AN IMPROVED EFFECTIVE WIDTH METHOD BASED ON THE THEORY OF PLASTICITY

Transcription

1 Advanced Steel Construction Vol. 6, No., pp () 55 AN IMPROVED EFFECTIVE WIDTH METHOD BASED ON THE THEORY OF PLASTICITY Thomas Hansen *, Jesper Gath and M.P. Nielsen ALECTIA A/S, Teknikeryen 34, DK-83 Virum, Denmark *(Corresponding author: Received: May 7; Revised: 7 July 8; Accepted: 7 Septemer 8 ABSTRACT: Currently, calculations of plates in compression are ased on the semi-empirical effective width method which was developed y Winter et al. The effective width method takes the post-uckling capacity into account. The aim of the paper is to estalish an effective width method, which is derived on the asis of a consistent theory. The method rests on the theory of plasticity, particularly the yield line theory. Emphasis is on uckling prolems related to plate girders. Two general cases are studied: Plates in uniaxial compression supported along all edges, cf. the compressed flange in a ox girder, and plates with one free edge, cf. the compressed flange and the transverse we stiffeners in an I-shaped girder. The results presented coincide closely with Winter s formulae and with tests. Keywords: Post-uckling, thin plates, theory of plasticity, yield line theory, effective width method, failure mechanisms, staility, in-plane loading. INTRODUCTION It was deduced many years ago that the elastic uckling theory is not ale to accurately account for the real strength of plates (Schuman and Back []). The main reason for this is that, in a large parameter interval, the ultimate load is reached after yielding of the plate. This fact was also pointed out y Kármán et al. [8], who suggested modifying the elastic solution y an empirical coefficient. This idea was taken up y Winter [5], who developed accurate formulae for the two important cases considered in the following. Winter s method is an the effective width method, on which calculations of plates in compression are still ased. Any design method thus has to take into account that plates in compression may carry loads much larger than the load for which elastic uckling will occur. The aim is here to estalish an effective width method, which is derived on the asis of a consistent theory. The method rests on the theory of plasticity, particularly the yield line theory. The new constriution of the paper lies in the way the yield line theory is applied on the deflected shape of the plate. The emphasis is attached to uckling prolems related to plate girders. Two general cases are studied: Plates in uniaxial compression supported along all edges, cf. the compressed flange in a ox girder, and plates with one free edge, cf. the compressed flange and the transverse we stiffeners in an I-shaped girder. The asics of the theory are descried in Chapter. How to take geometrical second order effects into account is illustrated y a column example. Solutions for plates in uniaxial compression supported along all edges are presented in Chapter 3, while solutions for plates with one free edge are treated in Chapter 4. Both chapters contain a comparison etween the theory and the experimental results, on which Winter s formulae were ased. The Hong Kong Institute of Steel Construction

2 56 An Improved Effective Width Method Based on the Theory of Plasticity The effect of imperfections is touched upon in Chapter 5 and other possile applications of the theory are mentioned in Chapter 6.. Historical Overview Bryan [] was the first to develop a solution for the elastic critical stress of a rectangular plate simply supported along all edges and sujected to a uniform longitudinal compressive stress. Later a large numer of solutions using Bryan s equation have een derived, see for instance (Timoshenko and Gere [4]). In general the elastic critical stress may e expressed as E t cr k () in which E is Young s modulus, Poisson s ratio, t/ the thickness-to-width ratio and k is a uckling coefficient, which is a function of plate geometry and oundary conditions. Useful information on k-factors may e found in a numer of references, e.g. (Timoshenko and Gere [4]). Tests y Schuman and Back [] on plates supported y V-grooves along the unloaded edges demonstrated that, for plates of the same thickness, an enhancement of the plate width eyond a certain value did not increase the ultimate load. Wider plates acted as though narrow side portions or effective load-carrying areas took most of the load. Furthermore, the ultimate load was found to e up to thirty times larger than the elastic critical uckling load determined y Bryan s equation, cf. Eq.. It is now well known that the post-uckling resistance of plates is due to redistriution of axial compressive stresses, and to a lesser extent, to tensile memrane effects and to shear that accompany the out-of-plane ending of the plate in oth longitudinal and transverse directions. The longitudinal stresses tend to concentrate in the vicinity of the longitudinally supported edges, which are the stiffer parts of the uckled plate. As a result, yielding egins along these edges, which limits the load-carrying capacity. Several researchers have een prompted y the tests of Schuman and Back [] to develop expressions for the ultimate strength of such plates. The first to use the effective width concept in handling this prolem was Kármán et al. [8]. They derived the following approximate formula for the effective width, e, of plates supported along all edges, ased on the assumption that two strips with total width, e, along the sides, each on the verge of uckling, carry the entire load: e 3 t E e () Here e is the edge stress along e and the remaining notation is as in Eq.. As a result of many tests and studies of post-uckling strength, Winter [5] suggested the following formula for the effective width: E t E e.9t..574 e e (3)

