American Journal o Engineering Research (AJER) 014 American Journal o Engineering Research (AJER) e-issn : 30-0847 p-issn : 30-0936 Volume-03, Issue-0, pp-09-16 www.ajer.org Research aper Open Access Critical Strength o steel Girder Web ratibha M Alandkar 1, Sandeep M Shiekar, Mukund R Shiekar 3 1 Assistant roessor, Sinhgad College o Engineering, une, Maharashtra, India, roessor, hd, 3 roessor, hd, Abstract: - When a member is subjected to combined action o bending moment, shear and axial orce, bending moment & axial orce is assumed to resist b whole section and shear is resisted b web onl. In such case web shall be designed or combined shear and axial orce. The present stud determines the strength o web o steel girder under the action o pure shear, pure axial orce and combination o it. The classical plate buckling theor is alread established to determine the itical buckling strength o web panel o the girder under pure axial compression and pure shear. Using Von Mises ield iteria interaction between axial compression and shear is presented. It has been observed that limit strength o web in combined axial compression and shear is governed b buckling o girder rather buckling o web panel. Kewords: - buckling, ielding, limit strength, compression, shear, interaction I. INTRODUCTION Nearl all members in a structure are subjected to bending moment, shear and axial load. articularl, while designing steel structural member, interactions o these actions need to be considered at limit state. Man literatures, codal provisions are available on interaction o Moment & axial compression (M-) as well as or moment & shear (M-V). As per the revised Indian codal provisions, in case o sections with web susceptible to shear buckling beore ielding, strength o member shall be calculated using one o the considerations as 1 The bending moment and axial orce assumed to be resisted b langes and onl shear is resisted b web. The bending moment and axial orce acting on the section ma be assumed to be resisted b the whole section. In such case web shall be designed or combined action o shear and normal stresses. It is observed that, the provision o interaction or shear and axial compression (V-) is less attended. The present stud determines the strength o web o steel girder under the action o pure shear, pure axial orce and combination o it. For the design o web plate two checks need to be applied, check against buckling and check against local ielding. The classical plate buckling theor is applied to determine the itical buckling strength o web o the girder and Von Mises ield iteria is also used to present interaction between axial compression and shear. II. BUCKLING OF WEB Web o rolled steel I section or built up plate girder behaves as a plate subjected to inplane uniorm axial compression and shear as shown in Fig 1. Buckling strength o web is aected b web panel dimensions, length to depth ratio called as aspect ratio (a/b), depth to thickness ratio (d/t) and boundar conditions. Depending upon whether the web panel is rom rolled steel section or rom built up plate girder, boundar conditions var. Web panel o rolled steel section is taken as simpl supported alongside b and builtin or ixed at the junction o web and lange abbreviated as SFSF, whereas web panel o plate girder is taken as simpl supported along all edges abbreviated as SSSS. Elastic buckling strength o such plates is known b classical plate theories. Classical plate theories are established b various researchers [1,3,4] or elastic buckling o plate under inplane action o pure compression, pure shear and combination o shear and axial compression. w w w. a j e r. o r g age 09
