2011-9-27同济大学 吴冲 Tongji University, Wu Chong01. Classification of cross sections1. Classification of cross sectionsFF Class 1 cross-sections Class 1 cross-sections欧洲的钢结构规范适用于所有钢结构欧洲的钢结构规范适用于所有钢结构欧洲的钢结构规范适用于所有钢结构欧洲的钢结构规范适用于所有钢结构Can form a plastic hinge with the rotation capacity required Can form a plastic hinge with the rotation capacity required from plastic analysis without eduction of the resistance.from plastic analysis without eduction of the resistance.FFClass 2 cross-sectionsClass 2 cross-sections Can develop their plastic moment resistance, but have limited Can develop their plastic moment resistance, but have limited rotation capacity because of local buckling.rotation capacity because of local buckling.FFClass 3 cross-sections Class 3 cross-sections The stress in the extreme compression fibre of the steel member The stress in the extreme compression fibre of the steel member assuming an elastic distribution of stresses can reach the yield assuming an elastic distribution of stresses can reach the yield strengthstrength边缘受压应变可以达到屈服应力边缘受压应变可以达到屈服应力边缘受压应变可以达到屈服应力边缘受压应变可以达到屈服应力 Local buckling is liable to prevent development of the plastic Local buckling is liable to prevent development of the plastic moment resistance.moment resistance.FFClass 4 cross-sections Class 4 cross-sections Local buckling will occur before the attainment of yield stress in Local buckling will occur before the attainment of yield stress in one or more parts of the cross-section.one or more parts of the cross-section.2011-9-27同济大学 吴冲 Tongji University, Wu Chong11. Classification of cross sections1. Classification of cross sections1.2.31.2.3表示截面表示截面表示截面表示截面分类。该图的分类。该图的分类。该图的分类。该图的大致意思就是大致意思就是大致意思就是大致意思就是对于某一类截对于某一类截对于某一类截对于某一类截面面面面 ，某一特定，某一特定，某一特定，某一特定的应力分布情的应力分布情的应力分布情的应力分布情况，宽厚比或况，宽厚比或况，宽厚比或况，宽厚比或长细比要满足长细比要满足长细比要满足长细比要满足的要求的要求的要求的要求2011-9-27同济大学 吴冲 Tongji University, Wu Chong21. Classification of cross sections1. Classification of cross sections自由边，即无支自由边，即无支自由边，即无支自由边，即无支撑的边缘撑的边缘撑的边缘撑的边缘2011-9-27同济大学 吴冲 Tongji University, Wu Chong31. Classification of cross sections1. Classification of cross sectionsTubularTubular表示表示表示表示圆管截面。对于圆管截面。对于圆管截面。对于圆管截面。对于圆管截面，满足圆管截面，满足圆管截面，满足圆管截面，满足d/td/t的条件之后，的条件之后，的条件之后，的条件之后，无需布置加劲肋无需布置加劲肋无需布置加劲肋无需布置加劲肋2011-9-27同济大学 吴冲 Tongji University, Wu Chong42. Shear lag in member design2. Shear lag in member designFF2.1 General 2.1 General shear lagshear lag表示剪力滞表示剪力滞表示剪力滞表示剪力滞Shear lag in flanges may be neglected if Shear lag in flanges may be neglected if b b0 0 L Le e/50/50 uub b0 0 is taken as the flange outstand or half the width of an is taken as the flange outstand or half the width of an internal elementinternal elementuuL Le e is the length between points of zero bending moment is the length between points of zero bending momentI If f b b0 0 L Le e/50/50 the effects due to shear lag in flanges should be the effects due to shear lag in flanges should be consideredconsidereduu A At t serviceabilityserviceability and and fatigue limit statefatigue limit state verifications verifications, , an an effectives width may be used.effectives width may be used.uu A At t ultimate limit state ultimate limit state verificationsverifications, , an effective area an effective area may be may be used.used.2011-9-27同济大学 吴冲 Tongji University, Wu Chong52. Shear lag in member design2. Shear lag in member designFF2 2. .2 2 Shear lag in global analysis Shear lag in global analysis (calculation of internal forces and moments, a simple rule to take (calculation of internal forces and moments, a simple rule to take shear lag into account in the global analysis of a multi-span I- or shear lag into account in the global analysis of a multi-span I- or box-girder.)box-girder.)The effectives flange width (on each side of the main web) is The effectives flange width (on each side of the main web) is assumed to be uniform over the length of each span. assumed to be uniform over the length of each span. For each span of the girder, this effectives flange width beff For each span of the girder, this effectives flange width beff should be taken as the lesser of the full width b0 (on each side of should be taken as the lesser of the full width b0 (on each side of the main web) and L/8, where L is the span of the girder. In case the main web) and L/8, where L is the span of the girder. In case of a cantilever girder, L is equal to twice the distance from the of a cantilever girder, L is equal to twice the distance from the support to the end of the cantilever.support to the end of the cantilever.The same effectives width