Shear design recommendations for stainless steel plate girders

Transcription

1 Shear design recommendations for stainless steel plate girders N. Saliba, E. Real and L. Gardner 3, 3 Depament of Civil and Environmental Engineering, Imperial College London, London SW7 Z, UK Depament of Construction Engineering, Universitat Politècnica de Catalunya, UPC, 834 Barcelona, Spain bstract The behaviour and design of stainless steel plate girders loaded in shear investigated in this paper. revie of existing methods for the design of stainless steel plate girders, including codified provisions, is first presented. collected database of thiy four experiments carried out on stainless steel plate girders of the austenitic, duplex and lean duplex grades is then repoed, and used to assess the current shear resistance design equations found in Eurocode 3: Pa.4, Eurocode 3: Pa.5 and those proposed by Estrada et al. The comparisons clearly indicate that the design provisions of Eurocode 3: Pa.4 are conservative and that improved results can be achieved by applying the Eurocode 3: Pa.5 and Estrada et al. design expressions. Hoever, yet fuher improvements are possible and, based on the available structural performance data, revised design expressions for the calculation of the ultimate shear capacity of stainless steel plate girders suitable for incorporation into future revisions of Eurocode 3: Pa.4 have been proposed and statistically verified. Unlike the current provisions of Eurocode 3: Pa.4, the design rules proposed herein differentiate beteen rigid and non-rigid end posts, and, offer enhancements in shear buckling capacity of around %. Keyords Design methods, Eurocode 3, Experimental data, Failure modes, Plate girders, Reliability analysis, Rigid and non-rigid end post, Rotated stress field method, Shear buckling, Stainless steel, Ultimate shear capacity,. Introduction Plate girders are idely used in the construction industry especially in bridge applications, as transfer beams and shear alls in buildings and in offshore structures, oing to their ability to ithstand heavy loads over long spans. For material efficiency, plate girder ebs are often of slender propoions, making them susceptible to a form of instability knon as shear buckling. This type of failure has been extensively studied over the past fe decades in carbon steel plate girders and a range of design methods have been established. more limited number of studies has been devoted to stainless steel plate girders, and current design provisions are knon to be conservative. Hence, the aims of this paper are to study the shear response of stainless steel plate girders, to collate and examine available structural performance data, to revie existing design methods and to develop and statistically verify revised design expressions suitable for inclusion in international design codes. total of thiy four experiments carried out on stainless steel plate girders of the austenitic, duplex and lean duplex grades, ith eb panel aspect ratios varying beteen. and 4. and ith rigid and non-rigid end posts, ere first collected and used to evaluate the shear resistance design equations of EN (6) []. Next, a comparative analysis of other design methods including EN (6) [] and the proposed design expressions of Estrada et al. [3] has been performed. Finally, based on the available experimental data, together ith suppoing numerical data, revised design expressions for the calculation of the ultimate shear capacity of stainless steel plate girders are proposed and a reliability analysis in accordance ith EN 99 () [4] as carried out to confirm their applicability. Literature revie. Introduction In this section, laboratory test data on stainless steel plate girders and existing design methods and proposals for assessing the shear buckling resistance of plate girders are presented and briefly revieed. The design methods discussed are the tension field method and the rotated stress field method.. Overvie of previous experimental studies The first experimental investigation of stainless steel plate girders as carried out by Carvalho et al. [5]. The results of this study served as the basis for the first codified proposals for determining the ultimate shear resistance of stainless steel beams in ENV (996) [6]. Folloing the introduction of ENV (996) [6], a series of N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London)