3 Thomas Hansen, Jesper Gath and M.P. Nielsen 57 This equation has een modified several times over the past. In the latest version, adopted in Eurocode for steel structures, EC3 [4], the formula is written as.. e cr cr e e (4) In the calculation of the ultimate compression load, the edge stress, e, is taken to e equal to the yield stress of the plate material. The elastic critical stress, cr, is determined y Eq.. Eq. 4 corresponds to Eq. 3 if the factor.574 is sustituted with a factor of.45. For plates supported along only one longitudinal edge, the effective width has een experimentally determined y Winter [5]. The original formula was E t E e.5t..333 e e (5) This equation has also een modified several times. In the latest version, adopted in EC3 [4], it is written as..88 e cr cr e e (6) In the 993-edition of EC3 [3], Eq. 4 referred to the case of plates supported along all edges and plates with one free edge. Hence, the difference etween the two cases is only the value of the uckling coefficient, k. It is practical and convenient to have only one expression. The reason for changing the factor. to.88, for plates with one free edge, is as yet not know to the authors. But the adjustment seems to e rather insignificant.. POST-BUCKLING THEORY FOR PLATES IN COMPRESSION The asic idea is to use plastic theory in the form of limit analysis on the deflected shape of the plate. The deflected shape has to e estimated which is done y using simple formulae from eam and plate theory. The plastic analysis is carried out using yield line theory. Hence, strain hardening is not taken into consideration. Furthermore, the effect of residual stresses is not considered since, according to the theory of plasticity, they have no influence on the load-carrying capacity. Imperfections may easily e taken into account if it is assumed that the imperfections have the same form as the deflected shape of the plate at maximum load. The contriution of the paper lies in the way the yield line theory is applied on the deflected shape of the plate. Since the deflected shape has to e estimated, the application of the method requires qualified engineering judgement.. Yield Line Theory Yield line theory is an upper-ound method. The mechanisms considered are a system of ending yield hinges along lines, the yield lines. The load-carrying capacity is determined y the work equation, equalising external work and dissipation in the yield lines.

4 58 An Improved Effective Width Method Based on the Theory of Plasticity It is assumed that the plane stress field existing efore uckling is known so that the principal normal forces may e found wherey the corresponding yield moments may e determined. In an unloaded direction, the ending capacity, m, reaches the full yield moment per unit length in ending, m p : (7) 4 m mp t fy where t is the thickness of the plate and f y the yield stress. In the direction of a principal compression or tension (normal force n per unit length), the ending capacity, m, is reduced due to the normal force as for eams sujected to comined ending and normal force, i.e. n mm p n p (8) where m p is given y Eq. 7 and n p = is the load-carrying capacity in pure compression or tension. By determining the yield moments in the two principal directions in this way, the simplification suggested y Johansen [6] may e used to calculate the ending moment in a yield line, m, as m m sin m cos (9) px py where is the angle etween the yield line and the x-axis. The x- and y-axes are directed along the principal directions, and m px and m py are the corresponding yield moments per unit length. The plastic yield moments, m px and m py, are determined y either Eq. 7 or Eq. 8. Eq. 9 is correct for reinforced concrete slas in general, which was shown y Nielsen []. It is also correct if the yield condition in principal moment space is square or rectangular. For steel plates, such assumptions are duious, when the sla is acted upon y torsion. Nevertheless, Eq. 9 is used in the following with surprisingly good results. A numer of different researchers have developed formulae for the plastic moment capacity of inclined yield lines, the first eing Murray []. Hiriyur and Schafer [5] and Zhao [6] have shown that the solutions otained y the different proposals vary widely, and Zhao [6] concludes that nothing etter than the method suggested y Murray [] has een found. The present paper aims at improving this state of affairs.. Effect of Deflections From the general theory of eam-columns, it is known that the equilirium equations may e derived for the undeformed structure if a fictitious load is included. The fictitious load is a moment per unit length of the eam, equal to du m N () dx where N is normal force and u the deflection, transverse to the eam axis, x. The statical equivalence of m may e expressed in several ways, ut for a given part of a eam sujected to a constant normal force, N, it may conveniently e expressed in the following simple way: The total moment, M, on a eam (A-B), when N is constant, is

5 Thomas Hansen, Jesper Gath and M.P. Nielsen 59 du du M mdx N dxn dxnu u dx dx Nu B B B A A A B A () which may e split into two forces, N u/l, in the two end points. The two forces are transverse to the eam as shown in Figure. The value of these forces is independent of the deflected shape etween the end points (A and B). Only the difference in the deflection at the end points is important. The work in a virtual displacement, where the eam considered moves as a rigid ody, may then e determined as the work done y the two forces, N u/l. In the simple cases considered in the present paper, the aove result may e used y considering the plate as eing sudivided into strips with constant normal force. In Figure, a simply supported square plate sujected to uniaxial compression is shown. N A ua ub x ub - ua = u L B N Figure. Part of a Beam Sujected to a Constant Normal Force, N, and Statically Equivalent Transverse Forces in the End Points n dy n n n A Deflected shape L B ub n n y x A Displacement increment B Figure. Plate Sudivided into Strips with Infinitesimal Widths

6 5 An Improved Effective Width Method Based on the Theory of Plasticity The strip, marked y the dashed lines, has a normal force, n, per unit length. The transverse forces are a uniform load, n, along the lines (A and B) acting upwards in (A) (i.e. perpendicular to the x,y-plane), and a uniform load acting downwards in (B). For a displacement increment,, along the line (B), the external work for the strip, d y, is given y dwe ndy () where the angular deflection increment, = u B /L. The dissipation contriution from the strip is determined y dwi m dy (3) L where m is calculated y Eq Column Example The procedure is illustrated y a simply supported column, see Figure 3. The deflected shape of the column is characterised y the deflection in the midpoint at maximum load, u m. The column is centrally loaded y a compressive normal force, N, and therey each half is sujected to the forces, N. In the figure they are only shown in the midpoint. The column is given a lateral displacement increment,, at the midpoint, as shown in Figure 4, where a plastic yield hinge is formed. The external work of the mechanism is um We N4N (4) L and the dissipation is W i M 4 L p (5) N um N N N = um / L Figure 3. Deflected Shape of a Simply Supported Column N N M p L Figure 4. Failure Mechanism for a Lateral Displacement Increment, M p

