ME 453/553 - Mechanics of Laminated Beams
Brian K. Bay – OSU School of Mechanical, Industrial & Manufacturing Engineering
General Thin Laminate
y
Cantilever Beam Top View
NX
x
z
Cantilever Beam Side View
MX
x
Note: This orientation of Mx is consistent with the formulations that follow (you may
be expecting an My). It is a convention followed for fiber-reinforced composites.
1
The structure is a thin fiber reinforced composite made up of individual lamina
(plys) numbered from bottom to top (the six layers is arbitrary, it could be any
number as long as the structure remains “thin”). It could be a plate or a beam; if it is
a beam let the long-axis extend to the right, along the x-axis. The y-axis lies in the
plane of the structure perpendicular to the x-axis, and the x-y plane is positioned at
the reference plane (geometric midpoint) of the laminate cross-section. The z-axis
defines positions perpendicular to the reference plane. XYZ is the geometric
(structural) coordinate system.
We will work toward the following equations that define strain and stress variations
through the beam cross-section:
x  x0   x 
   0   
y  y  z y 
0

 xy 
 
 xy 

 xy 
 


 x  Qx,x
  
 y   Qy,x

 xy 
k 
Qxy,x
Qx,y
Qy,y
Qxy,y
Eq.1
Qx,xy  x 
  
Qy,xy  y 
Qxy,xy 
k 
 xy 
k
Eq.2
First of all note that these equations assume “cylindrical curvature” of the structure.
If this condition is not met, then, as with simple beam theory, assume the equations
describe conditions locally and integrate for cumulative effects.
Detailed descriptions of each term follow:


 x 
 
 y 

 xy 

The x-y strain state at any point through the laminate thickness. Strains
vary linearly through the laminate thickness, and smoothly across layer
boundaries.
 x 
 
 y 

 xy 
k
The x-y stress state at any point within the kth layer. Stresses vary linearly
within a layer, but transition abruptly at the layer boundaries if the layer
stiffness matrices change.
Qx,x

Qy,x

Qxy,x
Qx,y
Qy,y
Qxy,y
th
Qx,xy  [Q] is the engineering stiffness matrix of the k layer in the
 geometric coordinate system. The symbol [Q] is used for the
Qy,xy  plane stress version of the general stiffness matrix [C].
Qxy,xy 
k
2

 x0 
 0 
 y 
0

 xy 

Reference plane strains.
z
Position within laminate (distance from reference plane).

 x 
 
 y 

 xy 

Curvatures in the x- and y-directions, and the plate twist.

Equation 1 states that if we know the strain on the reference plane and the
curvature/twist of the structure, we can calculate the strain at any location
(including at the top where we might have a strain gauge attached). With Equation 2
we can calculate stress at any position within a layer as long as we know the
stiffness matrix of that layer.

Laminate Combined Compliance and Stiffness Matrices:
In order to find the reference plane strains and the curvature/twist of the structure,
we have to establish the following relationship:
xo  ax,x ax,y ax,xy
  
y  ay,x ay,y ay,xy
 xyo  axy,x axy,y axy,xy
L  L L L L L L L
  
 x  c x,x c x,y c x,xy
 y  c y,x c y,y c y,xy
  
 xy  c xy,x c xy,y c xy,xy

- or -
bx,xy N x 
 
by,xy N y 
M bxy,x bxy,y bxy,xy N xy 
L L L L L L L L L 
M dx,x dx,y dx,xy M x 
 
M dy,x dy,y dy,xy M y 
 
M dxy,x dxy,y dxy,xy 
M xy 
M
M
bx,x
by,x
bx,y
by,y
Eq. 3
 o  a M b N 
  
 
L  L L L L 

c M d 

M

 
 
The partitioned vector on the left contains the reference plane strains and the local
curvatures of the structure. The partitioned vector on the right contains a

description of the loads applied to the structure: normal forces and in-plane shear
(expressed per unit length), and bending/twist moments, also expressed per unit
length.
3
The partitioned coefficient matrix that relates the two (the combined laminate
compliance matrix) is determined by inversion of the combined laminate stiffness
matrix. That matrix contains information about the individual ply material
characteristics, orientation with respect to the axes of analysis, stacking sequence,
and thickness. It is defined as:
N x  Ax,x Ax,y Ax,xy
  