2 experimental studies ere carried out by Olsson [7], Real et al. [8] and Estrada et al. [9], hich highlighted deficiencies in the existing design methods.. The main objectives of these investigations ere to develop a better understanding of the behaviour of stainless steel plate girders under shear and to propose design expressions capable of predicting accurately the shear resistance of stainless steel plate girders. Most recently, a detailed experimental and numerical study of lean duplex stainless steel plate girders as conducted [], bringing the total pool of laboratory test data to 34. The findings of these studies are discussed in Section 4 of this paper, hile the experimental data are used to verify a revised design treatment..3 Overvie of theoretical models for predicting shear buckling resistance The first attempt to estimate the shear resistance of slender plate girders as made by Basler et al. [, ]. ccording to Basler, once shear buckling had occurred in a plate girder eb, a theoretical tension field ould extend over the hole depth of the eb and the shear resistance could be expressed as the sum of the buckling and postbuckling resistances of the eb but ith no flange contribution. lthough there ere limitations to the Basler theory [3, 4], the basic tension field concept as able to represent observed physical behaviour and as fuher developed by Rockey et al. [5, 6]. draback to the tension field approach lies in its inability to predict accurately the shear buckling resistance of plate girders ith idely spread transverse eb stiffeners. solution, termed the rotated stress field method [7], as proposed by Höglund [7, 8]. This method as able to represent the postbuckling shear strength of both stiffened and unstiffened ebs. The rotated stress field method assumes that, prior to buckling, the eb is in a state of pure shear stress and the principal planes are inclined at an angle φ 45 to the horizontal. Hoever, once buckling has occurred, it is assumed that the principal compressive stress, remains equal to the shear buckling stress τ cr, and that fuher increases in load are resisted by an increase in the tensile stress. This causes the major principal plane to rotate toards the horizontal, and the ultimate resistance is said to be reached hen the von Mises yield criterion [7] is met. detailed description of the rotated stress field theory can be found in [9]. 3 Current and proposed design methods 3. Introduction In this section, the design methods for assessing the shear buckling resistance of plate girders in Eurocode 3 are revieed, including the evolution from the ENV prestandard to final EN standard. The provisions for both carbon steel and stainless steel, as ell as proposed changes to the latter, are considered. 3. Carbon steel design provisions 3.. ENV (99) To methods ere provided in ENV (99) [] for determining the design shear resistance of carbon steel plate girders: () the simple post critical method and () the tension field method. The first design method as developed by Dubas [] based on the rotated stress field theory, and applied to plate girders ith and ithout transverse stiffeners. The design shear resistance ignored any flange contribution, and as later found by Höglund [9] and Davies and Griffith [], to be unduly conservative. The second method, the tension field method, as found to be only appropriate for transversely stiffened ebs ith eb panel aspect ratios ranging beteen. and 3. [3]. Fuhermore, numerical studies by Presta et al. [4] shoed that the forces in the transverse stiffeners implied by the tension field method ere inaccurate. Limitations in both methods given in ENV (99) [], lead to revised design rules being provided upon conversion to the EN standards, at hich point the design provisions for shear buckling ere also moved to Pa.5 of the code. 3.. EN (6) The rotated stress field method developed by Höglund [7, 8] forms the basis of the shear design rules given in EN (6) []. In the EN (6) [] provisions, ultimate shear resistance V b,rd is expressed as the sum of the eb shear buckling resistance V b,rd (Eq. ()) and the flange contribution (Eq. (3)) V bf,rd, as set out in Eq. (). V b,rd V b,rd + V bf,rd ηf y h 3γ t M () here f y is the yield strength of the eb, f yf is the yield strength of the flanges, η is a parameter that approximates the influence of strain hardening, h is the depth of the eb, t is the thickness of the eb, b f is overall the flange idth, t f is the flange thickness, and γ M is a paial safety factor. N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London)