7 Thomas Hansen, Jesper Gath and M.P. Nielsen 5 Here, M p is the plastic yield moment, which must e reduced due to the normal force. The work equation renders: Nu M (6) m p In this case just, the usual equilirium equation is found. A constant, solid and rectangular cross-section of the column is assumed in order to compare the result with the solutions derived later for plates. Hence, the following solution also corresponds to a rectangular plate with two free unloaded edges. The sides of the rectangular cross-section are denoted and t, respectively, where t is assumed. The plastic yield moment, reduced due to the normal force, is then M t f p y 4 N p N (7) N is the normal force, f y the yield stress and N p = is the load-carrying capacity in pure compression or tension. The next step is to find a good estimate for the deflected shape of the column. The maximum deflection for a eam may in general e determined as um L (8) where is the curvature in a selected point, L the length and is a parameter depending on the shape of the curvature function. In the following, = 8, corresponding to a constant curvature function, is used. In a eam or column with yielding, the deflection corresponding to maximum load tends to e reached when the yield strain, y, is otained in one or oth faces. Here it is assumed that the deflection at maximum load may e found y assuming that the yield strain, y = f y /E (f y is yield stress and E Young s modulus), is reached in oth faces. Thus the curvature is determined y y /t. The resulting formula is modified y a parameter,, in the following way: y t (9) The parameter,, is an empirical coefficient accounting for the effect of imperfections and residual stresses. Hence for the column in Figure 3, the deflection equals: u m 8 t 4 y L f y L E t () Inserting Eq. 7 and Eq. into Eq. 6, the load-carrying capacity expressed y the non-dimensional value, N / ( ), is found to e

8 5 An Improved Effective Width Method Based on the Theory of Plasticity N t f y 4 4 () where the parameter,, has een introduced: L t f y E () Now this result is compared with the EC3 [4] formulae. Here a non-dimensional slenderness ratio, r, given y r Af N cr y (3) is introduced. In this formula, A is the cross-sectional area and N cr is the elastic critical uckling force (the Euler-load), see the following Eq. 7. For a solid, rectangular cross-section where is given y Eq., r may e expressed as, f y L r E t (4) The load-carrying capacity according to EC3 [4] is given y Ncr t f y r (5) where.5. r r (6) The so-called geometric equivalent imperfection factor in this equation may e otained from Tale. Notice that here is not the same as the introduced in Eq. 8. The uckling curves according to EC3 [4], cf. Eq. 5, and the curve given y Eq. for =.4 are shown in Figure 5. It turns out that y choosing a value of etween. and.6, all uckling curves in EC3 [4] may e well represented. For =.4, the result coincides very closely with the uckling curve. The figure also shows the Euler curve, which for a solid rectangular cross-section is given y Ncr t f y r Tale. Imperfection Factors for Buckling Curves According to EC3 Buckling curve a a c d Imperfection factor (7)

9 Thomas Hansen, Jesper Gath and M.P. Nielsen 53 N / ( ) Euler Theory μ =.4 EC3 a EC3 a EC3 EC3 c EC3 d Figure 5. N / ( ) as a Function, Plastic Theory and EC3 λ If the column in Figure 3 is also sujected to a lateral load, q, per unit length along the entire length, L, then this load is easily included in the calculations. The work done y the lateral load is simply added to the external work, thus the work equation ecomes, cf. Eq. 6, (8) 8 Num ql Mp Other column cases may e treated in a similar way ut this is not our purpose here. 3 PLATES SUPPORTED ALONG ALL EDGES The compression flange in a ox-girder may e considered as a plate simply supported along all edges. How to determine the post-uckling strength of plates supported along all edges is presented elow. 3. Square Plates The square plate in Figure 6 is considered. It is simply supported along all four edges and sujected to a uniform load per unit length, n, along two opposite edges. X n Y Y n X n n um n n = um / n n Figure 6. Failure Mechanism for a Simply Supported Square Plate

10 54 An Improved Effective Width Method Based on the Theory of Plasticity The first step is to find a good estimate for the deflected shape of the plate. As stated in Section.3, the maximum deflection for a eam may in general e determined as um L (9) where is the curvature, L the length and is a parameter depending on the shape of the curvature function. In the following = 8 is used, corresponding to a constant curvature function. In a plate with yielding, the deflection corresponding to maximum load, as mentioned aove, tends to e reached when the yield strain, y, is otained in one or oth faces. As in the case of a column, cf. Section.3, the deflection at maximum load may e found y assuming that the yield strain, y = f y /E, is reached in oth faces of the plate. Here, f y is the yield stress and E is the Young s modulus. In this case = is applied, cf. Eq. 9, thus the curvature is determined y (t eing the plate thickness) y (3) t For a square plate, L is taken equal to the side length,. Hence, for the plate in Figure 6, the deflection at maximum load, u m, is u m f 8 t 4 E t y y (3) The next step is to find the optimal failure mechanism (yield line pattern). In Figure 6, two different yield line patterns are shown, one with the free parameter, X, and one with the free parameter, Y (shown with dashed lines), respectively. For the mechanism with the free parameter, X, the external work for a displacement increment,, equals u 4 m um We X n X n X We num44 (3) and the dissipation is n Wi 4mp mp n p X (33) Here, the plastic yield moment, m p, per unit length is given y Eq. 7 and n p = is the load-carrying capacity in pure compression or tension. Equalising the external work and the dissipation and inserting Eq. 7 and Eq. 3, the load-carrying capacity, expressed y the non-dimensional parameter, n/( ), may e written as

