MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 8 potx

258 Structural elements Figure 4.54. Collision of two beams, asymmetrical case which start with equal and opposite velocities at t = 0. The contact time is equal to the back and forth travelling time along an equivalent beam of length 2L.No waves are excited and the velocities of the beams are simply interchanged by the collision. The last case corresponds to an asymmetrical system. The upper plot of Figure 4.54 shows a global view of the impact, which agrees satisfactorily with the conditions [4.100] and the lower plot focuses on the collision itself. Contact time depends on the beam properties and it may be noticed that waves are excited by the impact, which are clearly visible in the trajectory of the second beam (dashed line in Figure 4.54). Chapter 5 Plates: in-plane motion Plates are structural components used as walls, roofs, panels, windows, etc. They are modelled as two-dimensional structures characterized by a plane geometry bounded by a contour comprising straight and/or curved lines. Plates are intended to resist various load conditions, broadly classified as in-plane and out-of-plane loads, that is forces either parallel or perpendicular to the plane of the plate, respectively. It is found convenient to study their response properties by separating first the in-plane and the out-of-plane motions. In-plane motions of plates are generally marked by the coupling between the two in-plane components of the displacement field. Such a coupling gives rise to new interesting features of in-plane motions with respect to the longitudinal motions of straight beams. Coupling between in-plane and out-of-plane motions occurs in the presence of in-plane preloads and will be considered in the next chapter. On the other hand, for mathematical convenience in using Cartesian coordinates, presentation will focus first on rectangular plates. Then use of curvilinear coordinates will be introduced to consider plates bounded by curved contours. 260 Structural elements 5.1. Introduction 5.1.1 Plate geometry As shown in Figure 5.1, structural elements called plates are characterized by the two following geometrical properties: 1. One dimension, termed the thickness, denoted h, is much smaller than the other two (length L 1 and width L 2 for a rectangular shape, diameter D for a circular one etc.). This allows one to model the mechanical properties of plates by using a two-dimensional solid medium. 2. The geometrical support of the 2D model is the midsurface, taken at h/2ifthe material is homogeneous. In contrast to the case of shells to be studied later, this surface is flat and called the midplane of the plate. The border lines of the con- tour of the midplane are called edges, which can be straight or/and curved lines. A few other definitions concerning plate geometry are useful. As a three-dimensional body, the plate is bounded by closed surfaces comprising the so called faces and edge surfaces. If the plate thickness is uniform, the faces are flat and parallel to the midplane, whereas the edge surfaces can be cylindrical or plane, depending on the plate geometry. Intersection of two edge surfaces defines a corner edge. Finally, it is also found convenient to orientate the contour by defining a posit- ive unit vector normal to the midplane, which points from the lower face to the upper face of the plate. The usual positive orientation is in the anticlockwise direction. 