JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 1, pp. 93-110, Warsaw 2005 ON A CERTAIN MODEL OF UNIPERIODIC MEDIUM THICKNESS PLATES SUBJECTED TO INITIAL STRESSES Eugeniusz Baron Department of Building Structures Theory, Silesian University of Technology e-mail: ckibud@polsl.gliwice.pl The aim of the paper is twofold. First, governing equations for me- dium thickness elastic plateswhichhaveaperiodicallynon-homogeneous structure in one direction (uniperiodic) and subjected to initial in-plane stresses are derived. In order to obtain the aforementioned equations, the toleranceaveragingtechnique is applied.This technique leads to equ- ationswith constant coefficients. Second, theaboveequationsareapplied to analysis of certain stability and dynamic problems. The stiffnesses of plates were calculated by treating them as structurally anisotrpic. An interesting result is that two values of the critical force can be obtained. This result can have a physical meaning for the stability of plates under compression in one direction and tension in the perpendicular direction. Key words: uniperiodic plates, modelling, stability 1. Introduction The subject of analysis are medium thickness rectangular uniperiodic ela- stic plates, i.e. plates with a periodic non-homogeneous structure in one direc- tion. The above plates are composed of a large number of repeated elements having an identical form, dimensions and material properties. The geometry of a uniperiodic plate, apart from the global mid-plane length dimensions L1, L2, is characterized by the length l which determines the period of structure inhomogeneity. In general, in the direction perpendicular to the direction of periodicity, the material parameters may be not constant. However, in most cases ”existing in engineering practice” uniperiodic plates have constant pro- perties in that direction. Fragments of the aforementioned plates are shown in Fig.1 and Fig.2. 94 E.Baron Fig. 1. Example of the plate with a uniperiodic structure A formulation of different approximate models for these plates is a rather complicated problem. In most cases, homogeneous models of these plates are taken as a basis for analysis of special problems. The homogenized equations have constant coefficients and constitute a certain approximation of uniperio- dic plate equations having highly oscillating and non-continuous coefficients, cf. Lewiński (1991). However, the homogenized equations cannot describe the effect of the periodicity length parameter l on the overall plate behaviour (the length-scale effect). In the work by Baron (2002), a new approximate model of medium thick- ness uniperiodic plates was proposed. This model, obtained by using the to- lerance averaging technique, cf. Woźniak and Wierzbicki (2000), includes the length scale effect. The aim of this contribution is an extension and a certain generalization of the 2Dmodel of amedium thickness plate derived byBaron (2002) and the analysis of a certain quasi-stationary and dynamic problem for a rectangular plate. In the above article, in the course of modelling in terms containing the initial stress, fluctuation of displacement were taken into consideration. The obtainedmodel will be referred to as the length-scale model, since it includes the effect of the length period l on the overall plate behaviour. The general averaged model equations obtained in this paper will