11 Thomas Hansen, Jesper Gath and M.P. Nielsen 55 4 n X X tf 4 X y (34) where the parameter,, f y t E (35) has een introduced. Eq. 34 is the post-uckling load in this case. The value of n/( ) as a function of is shown for different values of X in Figure 7. It is seen that n/( ) > for small values of, which is not possile in reality, hence a cut-off at n/( ) = must e introduced. With this cut-off, it appears that almost the same load-carrying capacity is otained for X = /, X = /5 and X = /3. The mechanism with the free parameter, Y, cf. Figure 6, leads to Y = when optimised. However, it must e rememered that the deflected shape is used oth as an estimate of the deflection at maximum load and as the asis for a choice of the deflection increment at the yield load. Thus a normal optimisation may not e appropriate. The optimised value for Y = is therefore disregarded and the mechanism corresponding to X = ½ (Y = ½ ) is chosen for the following calculations. In Figure 8, the ratio, n/( ), as a function of for X = / is shown together with the elastic solution, which may e written as, cf. Eq., n tf y k (36) where k is the uckling coefficient and is Poisson s ratio. Here k = 4 and =.3 are used. The parameter,, is given y Eq X = / X = /5 X = /3 X = /4 X = /5 X = / n / ( ) λ Figure 7. n/( ) as a Function of for Different Values of X

12 56 An Improved Effective Width Method Based on the Theory of Plasticity The two curves coincide very closely, which would indicate that no extra post-uckling reserve is found from this approach. However, the result is incorrect, which is due to the fact that a redistriution of the stresses occurs when the uckling mechanism develops. This is illustrated in Figure 9. As previously mentioned, Kármán et al. [8] suggested that the redistriution may e taken into account y using an effective area sujected to a uniform stress equal to the yield stress, i.e. = f y in Figure 9. The same simplification will e used here. Hence the considered plate is sujected to a load per unit length equal to the yield stress multiplied y the thickness of the plate, i.e. n = n p = tf y, which is applied along two strips with the width equal to the unknown width s, see Figure..9 Theory Elastic.8.7 n / ( ) Figure 8. n/( ) as a Function of for X = ½ λ Figure 9. Redistriution of Edge Stresses During Buckling s s um = um / Figure. Simply Supported Square Plate Sujected to Uniaxial Compression Along Strips of width s

13 Thomas Hansen, Jesper Gath and M.P. Nielsen 57 The external work, W e, and the dissipation, W i, are for =, respectively: W t f u (37) s e 8 y m tf s y s s Wi 8m p 8mp 8mp n p (38) Again, u m is the deflection at maximum load, m p is the plastic yield moment per unit length given y Eq. 7 and n p = is the load-carrying capacity in pure compression or tension. The contriution from the part of the yield lines running in the widths, s, is equal to zero, as n p =. Inserting Eq. 7 and Eq. 3, and using W i = W e, s is found to e determined y f y t s s t (39) E Denoting the total effective width e = s, we find e 4 4 (4) where is given y Eq. 35. The average stress relative to the yield stress, f y, along the whole width,, from the yield force, e t f y, along e is e /( ) = e /. For a solution with uniform load, n, along the whole width,, the average stress relative to the yield stress is n/( ). In Figure, e /, cf. Eq. 4, is shown as a function of. The elastic uckling load characterised y n/( ), cf. Eq. 36, is also shown. It appears that the plate may carry compressive forces consideraly aove the elastic uckling load. The first to demonstrate this through tests appear to e Schuman and Back []..9.8 Theory Elastic Winter Kármán.7.6 e / Figure. e / as a function of λ

14 58 An Improved Effective Width Method Based on the Theory of Plasticity The third curve plotted in the figure is the semi-empirical expression derived y Winter [5]. With the introduced parameter,, given y Eq. 35, it may e written as, cf. Eq. 4, e k k. (4) Finally, a cut-off at e / = n/( ) = has een introduced. Winter s formula is a modification of the original formula introduced y Kármán et al. [8]: t E C f e (4) y C where C is an empirical constant. Based on the tests made y Schuman and Back [], C =.9 was found. Eq. 4 is also shown in the figure. Regarding Eq. 4, Winter [5] argued that C should depend on the parameter -. Based on his own tests and those made y Sechler (Winter [5]), he found the est fit to e C t E f (43) y As previously mentioned, this formula has later een modified several times. Eq. 4 is the newest modification and it is adopted in EC3 [4]. Eq. 4 corresponds, for k = 4 and for =.3, to C.9.4 (44) Figure shows that the present theoretical result closely follows Winter s semi-empirical solution..9 α = 8 α = e / Figure. e / as a Function of for = 8 and =, respectively λ

15 Thomas Hansen, Jesper Gath and M.P. Nielsen 59 In order to investigate the sensitiveness of the solution, Eq. 4, to the estimated deflection at maximum load, it is compared with the solution using = in Figure. Choosing = in Eq. 9 approximately corresponds to a sinusoidal curvature function ( = ). For =, the effective width is determined y e (45) where is still given y Eq. 35. Eq. 45 gives a post-uckling strength up to % larger than that otained y Eq. 4 for the -interval shown in Figure. Hence, the theory seems to e relatively insensitive to the estimated deflection at maximum load. When von Mises yield criterion is applied, m p in Eq. 38 is replaced y m p /3 if uniaxial strain is supposed. In that case also n p should e increased and e / then will measure n relative to the increased value of n p. Thus it turns out that nothing is changed, since e / is still given y, cf. Eq. 4, e 4 4 (46) In practise one is on the safe side y using the uniaxial yield stress. Thus the former solution is applied. 3. Rectangular Plates It is a well-known fact that a long rectangular elastic plate sujected to compression in the longitudinal direction uckles into a shape of half waves with a length equal to the plate width, see Figure 3. This result is applied when estimating the post-uckling strength. Thus a long plate is sudivided into a numer of square plates where the previous yield line patterns may e applied, cf. Section 3.. The vertical lines etween the square regions will act as simple supports, since if one square region forms a wave downwards, then the adjacent regions will form a wave upwards. Therefore, the post-uckling strength or the effective width for a long rectangular plate is in general given y Eq. 4. Figure 3. Deflection Shape and Failure Mechanism for Long Simply Supported Rectangular Plates

