Introduction to Plates

Transcription

1 Chapter Introduction to Plates Plate is a flat surface having considerabl large dimensions as compared to its thickness. Common eamples of plates in civil engineering are. Slab in a building.. Base slab and wall of water tanks. 3. Stem of retaining wall. A plate ma have different shapes e.g. rectangular, triangular, elliptic, circular etc. as shown in Fig... Fig.. Shapes of plates A plate ma have edge conditions like free, simpl supported, fied or elasticall supported as shown in Fig... Beam Fig.. Edge conditions

2 THEORY OF PLATES AND SHELLS In this chapter the coordinate sstem selected is clearl eplained first and then various forces to be considered on an element of plate are eplained and sign conventions are made clear. At the end a brief introduction is given to different theories available for the analsis of plates.. COORDINATE SYSTEMS In the analsis of plates, cartesian coordinate sstem with right hand rule is used. According to this when thumb, inde finger and middle finger are stretched to show three mutuall perpendicular directions, thumb indicates -coordinate direction, inde finger shows -coordinate direction and middle finger indicates z-coordinate direction. Figure.3 shows different orientation of,, z directions. The equations derived for plate analsis with cartesian coordinate sstem with right hand rule hold good for all these orientations. The commonl used orientation is that shown in Fig..3(a), since slabs are usuall subjected to downward loads and the analst is interested in downward deflections. o z ( a) z ( b) z z ( c ) ( d) Fig..3 Different orientation of coordinates with right hand rule For the analsis of circular plates, polar coordinate sstem shown in Fig..4 ma be used advantageousl. Reference ais d r dr z is downward direction Fig..4 Polar coordinates for circular plate

3 INTRODUCTION TO PLATES 3. STRESSES ON AN ELEMENT In cartesian sstem an element of size d d dz is selected at a point (, ), distance z below the middle surface [Refer Fig..5(a)]. The stresses acting on the element are shown in Fig..5(b), in their positive senses. Note that the sign convention used is that a stress on positive face in positive direction or on negative face in negative direction is positive stress. It means the direct tensile stress is positive. For shear stresses, the positive senses are as shown in Fig..5(b). It ma be noted that for shear stresses first subscript indicates the face and the second the direction. Thus τ z is shear stress on face and in direction z. d d dz z z z z z (a) An element at point (,, z) Fig..5 Stresses on an element z (b) Stresses on the element.3 TYPES OF THEORIES OF PLATES The theories that are available for the analsis of plates are. Thin plates with small deflections.. Thin plates with large deflections and 3. Thick plates..3. Theor of Thin Plates with Small Deflections This theor is satisfactor for plates with thickness less than th 0 of its lateral dimension and having deflection less than th of its thickness. In this theor the following three assumptions are made: 5. Points on the plate ling initiall on a normal to the middle surface of the plate remain on the normal to the middle surface of the plate even after bending.. The normal stresses in the direction transversal to the plate can be neglected i.e. Take σ z, τ z, τ z =0. 3. There is no deformation in the middle surface of the plate. This plane remains neutral during bending. Assumption means shear deformations are neglected. This assumption is generall satisfactor, but in some cases e.g. in case of holes in the plate, the effect of shear becomes considerable and hence corrections to the theor of thin plates are to be applied.

4 4 THEORY OF PLATES AND SHELLS Assumption is valid for thin plates, since the stresses are zero in z-direction at top and bottom of plates, as the are free edges. There ma be small variation inside the plate at an depth z, but it is negligible. Assumption 3 holds good if the deflections are small. However in actual structure when the plate bends, small forces ma develop in the middle surface. This inplane stress in the middle of plate reduces the bending moment at an other point. Hence neglecting this force is an assumption on safer side..3. Theor of Thin Plates with Large Deflections If the deflections are not small in comparison with its thickness, strains and stresses are introduced in the middle surface of the plate. These stresses are to be considered in deriving equilibrium equations. Inclusion of these stresses results into non-linear equations. This is called geometric nonlinearit. When this non-linearit is considered, the solution becomes more complicated..3.3 Theor of Thick Plates The first two theories discussed above become unrealistic in the case of plates of larger thicknesses, especiall in the case of highl concentrated loads. In such cases thick plate theor should be used. This theor considers analsis as a three dimensional problem of elasticit. The analsis becomes length and more complicated. Till toda the problems are solved onl for a few particular cases. QUESTIONS. Draw an element of plate in Cartesian sstem and show the stresses acting on it in their positive senses. Make the sign convention clear.. Briefl write on the following theories of plates to bring out differences among them. (a) Thin plates with small deflections. (b) Thin plates with large deflections. (c) Thick plates.