be transformed into a form which would enable investigation of dynamic and stability problems. A newexpression for the critical forcewill be comparedwith those obtained from the homogenized model of uniperiodic plates. It will be shown that in some special cases related to compression in the mid-plane in the direction along a certain axis and tension in the perpendicular direction, the homogenized model leads to higher values of critical forces than in the length-scale model introduced in this paper. Throughout the paper the subscripts α,β,. . . run over 1,2, subscripts i,j, . . . over 1,2,3 and superscripts A,B,. . . over 1,2, . . . ,N; summation co- nvention holds for all aforementioned indices. On a certain model of uniperiodic... 95 2. Basic assumptions and notations By x = (x1,x2) we denote Cartesian coordinates of a point on the plate mid-plane Π = (0,L1)× (0,L2), and by z a Cartesian coordinate in the direction normal to themid-plane. By z =±δ(x),x∈ Π we denote functions representing the upper and lower plate boundary, respectively; hence 2δ(x) is the plate thickness in a point x ∈ Π. By ρ = ρ(x,z) and Aijkl(x,z) we denote mass density and the tensor of elastic moduli of the plate material and assume that every z = const is an elastic symmetry plane. We also define Cαβγδ := Aαβγδ−Aαβ33A33γδ(A3333)−1,Bαβ := Aα3β3.We shall assume that the functions δ(·), ρ(·), Aijkl(·) are l-periodic with respect to the x1- coordinate, and are sufficiently regular with respect to z. Let p+ and p− be loadings (in the z-axis direction) on the upper and bottom surfaces of the plate, respectively. Let σoαβ be a tensor of the initial stress and b be a constant body force acting in the z-axis direction. Furthermore, let t be the time coordinate. The averaged value of an arbitrary integrable function ϕ(x1,x2, t) in the periodicity interval (x1− l/2,x1+ l/2) will be denoted by 〈ϕ〉(x,t) = 1 l x1+l/2∫ x1−l/2 ϕ(ξ,x2, t) dξ x=(x1,x2) (2.1) For an uniperiodic function ϕ(·), the above averaged value is independent of x1. 3. Modelling procedure. Model equations Setting µ(x)= δ∫ −δ ρ(x,z) dz p(x)= p+(x)+p−(x)+ b〈µ〉(x) J(x)= δ∫ −δ z2ρ(x,z) dz Gαβγδ(x)= δ∫ −δ z2Cαβγδ(x,z) dz Noαββ = δ∫ −δ σoαβ dz Dαβ(x)= δ∫ −δ KαβBαβ(x,z) dz 96 E.Baron where in the expression summation convention with respect to α and β does not hold for Dαβ, and Kαβ is a shear coefficient (introduced by Jemielita (2001)), we obtain the system of equations (Gαβγδϑ(γ,δ)),β −Dαβϑβ −Jϑ̈α =0 (3.1) Noαβw,αβ +[Dαβ(ϑβ +w,β)],α −µẅ+p =0 in which the deflection w and rotation ϑα are basic unknowns. Theabove equations represent themediumthickness 2D-platemodel of the Hencky-Boole type. For an uniperiodic plate, the above system of equations has functional coefficients which are periodicwith respect to the argument x1. These coefficients are certain highly oscillating and non-continuous functions. The exact solution to boundaryvalue problems formulated for these equations is, inmost cases, rather complicated. That iswhy various approximatemodels leading to equations with constant coefficients are proposed.We canmention here a knownhomogenizedmodel. However, thismodel is not able to describe the effect of the period length on the overall plate behaviour. In the paper by Baron (2002), a new non-asymptotic model was proposed. Thismodel was obtained by using the tolerance averaging method summarized by Woźniak andWierzbicki (2000). In accordance with the tolerance averaging procedure, the unknown de- flection w and rotations ϑα are assumed in the form ϑα(x, t)= ϑ o α(x, t)+ϑ ∗ α(x, t) w(x, t)= wo(x, t)+w∗(x, t) where wo(·), ϑoα(·) are the averaged deflection and rotations, and w∗(·), ϑ∗α(·) describefluctuationsof thefields ϑα(x, t),w(x, t) causedby the inhomogeneity of the plate. At the same time, the functions wo(·), ϑoα(·) have to be slowly varying and w∗(·), ϑ∗α(·) have to be periodic-like functions, cf. Woźniak and Wierzbicki (2000).We shall also assume that the fluctuations w∗(·), ϑ∗α(·) can be approximated by ϑ∗α(x, t) ∼=ha(x1)Θaα(x, t) a =1,2, . . . ,n w∗(x, t)∼= gA(x1)WA(x, t) A =1,2, . . . ,N (3.2) where WA(·), Θaα(·) are new slowly varying unknowns. At the same time, ha(x1), g A(x1) represent two systems of linear independent periodic shape functions, postulated a priori in every special problem under consideration. On a certain model of uniperiodic... 97 These functions are called mode-shape functions and they have to approxi- mate the expected form of the oscillating part of free vibration modes of the periodicity cell. The above functions have to satisfy the conditions 〈Jha〉=0, 〈µgA〉 = 0, ha(x1) ∈ O(l), gA(x1) ∈ O(l), lha,1(x1) ∈ O(l), lgA,1(x1) ∈ O(l). Taking into account the aforementioned conditions, we shall also introduce functions h a = l−1ha gA = l−1gA which are of the order O(1) when l → 0. In the subsequent considerations, slowly varying functions wo, ϑoα, W A, Θaα are basic kinematics unknowns. In order to obtain a system of equations for these unknowns, we shall apply a procedure similar to that discussed in Baron (2002), however, in terms containing the initial stress Noαβ, the fluctu- ation of displacement will not be neglected. That means that the assumption Noαβw,αβ ≈ Noαβwo,αβ has been substituted by the relation Noαβw,αβ = N o αβ(w o +w∗),αβ Setting aside all transformations, which are similar to those presented in Baron (2002), we arrive at the equations: — equations of motion Mαβ,β −Qα −〈J〉ϑ̈oα =0 (3.3) Noαβw o ,αβ + lN o α2〈gA〉WA,α2+Qα,α −〈µ〉ẅo +p =0 —kinematic equations for Θa, WA l2〈Jhahb〉Θ̈bα +Maα − lM̃aα,2 =0 (3.4) l2〈µgAgB〉ẄB +QA − lQ̃A,2+ −Noα2(l〈gA〉wo,α2+ l2〈gAgB〉WB,α2)+No11〈gA,1gB,1〉WB − l〈gAp〉=0 98 E.Baron —constitutive equations Mαβ = 〈Gαβγδ〉ϑo(γ,δ)+ 〈h a ,1Gαβ1δ〉Θaδ + l〈h a Gαβ2δ〉Θaδ,2 Qα = 〈Dαβ〉(ϑoβ +wo,β)+ l〈h a Dαβ〉Θaβ + 〈gA,1Dα1〉WA + l〈gADα2〉WA,2 Maα = 〈ha,1hb,1Gα11δ〉Θbδ + 〈h,1Gα1γδ〉ϑo(γ,δ)+ l〈h a ,1h b Gα12δ〉Θbδ,2+ +l2〈hahbDαβ〉Θbβ + l〈h a Dαβ〉(ϑoβ +wo,β)+ l〈h a gA,1Dα1〉WA + +l2〈hagADα2〉WA,2 (3.5) M̃aα = 〈h a hb,1Gα21δ〉Θbδ + 〈h a Gα2γδ〉ϑo(γ,δ)+ +l〈hahbGα22δ〉Θbδ,2 QA = 〈gA,1gB,1D11〉WB + 〈gA,1D1β〉(vtoβ +wo,β)+ l〈gA,1h a D1β〉Θaβ + +l〈gA,1gBD12〉WB,2 Q̃A = 〈gAgB,1D21〉WB + 〈gAD2β〉(ϑoβ +wo,β)+ l〈gAhaD2β〉Θaβ + +l〈gAgBD22〉WB,2 Averaged 2D-model equations (3.3)-(3.5) constitute the starting point for the subsequent analysis. The underlined terms in the above equations describe the influence of fluctuation displacement neglected in Baron (2002). In most cases, we deal with plates having a homogeneous structure in the x2-axis direction (cf. Fig.2). For such a type of uniperiodic plates, all coefficients in equations (3.3)-(3.5) are constant, and the subsequent considerations will be restricted to the aforementioned type of plates. 