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The stability of eccentrically compressed thin plates with a longitudinal free edge and with stress variation in the longitudinal direction
ARTICLE IN PRESS
Thin-Walled Structures 46 (2008) 494–505 www.elsevier.com/locate/tws
The stability of eccentrically compressed thin plates with a longitudinal free edge and with stress variation in the longitudinal direction Andrzej Szychowski Faculty of Civil and Environmental Engineering, Kielce University of Technology, Al. Tysia˛clecia Pan´stwa Polskiego 7, 25-314 Kielce, Poland Received 24 April 2007; received in revised form 23 October 2007; accepted 23 October 2007 Available online 3 December 2007
Abstract This paper presents results of the investigation of the stability of eccentrically compressed thin plates with one longitudinal free edge with the participation of loadings generating the variation of stresses in the direction of the length of a plate. A deflection function (5) is proposed to enable modelling of boundary conditions on the second longitudinal edge from a simply supported, through elastically restrained, to a built-in edge. Formulas are derived for work done by external force under loading generating the variation of stress according to a linear function and according to parabola 21. Tables and plots of buckling coefficient (k) for variously supported and variously loaded plates with a free edge, which are not found in the literature, are determined. The effect of a longitudinal stress variation on the plate buckling modes is analyzed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Structural stability; Longitudinal stress variation; Thin plates
1. Introduction In thin-walled bars with an open cross-section built from thin plates at composite loads there occur incidents in which component plates (with a free edge) are axially or eccentrically compressed by the simultaneous variation of stresses in the direction of the length of a segment. A thinwalled bar segment is defined here as a bar section between transverse stiffening (ribs, diaphragms, etc.), which assures a stiff cross-section contour. A number of local buckling ‘‘half-wavelengths’’ with varied length and amplitude can be formed on the bar segment length. A proper determination of buckling stresses for thus loaded component plates of the thin-walled bar serves to obtain a more accurate determination of limit load-carrying capacity on the basis of the effective width theory. Yu and Schafer [1] investigated the effect of a longitudinal stress gradient on the limit load-carrying capacity of plates axially compressed on one side, which are component walls of thin-walled bars. Plates simply Tel.: +4841 342 4575; fax: +4841 344 3784.
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supported on all edges and plates simply supported on three edges with a longitudinal free edge were investigated. The results were compared with the effective width theory by the finite element method (ABAQUS) according to Winter’s formula [2]. Buckling stresses of axially compressed plates at a longitudinal stress gradient, necessary to determine effective width, were determined on the basis of the proposal contained in [3]. In the literature there are no studies concerning the stability of eccentrically compressed plates (in their plane) supported on three edges (with a longitudinal free edge) with the participation of loadings generating the variation of stress in the direction of the length of a plate. Plates supported and loaded in this way occur, amongst others, as walls in thin-walled bars with open cross-sections in composite load states. However, there are studies concerning stability separated from the segment of the thin-walled bar of a rectangular plate supported on all edges by loadings generating variation of stress in the direction of the length of a plate, e.g. [1,3,4–9]. In [8,10,11], Jakubowski determined, amongst others, buckling stresses from the condition of local stability loss of variously loaded bars with closed rectangular and
ARTICLE IN PRESS A. Szychowski / Thin-Walled Structures 46 (2008) 494–505
Nomenclature
Qky
Aink, Ajqr, Ainp coefficients of power polynomials bs, ls, ts width, length, thickness of a plate (wall s) Ds plate flexural rigidity E Young’s modulus of elasticity fin, fjq dimensionless, free parameters of the deflection function i, j, n, q, p natural number index io number of half-wavelength of the function sine in the direction of the length of a plate k plate buckling coefficient Ls work done by external forces Linjq component elements of the work done by the external forces function m coefficient that characterizes the longitudinal stresses variation according to (13) My moment on the free edge of a plate no degree of the polynomial n¯ injq ; o¯ injq ; p¯ injq ; r¯injq ; ¯tinjq ; v¯ injq ; u¯ injq expressions which occur in the elements Linjq
Us Vs Vinjq
triangular sections. The starting point of the study [8] is the analysis of the stability of a rectangular plate supported on all edges and eccentric compression on transverse edges and shear forces acting in the plane of the plate. This kind of load produces a variation of normal stress in the direction of the length of a plate. The deflection function (in the direction of the width of a plate) was assumed to be in the form of power polynomials with a simple physical interpretation. The application of polynomials considerably facilitates the mathematical side of the problem, and their applicability in the investigations of local stability of plates and closed cross-sections has been confirmed in numerous investigations of the research team at the University of Technology in Ło´dz´ (Poland), which are listed in the monograph [9]. Another way of obtaining the variation of the stress in the direction of the length of the bar segment with a closed cross-section was applied by Kowal in [4,5]. The problem of stability of a transversely bended plate girder with a box section, at different transverse load variants, was reduced to the problem of stability of a rectangular plate (compressed flange) simply supported on all edges with the length of a bar segment. The plate was loaded with uniformly distributed forces (N0, N1) on the transverse edges and body forces (X) acting in the middle plane of a plate. The distribution of body forces was chosen depending on the longitudinal variation of the stress generated transverse load of the girder (Fig. 1). The problem was solved with the use of Galerkin’s method by calculating buckling coefficients of plates with different
ws xs, ys, zs X Yin as b(xs) dij n sx, txy scr sE s0, s1 r2
495
Kirchhoff’s substitute force on the free edge of a plate total potential energy of a system strain energy of the bending of a plate component elements of the bending strain energy function deflection function of a plate (wall s) Cartesian coordinates of a plate (wall s) longitudinal body forces power polynomials (8) with previously determined coefficients Ainp according to [8] coefficient stress distribution in the direction of the width of a plate function of normal stress distribution in the direction of the length of a plate Kronecker’s delta Poisson’s ratio membrane stresses critical stress Euler’s stress edge normal stresses (positive, if compressive) Laplace operator
ratios of side lengths and different variants of girder transverse load. Kro´lak [7] analyzed a similar case of longitudinal stress variation for an axially compressed plate. The variation of normal stresses in the direction of the length of a plate was obtained by introducing shear stress to. Displacement functions were written in the form of a double, finite sinus series. The solution of the problem was obtained by Galerkin’s method. The presented coefficients of critical stresses for plates of aspect ratio ls/bs from 0.5 to 4 in [7] and from 0.5 to 2 in [9] are, however, greater than those obtained by Kowal [4,5]. The technical solution of many problems of local buckling and limit load-carrying capacity of thinwalled bars with an open cross-section, built from flat walls, needs solutions of the critical state of plates with a longitudinal free edge at eccentric compression and the variation of stresses in the direction of the length of a plate. The aim of this study is to determine buckling stresses and buckling modes of rectangular plates supported on three edges with one longitudinal free edge and loaded by eccentrical compression (in the plate plane) with the participation of loadings generating the variation of stresses in the direction of the length of a plate. The following cases are considered: (1) a plate simply supported on three edges (Fig. 2a), and (2) a plate built-in on the longitudinal edge and simply supported on transverse edges (Fig. 2b). The above boundary conditions on the longitudinal-supported edge are cases of limit support of
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Fig. 1. Exemplary diagram of the plate support and load considered in study [4] (where: N0 ¼ s0ts; N1 ¼ s1ts).
Fig. 2. Limit boundary conditions of plates supported on three edges.
walls with a free edge in the segment of a thin-walled bar with an open cross-section. 2. Function of plate deflection The stability of axially or eccentrically compressed plates with one longitudinal free edge at a constant normal stress intensity in the direction of the length of a plate and boundary limit conditions on a longitudinal-supported edge (simply, built-in) and transverse edges (xs ¼ 0, xs ¼ ls) simply supported (Fig. 2) was the subject of investigation of many authors, including [6,12,13]. A comprehensive list of results is contained in Bulson’s monograph [6], which presents different methods for the solution of the problem. In the case of eccentric compression or bending in the middle plane of the plate with a longitudinal free edge, it is not possible to obtain an exact solution resulting from integration of the known differential equation of thin-plate stability. Using the energy method, Bulson [6] determined coefficient k of edge buckling stress for an eccentrically compressed plate with a free edge (Fig. 2a) at a constant normal stress intensity in the direction of the length of a plate, assuming the deflection function in the following form: ys ipxs ws ¼ f in sin . (1) bs ls Deflection function (1) satisfies the boundary conditions on the longitudinal-supported edge, whereas conditions (2) of disappearance moment My and conditions (3) of disappearing Kirchoff’s substitute force Qky on the free
Fig. 3. Diagram of moment (My) on the free edge of the plate axially compressed (after stability loss), whose deflection function is expressed by formula (1).
edge are fulfilled identically q2 ws q2 w s þ n 2 ¼ 0; 2 qys qxs
for ys ¼ bs ,
q3 ws q3 w s þ ð2 nÞ ¼ 0; qy3s qx2s qys
for ys ¼ bs .