3 The eb contribution V b,rd is given by χ V b,rd f y 3γ h M t here χ is the eb shear buckling reduction. The flange contribution V bf,rd is given by: () V bf,rd bf t f f cγ M yf M M Ed f,rd in hich M Ed is the coexistent design bending moment, M f,rd is the moment resistance of the cross-section considering only the flanges and the distance c hich defines the location of the plastic hinges that form in the flanges is given by:.6b t f f f yf c.5 + a (4) t h f y here a is the spacing of the transverse stiffeners. 3.3 Stainless steel design provisions 3.3. ENV (996) t the time of the development of ENV (996) [6], the only experimental research into the shear resistance of stainless steel members as that carried out by Carvalho et al. [5]. Carvalho et al. [5] performed a series of three point bending tests on cold-formed austenitic and ferritic stainless steel sections. The obtained results ere used in the formulation of the design provisions of ENV (996) [6], hich ere based on the simple post critical method of ENV (99) [], but ith modifications to reflect the material nonlinearity of stainless steel. Subsequent experimental studies on stainless steel plate girders [3, 7-9] shoed that the ENV (996) [6] provisions ere conservative, raised questions of the quality of the earlier test data [5] and emphasized the need to consider the flange contribution to the shear buckling capacity. 3.4 EN (6) Folloing the adoption of the simple post critical method in ENV (996) [6], Olsson [7] performed an experimental and analytical study to underpin the development of improved ne design expressions for stainless steel plate girders. Olsson s design equations ere based on the rotated stress field method, took into consideration both the eb and flange contributions to the shear resistance, and ere of the same basic form as EN (6) [], given by Eqs (-3) of the present paper. Deviation from the EN (6) [] provisions appeared in the expressions for the shear buckling reduction factor χ and in the definition of the distance c. In Olsson s [7] proposal χ η for λ.6/η and χ. +.64/ λ.5/ 3.5b t f f f yf c.7 + a ith.65 t h f y a λ for λ >.6/η, ith η., and c is defined by Eq. (5). c (5) Olsson s method as included in the second edition of the SCI/Euro Inox Design Manual for Structural Stainless Steel [5] and as later incorporated into the final version of EN (6) []. 3.5 Estrada et al. s proposal (7) Olsson s design expressions offered clear benefits over those given in ENV (996) [6]. Hoever, the improved rules still did not distinguish beteen rigid and non-rigid end posts. Hence, fuher research as carried out by Estrada et al. [3, 9] here the influence of end post rigidity as evaluated over a ide range of eb panel aspect ratios and slendernesses. Based on their findings, revised design expressions ere proposed. The proposed expressions ere again based on the rotated stress field method and the total shear buckling resistance comprised a eb and flange contribution, as set out in Eqs (), () and (3). The flange contribution as the same as that proposed by Olsson, hile the eb contribution as revised. s explained by Estrada et al. [3], different design expressions ere given for eb panel aspect ratios less than and greater than unity. For the former case, end post rigidity as taken into account, providing more accurate prediction of test behaviour. (3) N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 3

4 4 nalysis of structural performance data on stainless steel plate girders 4. Introduction In order to evaluate the provisions outlined in Section 3 for the design of stainless steel plate girders, the results from previously conducted laboratory tests on stainless steel plate girders ere collected and analysed. In this section, the available test data are compared to the shear design rules of EN (6) [], EN (6) [] and those proposed by Estrada et al. [3]. 4. Collected experimental data on stainless steel plate girders total of 34 experiments on stainless steel plate girders have been conducted- see Table. Of these, (labelled NR to NR) had non-rigid end posts and 3 (labelled R to R3) had rigid end posts. The tested eb panel aspect ratio a/h ranged beteen. and 3.5, hile the non-dimensional eb slenderness ranged beteen.44 and The collected experimental data is summarised in Table, here L is the specimen length, a is the shear panel length, b is the overall flange idth, h is the eb depth, t f is the thickness of the flange, t is the thickness of the eb, t s and t s are the thicknesses of the stiffeners, σ., and σ.,f are the.% proof stresses of the eb and flanges, a/h is the aspect ratio of the eb panel, λ is the non-dimensional eb slenderness and V u,test is the ultimate shear capacity from the experiments. The definitions of those symbols are also illustrated in Fig nalyses of results 4.3. General In analysing the test results, to cases ere considered - case : plate girders exhibiting a shear dominant failure defined as those here the ratio of shear force to bending moment in the test V u,test /M u,test > V b,rd /M f,rd and case : plate girders exhibiting a bending dominant failure or a combined bending plus shear failure (i.e. V u,test /M u,test V b,rd /M f,rd ). These to cases are illustrated in Fig.. Test results in case only ere used to assess the shear buckling resistance design expressions, hile all data (cases and ) ere used to investigate the provisions for moment-shear interaction. The collected test data are plotted in Fig. 3, together ith the normalised moment-shear interaction diagram, calculated according to EN (6) []. Three interaction curves are shon in Fig. 3. ll have the same form, but since the precise shape of the curves varies ith V b,rd /V bf,rd and M f,rd /M c,rd, and since these ratios are different for each test specimen, clearly there is no single curve against hich to compare. The three curves shon are for the average, minimum and maximum values of the to ratios. Note that, in all comparisons shon, the measured geometric and material propeies from the test specimens are used, and all paial safety factors are set equal to unity. Fuhermore, in the folloing sections of this paper, case and plate girders ere identified by studying the momentshear interaction diagram for each cross-section separately Comparison of existing test data ith EN (6) The results collected from the experiments carried out on stainless steel plate girders ere used to evaluate the current shear design provisions of EN (6) []. The test results of the plate girders exhibiting a shear dominant failure (i.e. case only) are plotted ith the EN design model in Fig. 4, in terms of shear buckling reduction factor χ versus eb slenderness λ. distinction has been made beteen the tested plate girders ith rigid and nonrigid end posts, though EN (6) [] makes no such distinction. In locating the test data points, the flange contribution calculated according to Eq. (5), has been deducted from the test ultimate shear resistance, and the result has been normalised by the yield capacity of the eb in shear. The high normalised shear capacities (beyond yield and indeed beyond. times yield) obtained at lo slendernesses are attributed to strain hardening. For assessing the design provisions for moment-shear interaction, all test data (i.e. cases and ) ere considered. ssuming propoional loading (i.e. the ratio of shear force to bending moment remains constant), the ratio by hich each test data point exceeded or fell sho of its respective design interaction curve as denoted U. value of U greater than unity indicates a safe result hereby the test data point lies outside the interaction curve. The results are shon in Table and illustrated in Fig. 5. The comparisons sho that the test results consistently lie above the codified predictions, ith a mean ratio of U EN of. and a coefficient of variation (COV) of Comparison of existing test data ith EN (6) The European design standard for carbon steel plate girders EN (6) provides shear design expressions of the same form as EN (6), but ith alternative coefficients to reflect the differences in material response. N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 4

