DESIGN OF ECCENTRICALLY LOADED SINGLE-ANGLE COMPRESSION MEMBERS

INTRODUCTION

Single-angle sections are often provided as web members in roof trusses, bracings in buildings and transmission line towers, and cross-frame members in plate girder bridges. They are often connected at the ends through one leg only, as shown in Fig.1. Due to this, these angle compression members are subjected to eccentric loading (Subramanian, 2016).

?Although equal angle sections may appear simple, their buckling behavior under compression is complex as they can be singly / mono-symmetric in which the centroidal (x-x, y-y) and principal axes (u-u, v-v) do not coincide (See Fig. 2). Also, the shear center ‘SC’ does not coincide with centroid ‘C’. The unsymmetric cross-section and how these angles are connected to the adjacent members are two important parameters, which make the analysis and design of these members complex. Due to these unique features, these members can’t be considered as pure axially loaded members. Hence, the designer may have to consider additional stresses to produce safe designs. Furthermore, the designer has to consider different failure modes such as local buckling, flexural buckling, and torsional-flexural buckling of the angles. Another difficult aspect for single angles is the determination of the effective slenderness ratio. The effective length factor can often be estimated about the geometric axes of the angle. However, as they are not the principal axes for the angle, the determination of the governing slenderness ratio is also a bit complex. Due to the difference in effective lengths of the individual geometric axes, the radius of gyration no longer represents the critical value (Bashar and Amanat, 2021). All these aspects make the analysis and design of steel angles rather complicated. Due to these difficulties, simplified assumptions are often made to design eccentrically loaded single-angle compression members. In addition, the design expressions provided in different international codes of practice differ widely in both assumptions and design procedures.

Design provisions in American, Euro, and Indian codes

Only the provisions of American, Euro, and Indian codes are provided here. For the design provisions of other countries and comparison of these provisions reference should be made to Bashar and Amanat, 2021.

AISC 360:22 Specification for Structural Steel Buildings[3]

According to AISC Specifications, angles that are individual members or web members of planer trusses with adjacent web members attached to the same side of the gusset plate or chord, the slenderness ratio for double bolted equal-leg angles or unequal-leg angles connected through the longer leg is given by Eqs. (2a) or (2b) as applicable. AISC specification does not have any explicit provision for single bolted angles. Considering that angles with single bolts at ends do not have any significant restraint against rotation, it is assumed that K = 1.0 for such cases. See Figure for these provisions.

Eurocode 3: Design of Steel Structures, EN 1993-1-1:2005[4]

???????? The provisions of Eurocode for compression capacity are similar to those of IS-800 especially the form of expression for the column curve. However, in specifying the effective length of double-bolted angle members, Eurocode provides different expressions for the slenderness parameter, λ. It also explicitly specifies that the effective length factor for single bolted angles shall be unity. See Figure for these provisions.

Design provisions of the Indian Code, IS 800 – 2007 [5]

?As per clause 7.5.1.2, the flexural-torsional buckling (FTB) strength of a single angle loaded eccentrically (through one leg) is to be evaluated considering the equivalent slenderness ratio, λe. The provisions contained in the code were based on the numerical work carried out by Sambasiva Rao et al. [2] at IIT Madras, India considering both bolted and welded end connections.

see Equations 8 to 14 in the top Figure.

????????Where, l = center – center length of the supporting member,???? rvv = radius of gyration about minor-principal axis, ?? b1, b2 = width of connected and outstanding legs respectively, t = thickness of leg of angle section, ?? ε = yield stress ratio = sq.root (250/fy), ??fy = yield stress or strength, ?? E = modulus of elasticity = 200 GPa = 2.0 x 105 MPa

?The imperfection factor α is to be taken as 0.49 as buckling class ‘c’ has to be considered for angle sections (as per clause 7. 1. 2. 2.)

?As per the code [5], the b / t and (b1+b2) / t ratios of the angle sections should not exceed 15.7ε and 25ε, to avoid local buckling. The constants k1, k2, and k3 account for different end connection fixities (as mentioned in Table 1) which were obtained by carrying out a multivariate regression analysis of the obtained data from the reported numerical investigation and available test results in the literature [6]. The bolted connection was classified into two or more bolts and single bolt cases (Figure 2 (a) and (b) respectively). In general, providing two or more bolts offers greater rotational restraint over the single bolt connection, whether in-plane of gusset or out-of-plane. Though the welded connection (Figure 2 (c)) was not mentioned separately, it was usually considered equivalent to two bolts.