(2)
(3)
Fig. 3 shows the distribution of moment My on the free edge of the plate axially compressed (after stability loss), whose deflection function is expressed by formula (1). Gra˛ dzki and Kowal–Michalska [14] applied the function in the form (1) and the polynomial: gðys Þ ¼ 6ðys =bs Þ2 4ðys =bs Þ3 þ ðys =bs Þ4 ,
(4)
ARTICLE IN PRESS A. Szychowski / Thin-Walled Structures 46 (2008) 494–505
to the description of deflection of the axially compressed plate with a longitudinal free edge and elastically restrained by another edge. Polynomial (4) according to [14] is a deflection function for a uniformly loaded cantilever beam. The same form of deflection is obtained for a cantilever plate strip (at infinite length) uniformly loaded on the whole surface. As can be concluded from the author’s investigation, the application of deflection function g(ys) resulting from the displacement form of variously loaded infinite plate strips can be a satisfactory approximation of the buckling modes of eccentrically compressed plates, built-in longitudinally on one side (cf. Fig. 2b) at a constant stress intensity in the direction of the length of a plate. However, it is not a universal approach because for different distributions of an eccentrical plate compression, one has to seek each time a deflection function which approximates the bulking shape well. Moreover, boundary conditions (2), (3) are generally not fulfilled on the free edge of the plate. However, the application of combinations of hyperbolic functions [6,12] to the description of plate deflection complicates considerably the mathematical side of the problem, particularly in these cases, when a plate with a free edge is a component element of a thin-walled bar. On the basis of the investigation of different forms of theoretical deflection functions, for the approximation of the buckling modes of eccentrically compressed plates with a longitudinal free edge with the participation of loadings generating the variation of stresses in the direction of the length of a plate, we propose to apply the function in the form of a series: n ! io no X X ys ipxs ws ðxs ; ys Þ ¼ ts f in . (5) sin bs ls i¼1 n¼1 The definite parameters fin of function (5) are given an initial value depending on boundary conditions on the longitudinal-supported edge, and therefore: fi1 ¼ 0, for a plate built-in on the longitudinal edge (Fig. 2b), fi2 ¼ 0, for a plate simply supported on the longitudinal edge (Fig. 2a). In the case of elastic restraint of a plate in the segment of a thin-walled bar, it is necessary to apply the full form of function (5) i.e. all fin are nonzero parameters of the deflection function, and their initial values result from static and kinematic boundary conditions on the connection of adjacent plates. The expression in the first bracket of the series (5) can be interpreted as an no degree polynomial, whose particular coefficients fin constitute parameters of the deflection function, and in boundary cases they are boundary conditions on the supported edge (simply, built-in), and some of them (fi1 and fi2) are given zero initial values. Such an approach enables one to adjust function (5) optimally to the buckling mode of the plate
497
Table 1 A comparison maximal ‘‘residual forces’’ (My, Qky) in relation to plate with ls/bs ¼ 1, (in %) ls/bs
My_1
Qky_1
My_3
Qky_3
My_4
Qky_4
My_5
Qky_5
1 2 3 4
100 25 11.1 6.25
100 25 11.1 6.25
96.7 23.7 10.4 5.86
103 25.1 11.1 6.21
32 5.1 2 1.05
70.3 10.6 4 2.09
8.2 0.58 0.12 0.04
30.3 2.1 0.43 0.14
Note: My_1, Qky_1-for a deflection function (1) according to [6]; My_3,4,5, Qky_3,4,5-for a deflection function according to formula (5) at no ¼ 3,4 or 5.
with one free edge irrespective of the load distribution and the manner of support on the second edge. Deflection function (5) with initial values of parameters (fi1, fi2) satisfies boundary conditions on the supported longitudinal edge, whereas conditions (2), (3) on the free edge, similar to that in [6,14], are not fulfilled identically. However, it is possible to demonstrate that the bending moment My and Kirchhoff’s substitute force Qky on the free edge of the plate tend to zero, minimizing thereby the complete potential energy of the system when the degree no of polynomial increases. The process of adjustment of the deflection function (5) to parameters fin from the condition of minimum energy to the shape of plate buckling occurs both in the formation of displacements and the reduction of ‘‘residual forces’’ (My, Qky) on the free edge. Moreover, the effect of ‘‘residual forces’’ generated by the adopted deflection function, which does not fulfill identically the boundary conditions (2), (3) on the free edge, decreases with increasing length of the plate. Table 1 presents the maximum values of ‘‘residual forces’’ (My, Qky) on the free edge of an axially compressed plate for different aspect ratio ls/bs and degree no of polynomial (5) from 1 to 5 with the initial value fi2 ¼ 0, which is equal to simply supported on the second edge (Fig. 3). The results are compared in percent with the values obtained at plate deflection approximation according to formula (1) according to [6] for a plate aspect ratio ls/bs ¼ 1. ‘‘Residual forces’’ on the free edge were determined after each standardization of displacements for the plate buckling mode described by a series (5). Plate buckling modes for ls/bs from 1 to 4 were determined by the energy method according to a procedure described in the latter part of this study. 3. Strain energy of plate bending In the case of a plate supported on three edges (Figs. 2 and 3) the strain energy of the bending of a plate, whose deflections are described by a series (5), can be determined from the formula: " Z l s Z bs ( D q2 ws q2 ws s V¯ s ¼ ðr2 ws Þ2 2ð1 nÞ 2 0 0 qx2s qy2s #) 2 2 q ws ð6Þ dxs dys . qxs qys
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Jakubowski in [8] provided formulas for the bending strain energy of a rectangular plate (bar walls with a closed cross-section), supported on four edges, whose deflection function can be represented in the form: ! io no X X ipxs ws ðxs ; ys Þ ¼ ts f in Y in sin , (7) ls i¼1 n¼1 where Yin—power polynomials (8) with previously determined coefficients Ainp according to [8] p 7 X y Y in ¼ Ainp s . (8) bs p¼0 The considerations of the practical local stability loss of a bar with a closed cross-section for a plate supported on all edges were reduced in [8–10] to polynomials of a maximum of the 7th degree, which fulfill conditions Yin(0)=Yin(1)=0. Function ws(xs, ys) according to (7) satisfies the boundary conditions of a plate supported on all edges and enables the complex shape of the buckling of the plate to be described in a general case, which occurs during compression, bending and shearing in the middle plane of the wall of the closed cross-section [8–10]. Proceeding analogously as in [8] and substituting deflection function ws(xs, ys) according to (5) for the dependence (6), we obtain: Vs ¼
io X io X no X no Ds t2s X V injq f in f jq . 4bs l s i¼1 j¼1 n¼1 q¼1
(9)
Elements Vinjq of the square form (9) depend on polynomial coefficients, Poisson’s ratio n and the plate aspect ratio ls/bs. Due to the fact that members (ys/bs)n in the series (5) can be treated in practice as one-dimensional polynomials in the form (8) with a coefficient Ainp ¼ 1, elements Vinjq in formula (9) can be calculated from the sequence of formulas derived for a plate supported on all edges in [8]. 4. Membrane stress state In the case of stability analysis of the segment of the thin-walled bar, in which the flat cross-section hypothesis or the section warping hypothesis is accepted (depending on the state of a bar load), the distribution of normal stress in a component plate supported on three edges (Fig. 4) can
be presented in the form: ys bðxs Þ. sx ¼ s0 1 as bs
(10)
The following cases are considered in this study: (1) linear variation of stress sx in the direction of the length of a plate, and (2) nonlinear variation of stress sx (according to parabola 21) in the direction of the length of a plate, for which function b(xs) of normal stress distribution in the direction of the length of a plate can be represented, respectively, in the form: xs bðxs Þ ¼ 1 m , (11) ls bðxs Þ ¼ 1 m
x2s , l 2s
(12)
where m ¼ 1 s1 =s0 .
(13)
Variation of normal stress in the direction of the length of a plate according to formulas (10), (11), (12) can be obtained by introducing shear stresses txy (Fig. 4a) or body forces X (Fig. 4b) with a distribution chosen according to that of a bar segment load. Taking into consideration the differential equilibrium equation: qsx qtxy þ þ X ¼ 0, qxs qys
(14)
and skipping body forces (X ¼ 0), one can present the shear stress distribution for a plate loaded according to Fig. 4a with a boundary condition txy(bs) ¼ 0 in the form: txy ¼
ðbs ys Þ½bs ðas 2Þ þ ys as s0 b0 ðxs Þ , 2bs
(15)
where b0 (xs)—first derivative of the normal stress distribution function in the direction of the length of a plate. Skipping in Eq. (14) shear stresses (txy ¼ 0), the distribution of body forces according to Fig. 4b assumes the form: qsx y X ¼ ¼ s0 1 as s b0 ðxs Þ. (16) @xs bs Introduction of longitudinal body forces, according to formula (16), which generate the variation of normal stress in the direction of the length of a plate according to
Fig. 4. The membrane stress state in a plate supported on three edges at eccentric compression and a longitudinal stress variation caused by (a) shear stresses txy, (b) longitudinal body forces X.
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formula (10), permits performing an analysis of the stability of a plate supported on three edges in those cases in which the distribution and intensity of shear stresses (txy) do not have a significant effect on the buckling mode. In the case of the secondary effect of shear stresses on the stability loss of a plate supported on three edges, load reduction according to Fig. 4a to normal stresses with participation of body forces according to Fig. 4b enables the deflection function (5) to be simplified by reducing the number of necessary members of series (5) necessary to approximate the buckling mode of the plate. In the case of transverse bending of thin-walled bars with a considerable participation of transverse forces in slender webs (plates supported on four edges), it is necessary to take into consideration shear stresses (txy), because their participation in the local stability loss of a web may be considerable [9,10]. Proper formulas for the work done by external forces taking into account of shear stresses (txy) for plates supported on all edges loaded by bending with shearing were introduced by Jakubowski in [8]. 5. Work done by external forces
The formulas for expressions n¯ injq ; o¯ injq ; r¯injq ; ¯tinjq ; p¯ injq ; v¯ injq and u¯ injq in formulas (19), (20) are shown in Section 5.1. 5.1. Components of the work done by external forces function (Ls) In this study formulas are derived for expressions n¯ injq ; o¯ injq ; p¯ injq ; r¯injq ; ¯tinjq ; v¯ injq and u¯ injq , which occur in the component elements Linjq of the work done by external forces function (18) in the following form: 8 0; for i ¼ j; > < 2 2 no P no A A inp jqr ; (21) n¯ injq ¼ s0 ijði þ j Þ P > ij 2 2 2 : ði j Þ p¼1 r¼1 1 þ p þ r
o¯ injq
p¯ injq
In this study the variation of normal stresses in the direction of the length of a plate according to formula (10) was obtained by introduction of longitudinal body forces X according to (16). The work done by external forces for a thus loaded plate (Fig. 4b) can be determined from the formula: 2 Z l s Z bs y qws ¯ s ¼ ts L s0 1 as s bðxs Þ dxs dys . (17) 2 0 0 bs qxs
8 0; > <
for i ¼ j 2 2 no P no A A inp jqr , ¼ s0 ijði þ j Þ P > : ij ði2 j 2 Þ2 p¼1 r¼1 2 þ p þ r
r¯ injq ¼
¯tinjq ¼ (18)
Elements Linjq depend on the coefficient as and function b(xs) of the normal stress distribution in the direction of the length of a plate. For linear variation of stress at b(xs) according to (11) elements Linjq of the work done by external forces function (18) were obtained in the form: 2 p ð1 m=2Þ¯vinjq m¯ninjq Linjq ¼ 4 p2 2 as u¯ injq mas o¯ injq p p¯ injq . ð19Þ 4 For a nonlinear variation of stress according to parabole 21 at b(xs) according to (12) elements Linjq of the function of the work done by external forces (18) are obtained in the form: 2 p Linjq ¼ ð1 m=3Þ¯vinjq m¯rinjq 4 p2 2p2 p¯ as u¯ injq mas ¯tinjq . ð20Þ 4 3 injq
(22)
8 > < 0;
for iaj no P no P Ainp Ajqr , ¼ ij > : p¼1 r¼1 8ð2 þ p þ rÞ
Substituting the deflection function (5) into formula (17), one obtains: io X io X no X no s0 bs t3s X Linjq f in f jq . Ls ¼ l s i¼1 j¼1 n¼1 q¼1
499
(23)
8 n n o Po P 1 > > Ainp Ajqr ; > > < p¼1 r¼1 8ð1 þ p þ rÞ
for i ¼ j ,
2 2 P no P no A A > inp jqr 00 ijði þ j Þ > > s > : ij ði2 j 2 Þ2 p¼1 r¼1 1 þ p þ r
8 n n o Po P Ainp Ajqr > > ; > > < p¼1 r¼1 8ð2 þ p þ rÞ
(24)
for i ¼ j
no P no A A ijði2 þ j 2 Þ P > inp jqr > > s00ij 2 > 2 2 : ði j Þ p¼1 r¼1 2 þ p þ r
,
(25)
u¯ injq ¼ ijdij
no X no X Ainp Ajqr , p þrþ2 p¼1 r¼1
(26)
v¯ injq ¼ ijdij
no X no X Ainp Ajqr . p þrþ1 p¼1 r¼1
(27)
In the above formulas are denoted by ( 0; for i ¼ j; i þ j even; s0ij ¼ , 1; for iaj; i þ j odd: ( 1; for i þ j even; 00 , sij ¼ 1; for i þ j odd: where Ainp, Ajqr—coefficients of power polynomials; for the deflection function described by formula (5) Ainp ¼ Ajqr ¼ 1. Formulas for expressions uinjq and vinjq have a form analogous to those shown in [8].