5 Unlike EN , EN differentiates beteen rigid and non-rigid end post plate girders for eb slenderness values λ >.8. s in Section 4.3., the test results that fall into case are used to compare ith the codified shear buckling resistance predictions (see Fig. 6), hile results for both cases and are used to assess the moment-shear interaction (see Table ). The comparisons sho that the predictions of EN (6) [] are noticeably higher than those of EN (6) [] and provide better agreement ith the test results, ith mean utilisation ratio U EN of.3 and a coefficient of variation (COV) of Comparison ith Estrada et al. [3] proposed design equations Similar comparisons to those described in Sections 4.3. and are made in the present section ith the proposals of Estrada et al. [3]. Unlike EN (6), Estrada et al. s [3] design expressions differentiate beteen rigid and non-rigid end post plate girders hen the eb panel aspect ratio is less than unity, hile for higher aspect ratios, revised design expressions are also provided. Test results for case only are compared ith the Estrada et al. [3] proposals for shear buckling resistance in Fig. 7, hile the utilisation ratio for all test results, considering combined bending and shear are shon in Table. The comparisons sho that Estrada et al. s predictions are higher than those of EN (6) [] and provide better agreement ith the test results, ith a mean utilisation ratio U Estrada of. and a coefficient of variation (COV) of Discussion Collected test data on stainless steel plate girders have been compared to the design provisions of EN (6) [], EN (6) [] and the proposed equations of Estrada et al. [3]. In general, it is observed that the results obtained from the estimations of EN (6) [] are conservative and better results can be obtained from EN (6) [] and the Estrada et al. [3] design expressions. Fuher improvements to the provisions, considering the recently available test and numerical data [], are proposed in the folloing section. 5 Design proposals 5. Introduction The comparisons of the previous section sho that the current provisions of EN (6) [] are conservative and better predictions of the test response are achieved ith the proposed design equations of Estrada et al. [3] or those of EN (6) []. In this section, fuher improvements are sought and ne design expressions for the calculation of the ultimate shear capacity of stainless steel plate girders are proposed. Statistical analyses, in accordance ith EN 99 [4], are also carried out to assess the reliability of the proposals. 5. Proposed design method Based on the collected test data repoed herein and the numerically generated plate girders from Saliba and Gardner [], ne design expressions are proposed to predict the ultimate shear capacity of stainless steel plate girders. The proposed expressions follo the same approach of EN (6) [] and EN (6) [] in hich the ultimate shear capacity of a stainless steel plate girder, V b,rd, consists of a eb contribution V b,rd and a flange contribution V bf,rd, as given by Eq. (). The flange contribution V bf,rd in the proposed approach is taken to be the same as that currently given in EN (6) [] see Eq. (3). The basis for this is that the expression (Eq. (5)) for the controlling parameter, c, hich defines the location of the plastic hinges in the ultimate shear collapse mechanism, has been confirmed to provide an accurate representation of recently generated test data [3, 9, ], as ell as Olsson s test data [7] upon hich the expression as originally derived. The eb contribution to shear resistance V b,rd, given by Eq. (), is controlled by the shear buckling reduction factor χ. Here, the proposed design expressions differ from previous provisions. The proposed expressions are developed on the folloing basis: () a larger body of test data (totalling 34 experiments) is considered than has previously been available, () a distinction is made beteen rigid and non-rigid end posts over the full spectrum of eb panel aspect ratios and (3) a reliability analysis in accordance ith EN 99 [4] is conducted. The proposed shear buckling reduction factors χ are presented in Table 3. The maximum value of χ adopted in the proposed design equations is the same as that of EN (6) [], at a value of.. This approximates the extra capacity beyond yield as a result of strain hardening. The results collected from the experiments on stainless steel plate girders ere used to assess the proposed equations. The utilisation ratios U Proposed (considering cases and ) are shon in Table and Fig. 8 ith a mean value of. and a coefficient of variation (COV) of.3. Note that the EN (6) [] moment-shear interaction expressions is N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 5

