Given• We will calculate the deflection of a simply supported
beam of PLA with dimensions of 6 inch width, 1/8 inch thick, 4 feet long. The distributed load of 0.25 pounds will be applied to the beam. We will do the calculation with multiple reinforcing fibers at 3 different fiber loadings.
Reinforcing materials– Carbon Nanotubes (CNT)– Carbon Fiber– Glass Fiber– No Reinforcement
Loadings– .5%– 5%– 20%
Moduli
• Pure PLA: 3.3 Gpa• Glass Fiber: 70 Gpa• Carbon Fiber: 400 Gpa• CNT: 1000 Gpa
Note: Gpa= Gigapascal= 1billion Pascals
Fiber Fraction Calculation
Calculation for .5% fibers– νf= .005– ν= 1
• So the proportion at:– .5% is .005– 5% is .05– 20% is .2
v
v ff
005.1
005.f
Composite Modulus
• E1= Composite Modulus
• ϕf = Proportion of fiber volume to overall volume• Ef = Modulus of fibers• Em= Modulus of matrix (polymer)
mfff EEE *)1(*1
Formula to calculate the modulus with perfect adhesion at the interface
Composite Modulus Calculation
Glass Fiber at .5% loading• ϕf = .005• Ef = 70 GPa
• Em= 3.3 GPa
mfff EEE *)1(*1
GPaE
GPaGPaE
6335.3
3.3*)005.1(70*005.
1
1
Beam Calculations
• Now that we have all the moduli for the different reinforcements and loadings we must calculate the moment of inertia of the beam in order to make the deflection calculations
Deflection
• ymax= max deflection• w = load• L = unsupported length• E = modulus• I = moment of inertia
4ft
.125in
Force: 0.25lbs distributed
IE
Lwy
**384
**5 4
max
iny
ininlb
ininlb
y
IE
Lwy
inLin
lbinlbsw
inI
psiE
833.6
0001.0*996,526*384
)48(*00521.0*5
**384
**5
48
00521.048/25.0
0001.0
996,526
max
42
4
max
4
max
4
1
Glass Fiber at .5%