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6. Elastic buckling stresses and plate buckling modes This section presents selected calculation results of the buckling stresses of eccentrically compressed plates supported on three edges with the participation of body forces generating the variation of normal stresses in the direction of the length of a plate (Fig. 4b). Buckling stress (scr) for a plate loaded in such a manner is defined as a maximal normal stress on the most compressed edge, associated with the stability loss of plate, and is expressed in the form of the classical formula: scr ¼ ksE ,
(28)
where sE ¼
2 p2 E ts . 2 12ð1 n Þ bs
(29)
The plate buckling coefficients k for formula (28) were determined by the energy method on the basis of dependencies given in Sections 2–5. After the calculation of elements Vinjq and Linjq the strain energy of bending of a plate Vs was determined according to formula (9) and work done by external forces Ls—according to (18). The total potential energy of the system is U s ¼ V s Ls .
(30)
Buckling stress and the plate buckling mode are determined from the equation set qU s ¼ 0, qf in
(31)
by reducing the problem to the classical problem of the determination of eigenvalues and eigenvectors. A computing program ‘‘Ncr_plate_free.nb’’ was used for the numerical determination of coefficients k of buckling stress and the plate buckling mode in the environment of
the package Mathematicas [15]. The program enables coefficients k to be tabled and the calculation results to be presented graphically (diagrams of coefficients, buckling modes). Presented in the latter section of the paper tables and the graphs of coefficients were marked for the value of parameters: E ¼ 205 GPa and n ¼ 0.3. The deflection function was approximated by a series (5) for polynomial degree no ¼ 5, when assuming initial values (fi1 ¼ 0) for a plate built-in on one longitudinal edge and simply supported on transverse edges, or (fi2 ¼ 0) for a plate simply supported on three edges. Parameter i0, which defines the number of half-wavelengths of the function sine in the direction of axis xs a series (5), was selected depending on the plate aspect ratio ls/bs and the variation of stress characterized by the coefficients m and as. A classical case of one-directional axially compression of a rectangular plate by uniform intensity of stress in the direction of its length occurs for m ¼ 0 and as ¼ 0. On the basis of numerical tests, it has been found that it is necessary to increase a greater number (io) of the members of series expansion (5) in the direction of axis xs with an increasing ratio ls/bs and increasing the longitudinal variation of stress (m-1). Fig. 5 presents diagrams of coefficient k for a plate built-in on the longitudinal edge bent in its plane (as ¼ 2) at linear variation of stress sx along axis xs according to (10), (11) for parameter m ¼ 1. Diagrams of coefficient k are represented for a plate with ls/bs ¼ 3 and ls/bs ¼ 6 in function io in the range from 2 to 10. It results from a comparison of the diagrams that for a plate with ratio ls/bs ¼ 3 coefficient k ¼ 2.994 sufficient accuracy is already obtained at io ¼ 6 of the expression of series extension (5) in the direction of axis xs (the difference between coefficient k determined for io ¼ 5 and io ¼ 6 is less than 0.04%). For a plate with ratio ls/bs ¼ 6, a comparable accuracy of coefficient (k ¼ 2.639) is obtained by using i0 ¼ 9 members of the series expansion (5).
Fig. 5. Diagram of the buckling stress coefficient k for a plate bended in the plane (as ¼ 2) and built-in on the longitudinal edge with linear variation of stress in the direction of the length of a plate for m ¼ 1 as a function of the parameter io.
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It has been found in tests that for parameters aso2 and mo1 the convergence of coefficient k in the function i0 is faster. In practice io ¼ 10 was assumed for further calculations of coefficients k of the buckling stress of a plate supported on three edges with ratio ls/bsp6 and the values of parameters asp2 and mp1, which gives sufficient accuracy from a technical point of view. Fig. 6 presents diagrams of coefficient k of the buckling stresses for variously supported plates axially compressed (as ¼ 0), at linear variation of normal stress in the direction of the length of a plate according to (10), (11) for m ¼ 0; 0.25; 0.5; 0.75; and 1 in function ls/bs. The dashed lines show diagrams of coefficient k determined from approximation formula according to [1,3] for parameter r=0 and 0.75, where r=1m. It results from a comparison of the diagrams that coefficient k of the buckling stresses of plates increases with increasing parameter m according to (13).
501
The ‘‘garland’’ shape of curves characteristic of m ¼ 0 also disappears in the case of a plate built-in on one side (Fig. 6a). Coefficients k decrease with an increasing ratio ls/bs for the constant value of parameters as and m. For a plate simply supported on three edges (Fig. 6b) the diagrams obtained are compatible with the results calculated from approximation formulas according to [1,3]. In the case of a plate built-in on one side for m=1 (Fig. 6a, r ¼ 0) the maximum differences in the range ls/bs from 2 to 6 are 2.6%. However, comparing diagrams for m ¼ 0.25 (r ¼ 0.75), it is possible to state that the approximation formulas proposed in [1,3] give good (conservative) results for a plate with ls/bsX3 (maximal differences do not exceed 4.7%). Tables 2 and 3 contain coefficient k for a built-in plate (Table 2) and a simply supported plate (Table 3) on the longitudinal edge at linear variation of normal stress in the
Fig. 6. Diagrams of the buckling stresses coefficients k for axially compressed plates (as ¼ 0) at the linear variation of stress in the direction of the length of a plate according to (10), (11) for m ¼ 0; 0.25; 0.5; 0.75; 1; in function ls/bs; dashed lines—plots determined according to formulas from [1] where: r ¼ 1m; (a) longitudinal built-in edge, (b) simply supported longitudinal edge.
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Table 2 Coefficients (k) of buckling stress of a plate with a longitudinal built-in edge at the linear variation of stress in the direction of the length of a plate according to (10), (11) ls/bs
m
as ¼ 1
as ¼ 0
as ¼ 104
as ¼ 2
as ¼ 4
2 2 2 3 3 3 4 4 4 5 5 5 6 6 6
0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0
6.3564 7.5374 8.6323 5.8862 7.0706 7.9079 5.9687 6.8646 7.5076 5.9438 6.7170 7.2600 5.8862 6.6169 7.0890
1.3360 1.6609 1.9718 1.2914 1.5754 1.7886 1.3360 1.5206 1.6838 1.2809 1.4838 1.6205 1.2914 1.4590 1.5769
1.6731 2.0866 2.4856 1.6265 1.9849 2.2546 1.6731 1.9138 2.1213 1.6096 1.8674 2.0410 1.6265 1.8360 1.9856
2.2170 2.7666 3.2949 2.1650 2.6391 2.9934 2.2170 2.5431 2.8178 2.1392 2.4821 2.7117 2.1650 2.4409 2.6386
1.8241 2.2764 2.7134 1.7765 2.1677 2.4618 1.8241 2.0895 2.3163 1.7569 2.0389 2.2285 1.7765 2.0045 2.1681
Table 3 Coefficients (k) of buckling stress of a plate simply supported on three edges at the linear variation of stress in the direction of the length of a plate according to (10), (11) ls/bs
m
as ¼ 1
as ¼ 0
as ¼ 104
as ¼ 2
as ¼ 4
2 2 2 3 3 3 4 4 4 5 5 5 6 6 6
0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0
2.5464 3.3141 4.3266 2.0844 2.6724 3.3241 1.9186 2.4155 2.8892 1.8411 2.2753 2.6459 1.7988 2.1850 2.4904
0.6681 0.8734 1.1621 0.5331 0.6870 0.8677 0.4860 0.6155 0.7451 0.4642 0.5773 0.6778 0.4524 0.5530 0.6352
0.8941 1.1687 1.5542 0.7125 0.9183 1.1601 0.6490 0.8221 0.9958 0.6196 0.7709 0.9056 0.6036 0.7383 0.8485
1.3263 1.7300 2.2793 1.0649 1.3701 1.7208 0.9719 1.2293 1.4834 0.9285 1.1537 1.3518 0.9049 1.1055 1.2681
1.0053 1.3136 1.7444 0.8016 1.0329 1.3040 0.7302 0.9249 1.1199 0.6971 0.8673 1.0186 0.6791 0.8306 0.9545
direction of the length of a plate according to (10), (11) for ls/bs ¼ 2, 3, 4, 5, 6, and parameters: as ¼ 1; 0; 104; 2; 4; and m ¼ 0; 0.5; 1; Note: for as ¼ 104; 2; 4 in formula (10) s0 was assumed with a minus sign (tension on the longitudinal supported edge, compression on the free edge). Fig. 7 compares diagrams of coefficient k for variously supported plates at nonlinear variation of normal stress in the direction of the length of a plate according to (10), (12) and a constant value of the parameter m ¼ 1 and different values of the parameter as ¼ 0; 104; 2 and 4. It results from diagrams that coefficient k increases with a change in transverse stress distribution from axial compression to bending in the plate plane. Higher values of k were obtained for a plate built-in on one side.