16 53 An Improved Effective Width Method Based on the Theory of Plasticity Now consider a rectangular plate with a length somewhat larger than the width. In Figure 4, the width is named and the length named a. The plate is sujected to uniaxial compression, n p =, acting on the two strips, s. The failure mechanism has the free parameter, X. The maximum deflection, u m, is assumed to e given y Eq. 3. The external work is then for = given y W e um s 4t fy (47) X and the dissipation s a Wi m p X X (48) Hence, s may e determined y f y t X a s s t E (49) The effective width, e ( e = s ), is then found to e e Xa 4 4 (5) where is given y Eq. 35. Furthermore, in the aove equations, t is the thickness, E is Young s modulus, f y the yield stress, m p is the plastic yield moment per unit length given y Eq. 7, and n p = is the load-carrying capacity in pure compression or tension. Assuming X = ½, the effective width is determined y e a 4 (5) When a is larger than, or equal to,, it appears that Eq. 5 gives the smallest post-uckling strength if a is equal to. Hence in practice, the yield line pattern for a square plate may e considered as the optimal solution, rather than the pattern shown in Figure 4. s = um / X X um X s Figure 4. Failure Mechanism for a Short Rectangular Plate

17 Thomas Hansen, Jesper Gath and M.P. Nielsen 53 Special attention is required for plates where a is smaller than, ut the theory may also easily e applied to this case. The yield line pattern shown in Figure 5 is considered, where a is still larger than ut the loaded strips are now vertical. The plate is sujected to a load per unit length equal to the yield stress, f y, multiplied y the thickness, t, along two strips of width s. The failure mechanism has the free parameter, X. The work equation must e derived separately for the two cases, s < X and s > X, respectively. In the case s < X, the external work for = is given y m We 4t f u y s (5) X and the dissipation a s Wi 4m p X (53) where m p is the plastic yield moment per unit length given y Eq. 7. The maximum deflection, u m, is taken as the deflection of a eam in the direction of the uniaxial compression with the curvature given y Eq. 3. Thus u m y f y (54) 8 t 4 E t Equalising the external work and the dissipation, the width of each strip, s, for s < X is found to e determined y f y t X ax s s t E (55) s s um X X = um / a Figure 5. Failure Mechanism for a We Plate with a > and s < X

18 53 An Improved Effective Width Method Based on the Theory of Plasticity In the case s > X, the external work for = is given y um We 4t fy s X (56) and the dissipation, which is identical to Eq. 53, is a s Wi 4m p X (57) Equalising the external work and the dissipation, the width of each strip, s, for s > X is found to e determined y f f E E X y t y a s X t (58) In Section 3., it was shown that inclined yield lines under an angle of 45 are normally a good choice, noting that minimising s with regard to X did not lead to any useful result. Hence, an angle of 45 is also chosen here, i.e. X = ½. Denoting the effective width e = s, the effective width for s < X ( e < ) is given y 4 e a a a a a (59) and for s > X ( e > ) e a a a (6) In oth Eq. 59 and Eq. 6, e / is required, and is given y Eq. 35. The parameter, e /a, through Eq. 59 and Eq. 6 as a function of is shown for different values of a/ in Figure 6. In Figure 7, the ratio, e /, as a function of is shown for different values of a/. For the two cases treated aove, the ratio, e /, may e written as: For e / : e 4 a (6) For e / : e a (6)

19 Thomas Hansen, Jesper Gath and M.P. Nielsen 533 By inserting e / = in either Eq. 6 or Eq. 6, it is found that the two curves, for any value of a/, in oth Figure 6 and Figure 7, intersect for a (63) Since, e a is required, a cut-off at e /a =, as shown in Figure 6, must e done for all curves. In Figure 7, it is shown that each curve has a cut-off for e / equal to the value of a/ corresponding to the actual curve. By inserting e = a into, for instance, Eq. 6, it is verified that the cut-off, in oth Figure 6 and Figure 7, takes place for a / (64).9.8 a/ = a/ = a/ = 3 a/ = 4 a/ =.7.6 e / a λ Figure 6. e /a as a Function of for Different Values of a/ a/ = a/ = a/ = 3 a/ = 4 a/ = e / Figure 7. e / as a Function of for Different Values of a/ λ

