7. semester Continuum mechanics Exercise 6

1**7.** **semester** **Continuum** **mechanics****Exercise** 6The exercise deals with calculations of strains in a plate with its mid-plane in the x − yplane. The thickness of the plate t is considerably smaller than the span width of the platein the other directions. Therefore it is assumed similar to the beam theory that "planesections remain plane". The plane sections are situated in the planes x − z and y − z.The displacements of the plate are described by the displacements of the mid-plane in thex−, y− and z−direction denoted ū, ¯v and ¯w respectively, and the rotations of the planesections around the x− and y−axis denoted θ x and θ y respectively. All displacements androtations are functions of x and y. In the following it is also assumed that z is positivein the upward direction and that w is positive downwards. The plate theory is calledReissner-Mindlin plate theory, and it takes the shear exibility into account.Question 1Sketch the deformations of the plate.Question 2Show that the displacements, u,v and w of the plate are dened as:u(x, y, z) = ū(x, y) + zθ y (x, y)v(x, y, z) = ¯v(x, y) − zθ x (x, y)w(x, y, z) = ¯w(x, y) (1)Question 3Find the strains in the x − y plane (ɛ xx ,ɛ yy and ɛ xy ) and explain how they can be dividedup into a membrane part and a bending part.Question 4Find ɛ zz and describe the limitations enforced by the plate theory.Question 5Find γ xz and γ yz .

2Question 6Analogue to the geometric restrictions in Bernouilli-Euler beam theory Kirchho-Love platetheory enforces geometric restrictions corresponding to "plane sections remain perpendicularto the mid-plane".Show that it is equivalent to θ x = − ¯w ,y and θ y = ¯w ,x .Question 7Express the strains in the x − y plane by means of ¯w and its derivatives.Question 8Find ɛ xz , ɛ yz , and explain the dierences between the two plate theories.Question 9In the stress calculation it is assumed that each layer parallel to the x − y-plane is in planestress.How does that match with the fact that the transverse displacement w does not dependon z.What would be the implications if the layers were assumed to be in plane strain?