4. An orthotropic plate with stiffeners Now let us assume that the plate is of constant thickness and is made of an orthotropic material, where the principal axis of orthotropy coincides with the Cartesian axis (x,z). Moreover, let us assume that the plate is reinforced by a certain system of periodically spaced stiffeners, cf. Fig.2.We also assume that the torsional stiffness of the stiffeners in the plane normal to the x2-axis is neglected. Let M be mass density of a stiffener and I be bending stiffness of the stiffener, respectively. Moreover, let G11 = G1111 G22 = G2222 G12 = G1122 = G2211 G = G1212 = G1221 = G2112 = G2121 D1 = D11 D2 = D22 On a certain model of uniperiodic... 99 be stiffness of the orthotropic plate under consideration. Fig. 2. A scheme of the uniperiodic plate under consideration Let us take exclusively two modal shape functions h(x1)= h 1(x1)= lh(x1) g(x1)= g 1(x1)= lg(x1) as the first approximation of the plate fluctuations caused by the uniperiodic plate structure. Let us consider the interval 〈0, l〉 as a representative plate segment. We assume that h(x1) is an odd function and g(x1) is an even function of x1. On the above assumptions, we obtain from (3.3)-(3.5) the following system of equations for the unknowns ϑα, Θ1 =Θ, W , w 〈G11〉ϑo1,11+ 〈G〉ϑo1,22+(〈G12+ 〈G〉)ϑo2,12−〈D1〉(ϑo1+wo,1)−〈J〉ϑ̈o1 =0 〈G22〉ϑo2,22+ 〈G〉ϑo2,11+(〈G12+ 〈G〉)ϑo1,12−〈D2〉(ϑo2+wo,2)− −l〈gD2〉W,2−〈J〉ϑ̈o2 =0 Noαβw o ,αβ + 〈D1〉(ϑo1+wo,1),1+ 〈D2〉(ϑo2+wo,2),2+ lNoα2〈g〉W,α2+ +l〈gD2〉W,22−〈µ〉ẅo +p =0 (4.1) −l2〈h2G〉Θ,22+(〈h2,1G11〉+ l2〈h 2 D1〉〉)Θ+ l〈hg,1D1〉W + l2〈h 2 J〉Θ̈ =0 −l2Noαβ〈g2〉W,αβ − l2〈g2D2〉W,22+No11〈g2,1〉W + 〈g2,1D1〉W − lNoα2〈g〉wo,α2+ −l〈gD2〉(ϑo2+wo,2〉),2+ l〈hg,1D1〉Θ+ l2〈g2µ〉Ẅ − l〈gp〉=0 and an independent equation for Θ2 −l2〈h2G22〉Θ2,22+(〈h2,1G〉+ l2〈h 2 D2〉)Θ2+ l2〈h 2 J〉Θ̈2 =0 (4.2) Equations (4.1) together with (4.2) have constant coefficients andwill be exa- mined together with appropriate boundary and initial conditions. 100 E.Baron The stiffnesses of theplatewill be calculated taking into account structural anisotropy. It means that the plate is made of a homogeneous and isotropic material and reinforcedbya systemof parallely spacedmaterial inclusions.By means of a particular way of calculating the stiffnesses, cf. Sokołowski (1957), this composite plate canbe treatedashomogeneousbutmadeof ananisotropic material. In this paper, in the inertial terms, the factual mass distribution is yet taken into consideration. Material properties of structurally anisotropic (strictly: orthotropic) plate shown in Fig.2 are represented by the Young modulae E1, E2 and by the Poisson ratios ν1, ν2. In this case, the plate stiffnesses are given by G11 = E1d 3 12(1−ν1ν2) G22 = E2d 3 12(1−ν1ν2) G12 = ν1G22 = ν2G11 G = √ G11G22 2(1+ν1) Setting for the plate material E = E1 and ν = ν1, it can be shown that G11 = 〈G11〉= Ed3 12(1−ν2) = Ho Similarly, taking into account averaging formula (2.1), we obtain G22 = 〈G22〉= Ho ( 1+ EsI Hol ) where Es is the Young modulus of the stiffener. Defining by ψ = EsI/(Hol) a constant which will be called the coefficient of nonhomogeneity related to uniperiodic plate structure, we obtain G22 = 〈G22〉= Ho(1+ψ) From the condition νG22 = ν2G11, cf. Sokołowski (1957), we conclude that ν2 = ν(1+ψ). Hence G22 = E2d 3 12(1−ν1ν2) = E2d 3 12[1−ν2(1+ψ)] = Ed3 12(1−ν2) (1+ψ) E2 = E 1−ν2(1+ψ) 1−ν2 (1+ψ) One should pay attention that if ν2 < 0.5, we obtain an additional condition for the coefficient ψ ψ < 1−2ν 2ν On a certain model of uniperiodic... 