The Elastic Properties of Multi-Ply Laminates

Displacements The displacements in the x, y and z directions are u,v and w respectively. It is assumed that plate displacements in the z-direction only arise from bending, there is no variation in thickness in the z-direction (ie. no through thickness strain).
Centre-Line The centerline is a line through the thickness of the laminate that divides the laminate vertically into two regions of equal thickness.
Bending When a uniform plate bends, as shown below, there is no extension at the centerline, but on the inside of the bend (above the centre line, z is positive) there is an increasing amount of compression (negative displacement = ) as we move away from the centre line; on the outside of the bend (below the centre-line, z is negative) there is an increasing amount of tension (positive displacement = ) as we move away from the centre-line to the outer surface of the laminate. For small angles .

When the plate is not elastically symmetric about the geometric centre line, the plane of zero bending strain will not co-incide with the plane that defines the geometric centre of the plate - in fact the plane of zero bending strain moves towards the stiffer side of the plate.

Definition of Strain in a Laminate

The in-plane displacements (u and v), which are functions of position (x,y,z) within the laminate and can be related to the centre-line displacements, u_o and v_o and the slopes by

Now that we have the displacements, we can get the normal strain. Recall that the normal strain is defined as the fractional change in length.

Next, we substitute for u, the function , and evaluate the derivative:-

The strain term is obtained in the same way. The engineering shear strain is just the change in the angle between two initially perpendicular sides. For small strains, .

Again we can substitute for u and v then differentiate with respect to x and y. The resulting strain matrix may be written as:-

Fortunately, the above equation can be written more simply as

where is the centre-line strains and the curvatures:-

and

Definition of Force and Stress in a Laminate

• The Fibre and Matrix Materials
  
• The Volume Fraction of Fibres
  
• The Orientation of the Fibres
  
  

where F_k is the force in the k_th ply of the laminate, _, is the stress, t_k, is the thickness of the k_th layer and w, the width of the laminate. By convention, when dealing with laminates, the force is described as N, the force per unit width of the laminate or the force resultant. Mathematically, the force resultant is defined as

for the force resultant in the x-direction. The term h is the total thickness of the laminate. We can write down both the force resultants and moment resultants (force per unit width of laminate x distance) in compact form

The integration of the total laminate thickness is actually very simple since an integral is actually a sum; so we can sum the stresses in each of the individual plies.
Remember, that if there is no bending then the stresses in each ply are constant. If there is bending, then the stresses in each ply will vary across the thickness of each ply.

where h_k is the position of the bottom of the k_th ply with respect to the centre-line of the laminate and h_k+1, the position of the top of the k_th ply with respect to the centre line of the laminate as shown below.

Relating force resultants {N} to strains {e} using the stiffness matrix Q

where the stiffness matrix is a function of orientation, fibre fraction and fibre and matrix materials. In a given ply, is constant hence

similarly for the moment resultants

The integrations are actually very simple since the stiffness matrices Q_k, the centre-line strains and curvatures are constant in each ply, so the only variable is z, the vertical position within each ply. Therefore

and

The stiffness of the laminate Q_L is simply [A]/h where h is the total thickness of the laminate. When B=[0], as occurs in symmetric laminates, then .