Question

I need helpnwith this Composites Homework Problem

3) What are the boundary conditions on the positive x-face and positive y-face of the simply- supported symmetric, (B)-0) laminated plate in terms of the displacements? Assume Kirchoff theory simply-supported simply-cappred

Answer 1

It is given that plate is laminated and symmetric, with its length along x-axis equal to 'a' and length along y-axis equal to 'b'. Also, the height of the composite plate is given 'H'.

Given the symmetry, we have assumed our origin of coordinate axis (0,0) such that x_max = a/2; y_max = b/2; z_max = +H/2 (downward). The figure representing the stated composite is shown below.

figure (1)

It is given that it follows the Kirchoff theory, which can be deduced in the following results relevant to our current laminate.

• In our figure, z=0 is an x-y plane called mid-plane. And any normal to the midplane remain straight before and after the deformation.  
• The length of the normal to the midplane which is H in our present case remains constant before and after the deformation.

undebouned mjdplane -ㄒㄧ召

figure (2)

Boundary conditions in terms of displacement of the laminate in accordance with Kirchoff's Law

1. According to the 1st deduction from the above stated Kirchoff's Law, which says "normal to un-deformed midplane remains normal to the deformed mid-plane", we can conclude that the transverse shear strain will be zero.
2. In y plane or x-z plane, there is a stretching action along the x-axis and there is a bending action along the z-axis, due to which a small displacement in both directions occur.
3. According to the 2nd deduction from the Kirchoff's Law, which says the length of the normal to the mid-plane does not change. This concludes that the transverse deflection at any point in the laminate is independent of its location in the z-axis. Let's say deflection of any point is defined by 'D'. Then

D(x,y,z)=D₀(x,y)

Let's say deformed midplane bends by an angle $\alpha$ then the slope can be written as

tan a -

For small deformation,

$\alpha=\frac{\delta D}{\delta x}$

Let's say there is a point 'P' in the midplane with the elevation z. After deformation, displacement of this point along the x-axis can be given, using the above-derived relation, as

$u(x,y,z)=u'(x,y)-z\tan\alpha=u'(x,y)-z\alpha=u'(x,y)-z\frac{\delta D}{\delta x}$

Similarly, in x-plane or y-z plane,

$v(x,y,z)=v'(x,y)-z\frac{\delta D}{\delta y}$

So, complete displacement of point 'P' can be defined by the following equations:

$u(x,y,z)=u'(x,y)-z\frac{\delta D}{\delta x}$

$v(x,y,z)=v'(x,y)-z\frac{\delta D}{\delta y}$

$d(x,y,z)=D(x,y)$

Now, using the above relation, we can set the boundary conditions for the deformation using the values

$x=\pm \frac{a}{2}$

$y=\pm \frac{b}{2}$

and $z=\pm \frac{D}{2}$