101 The shear stiffness will be calculated from the formula 〈D1〉= Ed 2(1+ν) K11 〈D2〉= E2d 2(1+ν2) K22 = Ed[1−ν2(1+ψ)] 2(1−ν2)[1+ν(1+ψ)] (1+ψ)K22 Following Jemielita (2001), for dynamic problems, we introduce the shear co- efficients K11 = 5 6−ν K22 = 5 6−ν(1+ψ) From assumptions on structural anisotropy (the plate can be treated as ho- mogeneous), we conclude that the stiffnesses, calculated by application of the mode-shape function h(x1), g(x1), are constant, i.e. 〈h2,1G11〉= G11〈h2,1〉. Equations (4.1), togetherwith the aforementioned procedure of calculating the coefficients, are the starting point for the analysis of special problems, which will be explained in the next section. 5. Applications Weare going to apply themodel equations obtained in theprevious section to the analysis of stability and a dynamic problem for a rectangular unipe- riodic plate. The plate is simply supported on its edges and subjected to the initial stress on the plate mid-plane, Fig.3. Taking into account the boundary conditions, for a plate simply supported on all edges, we look for the solution to equations (4.1) in the form ϑ1 =e iωt ∞∑ m=1 ∞∑ n=1 ϑ1mn cosαmx1 sinβnx2 ϑ2 =e iωt ∞∑ m=1 ∞∑ n=1 ϑ2mn sinαmx1cosβnx2 Θ1 =e iωt ∞∑ m=1 ∞∑ n=1 Θ1mn cosαmx1 sinβnx2 (5.1) w =eiωt ∞∑ m=1 ∞∑ n=1 wmn sinαmx1 sinβnx2 W =eiωt ∞∑ m=1 ∞∑ n=1 Wmn sinαmx1 sinβnx2 102 E.Baron where: αm = mπ/L1, βn = nπ/L2, m,n = 1,2, . . . and ϑ1mn, ϑ2mn, Wmn, wmn are constant amplitudes, ω is a vibrations frequency. Fig. 3. A scheme of a uniperiodic plate subjected to an edge loading Let us denote H1 = α 2 n〈G11〉+β2n〈G〉+ 〈D1〉 H2 = β 2 n〈G22〉+α2n〈G〉+ 〈D2〉 G = 〈G12〉+ 〈G〉 B = α2mβ 2 n〈g2〉〈D2〉G H1H2−α2mβ2nG 2 + 〈hg,1〉2〈D1〉 〈g2,1G11〉+ l2〈h 2 D1〉+ l2β2n〈h 2 G〉 and introduce non-dimensional stiffness and forces D1 =1− H2〈D1〉−β2n〈D2〉G H1H2−α2mβ2nG 2 D2 = 〈D2〉 〈D1〉 ( 1− H1〈D2〉−α 2 m〈D1〉G H1H2−α2mβ2nG 2 ) N1 = N11 〈D1〉 N2 = N22 〈D1〉 (N12 =0) Substituting (5.1) into (4.1) and taking into account the aforementioned denotations, after some transformations we obtain the following equations for the unknowns wmn and Wmn [ a11 l〈g〉β2n(N2+D2) l〈g〉β2n(N2+D2) a22 ][ wmn Wmn ] = [ 0 0 ] (5.2) On a certain model of uniperiodic... 103 where a11 = α 2 m(N1+D1)+β 2 n(N2+D2)− 〈µ〉 〈D1〉 ω2 a22 = 〈g2,1〉(N1+1)+ l2β2n〈g2〉(N2+D2)− l2B− l2 〈g2µ〉 〈D1〉 ω2 Equations (5.2) constitute the starting point for the subsequent examples. Numerical calculationswill be carried out for a constant-thickness concrete plate with E = 29000MPa (concrete B25), ν = 0.20, reinforced by periodi- cally spaced rolled steel sections (I-bar) with Es = 205000MPa, as it shown in Fig.2. We assumemode-shape functions in the form h(x1)= lsin 2π l x1 g(x1)= l ( c+cos 2π l x1 ) (5.3) The constant c can be calculated from the condition 〈µg〉=0 c =− ϕM 1+ϕM ϕM = M ρdl For the above functions, we obtain 〈g〉= c 〈g2〉= 1 2 + c2 〈g2,1〉=2π2 〈h2,1〉=2π2 〈h 2〉= 1 2 and 〈µ〉= ρd(1+ϕM)= µo 1 1+ c µo = ρd (5.4) 〈g2µ〉= µo[〈g2〉+(1+ 〈g〉)2ϕM] =µo (1 2 − c ) In the course of calculations, the influence of slenderness ratio λ, the parame- ters ε = l/L2,κ = L2/L1 andthe coefficient ofnon-homogeneity ψ = EsI/H0l have been taken into account. 5.1. Dynamic problem In this subsection, free vibrations in the long-wave propagation problem will be discussed. 