90 / 0
mm., for each orientation
You are given two laminates made with this material, which are subjected to two load cases:
Laminates: a)
[
]
s
2
2
90
/
0
, b)
[
]
s
2
2
45
/
45
Load cases:
a)
Nx = 1 N/mm, every other applied (i.e. non-thermal) load and moment component is zero;
T = -150 癈
b)
Mx = 1 N*mm/mm, every other applied (i.e. non-thermal) load and moment component is zero;
T = -150 癈
Calculate the A, B, D matrices for each laminate.
I used the stress_ABD.m code by Swanson. Values for A, B and D do not depend on the applied loading, and
are given below:
a) Cross-ply laminate,
[
]
s
2
2
90
/
0
:
A matrix, Pa-m :
A =
1.0e+007 *
8.4277 0.3350 0.0000
0.3350 8.4277 0.0000
0.0000 0.0000 0.5928
B matrix, Pa-m^2 :
B =
1.0e-011 *
0.3638 0.0199 0.0000
0.0199 0.4775 0.0000
0.0000 0.0000 0.0227
So, it is confirmed that B is a zero matrix (of a much lower order of magnitude than A and D). The laminate is
symmetric, therefore B should be zero.
D matrix, Pa-m^3 :
D =
2
12.9429 0.3117 0.0000
0.3117 2.7395 0.0000
0.0000 0.0000 0.5515
b) +/- 45 deg laminate,
[
]
s
2
2
45
/
45
:
A matrix, Pa-m :
A =
1.0e+007 *
4.9741 3.7886 0.0000
3.7886 4.9741 0.0000
0.0000 0.0000 4.0464
B matrix, Pa-m^2 :
B =
1.0e-011 *
0.3638 0.2274 0
0.1819 0.3183 0.1364
0 0 0.1819
D matrix, Pa-m^3 :
D =
4.6279 3.5249 2.5509
3.5249 4.6279 2.5509
2.5509 2.5509 3.7648
Calculate
11, 22 and 12 in each layer. These results will be used later in the course.
To calculate the local stress components for each layer, it is important to have consistent units across the code.
I chose to have the material properties in Pa, the thickness of each layer in m, and the applied loads in N/m and
N m/m. Therefore, load case a) becomes Nx = 1000 N/m,
T = -150 癈, and load case b) is Mx = 1 N m/m, T
= -150 癈.
The behavior of the entire laminate is studied in the global reference system, all the layers are supposed to be
bonded together and are subject to the applied load and temperature changes. The plates behavior is affected by
the compounded stiffness of the layers. This leads to Equation 3.30 p. 84 of Swansons handout:
=
+
o
th
th
D
B
B
A
M
N
M
N
The unknown vector of this matrix equation is the strain vector
o
. The vectors on the left hand side are
known. In particular,
th
th
M
N
is known because it is calculated using the coefficients of thermal expansion
(CTEs) in the global reference system, the stiffness matrix
Q
and geometry of the layup (h
k
, h
k-1
terms)
3
(Equations 3.31 and 3.32 from Swanson). Once
o
is calculated, then it is used for the overall strain of each
layer (due to the midplane contribution and the curvature about the midplane):
=
+
=
xy
y
x
o
z
. z values
are assigned to be equal to h
k
and h
k-1
for each layer. This global strain vector is then used to calculate the local
strains (through a transformation of coordinates) and the local stresses in each ply (through the constitutive
equations for each layer).
Laminate a), load case a)
Calculation of local stresses in an AS4/3501-6 laminate, under loads (N/m and N*m/m) and temperature (deg.
C)
Nx=1000
Ny=0 Nxy=0
Mx=0 My=0 Mxy=0 DeltaT=-150
angle(deg.) sig11(Pa)
sig22(Pa)
tau12(Pa)
0
-3.91e+007
4.09e+007
2.50e-009
0
-3.91e+007
4.09e+007
2.50e-009
90
-4.09e+007
4.09e+007
-2.51e-009
90
-4.09e+007
4.09e+007
-2.51e-009
90
-4.09e+007
4.09e+007
-2.51e-009
90
-4.09e+007
4.09e+007
-2.51e-009
0
-3.91e+007
4.09e+007
2.50e-009
0
-3.91e+007
4.09e+007
2.50e-009
Laminate a), load case b)
Calculation of local stresses in an AS4/3501-6 laminate, under loads (N/m and N*m/m) and temperature (deg.
C)
Nx=0 Ny=0 Nxy=0
Mx=1 My=0 Mxy=0 DeltaT=-150
angle(deg.) sig11(Pa)
sig22(Pa)
tau12(Pa)
0
-4.69e+007
4.07e+007
2.50e-009
0
-4.54e+007
4.08e+007
2.50e-009
90
-4.05e+007
4.06e+007
-2.48e-009
90
-4.07e+007
4.07e+007
-2.49e-009
90
-4.10e+007
4.09e+007
-2.51e-009
90
-4.11e+007
4.10e+007
-2.52e-009
0
-3.63e+007
4.09e+007
2.50e-009
0
-3.47e+007
4.09e+007
2.50e-009
Laminate b), load case a)
Calculation of local stresses in an AS4/3501-6 laminate, under loads (N/m and N*m/m) and temperature (deg.