Design provisions of IS 800-2007 Amendment 2 [7]

As per Amendment 2 to IS 800 – 2007, the single angles loaded eccentrically, and the combined effect of both flexural torsional buckling and bending has to be accounted for in the design. The design compressive strength in such cases can be determined using the following method, instead of adopting a more precise second-order analysis and design for combined bending and compression.

Where? λ? ?is the same as expressed in Equation (10)

laa = center-to-center length of lateral support preventing translation of ?member perpendicular to a-a axis, ?raa = radius of gyration of the angle member about the a-a axis

The values of ?k1, k2, and k3, as per the IS 800 Amendment 2 are given in Table 2

?The imperfection factor α is to be taken as 0.34, as buckling class ‘b’ has to be considered for angles.

Upon substituting the obtained in Equation (15) and thereafter substituting in place of fcd in Equation (14), the design strength Pd of the angle section is obtained.

Major changes incorporated in Amendment 2

Upon careful examination of the design provisions of both IS 800 – 2007 [5] and Amendment 2 to IS 800-2007 [7], the following major changes were observed:

1) Amendment 2 considers both flexural-torsional buckling FTB and bending effects in the latest design provisions, contrary to FTB alone considered earlier.

2)????? In the design calculations, the radius of gyration of the minor-principal axis is no longer required and the emphasis is on the centroidal axis parallel to the connected leg or the plane of the end gusset (designated as a-a axis). That is, considering out-of-plane buckling (buckling in the direction perpendicular to the plane of the gusset or structural system). Thus, it presumes much greater rotational restraint offered by the restraining member against in-plane buckling i.e., buckling of the angle about the axis perpendicular to the plane of the gusset [6,8]. However, the latest tests [9] reported that in-plane buckling and combined in-plane and out-of-plane buckling are also possible. Based on these experiments and numerical analysis a new set of equations to determine the rotational stiffness for both in-plane and out-of-plane buckling have been proposed [9-13].

3)????? The buckling class has been upgraded from curve ‘c’ to ‘b’, thereby resulting in a lower imperfection factor of 0.34 (compared to that of 0.49 adopted previously, corresponding to buckling class ‘c’), as adopted in EN 1993-1-1: EC3 [4].

4)????? A set of new values has been presented for constants k1, k2, and k3.

5)????? A new modification factor Kf has been introduced, which accounts for the influence of end connection fixity on the slenderness ratios λaa and λ?.

EXAMPLE

?Consider the example 9.16 of the book(Subramanian, 2016) with Angle 75×75×6

From IS 800:2021,? b1 = b2 = 75 mm, t = 6 mm, A = 866 mm2, ryy = rzz =23 mm

Length of member L =2.5 m

?Check for slenderness

ε = 1, b1/t = b2/t = 75/6 =12.5 < 15ε, (b1+b2) / t =150/6 =25= 25. Hence the section is not slender

Assuming that the gusset plate is parallel to the Z-axis,

L/rzz = 2500/23 =108.7

λaa = 108.7/88.86 = 1.22

λφ = ?2 x 75/(2x6)/88.86 =??0.141??????

(a)??? Assuming 2 bolts are provided at the end-fixed condition,?????

From Table 2, k1 = 0.798, k2 = 0.563, k3 = -2.072?

Kf = k1 + k2 λaa + k3 λ? ????= 0.798 +0.563×1.22-2.072×0.141 =1.193

The buckling class is b, hence, α = 0.34??

??? ? = 0.5 [ 1 + α (λaa – 0.2) + ? = 0.5 [1+ 0.34(1.22- 0.2) +1.222 = 1.418

Stress reduction factor, χ = ?1/[1.418 +?(sq. root (1.418^2 -1.22^2))] =? 0.467

??????? ?fcde = 1.193 x 0.467 x 250/1.1 = 126.62 N/mm2

?Design compressive strength = 126.62 x 866/1000 = ?109.65 kN

From the book, using the equations given in IS 800:2007, it was only 70.497 kN????????????????????????????

(b)?? ?If the ends are considered hinged,

From Table 2, k1 = 0.401, k2 = 0.420, k3 = -1.040?

Kf = k1 + k2 λaa + k3 λ? ????= 0.401 +0.420×1.22-1.040×0.141 =0.767

? = 0.5 [ 1 + α (λaa – 0.2) + ? = 0.5 [1+ 0.34(1.22- 0.2) +1.222 = 1.418

Stress reduction factor, χ = ???1/[1.418 +?(sq. root (1.418^2 -1.22^2))] =0.467

??????? ?fcde = 0.467 x 0.767 x 250 / 1.1 = 86.41 N/mm2

?Design compressive strength = 86.41x 866/1000= ?70.5 kN????????????????????????????

