy in flexural capacity between numerical and AISC results may be attributed mainly to the conservatism of AISC Specification 2005, especially for case with λ < 0.8. Fig. 10 further demonstrates the normalized flexural capacities of angles with b/t = 8.00 about minor-axis bending with either toes or heel in compression. It is noted that AISC Specification 2005 specifies the flexural strength for minor-axis bending as a shape factor of 1.5 applied to the yield moment regardless of the slenderness although some research [21] showed that the shape factor for L51 × 51 × 6.4 with respect to minor principal axis can reach 1.80. Numerical results show that this limit is markedly conservative for angles L51 × 51 × 6.4 in practical slenderness range and is reasonable when slenderness exceeds 2.0. Since AISC Specification 2005 limits the flexural capacity to be 1.5My for all the slenderness ratios and the application of eccentricity causing either principal axis bending or geometric axis bending, it is felt that 1.5My represents a general lower bound value of angle flexure capacity which can be expected in practice [22]. 7. Proposal of the new equation The previous sections have demonstrated the inconsistencies and overall conservatism inherent in the interaction equations for single angle design suggested in the current AISC Specification 2005. The possible sources of this conservatism were also identified and discussed. In an effort to provide a better estimate of the capacity of single angle beam–columns, a new interaction equation (Eq. (3)) for combined compression and principal axis bending is
Fig. 9. Comparison of normalized load–moment interaction diagram of analytical and AISC Specification 2005 results (minor axis bending): (a) toes in compression; (b) heel in compression.
proposed based on approximately 2000 finite element model results and curve-fitting technique. While the equation maintained the format of interaction between axial load and moment, two key differences are noted including the incorporation of the critical eccentricity concept and Earls’ flexural capacity equation (Eq. (2)) for the determination of major axis flexural capacity, Mnx . Pr
φ c Pn
+
Pr (ex )m
φb Mnx
+
8 Pr ey 9 φb Mny
= 1.
(3)
The critical eccentricity concept is incorporated in the term (ex )m . Defined as virtual eccentricity, it can be evaluated as (ex )m = n 0 ex − exo
(ex ≤ exo ) (ex > exo ) , where exo
= mo xo and mo is the critical ec-
centricity ratio expressed as mo = 0.3λ2.26 . The determination of Pn and Mny are kept to be the same as in AISC Specification 2005. Fig. 11 shows a comparison of load–moment interaction curves generated using the proposed equation (3) and the finite element model for major axis bending. As can be seen, the interaction curves obtained from Eq. (3) are in a good agreement but on the conservative side compared with finite element curves. Comparing with AISC Specification 2005, the proposed equation improves the load carrying capacity predictions, in particular, around the critical eccentricity region. 7.1. Evaluation of the proposed equation using available experimental database The experimental database used to verify the finite element model was used to evaluate the performance of the proposed equation. Resistance factors for compression and flexure were
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a
Table 4 Summary comparison of test-to-prediction ratios. Source
AISC Specification 2005
Proposed equation (3)
Avg.
C.O.V. (%)
Avg.
C.O.V. (%)
Wakabayashi and Nonaka [15] Ishida [16] Mueller and Erzurumlu [17] Hui [12]
1.18 1.07 1.10 1.23
11.7 7.8 11.9 11.4
1.12 1.07 1.04 1.08
7.7 7.8 11.6 7.4
Total
1.17
14.0
1.09
9.5
Table 5 Reliability index for studied angles.
b
All experimental data
Experimental data (major axis bending only)
Rm /Rn AISC Specification 2005 1.17 Eq. (3) 1.09
Vr (%)
β
Rm /Rn
Vr (%)
β
14.0 9.5
3.4 3.7
1.26 1.07
19.5 7.5
3.1 4.2
This indicates an average increase in the predicted load carrying capacity of 8.0%, and less scattered predicted values by using the proposed equation. It is noted that in the case of comparison with results by Ishida [16], the test-to-prediction ratios and associated COVs were identical obtained using either the proposed equation or AISC Specification 2005. This is attributed to the fact that for angles under eccentric load causing minor axis bending only, equations from both procedures were essentially the same. However, in the case of angles under eccentric loading causing major axis bending, the capacities predicted using AISC Specification 2005 were usually significantly lower than the test results and results from the proposed equation. 7.2. Reliability index, β
Fig. 10. Normalized flexural strength vs. slenderness parameter λ curves for minor axis bending of angles with b/t = 8.00: (a) toes in compression; (b) heel in compression.