applies for calculating the internal The same effectives width applies for calculating the internal forces and moments at forces and moments at SLS and at ULS.SLS and at ULS.2011-9-27同济大学 吴冲 Tongji University, Wu Chong62. Shear lag in member design2. Shear lag in member designFF2.32.3 Elastic shear lag in section analysis Elastic shear lag in section analysis (calculation of stresses at serviceability and fatigue limit states (calculation of stresses at serviceability and fatigue limit states ) )b beffeff = = b b b b b b0 0Fig. 2.5: Definition of notations (b0, Asl)2011-9-27同济大学 吴冲 Tongji University, Wu Chong72. Shear lag in member design2. Shear lag in member design2011-9-27同济大学 吴冲 Tongji University, Wu Chong82. Shear lag in member design2. Shear lag in member design2011-9-27同济大学 吴冲 Tongji University, Wu Chong92. Shear lag in member design2. Shear lag in member designFF要记得大致趋势要记得大致趋势要记得大致趋势要记得大致趋势Fig. 2.8: Influence of the flange stiffening ratio on the elastic effectives width2011-9-27同济大学 吴冲 Tongji University, Wu Chong102. Shear lag in member design2. Shear lag in member designFig. 2.8: Influence of the flange stiffening ratio on the elastic effectives width2011-9-27同济大学 吴冲 Tongji University, Wu Chong112. Shear lag in member design2. Shear lag in member design四四四四次次次次曲曲曲曲线线线线2011-9-27同济大学 吴冲 Tongji University, Wu Chong122. Shear lag in member design2. Shear lag in member designFF2.4 Elasto-plastic shear lag in section analysis 2.4 Elasto-plastic shear lag in section analysis (calculation of stresses at ultimate limit state (ULS) (calculation of stresses at ultimate limit state (ULS)Use the SLS elastic shear lag (conservative approach which does Use the SLS elastic shear lag (conservative approach which does not consider the elasto-plastic redistribution at all),not consider the elasto-plastic redistribution at all),Use the elastic-plastic shear lag effects allowing for limited Use the elastic-plastic shear lag effects allowing for limited plastic strainsplastic strainsuu A Aeffeff = = b b b bk k k kA Ac,effc,eff b b b b A Ac,effc,eff b b b b and and k k k k are taken from Table 3.1. are taken from Table 3.1. 2011-9-27同济大学 吴冲 Tongji University, Wu Chong131. Shear lag in member design1. Shear lag in member design弹塑性分析的剪力滞导致的有效宽度大，负弯矩区折减多弹塑性分析的剪力滞导致的有效宽度大，负弯矩区折减多弹塑性分析的剪力滞导致的有效宽度大，负弯矩区折减多弹塑性分析的剪力滞导致的有效宽度大，负弯矩区折减多Fig. 2.10: Influence of the considered limit state and of the bending zoneon the effectives width (for the case 0 = 1)2011-9-27同济大学 吴冲 Tongji University, Wu Chong142. Shear lag in member design2. Shear lag in member designFF2.5 Combined effects of shear lag and of plate buckling at ULS2.5 Combined effects of shear lag and of plate buckling at ULS A Aeffeff = = b b b bultult A Ac,effc,eff uu A Ac,effc,eff is the effectiveis the effectivep p area of the compression flange due to area of the compression flange due to plate buckling plate buckling uu b b b bultult is the effectiveis the effectives s width factor for the effect of shear lag at width factor for the effect of shear lag at the ultimate limit state, which may be taken as the ultimate limit state, which may be taken as b b b b determined determined from Table 3.1 with from Table 3.1 with 0 0 replaced by replaced by2011-9-27同济大学 吴冲 Tongji University, Wu Chong152. Shear lag in member design2. Shear lag in member designFF2.6 Shear lag in flanges in tension 2.6 Shear lag in flanges in tension The expressions above may also be applied for flanges in tension The expressions above may also be applied for flanges in tension in which case in which case A Ac,effc,eff should be replaced by the gross area of the should be replaced by the gross area of the tension flange.tension flange.uu A Aeffeff =max =max b b b bultult A, A, b b b bk k k kA, A, b b b b A A uu b b b b and and k k k k are taken from Table 3.1. are taken from Table 3.1. uub b b bultult is the effectiveis the effectives s width factor for the effect of shear lag at width factor for the effect of shear lag at the ultimate limit state, which may be taken as the ultimate limit state, which may be taken as b b b b determined determined from Table 3.1 with from Table 3.1 with 0 0 replaced by replaced by2011-9-27同济大学 吴冲 Tongji University, Wu Chong163. 3. P Plate buckling late buckling effects due to direct stresseseffects due to direct stressesFF3.1 P3.1 Plate bucklinglate bucklingFig. 2.12: Post-critical response of slender plates in compression 2011-9-27同济大学 吴冲 Tongji University, Wu Chong173. 