20 534 An Improved Effective Width Method Based on the Theory of Plasticity 3.3 Comparison with Experimental Results The Schuman and Back [] tests dealt with separate rectangular flat plates of four different metals, i.e. Duralumin, Stainless Iron, Monel Metal and Nickel. Eq. 4 is compared to these tests in Figure 8. The figure also shows the elastic solution, cf. Eq. 36, Winter s solution cf. Eq. 4 and the solution y Kármán et al. cf. Eq. 4. From the figure it appears that all formulae, except the elastic solution, overestimate the post-uckling strength, in some cases consideraly. However, it is generally accepted that these experiments are unreliale ecause of the duious V-groove supports. Hence, these experiments are not treated further here. In Figure 9, the theories are compared with more reliale experiments. The U-eams and I-eams tests made y Winter [5] oth consisted of specimens made y olting or welding U-sections together. Winter also used the tests made y Sechler (Winter [5]) to verify his method. The specimens in these tests were single plates, unconnected to any adjacent elements. These tests have een shown in Figure 9. Furthermore, newer tests y Moxham [9] are included in the figure. He conducted three test series, denoted Welded, Unwelded and Short in the following. Also, in these tests, all specimens were separate plates. He developed a new test rig, where he could estalish the simple support conditions in a reliale way. In the Welded series, the longitudinal edges were heat treated in order to induce residual stresses. The Short series was conducted on specimens where the loaded edges were slightly longer than the unloaded edges (length-to-width ratio:.875). The theoretical effective width of the short specimens is calculated y Eq. 59 and Eq. 6. The specimens in the Welded and Unwelded series all had a length-to-width ratio of 4.. The agreement with all the tests seems to e very good. In Figure, the correlation etween the present theory and tests is shown in a more illustrative way. For all tests, a mean value of.88 and a standard deviation of 6.9 % are otained. For the separate test series, the following results are otained: Sechler: Mean., standard deviation 7.9 %. Winter, U-eams: Mean.98, standard deviation 8.9 %. Winter, I-eams: Mean.9, standard deviation 4.6 %. Moxham, Welded: Mean.97, standard deviation. %. Moxham, Unwelded: Mean.93, standard deviation 5.8 %. Moxham, Short: Mean.946, standard deviation 6.4 %. It may e seen that the tests y Sechler deviate somewhat from the theory, especially for e / close to unity. One explanation for this might e initial imperfections, see Chapter 5. Moreover, he may have applied the same doutful V-groove supports as Schuman and Back []. Without Sechler s tests, a mean value of.45 and a standard deviation of 4.4 % are otained.

21 Thomas Hansen, Jesper Gath and M.P. Nielsen Theory Elastic Winter Kármán Duralumin Stainless Iron Monel Metal Nickel e / Figure 8. e / as a Function of, Theories Compared with Tests y Schuman and Back λ e / Theory Elastic Winter Kármán Sechler U eams I eams Welded Unwelded Short Figure 9. e / as a Function of, Theories and Tests λ.8 Sechler U eams I eams Welded Unwelded Short e / (theory) e / (test) Figure. Theory Versus Tests

22 536 An Improved Effective Width Method Based on the Theory of Plasticity 4. PLATES WITH ONE FREE EDGE The compressive flange and the transverse we stiffeners in an I-shaped steel plate girder may e considered to e long, rectangular plates simply supported along three of the edges, and free along one longitudinal edge. How to determine the post-uckling strength of such plates is presented elow. 4. Rectangular Plates The considered plate with length a, width and thickness t is sujected to uniaxial compression, n p =, along an effective width, e, near the supported longitudinal edge. The est failure mechanism, with free parameter, X, is shown in Figure. It is easily verifiale that plates with length, a, in the interval, X a, will have the same post-uckling strength. Plates with a < X require another failure mechanism, which will e treated in Section 4.3. The two regions with area, X, will approximately e sujected to pure torsion. Thus the principal directions are under 45 with the edges, and the principal curvatures are equal, ut with opposite sign. In the principal directions, the curvatures are estimated to e, in asolute value, the same as in Eq. 3. Then the torsional curvature, xy, will also e xy = y / t ( y = f y /E is the yield strain and t the thickness), which means that the deflection, u m, at the end point of the yield line in the free edge will e y f y X um xy X X (65) t E t For the mechanism shown in Figure, the external work is for = given y W e u X m e t fy (66) and the dissipation is X e Wi mp X X (67) e X um um X = um / X Figure. Failure Mechanism for a Rectangular Plate with One Free Edge

23 Thomas Hansen, Jesper Gath and M.P. Nielsen 537 where m p is given y Eq. 7. Hence the effective width, e, may e determined y f y t X e e t E X X (68) The non-dimensional value, e /, is then given y 4 e X 8 X 64 X 4 X (69) Eq. 69 is shown for different values of X in Figure. A cut-off at e / = must again e introduced. With this cut-off, it is seen that almost the same load-carrying capacity is otained for X =, X = 4/5 and X = 3/4 ; hence X = is used in the following calculations. The effective width for X equal to is found to e e (7) which is shown in Figure 3. The figure also shows the elastic uckling curve given y Eq. 36 with k =.43, cf. (Timoshenko and Gere [4]). Again, Poisson s ratio =.3 is assumed. The third curve shown is Winter s formula in the form given in EC3 [4], which may e written as, cf. Eq. 6, e k k.88 (7) Here, k =.43 and =.3 are again used. It is found that the theory gives a slightly larger post-uckling strength than Winter s solution x= x=4/5 x=3/4 x=/3 x=/ x=/5. e / Figure. e / as a Function of for Different Values of X λ

24 538 An Improved Effective Width Method Based on the Theory of Plasticity.9 Theory Elastic Winter e / Figure 3. e / as a Function of λ 4. Comparison with Experimental Results Figure 4 shows four test series compared with Eq. 7. The figure also shows the elastic solution cf. Eq. 36 and Winter s solution cf. Eq. 7. In the test series Plates (Bamach and Rasmussen []), the specimens were single plates, unconnected to any adjacent elements. These authors applied the test rig suggested y Moxham [9] which enaled them to estalish simple support conditions in a reliale way. The series Stu-column, Beams (Kalyanaraman et al. [7]) and I-eams (Winter [5]) all consisted of specimens made y olting or welding U-sections together. In Figure 5, the correlation etween theory and test is shown. For all tests, a mean value of.877 and a standard deviation of 5.7 % are otained. For the separate test series, the following results are otained: Bamach & Rasmussen, Plates: Mean.98, standard deviation 8.5 %. Kalyanaraman et al., Stu-column: Mean.77, standard deviation 6. %. Kalyanaraman et al., Beams: Mean.79, standard deviation. %. Winter, I-eams: Mean.95, standard deviation 6.7 %. The theory seems to underestimate the post-uckling strength for the tests Stu-column and Beams. Kalyanaraman et al. [7]) stated that the mean value of the elastic uckling coefficient was measured to e around k =.85. This numer shows that the longitudinal edges cannot have een simply supported, since for a plate with a simply supported longitudinal edge k =.43. For a fixed edge k =.8 according to the elastic theory, cf. (Timoshenko & Gere [4]). The test series I-eams is defected y a large scatter. Winter [5] gives several explanations for the large scatter. On the other hand, the test series Plates, with the most relialy estalished oundary conditions, shows a very good agreement with the theory.