5.1.2 Incidence of plate geometry on the mechanical response As a preliminary, it is useful to outline a few generic features of the response properties of plates which serve as a guideline to organize the presentation of this Figure 5.1. Plates: geometrical definitions Plates: in-plane motion 261 and the next chapters. As will be verified ‘a posteriori’, they arise as a consequence of the concept of midplane. Considering first linear motions about an equilibrium position with zero or negligible stress level, it can be stated that: 1. In-plane loads, i.e. parallel to the midplane, are balanced by in-plane stresses, which can be normal (compressive or tensile forces) and/or tangential (in-plane shear forces). 2. Out-of-plane loads, i.e. perpendicular to the midplane, are balanced by bending and torsional moments and by transverse shear forces. 3. If the load is oblique, all the previous stresses participate with the equi- librium. However, in the absence of initial stresses, coupling between the in-plane and the out-of-plane motions of the plate can be discarded. So the response to an oblique load is obtained by superposing the responses on the in-plane and to the out-of-plane components of the load, which can be studied separately. Considering then linear motions about a prestressed equilibrium position, it can be stated that: 1. In-plane initial stresses contribute to the out-of-plane equilibrium of a plate through restoring, or alternatively, destabilizing forces, similar to those arising in straight beams when prestressed axially. 2. As a limit case, the forces that contribute to the out-of-plane equilibrium of suitably stretched skin structures originate almost completely from the in-plane stresses which are prescribed initially. Such skin structures are referred to as membranes. A membrane can be seen as an idealized two-dimensional medium which has no flexural rigidity, in contrast with plates and shells. 3. Like cables, or strings, membranes belong to the general class of the so called tension structures which can support mechanical loads by tensile stresses only. In agreement with these preliminary remarks, it is found appropriate to invest- igate first the motions of plates which involve in-plane components of force and displacement fields only. Such in-plane fields are termed membrane components,or fields, as their nature is basically the same as those induced when a skin is stretched in its own plane. On the other hand, it is also found convenient to study first the case of rectangular plates as they can be described using Cartesian coordinates. Non rectangular plates have to be analysed by using oblique or curved coordin- ates, depending on the specificities of their shapes, which are more difficult to manipulate mathematically. 262 Structural elements 5.2. Kirchhoff–Love model 5.2.1 Love simplifications Let us consider a rectangular plate of uniform thickness h. The coordinates of a material point are expressed in a direct Cartesian frame Oxyz, where Oxy is the midplane of the plate, the origin O is at a corner, the axes Ox and Oy are parallel to the edges of lengths L x and L y respectively. The unit vectors of the Ox, Oy, Oz axes are denoted  i,  j,  k respectively. A main cross-section is obtained by cutting mentally the plate with a plane which is perpendicular to the midplane and parallel either to Ox or to Oy. An ordinary cross-section is neither parallel to Ox nor to Oy, P(x, y, 0) or P(x, y) designating a point of the midplane; the intersection of two cross-sections passing through P defines a normal fibre (see Figure 5.2). As h is very small with respect to L x and to L y , the following simplifying assumptions can be made, which hold in the case of small deformations: 1. The normal fibres behave as a rigid body. 2. As the midplane is deformed, the normal fibres remain perpendicular to it. 3. The normal stresses acting on the planes parallel to the midplane are consistent with the first hypothesis of rigidity of the normal fibres. These assumptions can be understood as a natural extension to the two- dimensional case of the basic beam model. In particular, the second hypothesis means that there is no shear between two neighbouring normal fibres as the plate bends, in agreement with the Bernoulli–Euler model. 