6 retained. In general, the utilization ratios for the proposed equations are significantly loer than those of EN (6) [], indicating better estimations of the ultimate shear resistance of the tested plate girders. The results of the case plate girders are also plotted ith the proposed design model in Fig. 9 here a better agreement ith the test data can be observed. Fuher statistical analysis is performed in the folloing section to verify the reliability of the proposed design expressions. 5.3 Reliability analysis The aim of the statistical analysis carried out in this section is to verify hether the developed design equations for the calculation of the shear resistance of stainless steel plate girders satisfy the Eurocode reliability requirements. The analysis folloed the standard statistical evaluation set out in nnex D of EN 99 () [4] and as applied to the collected test data repoed herein and the numerically generated results from Saliba and Gardner []. The analysis focused on shear buckling resistance and therefore considered the case results only. key assumption of the standard evaluation procedure is that the resistance expression is a function of independent variables. The dominant component of the shear buckling resistance (i.e. the eb shear buckling function resistance) V b,rd, hich generally appears in the form of Eq. () may alternatively be expressed as V b, Rd Xτ a y b (6) here f τ is equal to τ y / 3 and h t are the to independent variables and X is a constant, hich does not depend on the other to parameters. The poers a and b vary for different slenderness and should be determined for each test specimen. The poer a is calculated by assuming to plate girders of the same geometrical propeies but ith different material strengths τ y, and τ y,. The ratio of the resistances of the to considered plate girders is given by Eq. (7): V V b,rd, b,rd, Xτ Xτ a y, a y, b b τ τ y, y, Therefore, the poer a may be calculated as follos: a (7) V ln V τ ln τ b,rd, b,rd, y, y, a (8) and the poer b may be subsequently obtained from: V ln V τ aln τ, ln, b,rd, b,rd, y, y, b (9) by considering to plate girders of different eb areas, and,. The relationship beteen the to poers a and b and the non-dimensional eb slenderness λ R is plotted in Fig.. The values of the parameters are calculated based on the proposed equations in Section 5.. t lo slenderness ( λ.56), χ equals. and V b,rd.τ y /γ M, ith a b, hile at high slendernesses, here the resistance is governed by buckling and post-buckling tension field action, the values of a and b alter in response to the varying influence of τ y and. In order to properly allo for the influence of the basic variables τ y and at different values of eb slenderness in the reliability analysis, the variability of τ y and needs to be modified. The coefficient of variation of the basic variables, V, for a resistance function that is not simply linearly dependent on the basic variables may be obtained according to Eq. (D.6b) from nnex D of EN 99 () [4] and presented as follos: N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 6