5 Chapter Pure Bending of Plates As the title suggests, in this theor stress resultants produced due to bending moments onl are considered. In other words, deformation of the membrane due to eternal loads is ignored. Naturall, in this tpe of bending, middle surface remains neutral surface. In this chapter, some of the properties of bent surface are discussed and epressions are derived for stresses and moments in terms of single unknown deflection w.. SLOPE IN SLIGHTLY BENT PLATE Figure.(a) shows the plan view of an element and Fig..(b) shows sectional view of slightl bent plate. o m (, ) a d d a n Fig.. (a) Plan view of element m middle surface before deformation n w w + dw middle surface after deformation Fig.. (b) Sectional view of element

6 6 THEORY OF PLATES AND SHELLS Consider an element of size d d at point (, ) in the middle surface of the plate. Figure.(b) shows the middle surface of the plate cut b plane mn parallel to z plane. Then, Slope along -ais =q =...eqn.. Similarl if a plane parallel to z plane is considered, Slope along -ais =q =...eqn.. Let aa make an angle α with -ais (Refer Fig..(a)). The difference between the deflections at a and a is due to slopes in and directions. Let it be dw. Then, dw = θ d + θ d = d + d...eqn..3 Slope along aa which is in n direction is given b dw = w d + w d n n dn = cosa+ sin a. Let the maimum slope be at an angle α to -ais. Hence...eqn..4 Êdwˆ a Ë dn a=a = 0 (-sin a ) + cosa = 0 or tan a = w Putting eqn..4 to zero, we get the direction of zero slope. Let it be α. Then 0 = cosa + sina...eqn..5 tan a = - w From eqns..5 and.6, we get,...eqn..6 tan tan. a a = - It means the direction of zero slope (α ) and the direction of maimum slope (α ) are at right angles to each other.

7 PURE BENDING OF PLATES 7 Epression for Maimum Slope: The value of maimum slope = n a=a = cosa + sin a Ê ˆ = + tan a cosa Ë Ê ˆ = + Ë seca ʈ ʈ + Ë Ë = w + tan a ʈ ʈ + Ë Ë = Ê ˆ + Ë Êˆ ʈ + Ë Ë = ʈ ʈ + Ë Ë Êˆ ʈ = + Ë Ë...eqn..7. CURVATURE OF SLIGHTLY BENT PLATE The curvature of a bent surface is numericall equal to the rate of change of slope. If the curvature is considered positive when it is a sagging surface (Refer Fig..), the curvature in -direction where r ʈ w = = - = - r Ë is radius of curvature, negative sign since as d increases curvature reduces....eqn..8

8 8 THEORY OF PLATES AND SHELLS Similarl, curvature in -direction = = - w. r...eqn..9 Fig.. Positive sense of curvature Curvature in an direction Consider a direction n which makes angle α with -ais. Then from the definition of curvature, w =- r n From eqn..3, = cosa+ sin a n i.e. = cosa+ sin a n n w ʈ =- =- Ë r n n n n Ê ˆ Ê ˆ =- cosa+ sina cosa+ sina Ë Ë È w w w w =-Í cos a+ sin a+ cosasin a+ sin a cosa ÍÎ Noting that, and w =- r w =- r and taking w =, r we get w = cos a+ sin a- sin acosa r r r n + cosa -cosa = + - sin a r r r Ê ˆ Ê ˆ cosa = sin a Ër r Ër r r...eqn..0

9 PURE BENDING OF PLATES 9 r t If t is the direction at right angles to n direction, the direction t is at α + 90 to n-direction. Hence can be obtained b changing α to α + 90 in eqn..9. Thus Ê ˆ Ê ˆ = cos ( a+ 90) - sin ( a+ 90) rt Ër r Ër r r Ê ˆ Ê ˆ = cosa+ sin a Ër Ë r r r r...eqn.. Adding eqns..0 and., we get + = + rn rt r r...eqn.. Hence we can conclude, the sum of curvatures in an two mutuall perpendicular directions in a slightl bent plate is constant. Twist of the surface w.r.t. n and t directions: It is given b, Now from eqn..4, w ʈ = = Ë r n t n t nt = cosa+ sin a n = cosa+ sin a n Since t is the direction at right angles to n, we get α i.e. = cos( a+ 90) + sin ( a+ 90) t t =- sin a+ cosa from the above epression b changing α to ʈ = Ë r n t nt Ê ˆÊ ˆ = cosa+ sin a- sin a+ cosa Ë Ë w w w w sin cos sin cos cos sin =- a a+ a a+ a- a

10 0 THEORY OF PLATES AND SHELLS Ê ˆ sina = - + cos a- sin a Ër r r ( ) Ê ˆ sina = - + cos a Ër r r...eqn..3.3 PRINCIPAL CURVATURE The two mutuall perpendicular directions n and t with respect to which twist of the surface is called the direction of principal curvatures. Hence from eqn..3, we get the direction of principal curvature α as r tan a=- Ê ˆ - Ër r r nt = 0,...eqn..4 It can be shown that in the direction of principal curvatures, the curvature is maimum/minimum. For this proof, differentiate eqn..0 with respect to α. It gives, Ê ˆ - (-sin a ) - cosa = 0 Ër r r r tan a=-, i.e. Ê ˆ - Ër r which is same as eqn..4. Thus we find the planes of principal curvatures are the planes of etreme curvatures also. Magnitude of Principal Curvatures For such planes, r tana=- - r r Referring to Fig..3 r 4 r r r r / r Fig..3