From the book, using the equations given in IS 800:2007, it was only 36 kN????????????????????????????

It is seen that the values obtained by provisions in Amendment 2 to IS 800? for fixed-end conditions result in a 55.5 % increase in strength over IS 800:2007 provisions and for the hinged condition the increase is 95.8%. Recently, Vivek, et al. (2024) have shown that such an increase in strength is obtained for almost all the cross-sections of angle sections, using the provisions of Amendment 2 of IS 800 over IS 800:2007 provisions. They also found that the nominal strengths obtained as per the latest provisions [7] did not correlate well with the available test strengths (also validated through numerical analysis) [9, 16] considered by them.?

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References

1.???????? Subramanian, N. (2016) Design of Steel Structures – Limit States Method, 2nd ed.; Oxford University Press, New Delhi, India.

2.???????? Bashar, I., and Amanat, K. M. (2021) Comparison of codes for axial compression capacity of eccentrically loaded single angles, Journal of Constructional Steel Research, 185, 106829. ??https://doi.org/10.1016/j.jcsr.2021.106829

3.???????? ANSI/AISC 360:22, Specification for Structural Steel Buildings, American Institute for Steel Construction (AISC), Chicago, 2016.

4.???????? EN 1993-1-1: EC3 (2005). Design of Steel Structures – Part 1-1: General rules and rules for buildings. European Committee for Standardization, Brussels, Europe.

5.???????? IS 800:2007. Indian Standard General Construction in Steel – Code of Practice, 3rd Revision, Bureau of Indian Standards, New Delhi, India.

6.???????? Sambasiva Rao, S.; Satish Kumar, S. R.; Kalyanaraman, V. (2003) Numerical study on eccentrically loaded hot rolled steel single angle struts. In Topping, B. H. V. (Editor), Proceedings of the Ninth International Conference on Civil and Structural Engineering Computing, Civil-Comp Press, UK, doi:10.4203/ccp.77.44

7.???????? IS 800 -Amendment No. 2 (2024). Indian Standard General Construction in Steel – Code of Practice, 3rd Revision, Bureau of Indian Standards, New Delhi, India.?

8.???????? Woolcock, S. T.; Kitipornchai, S. (1987) Design of single angle web struts in trusses. Journal of Structural Engineering, ASCE Vol. 112, pp.1327-1345. https://doi.org/10.1061/(ASCE)0733-9445(1986)112:6(1327)

9.???????? Kettler, M.; Lichtl. G.; Unterweger, H. (2019) Experimental tests on bolted steel angles in compression with varying end supports, Journal of Constructional Steel Research, Vol. 155, pp. 301-315. https://doi.org/10.1016/j.jcsr.2018.12.030

10.????? Kettler, M.; Unterweger, H.; Zauchner, P. (2022) Design model for the compressive strength of angle members including welded end-joints. Thin-Walled Structures, Vol. 175, 109250. https://doi.org/10.1016/j.tws.2022.109250

11.????? Kettler, M.; Unterweger, H.; Harringer, T. (2019) Appropriate spring stiffness models for the end supports of bolted angle compression members. Steel Construction, pp. 291-298. 10.1002/stco.201900028

12.????? Kettler, M.; Taras, A.; Unterweger, H. (2017) Member capacity of bolted steel angles in compression – Influence of realistic end supports, Journal of Constructional Steel Research, Vol. 130, pp. 22-35. https://dx.doi.org/10.1016/j.jcsr.2016.11.021

13.????? Kettler, M.; Unterweger, H.; Zauchner, P. (2021) Design model for bolted angle members in compression including joint stiffness. Journal of Constructional Steel Research, Vol. 184, 106778. https://doi.org/10.1016/j.jcsr.2021.106778?

14.????? IS 808:2021. Hot Rolled Steel Beam, Column, Channel, and Angle Sections – Dimensions and Properties. 4th Revision, Bureau of Indian Standards, New Delhi, India.?

15.????? Vivek, K.S., Adil Dar, M., and Subramanian, N. (2024), "Efficacy of IS Code Provisions for Design of Eccentrically Loaded Single Angle Compression Members", Buildings, Vol. 4, 14, 2990. https:// doi.org/10.3390/buildings14092990

16.?????? Bhilawe, J. Study of equal angle subjected to compression for bolted end connection. J. Inst. Eng. India Ser. A 2017, 99, 123-132. https://doi.org/10.1007/s40030-017-0245-8

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