Compression Members Local Buckling and Section …

1 Compression Members Local Buckling and Section classification Summary: Structural sections may be considered as an assembly of individual plate elements. Plate elements may be internal ( the webs of open beams or the flanges of boxes) and others are outstand ( the flanges of open sections and the legs of angles). Loaded in Compression these plates may buckle locally. Local Buckling may limit the Section capacity by preventing the attainment of yield strength. Premature failure (by Local Buckling ) may be avoided by limiting the width to thickness ratio (or slenderness) of individual elements within the cross Section . This is the basis of the Section classification approach. EC3 defines four classes of cross- Section . The class into which a particular cross- Section falls depends on the slenderness of each element and the compressive stress distribution. Objectives: Sections may fail by compressive Buckling of plates within the Section .

2 Distinguish between internal and outstand elements. Demonstrate that plate slenderness and edge restraints control the Buckling behaviour. Sketch the relationship between normalised ultimate compressive stress and normalised plate slenderness Explain the meaning of different Section classifications. Derive a result from EC3 Tables for hot rolled sections. Use the Section classification method to choose appropriate sections. Describe the effective width approach for Class 4 sections. References: Eurocode 3: Design of steel structures Part General rules and rules for buildings The Behaviour and Design of Steel Structures, Chapter 4- Local Buckling of thin plate elements, N S Trahair and M A Bradford, E & FN Spon, Revised Second Edition 1994 Contents: Introduction classification Behaviour of plate elements in Compression Effective width approach to design of Class 4 sections Concluding summary 21.

3 Introduction Structural sections, rolled or welded, may be considered as an assembly of individual plate elements. Most of these elements (figure 1), if in Compression , can be separated into two categories: Internal or stiffened elements: these elements are considered to be simply supported along two edges parallel to the direction of compressive stress. Outstand or unstiffened elements; these elements are considered to be simply supported along one edge and free on the other edge parallel to the direction of compressive stress. OutstandInternalWebFlangeWebInternalFlan ge(a) Rolled I- Section (b) Hollow sectionFlange(c) Welded box sectionInternalOutstandInternalWeb Figure 1 - Internal or outstand elements As the plate elements in structural sections are relatively thin compared with their width, when loaded in Compression (as a result of axial loads and/or from bending) they may buckle locally.

4 The disposition of any plate element within the cross Section to buckle may limit the axial load carrying capacity, or the bending resistance of the Section , by preventing the attainment of yield. Avoidance of premature failure arising from the effects of Local Buckling may be achieved by limiting the width-to-thickness ratio for individual elements within the cross Section . 2. classification EC3 defines four classes of cross Section . The cross Section class depends upon the slenderness of each element (defined by a width-to-thickness ratio) and the compressive stress distribution uniform or linear. The classes are defined as performance requirements for bending moment resistance: Eurocode 3 5. (1) or Class 1 - cross-sections that can form a plastic hinge with the required rotational capacity for plastic analysis. Class 2 - cross-sections that, although able to develop a plastic moment, have limited rotational capacity and are therefore unsuitable for plastic design.

5 Class 3 - cross-sections that the calculated stress in the extreme Compression fibre can reach yield but Local Buckling prevents the development of the plastic moment resistance. Class 4 - cross-sections that in which Local Buckling limits the moment resistance (or Compression resistance for axially loaded Members ). Explicit allowance for the effects of Local Buckling is necessary. 3 Table 1 summarises the classes in terms of behaviour, moment capacity and rotational capacity. fyMomentLocalBucklingfyMomentLocalBuckli ngfyMomentLocalBucklingMelfyMomentLocalB ucklingMelModel ofBehaviourMomentResistanceRotation CapacityClass11111111 SufficientLimitedNoneNoneMMplMMplMMplMMp l1234 Plastic momenton gross sectionPlastic momenton gross sectionElastic momenton gross sectionPlastic moment oneffective sectionMplMplMplMplMel elastic moment resistance of cross-sectionMpl plastic moment resistance of cross-sectionM applied moment rotation (curvature) of Section rotation (curvature) of Section required to generate fully plastic stress distribution across sectionpl pl rotplplplpl Table 1 - Cross- Section classifications in terms of moment resistance and rotation capacity.

6 The moment resistances for the four classes defined above are: for Classes 1 and 2: the plastic moment (Mpl = Wpl . fy) for Class 3: the elastic moment (Mel = Wel . fy) for Class 4: the Local Buckling moment (Mo < Mel). 43. Behaviour of plate elements in Compression A thin flat rectangular plate subjected to compressive forces along its short edges has an elastic critical Buckling stress ( cr ) given by: ()222112 =btEkcr (1) Where k is the plate Buckling parameter which accounts for edge support conditions, stress distribution and aspect ratio of the plate - see figure 2a. (1) = Poisson s coefficient, E = Young s modulus (1) (d)(c)(b)(a)Simply supported onall four edgestLbSimply supportededgeFreeedgebL12345123045 Plate aspect ratio L / bBuckling coefficient kbLFreeExactk = + (b/L) Figure 2 - Behaviour of plate elements in Compression .