104 E.Baron The system of two linear equations for amplitudes wmn, Wmn (5.2) has nontrivial solutions provided that its determinant is equal to zero. In thisway, we obtain the characteristic equation for free vibration frequencies l2 〈µ〉〈g2µ〉 〈D1〉2 ω4− 1 〈D1〉 { 〈µ〉〈g2,1〉(N1+1)+ +l2β2n [ 〈g2µ〉No + 〈µ〉〈g2〉(N2+D2)− 〈µ〉 β2n B ]} ω2+β2n〈g2,1〉(N1+1)No + +l2β2n[β 2 n〈g2〉(N2+D2)No −β2n〈g〉2(N2+D2)2−BNo] = 0 (5.5) where β2nNo = α 2 m(N1+D1)+β 2 n(N2+D2) From (5.5), we arrive at the following approximate formulae for the lower ω1 and higher ω2 free vibration frequencies ω21 = β2n〈D1〉No 〈µ〉 − l2β 4 n〈g〉2〈D1〉(N2+D2)2 〈g2,1〉〈µ〉(N1+1) (5.6) ω22 = 〈g2,1〉〈D1〉(N1+1) l2〈g2µ〉 + 〈D1〉[β2n〈g2〉(N2+D2)−B] 〈g2µ〉 Commenting on the obtained results, it should be admitted that, contrary to the asymptotic homogenisationmethod, two basic free vibration frequencies have been obtained. The higher frequency ω2 depends on the period-length l and cannot be derived from the homogenized model. In further analysis, formulae for frequencies (5.6) will be transformed into a dimensionless form. To this end, we will introduce the denotations a1 = λ 230(1−ν) 6−ν a2 = λ 21−ν2(1+ψ) 1+ν(1+ψ) 30(1+ψ) 6−ν(1+ψ) e = ν(1+ν)+ √ 1+ψ 2(1+ν) h1 = π 2 [ m2κ2+n2 √ 1+ψ 2(1+ν) ] +a1 h2 = π 2 [ n2(1+ψ)+m2κ2 √ 1+ψ 2(1+ν) ] +a2 On a certain model of uniperiodic... 105 D1 =1− h2a1−π2n2a2e h1h2−π4m2n2κ2e2 (5.7) D2 = a2 a1 ( 1− h1a2−π2m2κ2a1e h1h2−π4m2n2κ2e2 ) B = n2 a2 2 a1 h1c 2 h1h2−π4m2n2κ2e2 + m2n2κ2e ( 1 2 + c2 ) 2π2+ 1 2 ε2 [ a1+π2n2 √ 1+ψ 2(1+ν) ] Multiplying both relations (5.6) by L22µo〈D1〉−1, and taking into account (5.4) and (5.7),weobtain the following formulae for thenon-dimensional frequencies Ω21 = n 2π2(1+ c) [ m2κ2(N1+D1)+N2+D2+ε 2c2 (N2+D2) 2 2(N1+1) ] (5.8) Ω22 = 2π2 1−2c [ 2 ε2 (N1+1)+n 2 (1 2 + c2 ) (N2+D2)−B ] In the course of numerical calculations, it has been assumed that the con- crete plate has mass density ρ =2200kg/m3, thickness d =0.15m and span L2 =6.00m. Three variants of reinforcing by rolled steel sections: I180, I220, I240 are taken into account. The I-bar is spaced every 0.75m; also l =0.75m and ε = 0.125. The shape of the mid-plane is characterized by the ratio κ = L2/L1, κ =0.5; 1.0; 2.0. The values of parameters ψ and c are placed in Table 1. Table 1 I-bar I M ϕ ψ c [10−8m4] [kg/m] I180 1450 21.9 0.0885 0.490 −0.0813 I220 3060 33.1 0.1337 1.030 −0.1179 I240 4250 36.2 0.1463 1.420 −0.1276 Diagrams representing the interrelation between non-dimensional free vi- bration frequencies Ω and forces N1, N2 are presented in Fig.4 and Fig.5. In these diagrams, the values of N1, N2, Ω should bemultiplied by 10 −3. Numerical calculationswere carried out for existing engineering structures. We into account a concrete plate reinforced by a system of periodically spaced I-bars. Thus, we dealt with a structure which has practical meaning in civil engineering.Thevalues of the in-plane stresses N11,N22 are restricted to those 106 E.Baron Fig. 4. Diagrams of interrelations between lower and higher free vibration frequencies Ω1, Ω2 and forces N1 and N2 Fig. 5. Interrelation between the lower and higher free vibration frequency Ω1 and stresses N1 and N2 which do not exceed the permissible stress. It has to be mentioned that the plates under consideration satisfy, in the exact manner, all assumptions of the theory proposed in this contribution. On a certain model of uniperiodic... 