3. P Plate buckling late buckling effects due to direct stresseseffects due to direct stressesFF3.2 B3.2 Buckling uckling behaviour of plates in compressionbehaviour of plates in compressionPlate-like Plate-like buckling buckling behaviourbehaviourColumn-like Column-like buckling buckling behaviourbehaviourFig. 2.19: Plate-like and column-like behaviour of plates in compression2011-9-27同济大学 吴冲 Tongji University, Wu Chong183. 3. P Plate buckling late buckling effects due to direct stresseseffects due to direct stressesFF3.2 B3.2 Buckling uckling behaviour of plates in compressionbehaviour of plates in compressionColumn-like Column-like buckling buckling behaviourbehaviouruuFor shorter plates with low aspect ratios For shorter plates with low aspect ratios = a/b, = a/b, the two-dimensional the two-dimensional plate-like behaviour has changed into one-dimensional column-like plate-like behaviour has changed into one-dimensional column-like behaviour that does not possess any behaviour that does not possess any postcritical postcritical resistance, see Fig. 2.19. resistance, see Fig. 2.19. 意思是屈曲之后不再有承载力意思是屈曲之后不再有承载力意思是屈曲之后不再有承载力意思是屈曲之后不再有承载力For unstiffened panels this occurs at aspect ratio For unstiffened panels this occurs at aspect ratio = a/b well below = a/b well below 1.0, 1.0, For longitudinally stiffened panels with pronounced orthotropic For longitudinally stiffened panels with pronounced orthotropic properties such behaviour may start at larger aspect ratios properties such behaviour may start at larger aspect ratios 1.0 (Fig. 1.0 (Fig. 2.19 c). 2.19 c). uuIn EN 1993-1-5 plates in In EN 1993-1-5 plates in column-like buckling are treated as unsupported column-like buckling are treated as unsupported along the longitudinal edges. Therefore it is evident that critical stresses along the longitudinal edges. Therefore it is evident that critical stresses for plate-like buckling are always larger than critical stresses for column-for plate-like buckling are always larger than critical stresses for column-like buckling. like buckling. Ultimate resistance of shorter plates depends on both types of buckling, plate-Ultimate resistance of shorter plates depends on both types of buckling, plate-like as well as column-like, and a suitable interpolation between both types of like as well as column-like, and a suitable interpolation between both types of behaviour was introduced in EN 1993-1-5 design proceduresbehaviour was introduced in EN 1993-1-5 design procedures2011-9-27同济大学 吴冲 Tongji University, Wu Chong193. 3. P Plate buckling late buckling effects due to effects due to direct stresses direct stresses 正应力正应力正应力正应力FF3.4 Simplified methods in practical design3.4 Simplified methods in practical designEffective width method or reduced cross-section methodEffective width method or reduced cross-section methodReduced stress method. Reduced stress method. Fig. 2.13: Basic ideas of reduced cross-section method and reduced stress method2011-9-27同济大学 吴冲 Tongji University, Wu Chong204. 4. Effective width methodEffective width methodFF4.1 Requirements of the effective width method to be applied 4.1 Requirements of the effective width method to be applied The panels and the sub-panels are rectangular or nearly The panels and the sub-panels are rectangular or nearly rectangular with flanges deviating from the horizontal line not rectangular with flanges deviating from the horizontal line not more than 10more than 10The panels may be unstiffened or stiffened with longitudinal or The panels may be unstiffened or stiffened with longitudinal or transverse stiffeners or in both directions. transverse stiffeners or in both directions. Unstiffened openings and cut-outs should be small, with Unstiffened openings and cut-outs should be small, with diameters not exceeding 0.05 diameters not exceeding 0.05 b b, where , where b b is the width of the plate is the width of the plate element. Properly stiffened holes may be larger, but EN 1993-1-5 element. Properly stiffened holes may be larger, but EN 1993-1-5 does not provide any design rules (see does not provide any design rules (see http:/lwo.steel-sci.orghttp:/lwo.steel-sci.org for for the documents on large web openings). the documents on large web openings). Members are of uniform cross-sections. When the thickness of Members are of uniform cross-sections. When the thickness of the panel is not constant, then the equivalent thickness may be the panel is not constant, then the equivalent thickness may be taken equal to the smallest one. taken equal to the smallest one. Flange induced web buckling is prevented by selecting Flange induced web buckling is prevented by selecting appropriate web slenderness. appropriate web slenderness. 