25 Thomas Hansen, Jesper Gath and M.P. Nielsen Theory Elastic Winter Plates Stu column Beams I eams e / Figure 4. e / as a Function of, Theories and Tests λ Plates Stu column Beams I eams.8 e / (theory) e / (test) Figure 5. Theory Versus Tests 4.3 Square Plates Square plates in uniaxial compression with one free edge require another failure mechanism than the one shown in Figure valid for rectangular plates. The optimal mechanism for a square plate is drawn in Figure 6. It has the free parameter, X, which is set at X = ½. The plate is again sujected to uniaxial compression, n p =, along an effective width, e, close to the support. In this case, two work equations must e derived; one corresponding to e < X and one corresponding to e > X, respectively. The deflection at maximum load, u m, is taken as the deflection of a eam in the direction of the uniaxial compression with the curvature given y Eq. 3. Thus u f 8 t 4 E t y y m (7)

26 54 An Improved Effective Width Method Based on the Theory of Plasticity e X um = um / Figure 6. Failure Mechanism for a Square Plate with One Free Edge The work equations show that the effective width is determined y: For e / X/ = ½: f y t 3 e e t (73) E e (74) For e / X/ = ½: f y t 3 f e t E 4 E y (75) e 3 4 (76) In the aove equations, is given y Eq. 35, is the total width, t the thickness, E Young s modulus and f y is the yield stress. Eq. 74 and Eq. 76 are shown in Figure 7 together with the elastic uckling solution, Eq. 36, and Winter s solution, Eq. 7.

27 Thomas Hansen, Jesper Gath and M.P. Nielsen 54.9 Theory Elastic Winter e / Figure 7. e / as a Function of λ Here, the uckling coefficient k =.43 is applied, cf. (Timoshenko and Gere [4]), and Poisson s ratio is taken to e =.3. Again, the theoretical solution gives almost the same result as Winter s semi-empirical solution. Unfortunately, no tests have een found in the literature to verify the theory. This however is not particularly important, since square plates with one free edge are seldom used in practice. 5. IMPERFECTIONS As previously mentioned residual stresses will have no influence on the calculations according to the theory of plasticity. Approximately imperfections may e taken into account y adding an initial deflection to the deflection used in the aove calculations. Thus it is tacitly assumed that the imperfection has the same form as the deflected shape used efore. When considering rectangular plates supported along all edges, the deflection at maximum load is determined y, cf. Eq. 3, u m f y ui (77) 4 E t where u i is the initial deflection, f y the yield stress, E Young s modulus, the total width and t is the thickness. Solving the work equation as shown in Section 3., leaves the result e u 4 4 t u t 4 i i 4 (78) For an imperfection proportional to the thickness, the term, u i /t, will e a constant. Another way of taking imperfections and other unknown parameters into account is simply to introduce an empirical coefficient,, in the deflection formula. The deflection at maximum load

28 54 An Improved Effective Width Method Based on the Theory of Plasticity may then e written u m f y (79) 4 E t The effective width is found to e e 4 4 (8) Imperfections may explain the large deviations in the tests y Sechler (Winter [5]), oth from Winter s formula and the present theoretical estimate. In Figure 5, it was shown that the tests y Sechler coincided closely with the theoretical estimate for small values of e /, i.e. large values of, while for low values of, the deviation was large. For the tests y Sechler shown in Figure 5, the mean value is.. By requiring the mean value. and y applying Eq. 79 to find the imperfection coefficient, =.5 is found. The correlation etween the tests and the theory, cf. Eq. 8, with =.5 is shown in Figure 8. In this way, a smaller deviation is achieved for low values of, ut the scatter is still large..8 e / (theory) e / (test) Figure 8. Theory Versus Tests for =.5.8 e / (theory) e / (test) Figure 9. Theory Versus Tests for u i /t =.94

29 Thomas Hansen, Jesper Gath and M.P. Nielsen 543 Unfortunately, the report y Sechler [3] referred to y Winter has not yet een availale to the authors. From Winter s reference it is only possile to deduce the value of, while the thickness of the specimens remains unknown. By applying Eq. 78 and require the mean value., u i /t =.94 is found. The correlation etween the tests and the theory with u i /t =.94, cf. Eq. 78, is shown in Figure 9. It appears that the effect is small for large values of, and large for low values of, which is precisely the requirement for removing the discrepancy. However, the effect on the result for low values of seems to e too great. Thus y applying Eq. 4, the theory overestimates the load-carrying capacity for low values of, while the opposite is achieved y applying Eq. 78. Hence it is not possile to verify Eq. 78 y these tests. Apart from initial imperfections, there may of course e other explanations for the large deviations found in the Sechler tests. In tests, it is very difficult to correctly estalish the ideal support conditions. The supports require the four edges of the initial mid-plane of the plate to remain in the same plane at all times. A solution was suggested y Schuman and Back [] for the case of single plate specimens. In their tests, the specimens were, as mentioned aove, supported y V-grooves, which cause any initial curvature of the edges to increase, which may again cause failure at a lower load than otherwise expected. The effect will e relatively larger for the thicker specimens. Therefore prolems related to V-grooves might also explain the deviations in the Sechler tests. However, this is quite hypothetical, since at the moment it is not known if this kind of support was even used in the tests. 6. OTHER APPLICATIONS OF THE THEORY The simple theory developed aove may e extended to apply to a large numer of practically important cases. Firstly, an external lateral load may e taken into account y simply adding the work done y the lateral load to the external work, calculated as aove. Further, iaxial compressive loads may e treated in the same manner without difficulties. Stiffeners and the compression flange in plate girders may e calculated y formulae given aove. Fixed supports may e treated y adding the contriution from the negative yield line at the fixed supports to the dissipation. However, the deflection at maximum load must also e changed, so that it corresponds to the fixed oundary conditions. Considering two cases of long rectangular plates with fixed unloaded edges (one fixed unloaded edge and oth unloaded edges fixed), the failure mechanisms may e as shown in Figure 3 and 3, respectively. In oth cases, the plate may e sudivided into square plates as shown in the figures. Here the deflection at maximum load is given y Eq. 3. The load-carrying capacity for a plate with one fixed unloaded edge is then e 4 5 (8) The load-carrying capacity for a plate with oth unloaded edges eing fixed equals