American Journal o Engineering Research (AJER) 014 Nx a x b Nx Nx x = 0 & x = a --- Simpl Supported = 0 & = b --- Simpl Supported or built in Figure1: Web panel subjected to inplane orces.1 Buckling Strength o Web: Classical plate theor approach.1.1 Web anel under Uniorm Compression: When a web panel subjected to uniorm compression (Nx) it will buckle in hal waves along its longitudinal direction. B graduall ineasing Nx and using equilibrium o the compressed plate, itical load at buckling or plate is given b D N x, ka b (1) Where, D is the lexural rigidit o plate. The buckling coeicient ka or all edges simpl supported, i.e. SSSS plate is given as [1], mb a k a a mb () Where, m is the number o hal waves o buckling. The buckling coeicient as per aspect ratio and transition o the buckling modes can be seen rom Fig. Buckling Coeicient, k a 18 16 14 1 10 8 6 4 0 m=1 m= m=3 0 0.5 0.5 0.75 1 1.5 1.5 1.75.5.5.75 3 3.5 Aspect Ratio (a/b) Figure : Buckling Coeicient ka as per aspect ratio or SSSS Condition Rudolph Szilard [3] presented the buckling coeicient ka or SFSF plate 1 16 a b k a 3m 8 3 m b a (3) The buckling coeicient or this condition as per aspect ratio and transition o the buckling modes can be seen rom Fig. 3 w w w. a j e r. o r g age 10
American Journal o Engineering Research (AJER) 014 k a 10 9.5 9 8.5 8 7.5 7 6.5 6 0 0.9 1.8.7 3.6 4.5 5.4 6.3 7. 8.1 9 9.9 10.8 11.7 1.6 13.5 k, m=1 k, m= k, m=3 k, m=4 k, m=5 k, m=6 k, m=7 k, m=8 k, m=9 a/b k, m=10 Figure 3: Buckling Coeicient ka as per aspect ratio or SFSF Condition Considering t as the thickness o plate, the itical value o the compressive stress is N t x, k a E t 1 1 b (4).1. Web anel under Uniorm Shear: When a plate is subjected to the action o shearing orces Nx uniorml distributed along the edges, the itical value o shearing stress, τ, at which buckling o the panel occurs is determined using the energ method. [1, 4] Critical shear stress, τ, at buckling is presented in equation (5) k v D bt (5) Where, kv is the shear buckling coeicient and its value with respect to aspect ratio are given in Table 1. [1] Table 1: Shear Buckling Coeicient a/b 1.0 1. 1.4 1.5 1.6 1.8.0.5 3.0 4.0 K v, SSSS plate 9.34 8.0 7.34 7.1 7.0 6.8 6.6 6.1 5.9 5.7 - K v, SFSF plate 1.8 - - 11.1 - - 10.1 9.81 9.61-8.99.1.3 Web anel under combined uniaxial compression and Shear: The interaction equation or determination o buckling strength under combined compression and shear using above plate theories is presented b S Bulson [4] as under x x, 1 III. ANALYTICAL VERIFICATION Veriication o results b numerical analsis using ANSYS is carried out. Based on the boundar conditions, two plate models are analzed using ANSYS or various (a/b) ratios under action o pure compression and pure shear sepratel. The boundar conditions considered are SSSS & SFSF. Following is the data considered or analsis in ANSYS b = 1. m, ν = 0.3, E = 10 Ga, t = 0.01 m 3.1 Web anel under Uniorm Compression erorming Eigen buckling solution and using element as SHELL93, the plate is checked or buckling resistance in ANSYS. For buckling o SSSS plate under pure axial compression, or minimum buckling load, buckling (6) w w w. a j e r. o r g age 11
American Journal o Engineering Research (AJER) 014 coeicient, ka is giving close relation with classical solution as shown in Table. The results o SFSF plate are tabulated in Table 3. 3. Web anel under ure Shear Similarl, using SHELL63, plate is checked or shear buckling resistance b ANSYS. The results or SSSS condition are tabulated in Table 4. In case o web o rolled steel section i.e. plate o SFSF condition, the results o classical plate theor are directl adopted. It is observed that or larger (a/b) ratio, buckling coeicient kv is almost constant to a value around 9. In practice ratio o (a/b) or rolled steel section is around 10, hence assumed value o kv = 9.00 is reasonabl acceptable. Table : Buckling Coeicients k a or SSSS Condition (ANSYS) a/b K a [1] Nx, [1] N/m Nx, ( ANSYS ) N/m % error K a ANSYS 1 4 75917 7510-0.94 3.96749 1.1 4.04 76676.17 75911-1.01 3.999684 1. 4.13 78384.3 778-0.7 4.100373 1.3 4.8 8131.19 80639-0.73 4.48798 1.4 4.47 84837.4 8435-0.71 4.43868 1.41 4.49 8516.83 84633-0.69 4.45939 Table 3: Buckling Coeicients k a or SFSF Condition (ANSYS) a/b K a [1] Nx, [1] N/m Nx, (ANSYS) N/m % error K a ANSYS 0.4 9.44 179164.1 169413 5.44 8.93 0.5 7.69 145950.4 13543 7.34 7.13 0.6 7.05 133803.7 1084 8.76 6.43 0.7 7 13854.7 119808 9.8 6.31 0.8 7.9 138358.7 14189 10.4 6.54 0.9 7.83 148607.5 133160 10.39 7.0 1 7.69 145950.4 145370 0.40 7.66 - - 1698-8.59 3 - - 118931-6.7 8 - - 1686-6.68 10 - - 14987-6.59 11 - - 14683-6.57 1 - - 19567-6.83 Table 4: Buckling Coeicients k v or SSSS Condition (ANSYS) a/b k v [1] Nx, [1] N/m Nx, (ANSYS) N/m % error k v ANSYS 1 9.34 131064 1.47E+06-19.0478 11.11906 1. 8 1054445 1059000-0.43199 8.034559 1.4 7.34 967453. 966471 0.10155 7.33548 1.5 7.1 935819.9 955360 -.0880 7.485 1.6 7 9639.3 939040-1.77759 7.14431 IV. YIELDING OF WEB Distortion energ theor o ailure arrives at expression or equivalent stress (σ e ), which is also reerred to as Von Mises stress which considers ailure b ielding. A ield iterion deines the limit o elastic behavior under an possible combination o stresses at a point in a given material. Equation based on the Von Mises ield iterion is expressed as w w w. a j e r. o r g age 1