N y  Ay,x Ay,y Ay,xy
N xy  Axy,x Axy,y Axy,xy
L  L L L L L L L
M x  B
Bx,y Bx,xy
   x,x
M
 y  By,x By,y By,xy
M xy  B
   xy,x Bxy,y Bxy,xy

Bx,xy xo 
 
By,xy y 
M Bxy,x Bxy,y Bxy,xy  xyo 
L L L L L L L L L 
 
M Dx,x Dx,y Dx,xy  x 

M Dy,x Dy,y Dy,xy  y 
 
M Dxy,x Dxy,y Dxy,xy 
 xy 
M Bx,x
M By,x
Bx,y
By,y
Eq. 4
N  A M B  o 
  
 
L  L L L L 

M
 
B M D

 

- or -
We can find the A, B, and D sub-matrices if we know about the layup. The terms of

each matrix are given by:
n
Aij   Qij k t k
Eq. 5
k1

n
Bij   Qij k t k zk
Eq. 6
k1

n
Dij   Qij k t k zk2  121 t k3 
Eq. 7
k1

where [Qij]k is the stiffness, tk the thickness, and zk the position of the centroid (with
respect to the reference plane) for the kth lamina. These terms are analogous to
modulus of elasticity, E, and second moment of the area, I, in simple isotropic beam
equations. The situation is more complicated here with potentially many lamina of
different thicknesses, positions in the stacking sequence, anisotropic in-plane
properties, and orientations of the principal material axes with respect to the
4
reference plane coordinate system. [A:B:B:D] and (after inversion) [a:b:c:d] take
care of it all.
From these summations the combined laminate stiffness matrix is formed, inverted
as in equation 8, and the analysis of strains and stresses in equations 1 and 2 is
possible.
1
a

L

c

M b  A M B 
 

L L  L L L 
M d 
 
B M D

Eq. 8
One final complication is that the ply (individual layer) stiffness matrices, [Q], must
be defined in the geometric (x,y) coordinate system, [Q]x,y. These are determined
from the individual ply compliance matrices defined in the principal material (1,2)
coordinate system as follows.
Let:
S1,2
= Compliance matrix in principal material coordinates (tensor shear),
S x,y
= Compliance matrix in geometric coordinates (tensor shear),
S x,y
= Compliance matrix in geometric coordinates (engineering shear),
T 
= Tensor transformation matrix (two-dimensional).
Qx,y
= Stiffness matrix in geometric coordinates (engineering shear).
ten
ten

eng




The first calculation involves filling in the ply principal compliance matrix:
S1,2
ten

 1
 21
 E11
E 22
12
1
 
E11
E 22

0
0





0 

1
2G12

0
Eq. 9
Note the 1/2G12 term; the one-half with the shear modulus identifies this as the
“tensor” form of the compliance matrix.
5
Next the compliance matrix is transformed (rotated by the ply alignment angle )
into the analysis coordinate system:
S x,y  T 1S1,2 T 
ten
ten
Eq. 10
where:

 m 2 n 2
2mn 
 2

T    n m 2 2mn L m  cos,n  sin 
2
2

mn mn m  n 

Eq. 11
and:


m 2 n 2
2mn 
 2

1
2
T  n m 2mn L m  cos,n  sin 
2
2

mn mn m  n 

The tensor form of the rotated compliance matrix has the unfortunate
characteristics that it is not symmetric, and it must be converted to the
“engineering” form for the calculation of the ply oriented stiffness matrix.
Sx,y
ten

Eq. 12
 Sx,x

  Sy,x
1

2 Sxy,x
Sx,y
Sy,y
1
2 Sxy,y
Sx,x
Sx,xy 


eng
Sy,xy  Sx,y  Sy,x
1


Sxy,x
2 Sxy,xy 
Sx,y
Sy,y
Sxy,y
Sx,xy 

Sy,xy 
Sxy,xy 

The conversion is not difficult: term-by-term multiply the bottom row of the
“tensor” form matrix by scalar 2 to get the “engineering” form matrix. Some authors
make use of an additional matrix multiply pair, often denoted [R], to perform the
conversion. Your choice.
The matrix we are after, the ply oriented stiffness matrix, is the inverse of the ply
oriented compliance matrix in “engineering” form:
Qx,y  S1x,y
eng
Eq. 12

6