C)
Nx=1000
Ny=0 Nxy=0
Mx=0 My=0 Mxy=0 DeltaT=-150
4
angle(deg.) sig11(Pa)
sig22(Pa)
tau12(Pa)
45
-4.00e+007
4.09e+007
-4.73e+005
45
-4.00e+007
4.09e+007
-4.73e+005
-45
-4.00e+007
4.09e+007
4.73e+005
-45
-4.00e+007
4.09e+007
4.73e+005
-45
-4.00e+007
4.09e+007
4.73e+005
-45
-4.00e+007
4.09e+007
4.73e+005
45
-4.00e+007
4.09e+007
-4.73e+005
45
-4.00e+007
4.09e+007
-4.73e+005
Laminate b), load case b)
Calculation of local stresses in an AS4/3501-6 laminate, under loads (N/m and N*m/m) and temperature (deg.
C)
Nx=0 Ny=0 Nxy=0
Mx=1 My=0 Mxy=0 DeltaT=-150
angle(deg.) sig11(Pa)
sig22(Pa)
tau12(Pa)
45
-4.38e+007
3.98e+007
2.69e+006
45
-4.31e+007
4.00e+007
2.02e+006
-45
-4.79e+007
4.06e+007
-1.34e+006
-45
-4.44e+007
4.07e+007
-6.72e+005
-45
-3.73e+007
4.09e+007
6.72e+005
-45
-3.38e+007
4.11e+007
1.34e+006
45
-3.86e+007
4.16e+007
-2.02e+006
45
-3.78e+007
4.19e+007
-2.69e+006
Calculate the stresses in the (x, y) reference for each layer of the laminates given below.
The calculation of the global stresses for each layer k is possible through 1) using Equation (3.27) p. 84 of
Swanson, or 2) converting the local stresses (result from previous part) in global stresses through
k
k
k
xy
yy
xx
T
=
12
22
11
1
, where T is the transformation matrix (also called T
1
in the textbook) for each layer k.
Note that global stresses and local stresses are equal to each other only for the 0 deg. layers of the cross-ply
laminate.
Laminate a), load case a)
Calculation of global stresses in an AS4/3501-6 laminate, under loads (N/m and N*m/m) and temperature (deg.
C)
Nx=1000
Ny=0 Nxy=0
Mx=0 My=0 Mxy=0 DeltaT=-150
angle(deg.) sigx(Pa)
sigy(Pa)
tauxy(Pa)
0
-3.91e+007
4.09e+007
2.50e-009
0
-3.91e+007
4.09e+007
2.50e-009
90
4.09e+007
-4.09e+007
-2.50e-009
5
90
4.09e+007
-4.09e+007
-2.50e-009
90
4.09e+007
-4.09e+007
-2.50e-009
90
4.09e+007
-4.09e+007
-2.50e-009
0
-3.91e+007
4.09e+007
2.50e-009
0
-3.91e+007
4.09e+007
2.50e-009
Laminate a), case b)
Calculation of global stresses in an AS4/3501-6 laminate, under loads (N/m and N*m/m) and temperature (deg.
C)
Nx=0 Ny=0 Nxy=0
Mx=1 My=0 Mxy=0 DeltaT=-150
angle(deg.) sigx(Pa)
sigy(Pa)
tauxy(Pa)
0
-4.69e+007
4.07e+007
2.50e-009
0
-4.54e+007
4.08e+007
2.50e-009
90
4.06e+007
-4.05e+007
-2.49e-009
90
4.07e+007
-4.07e+007
-2.49e-009
90
4.09e+007
-4.10e+007
-2.51e-009
90
4.10e+007
-4.11e+007
-2.51e-009
0
-3.63e+007
4.09e+007
2.50e-009
0
-3.47e+007
4.09e+007
2.50e-009
Laminate b), load case a)
Calculation of global stresses in an AS4/3501-6 laminate, under loads (N/m and N*m/m) and temperature (deg.
C)
Nx=1000
Ny=0 Nxy=0
Mx=0 My=0 Mxy=0 DeltaT=-150
angle(deg.) sigx(Pa)
sigy(Pa)
tauxy(Pa)
45
9.46e+005
3.57e-008
-4.04e+007
45
9.46e+005
3.54e-008
-4.04e+007
-45
9.46e+005
3.17e-008
4.04e+007
-45
9.46e+005
3.14e-008
4.04e+007
-45
9.46e+005
3.14e-008
4.04e+007
-45
9.46e+005
3.49e-008
4.04e+007
45
9.46e+005
4.23e-008
-4.04e+007
45
9.46e+005
4.58e-008
-4.04e+007
Laminate b), load case b)
Calculation of global stresses in an AS4/3501-6 laminate, under loads (N/m and N*m/m) and temperature (deg.
C)
Nx=0 Ny=0 Nxy=0
Mx=1 My=0 Mxy=0 DeltaT=-150
angle(deg.) sigx(Pa)
sigy(Pa)
tauxy(Pa)
45
-4.71e+006
6.59e+005
-4.18e+007
45
-3.54e+006
4.94e+005
-4.16e+007
-45
-4.99e+006
-2.31e+006
4.42e+007
-45
-2.50e+006
-1.15e+006
4.25e+007
-45
2.50e+006
1.15e+006
3.91e+007
6
-45
4.99e+006
2.31e+006
3.74e+007
45
3.54e+006
-4.94e+005
-4.01e+007
45
4.71e+006
-6.59e+005
-3.99e+007
Plot or draw
x, y and xy for each layer as a function of the thickness (see example below).
Normalize the stress component with respect to the max value for each stress in the laminate: for
example, when plotting
x, see what is the max value of the stresses x for all layers, and use that as
xmax. Divide the x for each layer by xmax when plotting, so that the x-axis varies between -1 a