2011-9-27同济大学 吴冲 Tongji University, Wu Chong214. 4. Effective width methodEffective width methodFF4.2 Principles of effective width calculation4.2 Principles of effective width calculationThe effectivep widths of each plate element in compression are The effectivep widths of each plate element in compression are calculated independently. calculated independently. Based on these effectivep widths effective cross-section area Based on these effectivep widths effective cross-section area A Aeffeff, , effective second moment of inertiaeffective second moment of inertia I Ieffeff and effective section and effective section modulus modulus WWeffeff are calculated (Fig. 2.14, Fig. 2.15, Fig. 2.16). are calculated (Fig. 2.14, Fig. 2.15, Fig. 2.16). For compression elements the effective widths are determined by For compression elements the effective widths are determined by taking into account combined effect of shear lag and plate taking into account combined effect of shear lag and plate buckling buckling For tension elements effectives widths come only from shear lag For tension elements effectives widths come only from shear lag effects. effects. The effective cross-section is then treated as an equivalent Class The effective cross-section is then treated as an equivalent Class 3 cross-section, assuming linear elastic strain and stress 3 cross-section, assuming linear elastic strain and stress distribution over the reduced cross-section. distribution over the reduced cross-section. The ultimate resistance is reached with the onset of yielding in The ultimate resistance is reached with the onset of yielding in the most compressed fibre. the most compressed fibre. 2011-9-27同济大学 吴冲 Tongji University, Wu Chong224. 4. Effective width methodEffective width methodFF4.3 Effective cross-section4.3 Effective cross-sectionFig. 2.14: Effective cross-section2011-9-27同济大学 吴冲 Tongji University, Wu Chong234. 4. Effective width methodEffective width methodFF4.3 Effective cross-section4.3 Effective cross-sectionFig. 2.15: Class 4 cross-sections in pure compression2011-9-27同济大学 吴冲 Tongji University, Wu Chong244. 4. Effective width methodEffective width methodFF4.3 Effective cross-section4.3 Effective cross-sectionFig. 2.16: Class 4 cross-section in pure bending2011-9-27同济大学 吴冲 Tongji University, Wu Chong254. 4. Effective width methodEffective width methodFF4.3 Effective cross-section4.3 Effective cross-sectionIn non-symmetrical cross-sections subjected to axial force In non-symmetrical cross-sections subjected to axial force N NededuuThe shift The shift e eN N occurs (of the centroid G of the effective area occurs (of the centroid G of the effective area A Aeffeff relative to relative to the centre of gravity of the gross cross-section G, see Fig. 2.15). the centre of gravity of the gross cross-section G, see Fig. 2.15). uuThis shift results in an additional bending moment This shift results in an additional bending moment M = eM = eN N N NEdEd that that should be taken into account in the cross-section verification should be taken into account in the cross-section verification uu Shift Shift e eMM (see Fig. 2.16) of the centre of gravity due to pure bending is not (see Fig. 2.16) of the centre of gravity due to pure bending is not included in the calculation of included in the calculation of MM, even if the cross-section is subjected to , even if the cross-section is subjected to the combination of axial force and bending moment. the combination of axial force and bending moment. If axial force and bending moment act simultaneouslyIf axial force and bending moment act simultaneouslyuuThe calculation of effectivep widths may be based on the resulting stress The calculation of effectivep widths may be based on the resulting stress distribution. distribution. uuEN 1993-1-5 allows a simplified approach where EN 1993-1-5 allows a simplified approach where A Aeffeff is calculated only for is calculated only for stresses coming from pure compression and stresses coming from pure compression and WWeffeff only for stresses coming only for stresses coming from pure bending. from pure bending. 2011-9-27同济大学 吴冲 Tongji University, Wu Chong264. 4. Effective width methodEffective width methodFF4.4 Effective4.4 Effectivep p widths by iterative procedure widths by iterative procedureGenerally the calculation of effectiveGenerally the calculation of effectivep p widths requires iterative widths requires iterative procedure shown in Fig. 2.17 that procedure shown in Fig. 2.17 that ends when the differences ends when the differences between two steps are sufficiently small. between two steps are sufficiently small. Fig. 2.17: Determination of effectivep area by iterative procedure2011-9-27同济大学 吴冲 Tongji University, Wu Chong274. 