AE 569
PROJECT FOR UNDERGRADUATE STUDENTS
(due June 17th, Friday)
DEMONSTRATION OF COUPLING EFFECTS AND TAILORING ISSUES IN
COMPOSITE STRUCTURES
Project aims at demonstrating the coupling effects and tailoring issues in composites structures
through three different applications. These applications are stand alone and they are not directly
related. The first application involves the demonstration of coupling effects analytically. The second
application is a combined analytical and finite element based demonstration of the coupling effects,
and their comparison. The third application is pure finite element based demonstration of the coupling
effects on a simple lifting surface. These three applications and detailed requirements are described
below in detail.
Application 1: Development of a computer program to calculate stiffness matrices of
composite laminates and investigation of coupling effects on representative laminates
For a general composite laminate write a computer code (in Matlab, Fortran or in any other
language you are comfortable with. Matlab is recommended because of the ease of performing
matrix operations) which will calculate the stiffness terms of the laminate, namely the elements of
A, B and D matrices and in-plane layer stresses  x , y , xy for in-plane force loading and edge


moment loading. Your program should be general and have the following properties:
 Read in the number of layers, layer material coefficients (E1, E2 etc.), layer thicknesses and
layer orientations  i from a data file






Should be able to calculate the elements of the A, B, D matrix
Should be able to output the elements of the A, B, D matrix in matrix format
Should be able to calculate the transverse shear stiffness coefficients A44, A45, A55 for the
laminate
Should be able to output layer stresses at a particular location through the thickness which is
specified by the user
In your code explain what you do through proper comment statements and make necessary
explanations
Verify your code by first calculating the stiffness coefficients of a laminate for which you
know the correct answer. You can use any result for a laminate given in composite books.
The material used in each layer is T300/N5208 graphite epoxy prepreg material with the
following properties:
Modulus in the fibre direction: E1 =132.38 GPa, Modulus transverse to fibre: E 2 =10.76GPa
Shear moduli: G12  G13  5.65 GPa, G23  3.38 Poisson’s ratio:  12 =0.24 ,  23 =0.49
Each layer has a thickness of 0.127 mm thickness.
Demonstrate coupling effects on the 4 layer laminate configurations for different in-plane and edge
moment load cases shown below.
In-plane loading
Load case 1: Nx=1, Ny=Nxy=Mx=My=Mxy=0
Load case 2: Ny=1, Nx=Nxy=Mx=My=Mxy=0
Load case 3: Nxy=1, Nx=Ny=Mx=My=Mxy=0
Edge moment loading
Load case 4: Mx=1, Nx=Ny=Nxy=My=Mxy=0
Load case 5: My=1, Nx=Ny=Nxy=Mx=Mxy=0
Load case 6: Mxy=1, Nx=Ny=Nxy=Mx=My=0
In-plane loading
Edge moment loading
z
z
Mxy
Ny
My
a
x
b
y
x
Mx
Nx
y
Nxy
Mxy
For the six load cases given above, make a table, as shown below, which will show the type of
coupling and magnitude of the coupling which is measured in terms of either as mid-plane strains or
curvatures. Note that you need to invert the stiffness matrices in each case. Therefore, Matlab is
recommended as the tool to use in performing the calculations. An example is given below. In the
coupling effect column, the first term corresponds load and the second term corresponds to the
coupling effect. If there is more than one coupling effect list them one after another. For each load
case, try to identify the laminate configurations with the most severe coupling effect. For instance,
[x(bottom layer)/x/x/x (top layer)] has the most severe bending-twisting coupling etc. For one of the
laminate configurations, which exhibit the most severe coupling effect, determine the in-plane stresses
at the mid-plane of the top layer.
Ex:
Load case: 4
In-plane stresses in midLaminate
configurations
Coupling effect
Magnitude
(  xo,yo,xyo, x ,y ,xy )
plane of the top layer of
one of the laminate
which exhibits most
severe coupling
0/0/0/0
None
-
Bending-twisting
 xy = xxx e-04
-
90/0/0/90
0/90/90/0
0/90/0/90
90/0/90/0
+45/-45/+45/-45
 x =aaa ,  y =bbb ,
 xy =ccc
90/45/0/-45
45/45/45/45
-45/-45/-45/-45
The aim of the first application is to familiarize you with the coupling effects. You can insert
additional laminate configurations to the tables as you wish.
Application 2: Investigation of extensional-bending coupling effect on axially loaded
beams
For the simply supported beam under axial loading shown below determine:
i) The lay-up configuration of four layered laminates which will give the most severe extensionalbending coupling effect analytically. Follow the example we solved in class and modify it as necessary
(We will complete this example after we finish the strength and failure of laminates section. In the
mean time you can prepare your FE model and perform sample runs). In other words, find the
configuration which will give the highest lateral displacement for the in plane loading shown below.