7 V g g g g g VR g g ( X ) m j [ ( X) ] ( X ) g ( X ) i [ ] ( ) ( ) ( ) a b a b aτ y σ τ + bτ y y σ X m ( X ) m ( X m ) ( X ) m m g τ y σ f τ m g + σ ag bg σ + y τ y τ σ τ σ y a + b y,m τ,m g X i σ σ i ( av ) + ( bv ) τ y here VR[g (X)] and g (X m ) are the variance and the mean, respectively, of the resistance function g (X), σ τy and σ are the standard deviations of the yield strength and eb area, respectively, τ y,m and,m are the mean values of the shear yield strength and eb area, respectively and V τy and V are the coefficients of variation of the shear yield strength and eb area, respectively. This modification as incorporated into Step 7 of the statistical analysis set out in nnex D of EN 99 () [4]. statistical evaluation based on the above described modified approach as then performed for the collected test and numerical data. Ceain statistical parameters ere assumed based on previous studies of the mechanical and geometrical propeies of structural sections. The ratio of mean to nominal yield strengths (i.e. the material overstrength) as taken as.33 for austenitic stainless steels and as. for duplex stainless steels and the coefficients of variation of yield strength V τy and geometric propeies V ere taken as.66 and.5 for austenitic stainless steel and.4 and.5 for duplex stainless steel, respectively [6, 7]. These values originate from industrial data obtained from European steel producers. The results of the analyses and a summary of the key statistical parameters are presented in Table 4. The folloing symbols are used: k d,n design (ultimate limit state) fractile factor for n tests, here n is the population of test data under consideration, b average ratio of experimental to model resistance based on a least squares fit to the test data; V δ coefficient of variation of the tests relative to the resistance model; and V r combined coefficient of variation incorporating both model and basic variable unceainties. Note that in accordance ith EN 99 () [4], the fractile factor for the full collection of data on stainless steel plate girders has been used in the statistical analysis. Considering the resistance function for austenitic stainless steel plate girders, the paial factor γ M as found to be.4 for the test data only and for the resistance function of lean duplex stainless steel plate girders, the paial factor γ M as found to be.6 for the test data plus FE results of Saliba and Gardner []. The obtained values of γ M for both austenitic and duplex/lean duplex stainless steel are less than., hich is the recommended paial factor for stainless steel shear buckling in EN (6) []. It is therefore recommended that the proposed equations for the calculation of the eb shear resistance in Section 5. can be safely applied to stainless steel plate girders, ith γ M.. 6 Conclusions The behaviour of stainless steel plate girders, ith an emphasis on the calculation of ultimate shear capacity as studied in this paper. revie of existing design methods and codified provisions as first presented. total of thiy four experiments carried out on stainless steel plate girders of the austenitic, duplex and lean duplex grades ere collected and used to assess the current shear design expressions of EN , EN and those proposed by Estrada et al [3]. It as found that the current EN shear design expressions are conservative and better results can be achieved by using the proposed design expressions of Estrada et al. and EN Hoever, fuher improvements ere possible and, on the basis of the available structural performance data revised design equations for the calculation of the ultimate shear capacity of stainless steel plate girders have been proposed. The proposed design expressions ere developed in a form similar to those of EN and EN to retain compatibility ith current provisions. Revised expressions for the shear buckling reduction factor χ for stainless steel plate girders that account for end post rigidity over the full range of eb panel aspect ratios ere proposed. The proposals ere subjected to a statistical analysis in accordance ith EN 99, here the reliability of the design recommendations ere verified. () N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 7