30 544 An Improved Effective Width Method Based on the Theory of Plasticity e 4 6 (8) Eq. 8 and 8 are shown in Figure 3 and 33, respectively. The figures also show the results determined e Winter s formulae. Again, the results presented coincide closely with Winter s formulae. Finally, it is proaly also possile to calculate in-plane ending loads. However, other mechanisms have to e used in this case. s s Figure 3. Failure Mechanism for a Plate with One Fixed Unloaded Edge s s Figure 3. Failure Mechanism for a Plate with Both Unloaded Edges Fixed.9 Theory Elastic Winter e / Figure 3. e / as a Function of, Fixed Supports Along One Unloaded Edge λ

31 Thomas Hansen, Jesper Gath and M.P. Nielsen Theory Elastic Winter e / Figure 33. e / as a Function of, Fixed Supports Along Both Unloaded Edges λ 7. CONCLUSION It is shown that extremely simple estimates of the post-uckling strength of plates with in-plane loading may e otained y using plastic solutions for the deflected shape. This shape must e known or estimated efore the calculation can e carried out. It seems that useful estimates of the deflected shape may e found using simple formulae from eam theory and plate theory. Initially, the calculation procedure is illustrated y simple solutions for columns. Solutions for plates are derived for the two practically important cases: Plates supported along all edges and plates with one free edge. In oth cases, the theory is applied on oth square and rectangular plates. Furthermore, it is shown how imperfections may e taken into account. Finally, it is shortly explained how the theory may e extended to a large numer of other practically important cases. The results have een compared with the well-known formulae of Winter and with tests. The agreement in oth cases is very good. NOTATION a plate length plate width e total effective width s width of effective strip f y yield stress k elastic uckling coefficient m ending capacity of a yield line per unit length; fictitious moment m ending moment per unit length in a yield line m p plastic yield moments per unit length n normal force per unit length n p load-carrying capacity per unit length in pure compression or tension m px, m py plastic yield moment in the x-direction and the y-direction, respectively p lateral load per unit area t thickness u, u A, u B deflection; deflection at point A and B, respectively u i deflection from imperfections

32 546 An Improved Effective Width Method Based on the Theory of Plasticity u m deflection at maximum load x, y coordinates in a Cartesian x,y -system of coordinates A cross-sectional area C empirical coefficient E Young s modulus L length M total moment M p plastic yield moment N normal force N cr critical uckling load N p load-carrying capacity in pure compression or tension W e, W i external work and dissipation, respectively X, Y free optimisation parameters parameter (shape of curvature function); imperfection factor displacement increment y yield strain, xy curvature and torsional curvature, respectively non-dimensional parameter r non-dimensional slenderness ratio according to EC3 empirical coefficient Poisson s ratio normal stress e edge stress angular deflection increment difference EC3 parameter for calculation of columns REFERENCES [] Bamach, M.R. and Rasmussen, K.J.R., Tests on Unstiffened Plate Elements under Comined Compression and Bending, J. Struct. Eng. ASCE, 4, Vol. 3, No., pp [] Bryan, G.H., On the Staility of a Plane Plate under Thrusts in Its Own Plane, with Applications to the Buckling of the Sides of a Ship, Proc. Lond. Math. Soc. 89, Vol.. [3] European Committee for Standardisation (EC3 993), Eurocode 3: Design of Steel Structures Part -: General Rules and Rules for Buildings, EN 993--, Brussels: CEN. [4] European Committee for Standardisation (EC3 5), Eurocode 3: Design of Steel Structures Part -5: Plated Structural Elements, ENV , Brussels: CEN. [5] Hiriyur, B.K.J. and Schafer, B.W., Yield-Line Analysis of Cold-Formed Steel Memers, Sumitted to International Journal of Steel Structures, August 4. [6] Johansen, K.W., Brudlinieteorier, Copenhagen: Gjellerup, 943. English translation: Johansen, K.W., Yield Line Theories, London: Cement and Concrete Association, 96. [7] Kalyanaraman, V., Winter, G. and Pekoz, T., Unstiffened Compressed Elements, J. Struct. Div. ASCE, 977, Vol. 3, No. 9, pp [8] Kármán, T. von, Sechler, E.E. and Donnell, L.H., The Strength of Thin Plates in Compression, Trans. ASME, 93, Vol. 54, pp [9] Moxham, K.E., Buckling Tests on Individual Welded Steel Plates in Compression, Report CUED/C-Struct/TR.3, Camridge: University of Camridge, 97.