American Journal o Engineering Research (AJER) 014 1 3 1 3 e (7) Where, σ 1, σ, σ 3 are the principal stresses and is the ield stress. Considering onl σ 1, σ the principal stresses acting at a point in the web panel and urther web panel under uniaxial compressive stress σ x and shear stress τ x, above expression (7) reduces knowing σ 3 = 0 1 1 In case o D elastic sstem, σ 1 and σ can be expressed in terms o σ x & τ x. x 3 x Hence, Converting this relation in nondimensional orm, equation (8) is arrived as, x x 3 1 V. DESIGN CONSIDERATIONS A conclusion section must be included and should indicate clearl the advantages, limitations, and possible applications o the paper. Although a conclusion ma review the main points o the paper, do not replicate the abstract as the conclusion. A conclusion might elaborate on the importance o the work or suggest applications and extentions. (10) 5.1 Interaction equation presented b Bulson [4], as per equation (6) is in the orm o stress. Modiing the equation (6) in nondimensional orm o compression & shear capacities, equation (9) is arrived, w, w V V 1 Where, w and V are the actual actions in the web &,w compression and shear respectivel or the web panel. w w w. a j e r. o r g age 13 (8) (9) and V are buckling capacities in axial 5. Interaction equation based on Von Mises ield iterion deines the limit o elastic behavior under an possible combination o stresses at a point in a given material. Thus the interaction gives the section capacit at a point. Modiing the equation (8) in nondimensional orm o compression and shear capacities, equation (10) is arrived, N w V V 1 dw d (10) Where w and V are the actual actions and Ndw & Vdw are the section capacities o the web in axial compression & shear respectivel. Section capacit or section strength as governed b material ailure b ielding or individual actions as per IS 800:007 are 1. Member subjected to pure axial compression N d A g / mo Where, Ag is the gross area o oss section and γmo is the partial saet actor or material.. Member subjected to pure Shear V A 3 d v mo Where, Av is the area o the web = b.t 5.3 Indian Standard Codal provisions / Limit state o strength in axial compression is given as under dw A e cd Where, Ae is the eective oss sectional area o the member and cd is the stress which accounts or the overall member buckling. In the case o girder subjected to axial compression, overall member buckling stress cd is based on governing slenderness ratio o the member and relevant buckling class o the section. Using this stress buckling capacit o web is evaluated as
American Journal o Engineering Research (AJER) 014 dw A v cd Where, Av is the area o the web = b.t Thus, interaction equation considering overall buckling o web is w dw V V 1 (11) VI. HILOSOHY OF INTERACTION EQUATIONS Above proposed interaction equations (9), (10) & (11) are used or webs o various Indian Standard rolled steel section and plate girder. The itical buckling strength is determined based on plate buckling theor applicable or web panel in equation (9). Taking stiened web o plate girder, a panel is considered as web portion between two adjacent stieners. Using practical spacing o stieners, panel aspect ratio (a/b) are taken as 0.9, 1.0, 1., 1.4 or plate girder. For unstiened web o rolled steel beam ratio (a/b) is span to depth o web & it is much larger than that o plate girder. In practice this ratio is around 10 or rolled steel beam. Shear buckling resistance o the web o rolled steel section is determined considering shear buckling coeicient as 9.00 (reer Table 1). Equation (10) based on Von Mises ield iteria. The strength in denominator is section strength at ielding o section which indicates ailure at a point. The buckling strength determined in equation (11) is based on overall member buckling in case o axial compression and panel dimensions in case o shear. Fig 4, Fig. 5 & Fig. 6 shows the interaction relationship as expressed b equation (9), (10), (11) w /,w 1. 