The current structural design philosophy adopted by either AISC Specification 2005 or CSA S16 is based on a first-order probabilistic design procedure [23]. The single angle design equations in AISC Specification 2005 were derived from beam–column theory and adjusted using experimental results to achieve an acceptable level of complexity. The load factors and resistance factors are arrived at by using a uniform ‘‘probability of failure’’ level for the full range of basic variables of all member types. The probability of failure, i.e. the level of safety, is defined by reliability index, β . To assess the safety margin of the AISC Specification 2005 procedure and the proposed equation for single angle design, the reliability indices β for both cases were examined using the following equation [23]:
φ=
Fig. 11. Normalized load–moment interaction diagram generated using the proposed equation and numerical results (major axis bending).
taken as 1.0 to facilitate the comparison between the equation and the raw experimental results. The test-to-prediction ratios based on AISC Specification 2005 and proposed equation (3) are summarized in Table 4 for experimental investigations conducted by various researchers. The average test-to-prediction ratios for AISC Specification 2005 and the proposed equation were found to be 1.17 and 1.09 with COVs equal to 14.0% and 9.5%, respectively.
Rm −αβ Vr e Rn
(4)
where φ is the resistance factor, Rm is the mean resistance obtained from experimental results, Rn is the averaged nominal resistance obtained from design equations, α is a constant taken as 0.55 as proposed by Galambos and Ravindra [24], and Vr is the coefficient of variation of Rm /Rn . Using experimental results as listed in Table 4 and resistance factor φ of 0.9, the reliability index β was calculated for (1) all experimental data from different sources, and (2) data concerning eccentric load causing major principal axis bending only. The results are summarized in Table 5. The table shows that the proposed equation provided results closer to the experimental results and improved coefficients of variation. The reliability indices based on either AISC Specification 2005 or proposed equations are greater than 3.0, which is acceptable for limit states design. It is however noted that the AISC Specification 2005 provides lower reliability indices for both cases of data sources than those of proposed equation.
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8. Conclusions
References
A finite element study was conducted to investigate the effects of several parameters on the load carrying capacity of equalleg single angles eccentrically loaded resulting in either major or minor principal axis bending. The performance of design equations suggested in AISC Specification 2005 and proposed in this paper was evaluated using both experimental and numerical results. The essential conclusions are listed as follows. For eccentric compression resulting in major axis bending, numerical results suggest that a critical eccentricity seemed to exist within which when the eccentricity was applied, the equal-leg single angles attained almost their concentric axial capacity. Values of the critical eccentricity were associated with slenderness and leg width-to-thickness ratio of the angle. A lower bound expression was proposed for the calculation of the critical eccentricity. For eccentric compression resulting in minor axis bending, the effects of all studied parameters on angle ultimate capacities showed negligible variations for cases of either the angle toes or the heel in compression. In contrast with the major axis bending, the critical eccentricity was not evident and the ultimate capacity consistently decreased with an increase in the eccentricity. Compared with the experimental and numerical results, AISC Specification 2005, in general, markedly underestimated the ultimate load capacity of single angles under eccentric loading causing major axis bending, especially in the region of critical eccentricity. Results also identified a few instances where AISC overestimated ultimate capacities of angles in the case of major axis bending. For eccentric loading causing minor axis bending, AISC Specification 2005 was shown to be markedly conservative for angles with a low λ value (λ = 0.27) but provide lower-bound and reasonably accurate predictions for angles with relatively large slenderness. A new interaction equation incorporating the concept of critical eccentricity and Earls’ flexural capacity equations was proposed and the comparison with the AISC results indicated that the proposed equation provided improved capacity estimates and acceptable reliability index for equal-leg angles with a wide range of parameters.
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Acknowledgements The authors wish to recognize the contribution of financial assistance by the Steel Structures Educational Foundation (SSEF) as well as in kind assistance from Marid Industries Ltd. for the donation of specimens.