8 References [] EN (6). Eurocode 3: Design of steel structures - Pa.4: General rules - Supplementary rules for stainless steel. CEN. [] EN (6). Eurocode 3: Design of steel structures - Pa.5: Plated structural elements. CEN. [3] Estrada, I., Real, E., and Mirambell, E. (7). General behaviour and effect of rigid and non-rigid end post in stainless steel plate girders loaded in shear. Pa II: Extended numerical study and design proposal. Journal of Constructional Steel Research. 63(7), [4] EN 99. (). Eurocode: basis of structural design. CEN. [5] Carvalho, E. C. G., Van den Berg, G. J. and Van der Mere, P. (99). Local shear buckling in cold-formed stainless steel beam ebs. Proceedings of the nnual Technical Session of the Structural Stability Research Council. [6] ENV (996). Eurocode 3: Design of steel structures - Pa.4: General rules - Supplementary rules for stainless steels. CEN. [7] Olsson,. (). Stainless steel plasticity-material modelling and structural applications. PhD Thesis. Lulea University of Technology, Seden. [8] Real, E., Mirambell, E., and Estrada, I. (7). Shear response of stainless steel plate girders. Engineering Structures. 9(7), [9] Estrada, I., Real, E., and Mirambell, E. (7). General behaviour and effect of rigid and non-rigid end post in stainless steel plate girders loaded in shear. Pa I: Experimental study. Journal of Constructional Steel Research. 63(7), [] Saliba, N. and Gardner, L. (3). Testing and design of lean duplex stainless steel plate girders. Engineering Structures. 46, [] Basler, K., Yen, B. T., Mueller, J.., and Thürlimann, B. (96). Web buckling tests on elded plate girders. Welded Plate Girders. Bulletin No. 64. [] Basler, K. (96). Strength of plate girders under combined bending and shear. Journal of Structural Engineering, SCE. 7, 5-8. [3] Calladine, C.R. (973). plastic theory for collapse of plate girders under combined shear force and bending moment. The Structural Engineer. 5(4), [4] Poer, D. M., Rockey, K.C., and Evans, H.R. (975). The collapse behaviour of plate girders loaded in shear. The Structural Engineer. 53(8), [5] Rockey, K. C., Evans, H. R., and Poer, D. M. (978). design method for predicting the collapse behaviour of plate girders. Proceeding of the Institution of Civil Engineers. (65), 85-. [6] Rockey, K. C., and Skaloud, M. (97). The ultimate load behaviour of plated girders loaded in shear. The Structural Engineer. 5(), [7] Höglund, T. (97). Behaviour and strength of the eb of thin plate I-girders. Bulletin No. 93, Division of building statics and structural engineering. The Royal Institute of Technology, Stockholm, Seden. [8] Höglund, T. (973). Design of thin plate I girders in shear and bending ith special reference to eb buckling. Bulletin No. 94, Division of building statics and structural engineering. Royal Institute of Technology, Stockholm, Seden. [9] Höglund, T. (998). Shear buckling resistance of steel and aluminium plate girders. Thin-Walled Structures 9( 4), 3 3. [] ENV (99). Eurocode 3: Design of steel structures - Pa.: General rules and rules for buildings. CEN. [] Dubas, P. (98). Reflexions sur ceains problems de seurite et stabilite en construction metallique. Memoires CERES. (55). Liege, Belgium. [] Davies,. W., and Griffith, D. S. C. (999). Shear strength of steel plate girders. Proceeding of the Institution of Civil Engineers Structures and Buildings. 34, N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 8

9 [3] Roca, P., Mirambell, E., and Costa, J. (996). Geometric and material nonlinearities in steel plates. Journal of Structural Engineering, SCE. (), [4] Presta, F., Hendy, C. R., and Turco, E. (8). Numerical validation of simplified theories for design rules of transversely stiffened plate girders. The Structural Engineer. 86(), [5] SCI/Euro Inox. (). Design manual for structural stainless steel. nd ed. Oxford (UK). The European Stainless Steel Development ssociation and the Steel Construction Institute. [6] Groth, H.L. and Johansson, R.E. (99). Statistics of the mechanical strength of stainless steels. Proceedings of the Nordic Symposium on Mechanical Propeies of Stainless Steels, Sigtuna, Seden (October 99), 7-3. [7] Leffler, B. (99). statistical study of the mechanical propeies of the hot-rolled stainless plate. Proceedings of the Nordic Symposium on Mechanical Propeies of Stainless Steels. Sigtuna, Seden, (October 99), 3-4. N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 9

10 Figures and Tables e t f Loading jack t f b t s t h t s h t b s t f Span L a Fig. : Geometry of the tested plate girders. V Test capacity (V u,test ) Case : V u,test /M u,test > V b,rd /M f,rd V b,rd Code capacity ith flange contribution V b,rd Code capacity ithout flange contribution Case : V u,test /M u,test V b,rd /M f,rd M f,rd M c,rd M Fig. : Moment-shear interaction diagrams and definition of cases and. N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London)