Tables 4 and 5 contain coefficients k for a built-in plate (Table 4) and a simply supported plate (Table 5) on the longitudinal edge at nonlinear variation of stress in the direction of the length of a plate according to (10), (12) for ls/bs ¼ 2, 3, 4, 5, 6, and parameters: as ¼ 1; 0; 104; 2; 4; and m ¼ 0.5; 1; Note: for as ¼ 104; 2; 4 in formula (10) s0 was assumed with a minus sign. Fig. 8 compares diagrams of coefficient k for a plate built-in on one side and load parameters: as ¼ 3, m ¼ 0.25; 1 at linear (dashed line) and nonlinear (continuous line) variations of stress in the direction of the length of a plate. Higher values of coefficients k were obtained for a linear stress distribution, while the difference between coefficients for a linear and a nonlinear distribution increases together with increasing parameter m. Fig. 9 compares diagrams of coefficient k for differently supported plates (simply, built-in) at linear variation of stresses for m ¼ 0.5 and as ¼ 0; 104. Higher values of coefficients k are obtained for plates built-in on one side. In both cases coefficients decrease with an increasing plate aspect ratio ls/bs. The variation of the distribution of stress, characterized by parameters m and as, has an effect on the buckling mode of a plate supported on three edges. For the classic case of eccentric compression of a plate built-in on one side (Fig. 2b), at constant stress intensity in the direction of the length of a plate (m ¼ 0), the spontaneously formed buckling half-wavelengths have an identical length and a constant amplitude, and their number depends on the ratio ls/bs. Symmetry or antisymmetry of displacements in relation to a transverse axis of the plate is maintained. In the case (m40) analyzed in this study, the longitudinal stress variation has an effect on the buckling mode of a plate built-in on one side. With an increase of parameter m, the buckling form is characterized by a variable length and different amplitudes of the successive ‘‘half-wavelengths’’ which are formed on the plate length. Moreover, conditions of symmetry or antisymmetry of displacements in relation to the transverse axis of the plate are not fulfilled. Fig. 10 compares the buckling modes of a plate built-in on one side with dimensions ls ¼ 500 mm, bs ¼ 100 mm and ts ¼ 1 mm at a linear stress distribution for parameters as ¼ 3 and m ¼ 0; 0.5; 1. Contour line diagrams were supplemented by the buckling shape of the free edge in the longitudinal cross-section. The eigenvector of the first mode of buckling was standardized so that the maximum deflection was 1. With the increase of parameter m, coefficient k of critical stresses increases and the number of significant ‘‘half-wavelengths’’ of buckling decreases along the plate length (from 3 for m ¼ 0 to 2 for m ¼ 1). Maximum deflections occur from the side of a more compressed edge, and amplitudes of subsequent ‘‘halfwavelengths’’ of buckling mode have a vanishing character. In the case of eccentric compression of a plate simply supported on three edges (Fig. 2a) at a constant stress intensity in the direction of the length of a plate (m ¼ 0), the buckling mode creates one half-wavelength when
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Fig. 7. Diagrams of the buckling stresses coefficients k at the nonlinear variation of stress in the direction of the length of a plate according to (10), (12) for parameters: m ¼ 1 and as ¼ 0; 104; 2; 4; in function ls/bs; (a) longitudinal built-in edge, (b) simply supported longitudinal edge. Table 4 Coefficients (k) of buckling stress of a plate with a longitudinal built-in edge at the nonlinear variation of stress in the direction of the length of a plate according to (10), (12)
Table 5 Coefficients (k) of buckling stress of a plate simply supported on three edges at the nonlinear variation of stress in the direction of the length of a plate according to (10), (12)
ls/bs
m
as ¼ 1
as ¼ 0
as ¼ 104
as ¼ 2
as ¼ 4
ls/bs
m
as ¼ 1
as ¼ 0
as ¼ 104
as ¼ 2
as ¼ 4
2 2 3 3 4 4 5 5 6 6
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
7.0464 7.5177 6.5603 6.9146 6.4068 6.6329 6.2924 6.4734 6.2229 6.3701
1.5512 1.7028 1.4571 1.5511 1.4143 1.4735 1.3841 1.4318 1.3667 1.4047
1.9485 2.1458 1.8351 1.9549 1.7797 1.8554 1.7416 1.8023 1.7193 1.7678
2.5839 2.8468 2.4418 2.5983 2.3652 2.4660 2.3152 2.3956 2.2854 2.3498
2.1258 2.3426 2.0050 2.1350 1.9430 2.0259 1.9015 1.9678 1.8770 1.9300
2 2 3 3 4 4 5 5 6 6
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
3.0847 3.6721 2.4917 2.8580 2.2568 2.5105 2.1303 2.3191 2.0499 2.1990
0.8125 0.9805 0.6402 0.7425 0.5746 0.6448 0.5401 0.5918 0.5183 0.5589
1.0873 1.3116 0.8556 0.9927 0.7675 0.8616 0.7211 0.7906 0.6920 0.7464
1.6098 1.9291 1.2769 1.4755 1.1479 1.2855 1.0795 1.1817 1.8364 1.1167
1.2221 1.4728 0.9625 1.1161 0.8635 0.9692 0.8113 0.8894 0.7785 0.8397
symmetry in relation to the transverse axis of the plate is preserved. For parameter m40 the longitudinal variation of the stress distribution generates an asymmetric shape of
displacements in relation to the transverse axis of the plate. The maximum deflections of one spontaneously formed asymmetric ‘‘half-wavelength’’ are displaced in the
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Fig. 8. A comparison of coefficient diagrams k for a plate built-in on one side (for as ¼ 3, m ¼ 0.25; 1) at linear (dashed line) and nonlinear (continuous line) variations of stress in the direction of the length of a plate.
Fig. 9. A comparison of coefficient diagrams k for variously supported plates at linear variation of stress in the direction of the length of a plate for m ¼ 0.5 and as ¼ 0; 104; continuous line—longitudinal edge built-in, dashed line—longitudinal edge simply supported.
direction of the transverse, more compressed edge. The asymmetry degree of displacements increases with increasing parameter m. 7. Conclusion The deflection function introduced in this study in the form of a polynomial-sinusoidal series (5) enables one to model the boundary conditions of a plate supported on the longitudinal edge by a simple support, by an elastic restraint in the segment of a thin-walled bar, until a built-in edge is obtained. The process of adjustment of the
deflection function (5) to parameters fin from the minimum energy condition to the plate buckling mode occurs both in displacement shaping and ‘‘residual forces’’ reduction on the free edge. An effective way to describe the variation of stress in the direction of the length of a plate supported on three edges is the introduction of longitudinal body forces operating in its middle plane. The distribution of body forces is selected depending on the manner of plate loading. Such an approach allows simplifying the deflection function by reducing the number of series expressions (5) necessary for the approximation of the buckling mode of a plate.
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Fig. 10. Buckling modes of a plate built-in on one side with ls/bs ¼ 5 at linear variation of stress in the direction of the length of a plate for parameters: as ¼ 3 and m ¼ 0; 0.5; 1.
The formulas derived in this study for the work done by external forces and the procedure of determining the strain energy of plate bending proposed by Jakubowski in [8] permit elaboration of effective computer programs which can be used to examine the stability of variously loaded plates which are components of thin-walled bars. The defined buckling stress of a plate loaded in the plane by a variable stress distribution enables one to construct effective diagrams and tables of buckling coefficients (k). This facilitates the assessment of the effective width and limit load-carrying capacity of this class of plates which are component parts of thin-walled bars with an open crosssection, e.g. in the manner demonstrated in [1]. The value of coefficient k of the buckling stresses of plates with a free edge increases with increasing parameter m of longitudinal stress variation. Smaller coefficients k by the same boundary conditions and the same values of parameters as, m and ls/bs were obtained for a nonlinear variation of normal stress in the direction of the length of a plate. The longitudinal stress variation also has an effect on the buckling modes of plates supported on three edges. With an increasing parameter m the buckling mode of a plate with a longitudinal built-in edge is characterized by a variable length and different amplitudes of subsequent buckling ‘‘half-wavelengths’’ which are formed on its length. For plates simply supported on three edges, the longitudinal stress variation causes deflection according to one nonsymmetric ‘‘half-wavelength’’ with maximal deflections displaced in the direction of the transverse edge which is more compressed.
References [1] Yu C, Schafer BW. Effect of longitudinal stress gradient on the ultimate strength of thin plates. Thin Walled Struct 2006;44:787–99. [2] NAS. North American specification for the design of cold-formed steel structural members. 2001 ed. Washington, DC, USA: American Iron and Steel Institute; 2001. [3] Yu C, Schafer BW. Effect of longitudinal stress gradients on elastic buckling of thin plates. J Eng Mech ASCE 2007;133(4):452–63. [4] Kowal Z. The stability of compressed flange of plate girder with a box section. Zeszyty Naukowe Politechniki Wroc"awskiej, Budownictwo 1965;122:73–85 (in Polish). [5] Kowal Z. The stability top metal plate of pontoon foundation. W˛egiel Brunatny 1966;4:331–3 (in Polish). [6] Bulson PS. The stability of flat plates. London: Chatto and Windus; 1970. [7] Kro´lak M. The buckling state and the analysis of the post-buckling behaviour of a rectangular plane subject to nonuniform compression and shear. Archiwum Budowy Maszyn 1978;4:633–43 (in Polish). [8] Jakubowski S. The matrix analysis of stability and free vibrations of walls of thin-walled girders. Archiwum Budowy Maszyn 1986; 4:357–76 (in Polish). [9] Kro´lak M, editor. Post-critical states and limit load-carrying capacity of thin-walled girders with flat walls. Warsaw, Lodz: State Scientific Publishers; 1990 (in Polish). [10] Jakubowski S. Buckling of thin-walled girders under compound load. Thin-Walled Struct 1988;6:129–50. [11] Jakubowski S. Local buckling of thin-walled girders of triangular cross-section. Thin-Walled Struct 1989;8:253–72. [12] Timoshenko SP, Gere JM. Theory of elastic stability. Part II. New York, NY: McGraw-Hill; 1961. [13] Handbook of structural stability. Edited by Column Research Committee of Japan, Tokyo: Corona Publishing Company; 1971. [14] Gra˛ dzki R, Kowal-Michalska K. Elastic and elasto-plastic buckling of thin-walled columns subjected to uniform compression. ThinWalled Struct 1985;3:93–108. [15] Wolfram S. Mathematica. Cambridge: Cambridge University Press.