You can use the output of the first application in deciding on the laminate configuration which will
result in the most severe extensional-bending coupling. This part may require either intelligent
guessing or writing a code which will determine the laminate configuration which will maximize the
lateral deflection (for instance peak deflection –decide on this yourself). Determine the axial and
lateral displacement for the configuration which gives the most severe extensional-bending coupling.
ii) For the configuration which gives the most severe extensional-bending coupling, determine the
axial and lateral displacement field by the finite element solution. Model the beam in Patran and carry
out FE solution. Give the vertical displacement plot. Compare with the analytical solution given in
part i.
iii) For the configuration which gives the most severe extensional-bending coupling, determine the
axial stresses at x=L/2 at the mid-planes of the layers by the analytical solution and finite element
solution and compare them.
z,w
GIVEN:
h
x,u
P=80 N
L
Geometric properties:
Beam length, L: 20 cm, Thickness, h=1.5 mm, Width, b:10 mm
Material properties: High modulus graphite epoxy
Modulus in the fibre direction: E1 =207.35GPa, Modulus transverse to fibre: E 2 =5.18GPa
Shear moduli: G12  G13  G23  3.11 GPa, Poisson’s ratio:  12 =0.25
Stacking sequence of the laminate: (to be decided by you [L1/L2/L3/L4] )
Thickness of each layer: 0.375 mm
Note: Model the beam by shell elements, not beam elements
Application 3: Investigation of the bending-twisting coupling effect on swept wings
In this application a swept forward flat plate wing will be investigated in terms of coupling bending
and twisting deformation. For this purpose, the rectangular flat plate wing shown below will be
analyzed. It is assumed that wing is connected to fuselage through front and rear spar connection as
shown by the light blue numbers. Front and rear spar fuselage connections prevent three translations
displacements. In addition, front spar-fuselage connection also prevents rolling motion (y axis
rotation). Assume that flat plate wing is composed of four layers which are made of high modulus
graphite epoxy with the material properties given below.
Free stream
Rear spar
1m
Front spar
1m
3m
Forward swept flat plate wing
Material properties: High modulus graphite epoxy
Modulus in the fibre direction: E1 =207.35GPa, Modulus transverse to fibre: E 2 =5.18GPa
Shear moduli: G12  G13  G23  3.11 GPa, Poisson’s ratio:  12 =0.25
Layer thicknesses: 5 mm
Note: In practice a 3 mm long flat plate with 4 mm thickness will be very flexible. Spars and stringers
would be needed to provide rigidity. However, for the purpose of demonstrating the coupling effects
we will only model the flat plate.
Loading on the wing:
The loading acting on the wing is simplified as shown below. It is assumed that on the strip between
the leading edge and the mid-chord a uniform pressure of 100 Pa is acting, and on the strip between
the mid-chord and the trailing edge a uniform pressure of 50 Pa is acting, as shown in the figure below.
These figures are arbitrary and in a real situation realistic air loading must be used.
Free stream
Free stream
Pressure distribution acting on the wing
Objective:
The objective of the third application is to come up with a stacking sequence (for the flat plate wing
with 4 layers) which will eliminate the twist induced due to bending. For the base configuration,
define material orientation in the x direction as shown below and assign 0 degree for the layer
orientations (which means that the base configuration has layers oriented in the direction of the
material orientation vector shown below).
i)
For the base forward swept wing configuration determine the twist distribution along the
wing by calculating the twist by using the vertical deflections of the nodes along the same
chord-line on the leading and trailing edges, as shown in the first figure on the next page.
Note that in the finite element solution there will be slight effect of chord-wise bending.
To see that chord-wise bending effect is very small, draw pictures of the vertical
deflection versus chord-wise position at three different wing span locations. Draw
conclusions on the adverse effects of bending-twisting coupling on the aeroelastic
behaviour of the flat plate wing.
ii)
Determine the twist distribution for the unswept and backward swept wings. Use the same
loading for comparison purposes. Draw conclusions on the effect of bending-twisting
coupling for forward/backward swept and unswept wings. (Note that such bending
twisting coupling that you observe in the swept wings is not due to the use of composite
material. Isotropic materials would also exhibit bending-twisting coupling)
iii)
In the figures below sample results are given for two different lay-ups which show
different twist effect. In this part come up with a stacking sequence which will eliminate
the twist due to bending for the forward swept wing. You can do several trial and errors or
read the input text file (.bdf) modify the stacking sequences automatically and perform
several Nastran runs to come up with a stacking sequence which will eliminate the twist.
Or, you can use the code you have written in the first application to decide on the stacking
sequence which will eliminate the bending twisting coupling.
High bending-twisting coupling
iv)
Low bending-twisting coupling
For the stacking sequence you have determined in part iii, make two additional runs on the
unswept and backward swept wings. Comment on the effect of the lay-up used on the
bending-twisting coupling effect.
Also comment on how you would use bending-twisting coupling effect (in general) to
gain advantage for unswept and backward swept wings. Assume that you are designing a
composite airplane and you want to use the effect of bending-twisting coupling to help
you with some aspects of your design. Give one or more examples.