4. Effective width methodEffective width methodFF4.4 Effective4.4 Effectivep p widths by iterative procedure widths by iterative procedureIterative procedureIterative procedureuuThe first iteration starts with the stress distribution on the The first iteration starts with the stress distribution on the gross cross-section gross cross-section A AG1G1. . uuThe effective area for the second iteration The effective area for the second iteration A Aeff2eff2 is calculated is calculated from this stress distributionfrom this stress distributionuuThe effective area for the third iteration The effective area for the third iteration A Aeff3eff3 from the from the stresses on stresses on A Aeff2eff2. . 2011-9-27同济大学 吴冲 Tongji University, Wu Chong284. 4. Effective width methodEffective width methodFF4.4 Effective4.4 Effectivep p widths by iterative procedure widths by iterative procedure For I-section and box cross-section in bending EN 1993-1-5 For I-section and box cross-section in bending EN 1993-1-5 allows a simplified approach that ends in two steps. allows a simplified approach that ends in two steps. uuIn the first step effectiveIn the first step effectivep p widths in flanges (if any) are widths in flanges (if any) are determined from the stress distribution on the gross cross-determined from the stress distribution on the gross cross-section. section. uuIn the second step the stresses are determined on the cross-In the second step the stresses are determined on the cross-section composed of the effectivesection composed of the effectivep p area of the compressed area of the compressed flanges and the gross areas of the web and the tensional flanges and the gross areas of the web and the tensional flanges. flanges. uuThe effectiveThe effectivep p width in the web is calculated based on these width in the web is calculated based on these stresses.stresses.2011-9-27同济大学 吴冲 Tongji University, Wu Chong294. 4. Effective width methodEffective width methodFF4.4 Effective4.4 Effectivep p widths by iterative procedure widths by iterative procedure When different stages of construction have to be considered, When different stages of construction have to be considered, which is a normal case in the design of composite bridges, the which is a normal case in the design of composite bridges, the following simplified approach proposed in a Note to clause 4.4 following simplified approach proposed in a Note to clause 4.4 (2) of EN 1993-1-5 may be used: (2) of EN 1993-1-5 may be used: uuin all relevant construction stages (e.g. concreting of the slab, in all relevant construction stages (e.g. concreting of the slab, normal use of a bridge) the stresses should be calculated on normal use of a bridge) the stresses should be calculated on the gross cross-section of the web and effective cross-section of the gross cross-section of the web and effective cross-section of the flanges (plate buckling and/or shear lag), when relevantthe flanges (plate buckling and/or shear lag), when relevantuuthe stresses from different construction stages are summed the stresses from different construction stages are summed up and used to determine a single effective cross-section of the up and used to determine a single effective cross-section of the web that is used for all construction stagesweb that is used for all construction stagesuufinally, the stresses for individual construction stages are finally, the stresses for individual construction stages are calculated on corresponding effective cross-sections and calculated on corresponding effective cross-sections and summed up to get the final cumulative stressessummed up to get the final cumulative stresses2011-9-27同济大学 吴冲 Tongji University, Wu Chong304. 4. Effective width method Effective width method 类似于混合梁的意思，指材料不同，无论横桥向还是纵桥向类似于混合梁的意思，指材料不同，无论横桥向还是纵桥向类似于混合梁的意思，指材料不同，无论横桥向还是纵桥向类似于混合梁的意思，指材料不同，无论横桥向还是纵桥向FF4.5 4.5 Hybrid girdersHybrid girdersFig. 2.18: Assumed stress distribution in a hybrid girder with h = 460 N/mm2 / 235 N/mm2 = 1.962011-9-27同济大学 吴冲 Tongji University, Wu Chong314. 