Chapter 4. Statics of Plate o Classical Linear Theory of Plate o Plane stress and Plane strain o Equations of Equilibrium and Boundary Conditions - Equilibrium Approach - Minimum Potential Energy Approach-Thin plate o Solution methods etc Basic Assumptions ~In the initial state, the surfaces of a flat plate are parallel planes, called the faces of the plate ~The distance between the faces is called the thickness of the plate. ~A plate is distinguished from a block by the fact that the thickness is small compared to the dimensions paralled to the faces. ~The plane midway between the faces is called the middle plane of the plates ~The bending properties of plate depend greatly on its thickness as compared with its other dimensions. :We will discuss the linear static theory of thin elastic plate in this chapter. ~ shell ? membrane ? plate ? 4.1 Classical linear theory of plate w Definition of small deformation : 1 : h Decoupled set of Bending and stretching ~ Isotropic material ~ Homogeneous materials ~ Isotropic or Anisotropic(Composite) 11 1 ~ Thick plate : hl/~ Thin plate hl/ 20 10 20 ~ Loading : o Bending action ~ deflection o Membrane action~stretching Configuration of curved surface ~ Displacements : ~ u(,),(,), x y v x y w x y (,)uw ( 1,2), : In-plane displacements uv, ~ deflection w 3-Dimensional linear elasticity ij, jq j 0 For plate bending problem, let q ( 1,2) u:0 , 3,3 ~ wF:3 , 33,3 3 with boundary conditions ! 4-1-1 Kinematics of Deformation The fundamental assumptions of the small deflection theory of bending or so called classical theory of isotropic homogeneous elastic thin plate ; Kirchhoff – Love hypothesis material property change Kinematic Assumptions 1. Deflection of mid-surface is small compared with the thickness of the plate. : Slope of the deflected surface is very small and the square of the slope is a negligible quantity in comparison with unity 2.The mid-plane remains unstrained subsequent to bending # Large deflection case ? 3. Plane sections initially normal to surface remains plane and normal to that surface after the bending : z 0( 1,2)~G ~ Before bending ~> After bending ? ? Higher-order theory ? Thick plate theory ~ Mindlin plate theory 4. No stretching or contraction of normal : zz 0 5.The stress normal to the mid-plane, zz :small compared with the other stress components and may be neglected : This assumption becomes unreliable in the vicinity of highly concentracted transverse loads. mm uxyzmm(,) uxy0 (,) uxyzuxyz 1 (,) 2 (,) 2 ... u (,,)(,) x y z u x y z m0 mmmm0 1 2 2 w(,,)(,) x y z u3 x y z u3(,) xyz uxy 3 (,) uxyzuxyz 3 (,) 3 (,) ... m0 ~ order of thickness effects ! w(,,) x y z 0:uo ( x , y , z ) w ( x , y ) zz z 3 Shear modulus : Composite layered plate ? 6. Developable surface This type of surface can be bent back to a plane without any variation in the distance between any two points on the surface. Under these assumptions for thin plate, 1 ()uu 2 ,, 11 30 (u ,3 u 3, ) ( u ,3 w , ) 3 22 33 0 1 (uw ) 0, 32 ,3 , : uw,3 , ~ u(,,)(,) x y z u x y zw , : ? 4-1.2. Plane stress and Plane strain For 3 D elasticity theory: ij ll2G ij EE with:, G 2(1 ) (1 )(1 2 ) Plane stress field for kinematic assumptions of 0 3 33 E () 11 (Load , Geometry ) 0 ~ z (i) Thin flat plate (ii) Applied load are acting on the surfaces and do not vary xy across the thickness. (iii) Upper and lower surface are stress free 0 zz z ~ through thickness. iz (iv) do not vary across the thickness Plane strain Dimension of z >> dimension of x,y 0 iz 4-1.3. Equation of Equilibrium & Boundary Conditions 3D ~> plate theory ? x3: Elimination of explicit dependency ! Stress, Moment resultants 1 2 , ) h (1,z ) dx3 (N M 2 ,:? MMxx yy :? M xy Shear force intensity h 2 h 33dx Q 2 According to basic assumptions and plane stress assumption, i30 0 uxyz(,,)(,)(,) uxy, zw xy u3(,,)(,) x y z w x y Total number of independent displacement components ? 1 z ()()u00 u w w 22 ,,,, ~ (1,zz )~ :(1, ) A . Equilibrium Approach 3-D elasticity (without body force) , 3,3 0 3 , 33,3q Model ! h hh 2 22 (a ): h (, 3,3 ) dx 3 N , 3 ]hh N , 3 ] 0 2 22 hhh 222 (b ):hh z (, 3,3 ) dx 3 M , z 3 ]h 3 dx 3 0 222 ~0MQ, (c) h h 2 2 h (3 , 33,3 )dx 3 Q , 33 ]h 0 2 2 ~Q (hh ) ( ) 0 , 3322 33 :0Qq, Substitute Eq.