107 Under the aforementioned restrictions, lower free vibration frequencies ne- arly coincide with those resulting from the homogenisation theory. Higher free vibration frequencies, which cannot be calculated by the homogenisation me- thod, do not have any meaning from the engineering point of view. However, discussion on formulae (5.6) leads to the conclusion that higher vibration fre- quencies can be calculated and applied provided that we shall deal with some new composite material having suitable material properties. 5.2. Stability problem Let us restrict the considerations to quasi-stationary processes and assu- me that the plate is subjected to compression N11 along the x1 axis. This compression have to be proportional to the stress N22 along the x2 axis; de- note then γ = N22/N11. We conclude that nontrivial solutions to (5.2) exist provided that [〈g2,1〉k+ l2β2nγ(k〈g2〉−β2n〈g〉2γ)]N 2 1+ −{〈g2,1〉(Do +k)+ l2[β2n〈g2〉(Do +kD2)−2β4n〈g〉2D2γ −kB]}N1+ (5.9) +〈g2,1〉Do + l2[β2n〈g2〉DoD2−β4n〈g〉2D 2 2−DoB] = 0 where Do = α 2 mD1+β 2 nD2 k = α 2 m +β 2 nγ Real roots of Eqs (5.9) represent critical values of the edge in-plane lo- adings for the stability problem under consideration. It can be observed that in the framework of the proposedmodel we deal with two values of the critical force N11,kr. This situation is quite different from those resulting from the well-known typical procedures leading to the evaluation of the critical force. Generalization of the well known analysis of a typical stability problem leads in the considered case to the following results • N11,kr = 〈D1〉Do/k for the homogenized model • N11,kr = 〈D1〉Do/k + O(l)2 for the model describing the length-scale effect. The aim of the foregoing numerical analysis of equation (5.9) is to deter- mine the interrelation between the non-dimensional critical force N11,kr and parameter γ = N22/N11. It is easy to verify that this interrelation depends on 108 E.Baron parameters: ν, κ = L2/L1, λ = L2/d, ε = l/L2. Having introduced (5.7) and bearing in mind (5.9), we arrive at the following formula { 2k+ε2n2γ [ k (1 2 + c2 ) −n2c2γ ]} N 2 1+ − { 2(Do +k)+ε 2 [ n2 (1 2 + c2 )( Doγ +k a2 a1 ) −2n4c2D2γ −kB ]} N1+ (5.10) +2Do +ε 2 [ n2 (1 2 + c2 ) Do a2 a1 −n4c2D22−DoB ] =0 Fig. 6. Diagrams of interrelations between the non-dimensional critical force and the parameter γ = N11/N22 On a certain model of uniperiodic... 109 The shape of the mid-plane is characterized here by the ratio κ = L2/L1, κ = 1; 0.5; 2 for two cases of the slenderness ratio λ = L2/d, λ = 20 and λ = 60. At the same time, the parameter ε = l/L is equal to 0.10. In both cases, the ratio ψ = ESI/(Hol) is 0.5; 1,0; 1,5. Subsequent calculations will be carried out for the aforementioned values of parameters. In Fig.6, diagrams representing the interrelation between N11 and γ = N22/N11 is presented. In these diagrams N I and NII denote solutions to equation (5.10), and No = Do/k is a non-dimensional critical force which can be derived also from the homogenized plate model. The diagrams presented in Fig.6 indicate that for the plate compressed in both directions, i.e. for γ > 0, the critical force is equal to N I, but the above value is close to No obtained from the homogenized model. In this case, the stability analysis based on the proposed model leads to similar results found fromclassical analysis. The above remark applies to a certain domain of γ ¬ 0 as well. Remarkable differences between the critical forces N0 and NI, NII appear for the parameter γ tending to −1/κ2. For example, for a square plate, if the value