5.2.2 Degrees of freedom and global displacements According to the first assumption made above, the motion of a normal fibre can be described by using five independent parameters, termed global displacements, Figure 5.2. Cross-sections and normal fibre in a plate Plates: in-plane motion 263 Figure 5.3. Global displacements: translation and rotation variables which are referred to a current point P(x, y) of the midplane. They are shown in Figure 5.3 and defined as follows: 1. the longitudinal displacement in the direction Ox: X(x, y; t)  i 2. the lateral displacement in the direction Oy: Y(x, y; t)  j 3. the transverse displacement in the direction Oz: Z(x, y; t)  k 4. the rotation around the Ox axis: ψ x (x, y; t)  i 5. the rotation around the Oy axis: ψ y (x, y; t)  j After deformation, the point P(x, y,0) is transformed into P(x +X, y +Y , Z). 5.2.3 Membrane displacements, strains and stresses 5.2.3.1 Global and local displacements The components X, Y are termed membrane displacements, which are sufficient to describe the in-plane motions of the plate. In such motions, all the material points lying on a same normal fibre have the same displacements. So, global and local displacements are also the same, see Figure 5.4.  ξ(x, y; t) =  X = X(x, y; t)  i + Y(x, y; t)  j [5.1] 5.2.3.2 Global and local strains In agreement with [5.1], the local and the global strains are also the same: ε = η. Restricting the study to the case of small deformations, by substituting [5.1] into 264 Structural elements Figure 5.4. Local and global displacements for in-plane motion Figure 5.5. Membrane strains the small strain tensor [1.25], we get: η xx = ∂X ∂x ; η yy = ∂Y ∂y ; η xy = η yx = 1 2  ∂X ∂y + ∂Y ∂x  [5.2] The geometrical meaning of these quantities is illustrated in Figure 5.5. They comprise two normal and one shear component. η xx is the relative longitudinal elongation (Ox direction), η yy is the relative lateral elongation (Oy direction) and η xy = η yx = γ/2 is the in-plane shear strain which is also expressed in terms of the shear angle γ . Finally, the strain tensor [5.2] can be conveniently written in the following matrix form: [ε]=[η]=     ∂X ∂x 1 2  ∂Y ∂x + ∂X ∂y  1 2  ∂X ∂y + ∂Y ∂x  ∂Y ∂y     [5.3] Plates: in-plane motion 265 Figure 5.6. Global stress components 5.2.3.3 Membrane stresses The membrane strains induce the membrane local stresses σ xx , σ yy , σ xy which are independent of z. These quantities are integrated over the plate thickness to produce the global membrane stresses which are suitably defined as forces per unit length:   N xx N yy N xy   =  h/2 −h/2   σ xx σ yy σ xy   dz = h   σ xx σ yy σ xy   [5.4] In [5.4] the stress components are used to form a stress vector instead of a stress tensor. To avoid confusion when using the matrix notation, the local and global stress vectors are denoted [σ ] and [  N ] respectively, whereas the local and global stress tensors are denoted [σ ] and [N ]. As shown in Figure 5.6, they are consistent with the sign convention adopted for the three-dimensional stresses (see Figure 1.6). N xx and N yy are the normal stresses, directed along the normal vectors of the edge surfaces parallel to Oy and Ox respectively, and N xy = N yx are the in-plane shear stresses directed in the tangential directions Oy and Ox respectively. The global stress tensor is written in matrix notation as: [N ]=  N xx N yx N xy N yy  [5.5] 5.3. Membrane equilibrium of rectangular plates 5.3.1 Equilibrium in terms of generalized stresses Because the geometry of rectangular plates is very simple, there is no difficulty in deriving the equilibrium equations using the Newtonian approach i.e. direct balancing of the forces acting on an infinitesimal rectangular element, as shown in the next subsection. However, it is also of interest to solve the problem by using Hamilton’s principle, which deals with scalar instead of vector quantities. It is 266 Structural elements applied first to the case of rectangular plates in subsection 5.3.1.2, and then to the case of orthogonal curvilinear coordinates in section 5.4. 