Thin-Walled Structures 46 (2008) 494–505 www.elsevier.com/locate/tws
The stability of eccentrically compressed thin plates with a longitudinal free edge and with stress variation in the longitudinal direction Andrzej Szychowski Faculty of Civil and Environmental Engineering, Kielce University of Technology, Al. Tysia˛clecia Pan´stwa Polskiego 7, 25-314 Kielce, Poland Received 24 April 2007; received in revised form 23 October 2007; accepted 23 October 2007 Available online 3 December 2007
Abstract This paper presents results of the investigation of the stability of eccentrically compressed thin plates with one longitudinal free edge with the participation of loadings generating the variation of stresses in the direction of the length of a plate. A deflection function (5) is proposed to enable modelling of boundary conditions on the second longitudinal edge from a simply supported, through elastically restrained, to a built-in edge. Formulas are derived for work done by external force under loading generating the variation of stress according to a linear function and according to parabola 21. Tables and plots of buckling coefficient (k) for variously supported and variously loaded plates with a free edge, which are not found in the literature, are determined. The effect of a longitudinal stress variation on the plate buckling modes is analyzed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Structural stability; Longitudinal stress variation; Thin plates
1. Introduction In thin-walled bars with an open cross-section built from thin plates at composite loads there occur incidents in which component plates (with a free edge) are axially or eccentrically compressed by the simultaneous variation of stresses in the direction of the length of a segment. A thinwalled bar segment is defined here as a bar section between transverse stiffening (ribs, diaphragms, etc.), which assures a stiff cross-section contour. A number of local buckling ‘‘half-wavelengths’’ with varied length and amplitude can be formed on the bar segment length. A proper determination of buckling stresses for thus loaded component plates of the thin-walled bar serves to obtain a more accurate determination of limit load-carrying capacity on the basis of the effective width theory. Yu and Schafer [1] investigated the effect of a longitudinal stress gradient on the limit load-carrying capacity of plates axially compressed on one side, which are component walls of thin-walled bars. Plates simply Tel.: +4841 342 4575; fax: +4841 344 3784.
E-mail address: [email protected] 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.10.009
supported on all edges and plates simply supported on three edges with a longitudinal free edge were investigated. The results were compared with the effective width theory by the finite element method (ABAQUS) according to Winter’s formula [2]. Buckling stresses of axially compressed plates at a longitudinal stress gradient, necessary to determine effective width, were determined on the basis of the proposal contained in [3]. In the literature there are no studies concerning the stability of eccentrically compressed plates (in their plane) supported on three edges (with a longitudinal free edge) with the participation of loadings generating the variation of stress in the direction of the length of a plate. Plates supported and loaded in this way occur, amongst others, as walls in thin-walled bars with open cross-sections in composite load states. However, there are studies concerning stability separated from the segment of the thin-walled bar of a rectangular plate supported on all edges by loadings generating variation of stress in the direction of the length of a plate, e.g. [1,3,4–9]. In [8,10,11], Jakubowski determined, amongst others, buckling stresses from the condition of local stability loss of variously loaded bars with closed rectangular and
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Nomenclature
Qky
Aink, Ajqr, Ainp coefficients of power polynomials bs, ls, ts width, length, thickness of a plate (wall s) Ds plate flexural rigidity E Young’s modulus of elasticity fin, fjq dimensionless, free parameters of the deflection function i, j, n, q, p natural number index io number of half-wavelength of the function sine in the direction of the length of a plate k plate buckling coefficient Ls work done by external forces Linjq component elements of the work done by the external forces function m coefficient that characterizes the longitudinal stresses variation according to (13) My moment on the free edge of a plate no degree of the polynomial n¯ injq ; o¯ injq ; p¯ injq ; r¯injq ; ¯tinjq ; v¯ injq ; u¯ injq expressions which occur in the elements Linjq
Us Vs Vinjq
triangular sections. The starting point of the study [8] is the analysis of the stability of a rectangular plate supported on all edges and eccentric compression on transverse edges and shear forces acting in the plane of the plate. This kind of load produces a variation of normal stress in the direction of the length of a plate. The deflection function (in the direction of the width of a plate) was assumed to be in the form of power polynomials with a simple physical interpretation. The application of polynomials considerably facilitates the mathematical side of the problem, and their applicability in the investigations of local stability of plates and closed cross-sections has been confirmed in numerous investigations of the research team at the University of Technology in Ło´dz´ (Poland), which are listed in the monograph [9]. Another way of obtaining the variation of the stress in the direction of the length of the bar segment with a closed cross-section was applied by Kowal in [4,5]. The problem of stability of a transversely bended plate girder with a box section, at different transverse load variants, was reduced to the problem of stability of a rectangular plate (compressed flange) simply supported on all edges with the length of a bar segment. The plate was loaded with uniformly distributed forces (N0, N1) on the transverse edges and body forces (X) acting in the middle plane of a plate. The distribution of body forces was chosen depending on the longitudinal variation of the stress generated transverse load of the girder (Fig. 1). The problem was solved with the use of Galerkin’s method by calculating buckling coefficients of plates with different
ws xs, ys, zs X Yin as b(xs) dij n sx, txy scr sE s0, s1 r2
495
Kirchhoff’s substitute force on the free edge of a plate total potential energy of a system strain energy of the bending of a plate component elements of the bending strain energy function deflection function of a plate (wall s) Cartesian coordinates of a plate (wall s) longitudinal body forces power polynomials (8) with previously determined coefficients Ainp according to [8] coefficient stress distribution in the direction of the width of a plate function of normal stress distribution in the direction of the length of a plate Kronecker’s delta Poisson’s ratio membrane stresses critical stress Euler’s stress edge normal stresses (positive, if compressive) Laplace operator
ratios of side lengths and different variants of girder transverse load. Kro´lak [7] analyzed a similar case of longitudinal stress variation for an axially compressed plate. The variation of normal stresses in the direction of the length of a plate was obtained by introducing shear stress to. Displacement functions were written in the form of a double, finite sinus series. The solution of the problem was obtained by Galerkin’s method. The presented coefficients of critical stresses for plates of aspect ratio ls/bs from 0.5 to 4 in [7] and from 0.5 to 2 in [9] are, however, greater than those obtained by Kowal [4,5]. The technical solution of many problems of local buckling and limit load-carrying capacity of thinwalled bars with an open cross-section, built from flat walls, needs solutions of the critical state of plates with a longitudinal free edge at eccentric compression and the variation of stresses in the direction of the length of a plate. The aim of this study is to determine buckling stresses and buckling modes of rectangular plates supported on three edges with one longitudinal free edge and loaded by eccentrical compression (in the plate plane) with the participation of loadings generating the variation of stresses in the direction of the length of a plate. The following cases are considered: (1) a plate simply supported on three edges (Fig. 2a), and (2) a plate built-in on the longitudinal edge and simply supported on transverse edges (Fig. 2b). The above boundary conditions on the longitudinal-supported edge are cases of limit support of
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Fig. 1. Exemplary diagram of the plate support and load considered in study [4] (where: N0 ¼ s0ts; N1 ¼ s1ts).
Fig. 2. Limit boundary conditions of plates supported on three edges.
walls with a free edge in the segment of a thin-walled bar with an open cross-section. 2. Function of plate deflection The stability of axially or eccentrically compressed plates with one longitudinal free edge at a constant normal stress intensity in the direction of the length of a plate and boundary limit conditions on a longitudinal-supported edge (simply, built-in) and transverse edges (xs ¼ 0, xs ¼ ls) simply supported (Fig. 2) was the subject of investigation of many authors, including [6,12,13]. A comprehensive list of results is contained in Bulson’s monograph [6], which presents different methods for the solution of the problem. In the case of eccentric compression or bending in the middle plane of the plate with a longitudinal free edge, it is not possible to obtain an exact solution resulting from integration of the known differential equation of thin-plate stability. Using the energy method, Bulson [6] determined coefficient k of edge buckling stress for an eccentrically compressed plate with a free edge (Fig. 2a) at a constant normal stress intensity in the direction of the length of a plate, assuming the deflection function in the following form: ys ipxs ws ¼ f in sin . (1) bs ls Deflection function (1) satisfies the boundary conditions on the longitudinal-supported edge, whereas conditions (2) of disappearance moment My and conditions (3) of disappearing Kirchoff’s substitute force Qky on the free
Fig. 3. Diagram of moment (My) on the free edge of the plate axially compressed (after stability loss), whose deflection function is expressed by formula (1).
edge are fulfilled identically q2 ws q2 w s þ n 2 ¼ 0; 2 qys qxs
for ys ¼ bs ,
q3 ws q3 w s þ ð2 nÞ ¼ 0; qy3s qx2s qys
for ys ¼ bs .