11 .6.4 Case Case M-V interaction (average) M-V interaction (minimum) M-V interaction (maximum). V/V b,rd.8.6 V b,rd /V b,rd.4 (V b,rd /)/V b,rd. M f,rd /M c,rd M/M c,rd Fig. 3: Test data and normalised moment-shear interaction diagram according to EN (6) [] Olsson non-rigid end post tests [7] Real et al. non-rigid end post tests [8] Estrada et al. non-rigid end post tests [9] Estrada et al. rigid end post tests [9] Saliba and Gardner rigid end post tests [] EN [] χ λ Fig. 4: Comparison beteen experimental results of case only and the shear resistance function of EN (6). N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London)

12 Olsson tests [7] Real et al. tests [8] Estrada et al. tests [9] Saliba and Gardner tests []. U EN λ Fig. 5: Utilization ratio (test/design resistance) as obtained from EN (6) Olsson non-rigid end post tests [7] Real et al. non-rigid end post tests [8] Estrada et al. non-rigid end post tests [9] Estrada et al. rigid end post tests [9] Saliba and Gardner rigid end post tests [] EN [] rigid EN [] non-rigid χ λ Fig. 6: Comparison beteen experimental results of case only and the shear resistance function of EN (6) []. N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London)

13 Olsson non-rigid end post tests [7] Real et al. non-rigid end post tests [8] Estrada et al. non-rigid end post tests [9] Estrada et al. rigid end post tests [9] Saliba and Gardner rigid end post tests [] Estrada et al. rigid a/h. [3] Estrada et al. non-rigid a/h. [3] Estrada et al. rigid and non-rigid a/h >. [3] χ λ Fig. 7: Comparison beteen experimental results of case only and the shear resistance function of Estrada et al. [3]..6.4 Olsson tests [7] Real et al. tests [8] Estrada et al. tests [9] Saliba and Gardner tests []. U Proposed λ Fig. 8: Utilization ratio (test/design resistance) as obtained from the proposed equations of this paper. N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 3

14 Olsson non-rigid end post tests [7] Real et al. non-rigid end post tests [8] Estrada et al. non-rigid end post tests [9] Estrada et al. rigid end post tests [9] Saliba and Gardner rigid end post tests [] Proposed rigid Proposed non-rigid EN [] χ λ Fig. 9: Comparison beteen experimental results of case only and the shear resistance functions of EN (6) [] and the proposed approach..8.6 a rigid b rigid a non-rigid b non-rigid.4 Values of poers a and b λ Fig. : The poers a and b for rigid and non-rigid end posts versus eb slenderness λ. N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 4

15 Table : Collected experimental data on stainless steel plate girders End-post Reference Label Grade σ., σ.,f L a h b t f t t s t s (N/mm ) (N/mm ) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) a/h λ V u,test (kn) NR NR NR Olsson [7] NR NR NR NR NR NR NR Non-rigid NR NR Real et al. [8] NR NR NR NR NR NR Estrada et al. [9] NR NR NR R Estrada et al. [9] R R R R R Rigid R Saliba and R Gardner [] R R R R R N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 5

16 Table : Utilization ratios (test/design resistance) for plate girders as obtained from EN (6), EN (6), Estrada et al. [3] and proposed design methods Reference Label U EN U EN U Estrada U Proposed NR NR NR Olsson [7] NR NR NR NR NR NR NR NR NR Real et al. [8] NR NR NR NR NR NR Estrada et al. [9] NR NR NR R Estrada et al. [9] R R R4.... R R R Saliba and R Gardner [] R R R R R3.... Mean..3.. COV Table 3: Proposed design expressions for the calculation of the eb contribution to the shear resistance χ for rigid end post χ for non-rigid end post λ R.56.. λ R > / λ λ./ / λ λ./ Table 4: Results of statistical analysis of test data for proposed eb shear resistance equations Data set No. of tests Fractile factor R test /R proposed Model scatter Resistance scatter n k d,n b V δ V r γ M ustenitic tests only Lean duplex tests + FE [] N Saliba (Imperial College London), E Real (UPC) and L Gardner (Imperial College London) 6