5.3.1.1 Local balance of forces Let us consider a rectangular plate loaded by the external force field:  f (e) (x, y; t) = f (e) x  i + f (e) y  j [5.6] where  f (e) is a force density per unit area of the midplane surface. In Figure 5.7, the balances of the longitudinal and lateral forces acting on an elementary rectangle dx, dy are sketched, which extend to the two-dimensional case the longitudinal force balance of straight beams shown in Figure 2.9. Of course, in the 2D case, it is appropriate to distinguish between a longitudinal and a lateral force balance, giving rise thus to two distinct equilibrium equations. Furthermore, the contribution of the tangential stresses due to the in-plane shear must be added to the in-plane normal stresses. As shear stresses are symmetric N xy = N yx , the moments are automatically balanced, not requiring any additional condition to describe the equilibrium of the rectangle. The two equilibrium equations, in the Ox, Oy directions respectively, are written as: ρh ¨ X −  ∂N xx ∂x + ∂N yx ∂y  = f (e) x ρh ¨ Y −  ∂N yy ∂y + ∂N xy ∂x  = f (e) y [5.7] Figure 5.7. In-plane forces acting on an infinitesimal element of the rectangular plate Plates: in-plane motion 267 In tensor notation, the system [5.7] is expressed as: ρh  ¨ X ·  ℓ −  div N  ·  ℓ =  f (e) ·  ℓ [5.8] where  ℓ denotes a unit vector in the plane of the plate. It may be noticed that equation [5.8] is similar to the general 3D equation [1.32] and is independent of the coordinate system (intrinsic form), in contrast with [5.7] which holds in the case of Cartesian coordinates only. 5.3.1.2 Hamilton’s principle Here, in addition to the surface force density defined in [5.7], an edge force density is also introduced:  t (e) (x, y; t) =  t (e) x  i + t (e) y  j  [5.9] The Lagrangian is written as: L =  L x 0  L y 0 {e κ − e s + w F }dx dy +  (C) w T ds [5.10] where e κ is the kinetic and e s the strain energy densities per unit area; w F is the work density of an external force field acting within the plate midplane and w T is the work density of an external force field acting on the contour (C) of the plate and s is the curvilinear abscissa along (C). Hamilton’s principle is first written as: δ[A]=  t 2 t 1   L x 0  L y 0 {δ[e κ ]−δ[e s ]+δ[w F ]}dx dy +  C δ[w T ]ds  dt = 0 [5.11] The remaining task is to evaluate the terms of [5.11] in a suitable way, as detailed just below. The kinetic energy density is found to be: e κ = 1 2 ρh( ˙ X 2 + ˙ Y 2 ) [5.12] [...]... (y); 2EI Y (x, y) = − νF (x − L)y 2 + Y0 (x) 2EI Substitution of these results into the shear equation leads to: F 2 = 8GI dX0 F y2 νFy 2 + − dy 2GI 2EI + F x(x − 2L) dY0 + dx 2EI 28 2 Structural elements so, (2 + ν)Fy 2 dX0 =a− ; dy 2EI where a + b = dY0 =b− dx F x(x − 2L) 2EI F 2 8GI The resulting displacement field is: F (2 + ν) 3 F y ; Y0 (x) = bx + d + (3Lx 2 − x 3 ) 6EI 6EI F xy(x − 2L) F (2 +... 28 9 In-plane, or membrane, natural modes of vibration 5.3.6.1 Solutions of the modal equations by variable separation The modal equations derived from the vibration equations [5.35] can be written as: ω c 2 ω c 2 X+ ∂ 2X 1 + ν ∂ 2Y 1 − ν ∂ 2X + =0 + 2 ∂x∂y 2 ∂y 2 ∂x 2 ∂ 2Y 1 − ν ∂ 2Y 1 + ν ∂ 2X Y+ + + =0 2 ∂x∂y 2 ∂x 2 ∂y 2 where c= [5.56] E ρ(1 − ν 2 ) Incidentally, it may be noted that the speed of. .. noted K The tangential stiffness of the joints is neglected Because of the symmetry of the structure it is advantageous to use the centre of the plate as the origin of the axes According to the thermoelastic stresses [5. 