(2)
(3)
Fig. 3 shows the distribution of moment My on the free edge of the plate axially compressed (after stability loss), whose deflection function is expressed by formula (1). Gra˛ dzki and Kowal–Michalska [14] applied the function in the form (1) and the polynomial: gðys Þ ¼ 6ðys =bs Þ2 4ðys =bs Þ3 þ ðys =bs Þ4 ,
(4)
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to the description of deflection of the axially compressed plate with a longitudinal free edge and elastically restrained by another edge. Polynomial (4) according to [14] is a deflection function for a uniformly loaded cantilever beam. The same form of deflection is obtained for a cantilever plate strip (at infinite length) uniformly loaded on the whole surface. As can be concluded from the author’s investigation, the application of deflection function g(ys) resulting from the displacement form of variously loaded infinite plate strips can be a satisfactory approximation of the buckling modes of eccentrically compressed plates, built-in longitudinally on one side (cf. Fig. 2b) at a constant stress intensity in the direction of the length of a plate. However, it is not a universal approach because for different distributions of an eccentrical plate compression, one has to seek each time a deflection function which approximates the bulking shape well. Moreover, boundary conditions (2), (3) are generally not fulfilled on the free edge of the plate. However, the application of combinations of hyperbolic functions [6,12] to the description of plate deflection complicates considerably the mathematical side of the problem, particularly in these cases, when a plate with a free edge is a component element of a thin-walled bar. On the basis of the investigation of different forms of theoretical deflection functions, for the approximation of the buckling modes of eccentrically compressed plates with a longitudinal free edge with the participation of loadings generating the variation of stresses in the direction of the length of a plate, we propose to apply the function in the form of a series: n ! io no X X ys ipxs ws ðxs ; ys Þ ¼ ts f in . (5) sin bs ls i¼1 n¼1 The definite parameters fin of function (5) are given an initial value depending on boundary conditions on the longitudinal-supported edge, and therefore: fi1 ¼ 0, for a plate built-in on the longitudinal edge (Fig. 2b), fi2 ¼ 0, for a plate simply supported on the longitudinal edge (Fig. 2a). In the case of elastic restraint of a plate in the segment of a thin-walled bar, it is necessary to apply the full form of function (5) i.e. all fin are nonzero parameters of the deflection function, and their initial values result from static and kinematic boundary conditions on the connection of adjacent plates. The expression in the first bracket of the series (5) can be interpreted as an no degree polynomial, whose particular coefficients fin constitute parameters of the deflection function, and in boundary cases they are boundary conditions on the supported edge (simply, built-in), and some of them (fi1 and fi2) are given zero initial values. Such an approach enables one to adjust function (5) optimally to the buckling mode of the plate
497
Table 1 A comparison maximal ‘‘residual forces’’ (My, Qky) in relation to plate with ls/bs ¼ 1, (in %) ls/bs
My_1
Qky_1
My_3
Qky_3
My_4
Qky_4
My_5
Qky_5
1 2 3 4
100 25 11.1 6.25
100 25 11.1 6.25
96.7 23.7 10.4 5.86
103 25.1 11.1 6.21
32 5.1 2 1.05
70.3 10.6 4 2.09
8.2 0.58 0.12 0.04
30.3 2.1 0.43 0.14
Note: My_1, Qky_1-for a deflection function (1) according to [6]; My_3,4,5, Qky_3,4,5-for a deflection function according to formula (5) at no ¼ 3,4 or 5.
with one free edge irrespective of the load distribution and the manner of support on the second edge. Deflection function (5) with initial values of parameters (fi1, fi2) satisfies boundary conditions on the supported longitudinal edge, whereas conditions (2), (3) on the free edge, similar to that in [6,14], are not fulfilled identically. However, it is possible to demonstrate that the bending moment My and Kirchhoff’s substitute force Qky on the free edge of the plate tend to zero, minimizing thereby the complete potential energy of the system when the degree no of polynomial increases. The process of adjustment of the deflection function (5) to parameters fin from the condition of minimum energy to the shape of plate buckling occurs both in the formation of displacements and the reduction of ‘‘residual forces’’ (My, Qky) on the free edge. Moreover, the effect of ‘‘residual forces’’ generated by the adopted deflection function, which does not fulfill identically the boundary conditions (2), (3) on the free edge, decreases with increasing length of the plate. Table 1 presents the maximum values of ‘‘residual forces’’ (My, Qky) on the free edge of an axially compressed plate for different aspect ratio ls/bs and degree no of polynomial (5) from 1 to 5 with the initial value fi2 ¼ 0, which is equal to simply supported on the second edge (Fig. 3). The results are compared in percent with the values obtained at plate deflection approximation according to formula (1) according to [6] for a plate aspect ratio ls/bs ¼ 1. ‘‘Residual forces’’ on the free edge were determined after each standardization of displacements for the plate buckling mode described by a series (5). Plate buckling modes for ls/bs from 1 to 4 were determined by the energy method according to a procedure described in the latter part of this study. 3. Strain energy of plate bending In the case of a plate supported on three edges (Figs. 2 and 3) the strain energy of the bending of a plate, whose deflections are described by a series (5), can be determined from the formula: " Z l s Z bs ( D q2 ws q2 ws s V¯ s ¼ ðr2 ws Þ2 2ð1 nÞ 2 0 0 qx2s qy2s #) 2 2 q ws ð6Þ dxs dys . qxs qys
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Jakubowski in [8] provided formulas for the bending strain energy of a rectangular plate (bar walls with a closed cross-section), supported on four edges, whose deflection function can be represented in the form: ! io no X X ipxs ws ðxs ; ys Þ ¼ ts f in Y in sin , (7) ls i¼1 n¼1 where Yin—power polynomials (8) with previously determined coefficients Ainp according to [8] p 7 X y Y in ¼ Ainp s . (8) bs p¼0 The considerations of the practical local stability loss of a bar with a closed cross-section for a plate supported on all edges were reduced in [8–10] to polynomials of a maximum of the 7th degree, which fulfill conditions Yin(0)=Yin(1)=0. Function ws(xs, ys) according to (7) satisfies the boundary conditions of a plate supported on all edges and enables the complex shape of the buckling of the plate to be described in a general case, which occurs during compression, bending and shearing in the middle plane of the wall of the closed cross-section [8–10]. Proceeding analogously as in [8] and substituting deflection function ws(xs, ys) according to (5) for the dependence (6), we obtain: Vs ¼
io X io X no X no Ds t2s X V injq f in f jq . 4bs l s i¼1 j¼1 n¼1 q¼1
(9)
Elements Vinjq of the square form (9) depend on polynomial coefficients, Poisson’s ratio n and the plate aspect ratio ls/bs. Due to the fact that members (ys/bs)n in the series (5) can be treated in practice as one-dimensional polynomials in the form (8) with a coefficient Ainp ¼ 1, elements Vinjq in formula (9) can be calculated from the sequence of formulas derived for a plate supported on all edges in [8]. 4. Membrane stress state In the case of stability analysis of the segment of the thin-walled bar, in which the flat cross-section hypothesis or the section warping hypothesis is accepted (depending on the state of a bar load), the distribution of normal stress in a component plate supported on three edges (Fig. 4) can
be presented in the form: ys bðxs Þ. sx ¼ s0 1 as bs
(10)
The following cases are considered in this study: (1) linear variation of stress sx in the direction of the length of a plate, and (2) nonlinear variation of stress sx (according to parabola 21) in the direction of the length of a plate, for which function b(xs) of normal stress distribution in the direction of the length of a plate can be represented, respectively, in the form: xs bðxs Þ ¼ 1 m , (11) ls bðxs Þ ¼ 1 m
x2s , l 2s
(12)
where m ¼ 1 s1 =s0 .
(13)
Variation of normal stress in the direction of the length of a plate according to formulas (10), (11), (12) can be obtained by introducing shear stresses txy (Fig. 4a) or body forces X (Fig. 4b) with a distribution chosen according to that of a bar segment load. Taking into consideration the differential equilibrium equation: qsx qtxy þ þ X ¼ 0, qxs qys
(14)
and skipping body forces (X ¼ 0), one can present the shear stress distribution for a plate loaded according to Fig. 4a with a boundary condition txy(bs) ¼ 0 in the form: txy ¼
ðbs ys Þ½bs ðas 2Þ þ ys as s0 b0 ðxs Þ , 2bs
(15)
where b0 (xs)—first derivative of the normal stress distribution function in the direction of the length of a plate. Skipping in Eq. (14) shear stresses (txy ¼ 0), the distribution of body forces according to Fig. 4b assumes the form: qsx y X ¼ ¼ s0 1 as s b0 ðxs Þ. (16) @xs bs Introduction of longitudinal body forces, according to formula (16), which generate the variation of normal stress in the direction of the length of a plate according to
Fig. 4. The membrane stress state in a plate supported on three edges at eccentric compression and a longitudinal stress variation caused by (a) shear stresses txy, (b) longitudinal body forces X.
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formula (10), permits performing an analysis of the stability of a plate supported on three edges in those cases in which the distribution and intensity of shear stresses (txy) do not have a significant effect on the buckling mode. In the case of the secondary effect of shear stresses on the stability loss of a plate supported on three edges, load reduction according to Fig. 4a to normal stresses with participation of body forces according to Fig. 4b enables the deflection function (5) to be simplified by reducing the number of necessary members of series (5) necessary to approximate the buckling mode of the plate. In the case of transverse bending of thin-walled bars with a considerable participation of transverse forces in slender webs (plates supported on four edges), it is necessary to take into consideration shear stresses (txy), because their participation in the local stability loss of a web may be considerable [9,10]. Proper formulas for the work done by external forces taking into account of shear stresses (txy) for plates supported on all edges loaded by bending with shearing were introduced by Jakubowski in [8]. 5. Work done by external forces
The formulas for expressions n¯ injq ; o¯ injq ; r¯injq ; ¯tinjq ; p¯ injq ; v¯ injq and u¯ injq in formulas (19), (20) are shown in Section 5.1. 5.1. Components of the work done by external forces function (Ls) In this study formulas are derived for expressions n¯ injq ; o¯ injq ; p¯ injq ; r¯injq ; ¯tinjq ; v¯ injq and u¯ injq , which occur in the component elements Linjq of the work done by external forces function (18) in the following form: 8 0; for i ¼ j; > < 2 2 no P no A A inp jqr ; (21) n¯ injq ¼ s0 ijði þ j Þ P > ij 2 2 2 : ði j Þ p¼1 r¼1 1 þ p þ r
o¯ injq
p¯ injq
In this study the variation of normal stresses in the direction of the length of a plate according to formula (10) was obtained by introduction of longitudinal body forces X according to (16). The work done by external forces for a thus loaded plate (Fig. 4b) can be determined from the formula: 2 Z l s Z bs y qws ¯ s ¼ ts L s0 1 as s bðxs Þ dxs dys . (17) 2 0 0 bs qxs
8 0; > <
for i ¼ j 2 2 no P no A A inp jqr , ¼ s0 ijði þ j Þ P > : ij ði2 j 2 Þ2 p¼1 r¼1 2 þ p þ r
r¯ injq ¼
¯tinjq ¼ (18)
Elements Linjq depend on the coefficient as and function b(xs) of the normal stress distribution in the direction of the length of a plate. For linear variation of stress at b(xs) according to (11) elements Linjq of the work done by external forces function (18) were obtained in the form: 2 p ð1 m=2Þ¯vinjq m¯ninjq Linjq ¼ 4 p2 2 as u¯ injq mas o¯ injq p p¯ injq . ð19Þ 4 For a nonlinear variation of stress according to parabole 21 at b(xs) according to (12) elements Linjq of the function of the work done by external forces (18) are obtained in the form: 2 p Linjq ¼ ð1 m=3Þ¯vinjq m¯rinjq 4 p2 2p2 p¯ as u¯ injq mas ¯tinjq . ð20Þ 4 3 injq
(22)
8 > < 0;
for iaj no P no P Ainp Ajqr , ¼ ij > : p¼1 r¼1 8ð2 þ p þ rÞ
Substituting the deflection function (5) into formula (17), one obtains: io X io X no X no s0 bs t3s X Linjq f in f jq . Ls ¼ l s i¼1 j¼1 n¼1 q¼1
499
(23)
8 n n o Po P 1 > > Ainp Ajqr ; > > < p¼1 r¼1 8ð1 þ p þ rÞ
for i ¼ j ,
2 2 P no P no A A > inp jqr 00 ijði þ j Þ > > s > : ij ði2 j 2 Þ2 p¼1 r¼1 1 þ p þ r
8 n n o Po P Ainp Ajqr > > ; > > < p¼1 r¼1 8ð2 þ p þ rÞ
(24)
for i ¼ j
no P no A A ijði2 þ j 2 Þ P > inp jqr > > s00ij 2 > 2 2 : ði j Þ p¼1 r¼1 2 þ p þ r
,
(25)
u¯ injq ¼ ijdij
no X no X Ainp Ajqr , p þrþ2 p¼1 r¼1
(26)
v¯ injq ¼ ijdij
no X no X Ainp Ajqr . p þrþ1 p¼1 r¼1
(27)
In the above formulas are denoted by ( 0; for i ¼ j; i þ j even; s0ij ¼ , 1; for iaj; i þ j odd: ( 1; for i þ j even; 00 , sij ¼ 1; for i þ j odd: where Ainp, Ajqr—coefficients of power polynomials; for the deflection function described by formula (5) Ainp ¼ Ajqr ¼ 1. Formulas for expressions uinjq and vinjq have a form analogous to those shown in [8].