(b) into Eq.(c) Mq, 0 Finally, Mxx,,, xx2 M xy xy M yy yy q NNxx,, x xy y 0 NNyx,, x yy y 0 Equations may be coupled set. For isotropic plate, Membrane stretching effect and Bending effect: Decoupled Mid-plane ? For rectangular plate: Obtain stress field in terms of strain components Obtain N,M in terms of u,v,w … then, ; Bending problem; 4th order eqn.( Hermite approximation:C1 continuity) 4 D( w ,xxxx 2 w , xyxy w, yyyy ) D w q ; and Stretching equations 11 u u, v , 0 ,xx22 yy xy 11 v v, u , 0 ,yy22 yy xy % MQ, 0 xyy Qx M xx,, x M xy y D( w , xxx w , yyx (1 ) w , ) D(,,)(,,)() wxxx w xyy D w xx w yy,, x D 2 w x ~ 2 , y Qy D() w B. Minimum Potential Energy Approach ~ Strain energy of entire plate with linear elastic behavior h 1 2 U dzdA ()i ij ij 2 h A 2 Replacing ij in terms of ij in conjunction with plane stress state h 1 2 U dzdA ()i 2 2(1 ) h A 2 h 1 2 22 2 2(1 ) 2 dzdA 2 xx xx yy yy xy 2(1 ) h A 2 Potential energy for the external load U()e A q(,)(,) x y w x y dA 1 Remembering, u u0 zw, and ()uu 2 ,, h 1 2 U( u0 zw ) 2 2 2 2(1 ) 2 dzdA ()i 2 ,,x xx xx yy yy xy 2(1 ) h A 2 Total potential energy UU()()()i U e U i A qxywxydA(,)(,) Integrating by part with respect to z: (~ Extensional energy + Bending Energy ! ) mB UUU()()()i i i where, mB00 U()()ii( u , v ), U ( w ) : Decoupling ! ? Obtain the expression & govering eqn for (Report !) We shall be concerned with the stresses and moments in plates as a result of transverse loadings of bending part mB UUU()()ii h 2 B D 2 2 2 ~U()i ()2(1)[ w w,xy w ,,] xx w yy dA qwdA 2 h AA 2 To extremize this functional, we get U 0 h D 2 2 2 2 [ ()2(1)[w w,xy w ,,] xx w yy dA qwdA ] 2 h AA 2 D 2(22w ) w 2(1 )[2 w w , w , w , w , w , ] dA 2 ,xy xy xx yy xx yy D 2( 22w ) w (1 )[2( w w , w w , ) 2 ,,xy xy yx yx 2w ,xx w , yy 2 w , xx w , yy ] dA D( 22 w ) w (1 )[2( w w , w w , ) ,,xy xy yx yx 2w ,xx w , yy 2 w , xx w , yy ] dA D(, w w w w,)(,,,, w w w w ,xx xx, yy yy yy xx xx yy (1 )(w w , w w , )] dA ,,xy xy yx yx Using Green’s formula(Report!) 4 (D w q ) w dA D ..... 0 Introducing the moment intensity, then the above equation becomes(Report) (DwqwdA4 ) Mwdy , ......... ( M M ) wdy 0 xx x xx,, x xy y We may rewrite the last two integrals in terms of shear force intensities. We then have ( Report ) (DwqwdA4 ) M wdy , ......... Qwdy 0 xx x x To simplify this functional further, let introduce the coordinate n, s which are the normal and tangential directions to contour, respectively. Then ( Report )… ww (D4 w q ) w dxdy M ds M ds . Q wds 0 nnns ns n The 3rd integral becomes, by integration by parts w M M ds ns wds...(?) ns ss We then get for the variation of the total potential w M (D4 w q ) w dA M ds ( Q ns ) wds 0 nnns n Euler-Lagrange equation for this problem is 4 D w q Report :Derive the governing equation with with variable bending rigidity which have been derived in previous section. Now we get as a result of the variational process two sets of boudary conditions, namely, the natural and the kinematic bound conditions. Then on the boundary we require that w M 0.. or .. .. is .. prescribed (a) nn n M Qns 0.. or .. w .. is .. prescribed (b) n s There are only 2 conditions, in spite of the fact the there 3 variables MQM,, nn n ns The 1st condition is acceptable by physical consideration and needs no further comment. We shall now examine the 2nd condition with the view reaching some physical explanation. : We have shown part of the edge of the plate when two panels of length S have been identified. _ Figure_ The twisting moments have been expressed in the first and second panel. A third panel of length may be imagined the center of two panels, which is shown in Fig AB. The shear force for thin panel is shown as QSn . Now we make use of the Saint-Venant principle by replacing the twisting moment distribution of the original two panels by two couples having forces M ns M M ns value ns and s respectively with a distance of between the forces as shown in the figure.