7 (Trahair and Bradford) The elastic critical Buckling stress ( cr ) is thus inversely proportional to (b/t)2 and analogous to the slenderness ratio (L/i) for column Buckling . Open structural sections comprise a number of plates that are free along one longitudinal edge (figure 2b) and tend to be very long compared with their width. These plates buckled shape is seen in figure 2c. The relationship between aspect ratio and Buckling parameter for a long thin outstand element of this type is shown in figure 2d. The Buckling parameter tends towards a limiting value of as the plate aspect ratio increases. For a Section to be classified as class 3 or better the elastic critical Buckling stress ( cr ) must exceed the yield stress fy . From equation (1) (substituting = and rearranging) this will be so if ()0,5yE/fk0,92<b/t (2) This expression is general as the effect of stress gradient, boundary conditions and aspect ratio are all encompassed within the Buckling parameter k.

8 5 Table 2 gives values for k for internal and outstand elements under various elastic stress distributions. Support conditions at long edgesClamped + clampedClamped + simply supportedSimply supported + simply supportedClamped + freeSimply supported + freeFree + freeBuckling factor k Various supportconditions6,975,404,001,280,43(b/ a)2aba/b >> 1 = is maximum stress, Compression = 2 / 1 +1 1 > > 0 0 0 > > -1 Case I Internal element 4,0 8,02 1,05 + 7,81 7,81+6,29 +9,78 2 23,9 Case II Outstand element 0,43 0,57-0,21 +0,07 2 0,57 0,57-0,21 +0,07 2 0,85 Case III Outstand element 0,43 0,578 +0,34 1,70 1,7-5 +17,1 2 23,8 Table 2 - Buckling factors and stress distribution. The elastic-plastic behaviour of a perfect plate element subject to uniform Compression may be represented by a normalised load-slenderness diagram where normalised ultimate load, Np, and normalised plate slenderness, p, are given by: Np = ult / fy (3) ()5,0/crypf = (4) Substituting equation (1) for cr into (4), and replacing fy with 235/ 2 (so that the expression may be used for any grade of material) the normalised plate slenderness, p, may be expressed as = = tbfcryp (5) where b is the appropriate width for the type of element and cross- Section type.

9 1 2 2 1 1 2 I II III 6 Figure 3 shows the relationship between Np and p. Class 1 Class 2 Class 3 Euler Buckling Stress0,5 0,60,911,0 pNfpuy= Figure 3 - Dimensionless representation of the elastic-plastic Buckling stress. For normalized plate slenderness less than one, the normalised ultimate load can reach its squash load. For greater values of p, Np decreases as the plate slenderness increases, the ultimate stress sustained being limited to the elastic critical Buckling stress, cr. The actual behaviour is somewhat different from the ideal elastic-plastic behaviour due to: i. initial geometrical and material imperfections, ii. strain-hardening of the material, iii. the postbuckling behaviour. These factors require p values to be reduced. This is made to delay the onset of Local Buckling until the requisite strain distribution through the Section (yield at the extreme fibre or fully plastic distribution) has been attained.

10 EC3 uses the following normalised plate slenderness as limits for classifications: (7) Class 1 p < 0,5 Class 2 p< 0,6 Class 3 p < 0,9 for elements under a stress gradient; this is further reduced to 0,74 for elements in Compression throughout. By substituting the appropriate values of k into equation (5) and noting the p to be used for each class, limiting b/t ratios can be calculated. Tables 4-7 are EC3 extracts giving the limiting proportions for Compression elements from class 1 to 3. When any of the Compression elements within a Section fail to satisfy the limit for class 3 the whole Section is classified as class 4 (commonly referred to as slender), and Local Buckling should be taken into account in the design using an effective cross Section . 7a. Webs: (internal elements perpendicular to axis of bending)twtwdtwdtwtfhdAxis ofBendingd = h-3t (t = tf = tw )ClassWeb subject tobendingWeb subject tocompressionWeb subject to bendingand compressionStressdistribution inelement(compressionpositive)+ fyfy -dh+ fy+ fyfy -fy -dhhd dwhen > 0,5:d/tw < 396 /(13 1)when < 0,5:d/tw < 36 / __1d/tw < 72 _d/tw < 33 _d/tw < 83 _d/tw < 38 _2when > 0,5:d/tw < 456 /(13 1)_when < 0,5:d/tw < 41,5 / _Stressdistribution inelement(compressionpositive)+ fyfy -+ fy+ fy fy -d/2d/2h+ dhdh3d/tw < 124 _d/tw < 42 _when > 1:d/tw < 42 /(0,67 + 0,33 )_when < 1:_d/tw < 62 /(1 )( ) =fy/235fy 23527535510,920,81_ Table 4 - Maximum width-to-thickness ratios for Compression elements.