Compression Memers Loal Bukling and Setion Classifiation Summary: Strutural setions may e onsidered as an assemly of individual plate elements. Plate elements may e internal (e.g. the wes of open eams or the flanges of oxes) and others are outstand (e.g. the flanges of open setions and the legs of angles). Loaded in ompression these plates may ukle loally. Loal ukling may limit the setion apaity y preventing the attainment oield strength. Premature failure (y loal ukling) may e avoided y limiting the width to thikness ratio (or slenderness) of individual elements within the ross setion. This is the asis of the setion lassifiation approah. EC3 defines four lasses of rosssetion. The lass into whih a partiular rosssetion falls depends on the slenderness of eah element and the ompressive stress distriution. Ojetives: Setions may fail y ompressive ukling of plates within the setion. Distinguish etween internal and outstand elements. Demonstrate that plate slenderness and edge restraints ontrol the ukling ehaviour. Sketh the relationship etween normalised ultimate ompressive stress and normalised plate slenderness Explain the meaning of different setion lassifiations. Derive a result from EC3 Tales for hot rolled setions. Use the setion lassifiation method to hoose appropriate setions. Desrie the effetive width approah for Class 4 setions. Referenes: Euroode 3: Design of steel strutures Part. General rules and rules for uildings The Behaviour and Design of Steel Strutures, Chapter 4 Loal ukling of thin plate elements, N S Trahair and M A Bradford, E & FN Spon, Revised Seond Edition 994 Contents: Introdution Classifiation Behaviour of plate elements in ompression Effetive width approah to design of Class 4 setions Conluding summary
. Introdution Strutural setions, rolled or welded, may e onsidered as an assemly of individual plate elements. Most of these elements (figure ), if in ompression, an e separated into two ategories: Internal or stiffened elements: these elements are onsidered to e simply supported along two edges parallel to the diretion of ompressive stress. Outstand or unstiffened elements; these elements are onsidered to e simply supported along one edge and free on the other edge parallel to the diretion of ompressive stress. Outstand Internal Outstand Internal We Internal We We Internal Flange (a) Rolled Isetion Flange () Hollow setion () Welded ox setion Flange Figure Internal or outstand elements As the plate elements in strutural setions are relatively thin ompared with their width, when loaded in ompression (as a result of axial loads and/or from ending) they may ukle loally. The disposition of any plate element within the ross setion to ukle may limit the axial load arrying apaity, or the ending resistane of the setion, y preventing the attainment oield. Avoidane of premature failure arising from the effets of loal ukling may e ahieved y limiting the widthtothikness ratio for individual elements within the ross setion.. Classifiation EC3 defines four lasses of ross setion. The ross setion lass depends upon the slenderness of eah element (defined y a widthtothikness ratio) and the ompressive stress distriution i.e. uniform or linear. The lasses are defined as performane requirements for ending moment resistane: Euroode 3 5. 3. () or 5.5. Class rosssetions that an form a plasti hinge with the required rotational apaity for plasti analysis. Class rosssetions that, although ale to develop a plasti moment, have limited rotational apaity and are therefore unsuitale for plasti design. Class 3 rosssetions that the alulated stress in the extreme ompression fire an reah yield ut loal ukling prevents the development of the plasti moment resistane. Class 4 rosssetions that in whih loal ukling limits the moment resistane (or ompression resistane for axially loaded memers). Expliit allowane for the effets of loal ukling is neessary.