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6. Elastic buckling stresses and plate buckling modes This section presents selected calculation results of the buckling stresses of eccentrically compressed plates supported on three edges with the participation of body forces generating the variation of normal stresses in the direction of the length of a plate (Fig. 4b). Buckling stress (scr) for a plate loaded in such a manner is defined as a maximal normal stress on the most compressed edge, associated with the stability loss of plate, and is expressed in the form of the classical formula: scr ¼ ksE ,
(28)
where sE ¼
2 p2 E ts . 2 12ð1 n Þ bs
(29)
The plate buckling coefficients k for formula (28) were determined by the energy method on the basis of dependencies given in Sections 2–5. After the calculation of elements Vinjq and Linjq the strain energy of bending of a plate Vs was determined according to formula (9) and work done by external forces Ls—according to (18). The total potential energy of the system is U s ¼ V s Ls .
(30)
Buckling stress and the plate buckling mode are determined from the equation set qU s ¼ 0, qf in
(31)
by reducing the problem to the classical problem of the determination of eigenvalues and eigenvectors. A computing program ‘‘Ncr_plate_free.nb’’ was used for the numerical determination of coefficients k of buckling stress and the plate buckling mode in the environment of
the package Mathematicas [15]. The program enables coefficients k to be tabled and the calculation results to be presented graphically (diagrams of coefficients, buckling modes). Presented in the latter section of the paper tables and the graphs of coefficients were marked for the value of parameters: E ¼ 205 GPa and n ¼ 0.3. The deflection function was approximated by a series (5) for polynomial degree no ¼ 5, when assuming initial values (fi1 ¼ 0) for a plate built-in on one longitudinal edge and simply supported on transverse edges, or (fi2 ¼ 0) for a plate simply supported on three edges. Parameter i0, which defines the number of half-wavelengths of the function sine in the direction of axis xs a series (5), was selected depending on the plate aspect ratio ls/bs and the variation of stress characterized by the coefficients m and as. A classical case of one-directional axially compression of a rectangular plate by uniform intensity of stress in the direction of its length occurs for m ¼ 0 and as ¼ 0. On the basis of numerical tests, it has been found that it is necessary to increase a greater number (io) of the members of series expansion (5) in the direction of axis xs with an increasing ratio ls/bs and increasing the longitudinal variation of stress (m-1). Fig. 5 presents diagrams of coefficient k for a plate built-in on the longitudinal edge bent in its plane (as ¼ 2) at linear variation of stress sx along axis xs according to (10), (11) for parameter m ¼ 1. Diagrams of coefficient k are represented for a plate with ls/bs ¼ 3 and ls/bs ¼ 6 in function io in the range from 2 to 10. It results from a comparison of the diagrams that for a plate with ratio ls/bs ¼ 3 coefficient k ¼ 2.994 sufficient accuracy is already obtained at io ¼ 6 of the expression of series extension (5) in the direction of axis xs (the difference between coefficient k determined for io ¼ 5 and io ¼ 6 is less than 0.04%). For a plate with ratio ls/bs ¼ 6, a comparable accuracy of coefficient (k ¼ 2.639) is obtained by using i0 ¼ 9 members of the series expansion (5).
Fig. 5. Diagram of the buckling stress coefficient k for a plate bended in the plane (as ¼ 2) and built-in on the longitudinal edge with linear variation of stress in the direction of the length of a plate for m ¼ 1 as a function of the parameter io.
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It has been found in tests that for parameters aso2 and mo1 the convergence of coefficient k in the function i0 is faster. In practice io ¼ 10 was assumed for further calculations of coefficients k of the buckling stress of a plate supported on three edges with ratio ls/bsp6 and the values of parameters asp2 and mp1, which gives sufficient accuracy from a technical point of view. Fig. 6 presents diagrams of coefficient k of the buckling stresses for variously supported plates axially compressed (as ¼ 0), at linear variation of normal stress in the direction of the length of a plate according to (10), (11) for m ¼ 0; 0.25; 0.5; 0.75; and 1 in function ls/bs. The dashed lines show diagrams of coefficient k determined from approximation formula according to [1,3] for parameter r=0 and 0.75, where r=1m. It results from a comparison of the diagrams that coefficient k of the buckling stresses of plates increases with increasing parameter m according to (13).
501
The ‘‘garland’’ shape of curves characteristic of m ¼ 0 also disappears in the case of a plate built-in on one side (Fig. 6a). Coefficients k decrease with an increasing ratio ls/bs for the constant value of parameters as and m. For a plate simply supported on three edges (Fig. 6b) the diagrams obtained are compatible with the results calculated from approximation formulas according to [1,3]. In the case of a plate built-in on one side for m=1 (Fig. 6a, r ¼ 0) the maximum differences in the range ls/bs from 2 to 6 are 2.6%. However, comparing diagrams for m ¼ 0.25 (r ¼ 0.75), it is possible to state that the approximation formulas proposed in [1,3] give good (conservative) results for a plate with ls/bsX3 (maximal differences do not exceed 4.7%). Tables 2 and 3 contain coefficient k for a built-in plate (Table 2) and a simply supported plate (Table 3) on the longitudinal edge at linear variation of normal stress in the
Fig. 6. Diagrams of the buckling stresses coefficients k for axially compressed plates (as ¼ 0) at the linear variation of stress in the direction of the length of a plate according to (10), (11) for m ¼ 0; 0.25; 0.5; 0.75; 1; in function ls/bs; dashed lines—plots determined according to formulas from [1] where: r ¼ 1m; (a) longitudinal built-in edge, (b) simply supported longitudinal edge.
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Table 2 Coefficients (k) of buckling stress of a plate with a longitudinal built-in edge at the linear variation of stress in the direction of the length of a plate according to (10), (11) ls/bs
m
as ¼ 1
as ¼ 0
as ¼ 104
as ¼ 2
as ¼ 4
2 2 2 3 3 3 4 4 4 5 5 5 6 6 6
0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0
6.3564 7.5374 8.6323 5.8862 7.0706 7.9079 5.9687 6.8646 7.5076 5.9438 6.7170 7.2600 5.8862 6.6169 7.0890
1.3360 1.6609 1.9718 1.2914 1.5754 1.7886 1.3360 1.5206 1.6838 1.2809 1.4838 1.6205 1.2914 1.4590 1.5769
1.6731 2.0866 2.4856 1.6265 1.9849 2.2546 1.6731 1.9138 2.1213 1.6096 1.8674 2.0410 1.6265 1.8360 1.9856
2.2170 2.7666 3.2949 2.1650 2.6391 2.9934 2.2170 2.5431 2.8178 2.1392 2.4821 2.7117 2.1650 2.4409 2.6386
1.8241 2.2764 2.7134 1.7765 2.1677 2.4618 1.8241 2.0895 2.3163 1.7569 2.0389 2.2285 1.7765 2.0045 2.1681
Table 3 Coefficients (k) of buckling stress of a plate simply supported on three edges at the linear variation of stress in the direction of the length of a plate according to (10), (11) ls/bs
m
as ¼ 1
as ¼ 0
as ¼ 104
as ¼ 2
as ¼ 4
2 2 2 3 3 3 4 4 4 5 5 5 6 6 6
0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0
2.5464 3.3141 4.3266 2.0844 2.6724 3.3241 1.9186 2.4155 2.8892 1.8411 2.2753 2.6459 1.7988 2.1850 2.4904
0.6681 0.8734 1.1621 0.5331 0.6870 0.8677 0.4860 0.6155 0.7451 0.4642 0.5773 0.6778 0.4524 0.5530 0.6352
0.8941 1.1687 1.5542 0.7125 0.9183 1.1601 0.6490 0.8221 0.9958 0.6196 0.7709 0.9056 0.6036 0.7383 0.8485
1.3263 1.7300 2.2793 1.0649 1.3701 1.7208 0.9719 1.2293 1.4834 0.9285 1.1537 1.3518 0.9049 1.1055 1.2681
1.0053 1.3136 1.7444 0.8016 1.0329 1.3040 0.7302 0.9249 1.1199 0.6971 0.8673 1.0186 0.6791 0.8306 0.9545
direction of the length of a plate according to (10), (11) for ls/bs ¼ 2, 3, 4, 5, 6, and parameters: as ¼ 1; 0; 104; 2; 4; and m ¼ 0; 0.5; 1; Note: for as ¼ 104; 2; 4 in formula (10) s0 was assumed with a minus sign (tension on the longitudinal supported edge, compression on the free edge). Fig. 7 compares diagrams of coefficient k for variously supported plates at nonlinear variation of normal stress in the direction of the length of a plate according to (10), (12) and a constant value of the parameter m ¼ 1 and different values of the parameter as ¼ 0; 104; 2 and 4. It results from diagrams that coefficient k increases with a change in transverse stress distribution from axial compression to bending in the plate plane. Higher values of k were obtained for a plate built-in on one side.