4. Plate-like bucklingPlate-like bucklingFF4.5 Hybrid girders4.5 Hybrid girdersRequirementsRequirementsuuto utilize the full strength of the flanges a partial plastic to utilize the full strength of the flanges a partial plastic redistribution of stresses is allowed in the web (see Fig. 2.18)redistribution of stresses is allowed in the web (see Fig. 2.18)uuto assure some minimum ductility requirements in the web, to assure some minimum ductility requirements in the web, the effectivethe effectivep p area of the web should be determined based on area of the web should be determined based on the flange yield strength the flange yield strength f fyfyf and not on the web yield strength and not on the web yield strength f fywywuuto limit the partial plastic stress distribution in the web, for to limit the partial plastic stress distribution in the web, for the flange to web yield strength ratio the flange to web yield strength ratio h h the following upper the following upper limit is recommended: limit is recommended: When calculating the location of the centre of gravity of When calculating the location of the centre of gravity of effectiveeffectivep p area area GG, linear stress distribution in the web may be , linear stress distribution in the web may be assumed, neglecting the limitation of the web stress . assumed, neglecting the limitation of the web stress . 2011-9-27同济大学 吴冲 Tongji University, Wu Chong325. Plate-like buckling5. Plate-like bucklingFF5.1 Unstiffened plates 5.1 Unstiffened plates The effectiveThe effectivep p widths widths b beffeff or effective areas or effective areas A Ac,effc,eff of the slender of the slender plate elements in compression are obtained with the help of the plate elements in compression are obtained with the help of the plate buckling reduction factorplate buckling reduction factor = = locloc 2011-9-27同济大学 吴冲 Tongji University, Wu Chong335. Plate-like buckling5. Plate-like bucklingFF5.1 Unstiffened plates 5.1 Unstiffened plates plate buckling reduction factorplate buckling reduction factorFig. 2.20: Examples of internal and outstand plate elements of cross sectionsinternal compression plate elements: 2011-9-27同济大学 吴冲 Tongji University, Wu Chong345. Plate-like buckling5. Plate-like bucklingFF5.1 Unstiffened plates 5.1 Unstiffened plates plate buckling reduction factorplate buckling reduction factoruuinternal compression plate elements internal compression plate elements where is the ratio of stresses at both edges of the plate withthe maximum compression stress in the denominatoruuoutstand compression plate elementsoutstand compression plate elements2011-9-27同济大学 吴冲 Tongji University, Wu Chong355. Plate-like buckling5. Plate-like bucklingFF5.1 Unstiffened plates 5.1 Unstiffened plates Relative plate slendernessRelative plate slendernessuuelastic critical plate buckling stress elastic critical plate buckling stress b is the appropriate plate width t is the plate thicknessE is elastic modulus of steel (E = 210000 N/mm2) is the Poisson coefficient of steel ( = 0.3)k is the plate buckling coefficient, dependant on the stress ratio and boundary conditions (for plates with = a/b 1.0 k is given in Table 2.2 and Table 2.3). For plates with 1.0 and subjected to uniform compression, k is given as 2011-9-27同济大学 吴冲 Tongji University, Wu Chong365. Plate-like buckling5. Plate-like bucklingFF5.1 Unstiffened plates 5.1 Unstiffened plates Effectivep width and plate buckling coefficient2011-9-27同济大学 吴冲 Tongji University, Wu Chong375. Plate-like buckling5. Plate-like bucklingFF5.1 Unstiffened plates 5.1 Unstiffened plates Effectivep width and plate buckling coefficient2011-9-27同济大学 吴冲 Tongji University, Wu Chong385. Plate-like buckling5. Plate-like bucklingFF5.1 Unstiffened plates5.1 Unstiffened platesIf maximum compressive stress If maximum compressive stress com,Edcom,Ed is less than is less than f fy y, effective , effective cross-section is larger and this beneficial effect may be accounted cross-section is larger and this beneficial effect may be accounted for by reducing relative plate slenderness. for by reducing relative plate slenderness. This procedure is This procedure is conservative and requires an iterative calculation. conservative and requires an iterative calculation. An alternative and more accurate procedure An alternative and more accurate procedure for internal compression elements for internal compression elements for outstand compression elementsfor outstand compression elements 2011-9-27同济大学 吴冲 Tongji University, Wu Chong395. Plate-like buckling5. Plate-like bucklingFF5.2 5.2 Longitudinally stiffened platesLongitudinally stiffened platesPlate-like buckling of longitudinally stiffened plates means a Plate-like buckling of longitudinally stiffened plates means a global buckling of the whole panel composed of a