1 0.8 0.6 0.4 w /N dw 1. 1 0.8 0.6 0.4 0. 0. 0 0 0. 0.4 0.6 0.8 1 1. V/V 0 0 0. 0.4 0.6 0.8 1 1. V/V dw Figure 4: Interaction Curve using Eq (9) Figure 5: Interaction Curve using eq. (10) 1. 1 w / dw 0.8 0.6 0.4 0. 0 0 0. 0.4 0.6 0.8 1 1. V/V Figure 6: Interaction Curve using Eq (11) w w w. a j e r. o r g age 14
American Journal o Engineering Research (AJER) 014 VII. ALICATION OF INTERACTION EQUATIONS Some o the cases where the member is subjected to combined action o shear and axial compression are hinged bases o column, hinged bases o arch bridge and at simpl supported end o plate girder o railwa bridge under longitudinal traction. The above proposed interaction equations are applied to various conigurations o plate girder b varing (a/b) and or the ratio o (w/v). arametric studies are perormed based on ollowing data is tabulated here, Web panel dimensions (a/b) = (1500/100) = 1.5, thickness o panel = 10 mm, modulus o elasticit = E = 00 Ga, ν = 0.3, = 50 Ma Table 5: Comparison o Interaction Equations Axial Compression Capacities Shaer Capacities Interaction Equations a/b w /V Nd d V Vd Eq. (9) Eq. (10) Eq. (11) 1.5 1 57. 77.7 407.35 1004.3 1574.59 0.419 0.0 0.588 1.5 57. 77.7 407.35 1004.3 1574.59 0.389 0.009 0.55 1.5 3 57. 77.7 407.35 1004.3 1574.59 0.384 0.007 0.545 1.5 4 57. 77.7 407.35 1004.3 1574.59 0.38 0.006 0.543 1.5 5 57. 77.7 407.35 1004.3 1574.59 0.381 0.006 0.54 VIII. CONCLUSION Interaction equation proposed b S Bulson [4] or combined action o & V is linear-quadratic which uses classical plate theor. Buckling strength under individual action o & V are evaluated on the basis o web panel dimensions and boundar condition o panel edges. Von Mises energ distortion theor which is known or ielding at the point o ailure, when used or web is not a unction o web panel dimensions. The equation basicall indicates section strength at ielding rather buckling. The interaction or combined action o & V is quadratic-quadratic. Considering buckling o member subjected to combined action o & V, the value o when acting alone & at which member buckles is dependent on I and eective length o the member. Thus, strength under compression is determined based on overall member buckling. However, buckling strength o web under pure shear depends upon web panel dimensions & its boundar condition. Interaction or combined action is linear-quadratic. At limit state, interaction o & V is governed b the limit strength in axial compression on the basis o member strength and that o shear on the basis o panel strength. Interaction values o & V under three dierent limit states such as web panel buckling, local ielding, & overall buckling o the member, it is observed that overall buckling o the member governs the limit state. ACKNOWLEDGEMENT The authors grateull acknowledge the support rom BCUD research roposal scheme, Universit o une, une, Maharashtra, India REFERENCES Journal apers: [1] Y Z Chen et.al. Concise ormula or the itical buckling stresses o an elastic plate under biaxial compression and shear, Journal o constructional steel research, 009, 1507 1510. [] Yukio Ueda et.al. Buckling and Ultimate Strength Interaction in lates and Stiened anels under Combined Inplane Biaxial and Shearing Forces, Marine Structures, 8, 1995,1-36 [3] C M Wang et.al. lastic Buckling o rectangular plates under combined Uniaxial and Shear Stresses, ASCE, Aug. 009, 89-895 [4] M Weaver, M Nemeth, Improved Design Formulae or Buckling o Orthotropic lates under Combined Loading, American Institute o Aeronautics and Astronautics [5] M A Bradord et.al. Semi-Compact Steel lates with unilateral restraint subjected to bending, compression and shear, Journal o Constructional Steel research, 56,000,47-67 w w w. a j e r. o r g age 15
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