of the tensile force N22 tends to the value of the compressive force N11, we obtain N I < N0. Thus, the critical force should be calculated from relations obtained within the proposed model, not from the homoge- nized one. 6. Conclusions In this contribution, a new averaged 2D-model of uniperiodic medium- thickness elastic plates is proposed. The model is described by a system of equations with constant coefficients. In contrast to the homogenized model, cf. Lewiński (1991), the proposed model is derived by using a tolerance ave- raging technique and describes the effect of the period-length on the overall plate behaviour.Moreover, thismodel takes into account the neweffect caused by the interrelation between the in-plane forces and displacement fluctuations due to uniperiodicity of the plate structure, and is a certain generalization of that introduced in Baron (2002), where the above effect was neglected. The obtained theoretical results were applied to stability analysis of a rectangular uniperiodic plate. It was shown that, for some special cases, the value of the critical force obtained from the proposed model were smaller than values de- rived from the homogenized plate model. However, the specification of those special cases has rather a qualitative than quantitative significance. At the 110 E.Baron same time, the effect of the coupling between the in-plane forces and displa- cement fluctuations due to uniperiodicity of the plate structure does not play any role as far as the plate stability is concerned. References 1. Baron E., 2002, Onmodelling of medium thickness plates with a uniperiodic structure, Journal of Theor. and Appl. Mech., 40, 1, 7-222 2. Jemielita G., 2001, Meandry teorii płyt, w: Mechanika Sprężystych Płyt i Powłok, edit. C.Woźniak, PWN,Warszawa 3. Jędrysiak J., 2000, On stability of thin periodic plates, Eur. J. Mech. A/Solids, 19, 487-502 4. Lewiński T., 1991, Effective models of composite periodic plates: I. Asymp- totic solutions, II. Simplifications due to symmetries, III. Two dimensional ap- proaches, Int. J. Solids Structures, 27, 1135-1203 5. Sokołowski M., 1957, Obliczanie stałych sprężystości dla płyt o ortotropii technicznej,Arch. Inż. Lądow., 3, 4, 457-485 6. Wierzbicki E., Woźniak C., 2002, Continuum modelling and the internal instability of certain periodic structures,Arch. of Appl. Mech., 72, 451-457 7. Woźniak C., Wierzbicki E., 2000,Averaging Techniques in Thermomecha- nics of Composite Solids, Wydawn. Pol. Częstochowskiej Pewien model wstępnie napiętych uniperidycznych płyt średniej grubości Streszczenie Celempracy jest rozszerzenie i pewne uogólnienie dwywymiarowegomodelu śred- niej grubości (typu Reissnera) sprężystych płyt o jednokierunkowej strukturze pe- riodycznej zaproponowanego przez Barona (2002). Zastosowanometodę uśredniania tolerancyjnego opisanego np. przezWoźniaka i Wierzbickiego (2000). Metoda ta po- zwala uwzględnić wpływwymiaru powtarzalnego segmentu płyty (okresu powtarzal- ności) na jej makromechaniczne własności, czyli tzw. efekt skali. Uzyskane ogólne równania przekształcono do postaci dogodnej do analizy płyt technicznie anizotropo- wych. Przeprowadzono analizę zagadnienia drgań swobodnych oraz stateczności płyt stalowo-betonowych stosowanych w budownictwie. Uzyskano dodatkową relację do obliczania sił krytycznych. Manuscript received April 27, 2004; accepted for print September 1, 2004