plate and global buckling of the whole panel composed of a plate and stiffeners (Fig. 2.21). stiffeners (Fig. 2.21). Fig. 2.21: Plate-like buckling of a stiffened plate2011-9-27同济大学 吴冲 Tongji University, Wu Chong405. Plate-like buckling5. Plate-like bucklingFF5.2 5.2 Longitudinally stiffened platesLongitudinally stiffened platesWhen sub-panels of width When sub-panels of width b bi i are slender and subjected to local are slender and subjected to local buckling, the interaction of local and global plate buckling buckling, the interaction of local and global plate buckling should be considered. This interaction is taken into account by should be considered. This interaction is taken into account by modified relative plate slenderness: modified relative plate slenderness: Fig. 2.22: Stiffened plate under uniform compression 2011-9-27同济大学 吴冲 Tongji University, Wu Chong415. Plate-like buckling5. Plate-like bucklingFF5.2 5.2 Longitudinally stiffened platesLongitudinally stiffened platesWhen the plate is under stress gradientWhen the plate is under stress gradientFig. 2.23: Stiffened web plate in bending 2011-9-27同济大学 吴冲 Tongji University, Wu Chong425. Plate-like buckling5. Plate-like bucklingFF5.2 5.2 Longitudinally stiffened platesLongitudinally stiffened platesWhen the plate is under stress gradient, effectiveWhen the plate is under stress gradient, effectivep p widths and contributing widths and contributing widths of the gross area are determined according to Table 4.1 and Table 4.2.widths of the gross area are determined according to Table 4.1 and Table 4.2. 2011-9-27同济大学 吴冲 Tongji University, Wu Chong435. Plate-like buckling5. Plate-like bucklingFF5.2 5.2 Longitudinally stiffened platesLongitudinally stiffened platesE Elastic critical stresslastic critical stress of of stiffened platesstiffened plates with three or more with three or more equally spaced stiffeners with aspect ratio equally spaced stiffeners with aspect ratio = =a/ba/b 0.5. 0.5. Isl is the second moment of area of the whole stiffened plate Ip = bt3/(12 (1-p2) = bt3/10.92 is the second moment area of the plate itselfAsl is the sum of the gross areas of individual longitudinal stiffenersAp is the gross area of the plate. 2011-9-27同济大学 吴冲 Tongji University, Wu Chong445. Plate-like buckling5. Plate-like bucklingFF5.2 5.2 Longitudinally stiffened platesLongitudinally stiffened platesElastic critical stress of stiffened plates with oElastic critical stress of stiffened plates with one stiffener in the ne stiffener in the compression zone compression zone Fig. 2.24: Web plate with a single stiffener in compression2011-9-27同济大学 吴冲 Tongji University, Wu Chong455. Plate-like buckling5. Plate-like bucklingFF5.2 5.2 Longitudinally stiffened platesLongitudinally stiffened platesElastic critical stress of stiffened plates with Elastic critical stress of stiffened plates with two stiffeners in the two stiffeners in the compression zone compression zone Fig. 2.25: Model with two longitudinal stiffeners in compression zone 2011-9-27同济大学 吴冲 Tongji University, Wu Chong465. Plate-like buckling5. Plate-like bucklingFF5.2 5.2 Longitudinally stiffened platesLongitudinally stiffened platesElastic critical stress of stiffened plates with Elastic critical stress of stiffened plates with two stiffeners in the two stiffeners in the compression zone compression zone uuAbove procedure for one stiffener is repeated three timesAbove procedure for one stiffener is repeated three timesfor each of the stiffeners considering the other stiffener as for each of the stiffeners considering the other stiffener as a rigid support to a web a rigid support to a web for a fictive lumped stiffener defined as follows: for a fictive lumped stiffener defined as follows: l lthe cross-section properties are the sum of the the cross-section properties are the sum of the properties of individual stiffenersproperties of individual stiffenersl lthe lumped stiffener is located in the point of the lumped stiffener is located in the point of application of the stress resultants acting in individual application of the stress resultants acting in individual stiffeners stiffeners may be calculated with may be calculated with uuThe smallest of the three results for The smallest of the three results for cr,pcr,p is taken into is taken into account.account.2011-9-27同济大学 吴冲 Tongji University, Wu Chong476. 