Tale summarises the lasses in terms of ehaviour, moment apaity and rotational apaity. Moment Moment Model of Behaviour Loal Bukling Moment Resistane Plasti moment on gross setion Plasti moment on gross setion M M Rotation Capaity Suffiient Limited rot pl pl Class Loal Bukling Moment M el Loal Bukling Elasti moment on gross setion M None pl 3 Moment Loal Bukling M el Plasti moment on effetive setion M None M el elasti moment resistane of rosssetion plasti moment resistane of rosssetion M applied moment rotation (urvature) of setion rotation (urvature) of setion required to generate fully plasti stress distriution pl aross setion pl pl 4 Tale Crosssetion lassifiations in terms of moment resistane and rotation apaity. The moment resistanes for the four lasses defined aove are: for Classes and : the plasti moment ( = W pl. ) for Class 3: the elasti moment (M el = W el. ) for Class 4: the loal ukling moment (M o < M el ). 3
3. Behaviour of plate elements in ompression A thin flat retangular plate sujeted to ompressive fores along its short edges has an elasti ritial ukling stress (σ r ) given y: k E σ = π σ r ( ν ) t () Where k σ is the plate ukling parameter whih aounts for edge support onditions, stress distriution and aspet ratio of the plate see figure a. 3..5 () ν= Poisson s oeffiient, E = Young s modulus 3..5 () t L (a) () Simply supported on all four edges Bukling oeffiient k 5 Simply supported edge 4 3 Exat L Free k = 0.45 (/L) Free edge L () (d) 0 0.45 3 4 5 Plate aspet ratio L / Figure Behaviour of plate elements in ompression. (Trahair and Bradford) The elasti ritial ukling stress (σ r ) is thus inversely proportional to (/t) and analogous to the slenderness ratio (L/i) for olumn ukling. Open strutural setions omprise a numer of plates that are free along one longitudinal edge (figure ) and tend to e very long ompared with their width. These plates ukled shape is seen in figure. The relationship etween aspet ratio and ukling parameter for a long thin outstand element of this type is shown in figure d. The ukling parameter tends towards a limiting value of 0.45 as the plate aspet ratio inreases. For a setion to e lassified as lass 3 or etter the elasti ritial ukling stress (σ r ) must exeed the yield stress. From equation () (sustituting ν = 0.3 and rearranging) this will e so if ( ) 0,5 /t < 0,9 kσ E/ () This expression is general as the effet of stress gradient, oundary onditions and aspet ratio are all enompassed within the ukling parameter k σ. 4
Tale gives values for k σ for internal and outstand elements under various elasti stress distriutions. Support onditions at long edges Clamped lamped Clamped simply supported Simply supported simply supported Clamped free Simply supported free Free free Bukling fator kσ 6,97 5,40 4,00,8 0,43 (/a) a Various support onditions a/ >> σ σ σ σ σ σ I II III σ = is maximum stress, ompression ψ = σ / σ > ψ > 0 0 0 > ψ > Case I Internal element Case II Outstand element Case III Outstand element 4,0 8,0,05 ψ 7,8 7,86,9ψ9,78ψ 3,9 0,43 0,570,ψ0,07ψ 0,57 0,570,ψ0,07ψ 0,85 0,43 0,578 ψ0,34,70,75ψ7,ψ 3,8 Tale Bukling fators and stress distriution. The elastiplasti ehaviour of a perfet plate element sujet to uniform ompression may e represented y a normalised loadslenderness diagram where normalised ultimate load, Np, and normalised plate slenderness, λ p, are given y: Np = σ ult / (3) ( ) 0, 5 λ p = f σ y / r Sustituting equation () for σ r into (4), and replaing with 35/ε (so that the expression may e used for any grade of material) the normalised plate slenderness, λ p, may e expressed as (4) 0.5 f y = / t λ p = σ r 8.4ε k σ (5) where is the appropriate width for the type of element and rosssetion type. 5
Figure 3 shows the relationship etween Np and λ p. u N p = σ f Class 3 y Class Class Euler Bukling Stress 0,5 0,6 0,9,0 λ p Figure 3 Dimensionless representation of the elastiplasti ukling stress. For normalized plate slenderness less than one, the normalised ultimate load an reah its squash load. For greater values of λ p, Np dereases as the plate slenderness inreases, the ultimate stress sustained eing limited to the elasti ritial ukling stress, σ r. The atual ehaviour is somewhat different from the ideal elastiplasti ehaviour due to: i. initial geometrial and material imperfetions, ii. strainhardening of the material, iii. the postukling ehaviour. These fators require λ p values to e