48] , the equilibrium equations are: ∂ 2X ∂ 2Y +ν 2 ∂x∂y ∂x Eh 1 − 2 + Gh ∂ 2Y ∂ 2X +ν ∂x∂y ∂y 2 Eh 1 − 2 ∂ 2Y ∂ 2X + 2 ∂x∂y ∂y ∂ 2X ∂ 2Y + + Gh ∂x∂y ∂x 2 =− αEh ∂ θ 1 − ν ∂x [5.51] αEh... θ0 1−ν αE E θ0 (2A1 L + C1 + νA2 ) = ⇒ 2 1−ν 1−ν αEh θ0 ∂Y Eh ∂X = +ν x 2 ∂y ∂x y=±ℓ 1−ν L 1−ν ⇒ E αE θ0 (A2 + ν(2A1 x + C1 )) = x 1−ν L 1 − 2 then, E αE θ0 (A2 + ν(2A1 x + C1 )) = x 2 1−ν L 1−ν α θ0 (1 + ν) ⇒ A2 + νC1 = 0; A1 = 2L E αE (2A1 L + C1 + νA2 ) = θ0 1−ν 1 − 2 ⇒ C1 + νA2 = 0 ⇒ C1 = A2 = 0 leading finally to the longitudinal displacement field: X(x) = α θ0 (1 + ν) 2 x ; 2L Y =0 [5.55] Such... Plates: in-plane motion 29 1 substituted into equations [5.59] to give: 2 ω c ω c 2 − mπ Ly 2 − 2 nπ Lx − 1−ν 2 1−ν 2 − nπ Lx 2 + 2 mπ Ly 1+ν 2 + nπ Lx 2 1+ν 2 mπ Ly α=0 2 1 =0 α [5.63] Elimination of the pulsation between the two equations is immediate It produces a quadratic equation which determines the modal values of α The result is: 2 nπ Lx (1) αn,m = ; mπ Ly (2) αn,m = − 2 [5.64] There are infinitely... one edge 27 6 Structural elements in the lateral direction Oy The other edges are free In terms of distributions the equilibrium equations are: − Eh 1 − 2 − Eh 1 − 2 ∂ 2X ∂ 2Y +ν ∂x∂y ∂x 2 − Gh ∂ 2Y ∂ 2X + ∂x∂y ∂y 2 (e) = tx δ(Lx − x) ∂ 2Y ∂ 2X +ν ∂x∂y ∂y 2 − Gh ∂ 2X ∂ 2Y + ∂x∂y ∂x 2 =0 The boundary conditions are: 1 Sliding edge x = 0: X(0, y) = 0; Nxy = 0 ⇐⇒ ∂X(0, y) ∂Y (0, y) + =0 ∂y ∂x 2 Free edge... inherent in 2D shear effect They can be put in evidence even more clearly by writing the equations [5.35] in matrix form: ¨ X 0 ¨ ρh Y  Eh 2 2  − 1 − ν 2 ∂x 2 − Gh ∂y 2  +  2 Ehν + Gh − 2 ∂x∂y 1−ν ρh 0 − − Ehν 2 + Gh ∂x∂y 1 − 2 Eh 1 − 2 2 ∂y 2 − Gh (e) × f X = x (e) Y fy 2 ∂x 2      [5.36] In [5.36] we recognize the canonical form [M][q] + [K][q] = [Q(e) ] of the ¨ linear equations of any... (L, 2 , h) is considered The edge x = 0 is a sliding support The problem consists in determining the response to the linear temperature distribution θ (x) = θ0 x/L, see Figure 5.13 It is governed by the Plates: in-plane motion 28 7 Figure 5.13 Plate loaded by a uniform thermal gradient following system of equations: Eh 1 − 2 Eh 1 − 2 ∂ 2X ∂ 2Y +ν ∂x∂y ∂x 2 + Gh ∂ 2Y ∂ 2X + ∂x∂y ∂y 2 = ∂ 2Y ∂ 2X +ν 2. .. y 2 + e1 y + f1 = 0 ⇒ c1 = e1 = f1 = 0 Y (x, 0) = 0 ⇒ a2 x + c2 = 0 ⇒ a2 = c2 = 0 28 8 Structural elements Changing the name of the constants, the solution is rewritten as X = A1 x 2 + B1 xy + C1 x; Y = A2 y The condition of no shear at the edges gives: ∂X ∂Y + = B1 = 0 ∂y ∂x As a consequence, shear vanishes everywhere in the plate The condition of zero normal stresses at the edges gives: Eh 1 − 2. .. c 2 cos nπ Lx mπ y Ly 2 + ; mπ Ly 2 Yn,m = ; n, m = 1, 2, mLx nπ x cos nLy Lx sin mπ y Ly [5.66] 29 2 Structural elements (2) Modes related to αn,m < 0 are the out -of- phase modes The natural frequencies and mode shapes are: fn,m = Xn,m = sin c 2 nπ x Lx 1−ν 2 cos mπ y Ly nπ Lx ; 2 + mπ Ly Yn,m = − 2 ; n, m = 1, 2, nLy nπ x cos mLx Lx sin mπ y Ly [5.67] It may be noted that the frequencies of . ν 2  ∂ 2 ∂x 2  − Gh  ∂ 2 ∂y 2  −  Ehν 1 − ν 2 + Gh  ∂ 2 ∂x∂y  −  Ehν 1 − ν 2 + Gh  ∂ 2 ∂x∂y  − Eh 1 − ν 2  ∂ 2 ∂y 2  − Gh  ∂ 2 ∂x 2       ×  X Y  =  f (e) x f (e) y  [5.36] In. ν 2  ∂ 2 X ∂x 2 + ν ∂ 2 Y ∂x∂y  − Gh  ∂ 2 X ∂y 2 + ∂ 2 Y ∂x∂y  = t (e) x δ(L x − x) − Eh 1 − ν 2  ∂ 2 Y ∂y 2 + ν ∂ 2 X ∂x∂y  − Gh  ∂ 2 Y ∂x 2 + ∂ 2 X ∂x∂y  = 0 The boundary conditions are: 1 be: e κ = 1 2 ρh( ˙ X 2 + ˙ Y 2 ) [5. 12] 26 8 Structural elements Its variation is integrated with respect to time, giving:  t 2 t 1   L x 0  L y 0 {δ[e κ ]}dx dy  dt =−  t 2 t 1   L x 0  L y 0 ρh{ ¨ XδX