Tables 4 and 5 contain coefficients k for a built-in plate (Table 4) and a simply supported plate (Table 5) on the longitudinal edge at nonlinear variation of stress in the direction of the length of a plate according to (10), (12) for ls/bs ¼ 2, 3, 4, 5, 6, and parameters: as ¼ 1; 0; 104; 2; 4; and m ¼ 0.5; 1; Note: for as ¼ 104; 2; 4 in formula (10) s0 was assumed with a minus sign. Fig. 8 compares diagrams of coefficient k for a plate built-in on one side and load parameters: as ¼ 3, m ¼ 0.25; 1 at linear (dashed line) and nonlinear (continuous line) variations of stress in the direction of the length of a plate. Higher values of coefficients k were obtained for a linear stress distribution, while the difference between coefficients for a linear and a nonlinear distribution increases together with increasing parameter m. Fig. 9 compares diagrams of coefficient k for differently supported plates (simply, built-in) at linear variation of stresses for m ¼ 0.5 and as ¼ 0; 104. Higher values of coefficients k are obtained for plates built-in on one side. In both cases coefficients decrease with an increasing plate aspect ratio ls/bs. The variation of the distribution of stress, characterized by parameters m and as, has an effect on the buckling mode of a plate supported on three edges. For the classic case of eccentric compression of a plate built-in on one side (Fig. 2b), at constant stress intensity in the direction of the length of a plate (m ¼ 0), the spontaneously formed buckling half-wavelengths have an identical length and a constant amplitude, and their number depends on the ratio ls/bs. Symmetry or antisymmetry of displacements in relation to a transverse axis of the plate is maintained. In the case (m40) analyzed in this study, the longitudinal stress variation has an effect on the buckling mode of a plate built-in on one side. With an increase of parameter m, the buckling form is characterized by a variable length and different amplitudes of the successive ‘‘half-wavelengths’’ which are formed on the plate length. Moreover, conditions of symmetry or antisymmetry of displacements in relation to the transverse axis of the plate are not fulfilled. Fig. 10 compares the buckling modes of a plate built-in on one side with dimensions ls ¼ 500 mm, bs ¼ 100 mm and ts ¼ 1 mm at a linear stress distribution for parameters as ¼ 3 and m ¼ 0; 0.5; 1. Contour line diagrams were supplemented by the buckling shape of the free edge in the longitudinal cross-section. The eigenvector of the first mode of buckling was standardized so that the maximum deflection was 1. With the increase of parameter m, coefficient k of critical stresses increases and the number of significant ‘‘half-wavelengths’’ of buckling decreases along the plate length (from 3 for m ¼ 0 to 2 for m ¼ 1). Maximum deflections occur from the side of a more compressed edge, and amplitudes of subsequent ‘‘halfwavelengths’’ of buckling mode have a vanishing character. In the case of eccentric compression of a plate simply supported on three edges (Fig. 2a) at a constant stress intensity in the direction of the length of a plate (m ¼ 0), the buckling mode creates one half-wavelength when
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Fig. 7. Diagrams of the buckling stresses coefficients k at the nonlinear variation of stress in the direction of the length of a plate according to (10), (12) for parameters: m ¼ 1 and as ¼ 0; 104; 2; 4; in function ls/bs; (a) longitudinal built-in edge, (b) simply supported longitudinal edge. Table 4 Coefficients (k) of buckling stress of a plate with a longitudinal built-in edge at the nonlinear variation of stress in the direction of the length of a plate according to (10), (12)
Table 5 Coefficients (k) of buckling stress of a plate simply supported on three edges at the nonlinear variation of stress in the direction of the length of a plate according to (10), (12)
ls/bs
m
as ¼ 1
as ¼ 0
as ¼ 104
as ¼ 2
as ¼ 4
ls/bs
m
as ¼ 1
as ¼ 0
as ¼ 104
as ¼ 2
as ¼ 4
2 2 3 3 4 4 5 5 6 6
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
7.0464 7.5177 6.5603 6.9146 6.4068 6.6329 6.2924 6.4734 6.2229 6.3701
1.5512 1.7028 1.4571 1.5511 1.4143 1.4735 1.3841 1.4318 1.3667 1.4047
1.9485 2.1458 1.8351 1.9549 1.7797 1.8554 1.7416 1.8023 1.7193 1.7678
2.5839 2.8468 2.4418 2.5983 2.3652 2.4660 2.3152 2.3956 2.2854 2.3498
2.1258 2.3426 2.0050 2.1350 1.9430 2.0259 1.9015 1.9678 1.8770 1.9300
2 2 3 3 4 4 5 5 6 6
0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0 0.5 1.0
3.0847 3.6721 2.4917 2.8580 2.2568 2.5105 2.1303 2.3191 2.0499 2.1990
0.8125 0.9805 0.6402 0.7425 0.5746 0.6448 0.5401 0.5918 0.5183 0.5589
1.0873 1.3116 0.8556 0.9927 0.7675 0.8616 0.7211 0.7906 0.6920 0.7464
1.6098 1.9291 1.2769 1.4755 1.1479 1.2855 1.0795 1.1817 1.8364 1.1167
1.2221 1.4728 0.9625 1.1161 0.8635 0.9692 0.8113 0.8894 0.7785 0.8397
symmetry in relation to the transverse axis of the plate is preserved. For parameter m40 the longitudinal variation of the stress distribution generates an asymmetric shape of
displacements in relation to the transverse axis of the plate. The maximum deflections of one spontaneously formed asymmetric ‘‘half-wavelength’’ are displaced in the
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Fig. 8. A comparison of coefficient diagrams k for a plate built-in on one side (for as ¼ 3, m ¼ 0.25; 1) at linear (dashed line) and nonlinear (continuous line) variations of stress in the direction of the length of a plate.
Fig. 9. A comparison of coefficient diagrams k for variously supported plates at linear variation of stress in the direction of the length of a plate for m ¼ 0.5 and as ¼ 0; 104; continuous line—longitudinal edge built-in, dashed line—longitudinal edge simply supported.
direction of the transverse, more compressed edge. The asymmetry degree of displacements increases with increasing parameter m. 7. Conclusion The deflection function introduced in this study in the form of a polynomial-sinusoidal series (5) enables one to model the boundary conditions of a plate supported on the longitudinal edge by a simple support, by an elastic restraint in the segment of a thin-walled bar, until a built-in edge is obtained. The process of adjustment of the
deflection function (5) to parameters fin from the minimum energy condition to the plate buckling mode occurs both in displacement shaping and ‘‘residual forces’’ reduction on the free edge. An effective way to describe the variation of stress in the direction of the length of a plate supported on three edges is the introduction of longitudinal body forces operating in its middle plane. The distribution of body forces is selected depending on the manner of plate loading. Such an approach allows simplifying the deflection function by reducing the number of series expressions (5) necessary for the approximation of the buckling mode of a plate.
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Fig. 10. Buckling modes of a plate built-in on one side with ls/bs ¼ 5 at linear variation of stress in the direction of the length of a plate for parameters: as ¼ 3 and m ¼ 0; 0.5; 1.
The formulas derived in this study for the work done by external forces and the procedure of determining the strain energy of plate bending proposed by Jakubowski in [8] permit elaboration of effective computer programs which can be used to examine the stability of variously loaded plates which are components of thin-walled bars. The defined buckling stress of a plate loaded in the plane by a variable stress distribution enables one to construct effective diagrams and tables of buckling coefficients (k). This facilitates the assessment of the effective width and limit load-carrying capacity of this class of plates which are component parts of thin-walled bars with an open crosssection, e.g. in the manner demonstrated in [1]. The value of coefficient k of the buckling stresses of plates with a free edge increases with increasing parameter m of longitudinal stress variation. Smaller coefficients k by the same boundary conditions and the same values of parameters as, m and ls/bs were obtained for a nonlinear variation of normal stress in the direction of the length of a plate. The longitudinal stress variation also has an effect on the buckling modes of plates supported on three edges. With an increasing parameter m the buckling mode of a plate with a longitudinal built-in edge is characterized by a variable length and different amplitudes of subsequent buckling ‘‘half-wavelengths’’ which are formed on its length. For plates simply supported on three edges, the longitudinal stress variation causes deflection according to one nonsymmetric ‘‘half-wavelength’’ with maximal deflections displaced in the direction of the transverse edge which is more compressed.
References [1] Yu C, Schafer BW. Effect of longitudinal stress gradient on the ultimate strength of thin plates. Thin Walled Struct 2006;44:787–99. [2] NAS. North American specification for the design of cold-formed steel structural members. 2001 ed. Washington, DC, USA: American Iron and Steel Institute; 2001. [3] Yu C, Schafer BW. Effect of longitudinal stress gradients on elastic buckling of thin plates. J Eng Mech ASCE 2007;133(4):452–63. [4] Kowal Z. The stability of compressed flange of plate girder with a box section. Zeszyty Naukowe Politechniki Wroc"awskiej, Budownictwo 1965;122:73–85 (in Polish). [5] Kowal Z. The stability top metal plate of pontoon foundation. W˛egiel Brunatny 1966;4:331–3 (in Polish). [6] Bulson PS. The stability of flat plates. London: Chatto and Windus; 1970. [7] Kro´lak M. The buckling state and the analysis of the post-buckling behaviour of a rectangular plane subject to nonuniform compression and shear. Archiwum Budowy Maszyn 1978;4:633–43 (in Polish). [8] Jakubowski S. The matrix analysis of stability and free vibrations of walls of thin-walled girders. Archiwum Budowy Maszyn 1986; 4:357–76 (in Polish). [9] Kro´lak M, editor. Post-critical states and limit load-carrying capacity of thin-walled girders with flat walls. Warsaw, Lodz: State Scientific Publishers; 1990 (in Polish). [10] Jakubowski S. Buckling of thin-walled girders under compound load. Thin-Walled Struct 1988;6:129–50. [11] Jakubowski S. Local buckling of thin-walled girders of triangular cross-section. Thin-Walled Struct 1989;8:253–72. [12] Timoshenko SP, Gere JM. Theory of elastic stability. Part II. New York, NY: McGraw-Hill; 1961. [13] Handbook of structural stability. Edited by Column Research Committee of Japan, Tokyo: Corona Publishing Company; 1971. [14] Gra˛ dzki R, Kowal-Michalska K. Elastic and elasto-plastic buckling of thin-walled columns subjected to uniform compression. ThinWalled Struct 1985;3:93–108. [15] Wolfram S. Mathematica. Cambridge: Cambridge University Press.