redued. This is made to delay the onset of loal ukling until the requisite strain distriution through the setion (yield at the extreme fire or fully plasti distriution) has een attained. EC3 uses the following normalised plate slenderness as limits for lassifiations: 5...4 (7) Class λ p < 0,5 Class λ p< 0,6 Class 3 λ p < 0,9 for elements under a stress gradient; this is further redued to 0,74 for elements in ompression throughout. By sustituting the appropriate values of k σ into equation (5) and noting the λ p to e used for eah lass, limiting /t ratios an e alulated. Tales 47 are EC3 extrats giving the limiting proportions for ompression elements from lass to 3. When any of the ompression elements within a setion fail to satisfy the limit for lass 3 the whole setion is lassified as lass 4 (ommonly referred to as slender), and loal ukling should e taken into aount in the design using an effetive ross setion. 6
a. Wes: (internal elements perpendiular to axis of ending) tf tw Axis of Bending d tw d tw d tw h d = h3t (t = tf = t w ) Class Stress distriution in element (ompression positive) We sujet to ending d/t w < _ 7ε Stress distriution in element (ompression positive) d/ d/t w < _ 83 ε d h h d/ 3 d/t w < _ 4 ε We sujet to ompression We sujet to ending and ompression fy fy αd d h d h when α > 0,5: d/t w < 396ε/(3α ) d/t _ w < _ 33 ε when α < 0,5: d/t w _ < 36ε/α d/t w < _ 38 ε when α > 0,5: d/t w < _ 456ε/(3α ) when α < 0,5: d/t w < _ 4,5ε/α d/t w < _ 4 ε d h d h ψ when ψ > : d/t w _ < 4ε/(0,67 0,33ψ) when ψ < _ : d/t w _ < 6ε/( ψ) ( ψ ) ε = 35 / fy ε 35 75 355 0,9 0,8 Tale 4 Maximum widthtothikness ratios for ompression elements.. Internal flange elements: (internal elements parallel to axis of ending) axis of ending t f tf tf tf Class Type Setion in ending Setion in ompression Stress distriution in element and aross setion (ompression positive) Rolled hollow setion Other Rolled hollow setion Other Stress distriution in element and aross setion (ompression positive) ( 3t f )/ t f <33ε _ / t f _ <33ε ( 3t f)/ tf _ <38ε / t f _ <38ε fy fy ( 3t f)/ tf / t f ( 3t f)/ tf / t f fy _ <4ε * <4ε _ <4ε _ * <4ε _ 3 ε = 35/ Rolled hollow setion Other fy ε ( 3t f)/ tf / t f _ <4ε _ <4ε ( 3t f)/ tf / t f 35 75 335 0,9 0,8 * For a ross setion in ompression with no ending the lassifiation,,3 are irrelevant and hene the limit is the same in eah ase. <4ε _ * _ <4ε Tale 5 Maximum widthtothikness ratios for ompression elements. 7
. Outstand flanges: t f t f t f t f Rolled setions Class Type of setion Flange sujet to ompression Stress distriution in element (ompression positive) Welded setions Flange sujet to ompression and ending Tip in Tip in ompression tension α α Rolled Welded /t f < _ 0ε /t f _ < 9ε /t f < _ 0ε α /t f <_ 9e α 0ε /t f <_ α α 9ε /t f <_ α α Rolled Welded /t f < _ ε /t f < _ 0ε /t f < _ ε α /t f <_ 0ε α ε /t f <_ α α /t 0ε f <_ α α Stress distriution in element (ompression positive) 3 Rolled Welded /t f < _ 5ε /t f < _ 4ε /t f <_ 3ε k σ /t f <_ 3ε k σ For k σ see figure d and tale 8 ε = 35/ fy ε 35 75 355 0,9 0,8 Tale 6 Maximum widthtothikness ratios for ompression elements. d. Angles: Refer also to. 'Outstand flanges' (Tale 6) t h (Does not apply to angles in ontinuous ontat with other omponents). t Class Setion in ompression Stress distriution aross setion (ompression positive) 3 h t e. Tuular setions: h 5 ε: 5, ε t t t d Class Setion in ending and/or ompression d / t 50ε d/ t 70ε 3 d/ t 90ε ε= 35/ ε 0,9 0,8 35 75 355 ε 0,85 0,66 Tale 7 Maximum widthtothikness ratios for ompression elements. 8
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4. Effetive width approah to design of Class 4 setions For memers with Class 4 setions the effet of loal ukling on gloal ehaviour at the ultimate limit state is suh that the elasti resistane, alulated on the assumption oielding of the extreme fires of the gross setion (riteria for Class 3 setions), annot e ahieved. Figure 4 gives the moment defletion urve for a point loaded eam (Class 4). Figure 4 Moment versus defletion urve of a pointed loaded eam. The reason for the redution in strength is that loal ukling ours at an early stage in parts of the ompression elements of the memer; the stiffness of these parts in ompression is therey redued and the stresses are distriuted to the stiffer edges, see Figure 5. Figure 5 Strain/stress distriution of a memer with dek plate loal ukling in ompression.