6. ColumnColumn-like buckling-like bucklingFF6.1 Unstiffened plates6.1 Unstiffened platesThe plate is considered completely unsupported along the The plate is considered completely unsupported along the longitudinal edges longitudinal edges The corresponding column reduction factor The corresponding column reduction factor c c is obtained from is obtained from 6.3.1.2 of EN 1993-1-1 by selecting a buckling curve 6.3.1.2 of EN 1993-1-1 by selecting a buckling curve a a ( ( = 0.21) = 0.21) 2011-9-27同济大学 吴冲 Tongji University, Wu Chong486. 6. ColumnColumn-like buckling-like bucklingFF6.2 6.2 Longitudinally stiffened plates Longitudinally stiffened plates 加劲是纵向加劲肋加劲是纵向加劲肋加劲是纵向加劲肋加劲是纵向加劲肋The relative column slenderness for stiffened plates is defined The relative column slenderness for stiffened plates is defined as: as: Asl,1 is the gross cross-sectional area of the stiffener and adjacent parts of the plateAsl,1,eff is the effectivep cross-sectional area of the stiffener and adjacent parts of the plate with due allowance for plate buckling of sub-panels2011-9-27同济大学 吴冲 Tongji University, Wu Chong496. 6. ColumnColumn-like buckling-like bucklingFF6.2 6.2 Longitudinally stiffened platesLongitudinally stiffened platesThe elastic critical column buckling stress The elastic critical column buckling stress cr,ccr,c is determined as is determined as the buckling stress the buckling stress cr.slcr.sl of a single stiffener closest to the panel of a single stiffener closest to the panel edge having the highest compression stressedge having the highest compression stress: : uuA Asl,1sl,1 , I , Isl,1sl,1 are the area and the second moment of area of the are the area and the second moment of area of the gross cross-section of the stiffener and the adjacent parts of gross cross-section of the stiffener and the adjacent parts of the plate (out-of-plane bending of the plate). the plate (out-of-plane bending of the plate). uua a is the buckling length of a stiffener normally equal to the is the buckling length of a stiffener normally equal to the distance between rigid transverse stiffeners (a panel length). distance between rigid transverse stiffeners (a panel length). Note that for large distances between transverse stiffeners this Note that for large distances between transverse stiffeners this leads to small leads to small cr,ccr,c and plate-like buckling completely prevails and plate-like buckling completely prevails (plate buckling in several half waves). (plate buckling in several half waves). 2011-9-27同济大学 吴冲 Tongji University, Wu Chong506. 6. ColumnColumn-like buckling-like bucklingFF6.2 6.2 Longitudinally stiffened platesLongitudinally stiffened platesThe corresponding column reduction factor The corresponding column reduction factor c c is obtained from is obtained from 6.3.12 of EN 1993-1-1. For stiffened plates increased values of 6.3.12 of EN 1993-1-1. For stiffened plates increased values of imperfection parameter imperfection parameter should be used: should be used: = 0.34 (buckling curve b) for closed stiffeners = 0.49 (buckling curve c) for open stiffenerse = max (e1, e2)e1 for single-sided stiffeners is the distance between the centres of gravity of the stiffener alone Gst and the stiffener with the contributing plating Gsl (see Fig. 2.26a) ande2 is the distance between the centres of gravity of the contributing plating alone Gp and the stiffener with the contributing plating Gsl. For double-sided symmetrical stiffeners e1 = e2 (see Fig. 2.26b). Fig. 2.26: Definition of distances e1 and e22011-9-27同济大学 吴冲 Tongji University, Wu Chong517. 7. Interaction between plate-like and column-like bucklingInteraction between plate-like and column-like bucklingFF7.1 7.1 Interaction relation to obtain the final reduction factor Interaction relation to obtain the final reduction factor c c and care plate buckling reduction factor and column buckling reduction factor of unstiffened or stiffened platescr,p and cr,c are elastic critical stresses for plate-like and column-like buckling of unstiffened or stiffened platesFig. 2.28: Interpolation between plate-like and column-like behaviour2011-9-27同济大学 吴冲 Tongji University, Wu Chong527. 7. Interaction between plate-like and column-like bucklingInteraction between plate-like and column-like bucklingFF7.2 7.2 The final effectivep area of the compression zone The final effectivep area of the compression zone A Ac,effc,eff Unstiffened plateUnstiffened plateLongitudinally stiffened plates Longitudinally stiffened plates If cfy/ M1 is smaller than the average stress com,Ed in the column (stiffener), the effectivep area of that column should be reduced accordingly:2011-9-27精品课件精品课件！2011-9-27精品课件精品课件！2011-9-27同济大学 吴冲 Tongji University, Wu Chong55Thank Youfor your attention