To allow for the redution in strength the atual non linear distriution of stress is taken into aount y a linear distriution of stress ating on a redued "effetive plate width" leaving an "effetive hole" where the ukle ours, Figure 5. By applying this model an "effetive rosssetion" is defined for whih the resistane is then alulated as for Class 3 setions (y limiting the stresses in the extreme fires to the yield strength). The effetive widths eff are alulated on the asis of the Winter formula: eff = ρ. Redution oeffiient ρ depends on the plate slenderness p defined y plate uking theory, Figure 6. Figure 6 Redution oeffiient ρ for the effetive width. Crosssetions with lass 4 elements may e replaed y an effetive rosssetion, taken as the gross setion minus holes where the ukles may our, and then designed in a similar manner to lass 3 setions using elasti rosssetional resistane limited y yielding in the extreme fires. Effetive widths of ompression elements may e alulated y use of a redution fator ρ whih is dependent on the normalised plate slenderness λ p (whih is in turn dependent on the stress distriution and element oundaries through appliation of the ukling parameter k σ ) as follows: ρ = ( λ p) ( λ p) 0, (6) The redution fator ρ may then e applied to outstand or internal element as shown in Tales 8 and 9.
Stress distriution (ompression positive) Effetive width eff eff σ > ψ 0: σ eff = ρ t ψ < 0: σ = ρ = ρ/( ψ) eff σ eff ψ = σ / σ 0 ψ Bukling fator k σ 0,43 0,57 0,85 0, 57 0, ψ 0, 07ψ eff > ψ 0: σ σ eff = ρ σ eff ψ<0: σ = ρ = ρ/( ψ) eff t ψ = σ / σ Bukling fator k σ 0,43 > ψ > 0 0, 578 ψ 034, 0,70 0> ψ > 7, 5ψ 7, ψ Tale 8 Effetive widths of outstand ompression elements. 3,8 3
Stress distriution (ompression positive) Effetive width eff σ σ e e σ σ ψ = : = 3t eff = ρ e = 0,5 eff e = 0,5 eff >ψ _ > 0: = 3t eff = ρ e e e = eff 5 ψ e = eff e t ψ < 0: σ σ = 3t = ρ = ρ/( ψ ) eff e e e = 0,4 eff e = 0,6 eff ψ = σ / σ > ψ > 0 0 0> ψ > > ψ > Bukling fator k σ 4,0 8, 05, ψ 7,8 7, 8 6, 9ψ 9, 78ψ 3,9 598, ( ψ ) 6 Alternatively, for _ > ψ _ > : k σ = ψ 0 ψ 0, 5 [( ), ( ) ] ( ψ ) Illustrated as rhs. For other setions = d for wes = for internal flange elements (exept rhs) Tale 9 Effetive widths of ompression elements Figure 7 shows examples of effetive rosssetions for memers in ompression or ending. Notie that the effetive rosssetion entroidal axis may shift relative to the gross rosssetion. For ending memers this will e onsidered when alulating the effetive setion properties. For axial fore memers the shift of the entroidal axis will give rise to a moment that should e aounted for in memer design. 4
Centroidal axis of gross rosssetion Centroidal axis of effetive rosssetion Centroidal axis of gross rosssetion e N Noneffetive zones Gross rosssetion (a) Class 4 rosssetions axial fore Centroidal axis e M Noneffetive zone Centroidal axis of effetive setion e M Noneffetive zone Centroidal axis Centroidal axis of effetive setion Gross rosssetion () Class 4 rosssetions ending moment Figure 7 Effetive rosssetions for lass 4 in ompression and ending 6. Conluding summary Strutural setions may e onsidered as an assemly of individual plate elements. When loaded in ompression these plates may ukle loally. Loal ukling within the rosssetion may limit the load arrying apaity of the setion y preventing the attainment oield strength. Premature failure (from loal ukling) may e avoided y limiting the width to thikness ratio or slenderness of individual elements within the ross setion. This is the asis of the setion lassifiation approah. EC3 defines 4 lasses of rosssetion. The lass into whih a partiular rosssetion falls depends upon the slenderness of eah element and the ompressive stress distriution. Additional reading [] Salmon, C.G., Johnson, J.E., "Steel Strutures. Design and Behaviour", Harper et Row, New York. [] Duas, P., Gehri, E., "Behaviour and Design of Steel Plated Strutures", Pu. 44, ECCS, TC8, 986. [3] Bulson, P.S., "The Staility of Flat Plates" Chatto and Windus, London. 5