6Units and Dimensions
Examples of Dimensionless Units- Specific Gravity and Strain
Parameters that are Dependent on Physical Dimensions- Area moment of inertia, Euler buckling load
Stress and Strain
Stress-Strain Diagrams
A- Least plastic deformation
B- Greatest yield strength
C- Greatest Young’s
Modulus, most necking
Example
A foam cushion is compressed from 6cm to 5.1cm. The area is 0.1m2. The foam has an ultimate tensile strength of 15kPa and a Young’s
Modulus of 20.8kPa.
Δl Calculate the strain in the cushion. Use the correct sign.
ϵ= (5.1 cm−6 cm)
l0 =−0.15=−15 %
change ∈length 6 cm
strain=
original length
σ =E ϵ Calculate the stress in the cushion. Any sign is acceptable.
stress=Young ' s Modulus × strain 20.8 ×0.15=−3.1 kPa
F Calculate the weight of the force on the cushion.
σ= −3.1 ×0.1=−0.31 kN=−310 N
A
Force
stress=
Area
Young’s Modulus(E)- A higher Young’s Modulus means the material is not as stretchy or durable, no matter what the ultimate tensile strength
is.
Area Moment of Inertia Equations
Euler Buckling Load Equation
2
Π EI K=2 (one fixed end, one free end) K=1 (both ends pinned) K = constant depends on how specimen is held
Pcr = 2 K=0.7 (one fixed end, one pinned end) I = area moment of inertia L = length of the specimen
( KL) K=0.5 (both ends fixed) E = Young’s modulus
Ultimate Tensile Strength Equation
2 yrE σmax = ultimate tensile strength y = deflection length r = radius of specimen E = Young’s modulus
σ max= 2 L = length
L 2
( ) +y
2
Bending Load Equation
48 yIE Pb = critical bending load y = deflection length I = area moment of inertia L = length
P b= 3 E = Young’s modulus
L
Adding and Subtracting Forces
, R A y =Asinθ R x =Acos θ A + Bcos θB
B
A x = Acos θ R y =Asin θ A❑ + Bsin θB
A B B y =Bsin θ R=√ ❑
A
y A
x
B
x
y Bx =Bcos θ
( )
θ R=tan −1 y
R
Rx
Truss Analysis
Supports- Fixed support has no movement, pinned support has rotation, and rolling support has translation.
Moments
M =F × d
Moment =Force× distance F 3 2 F M 1−M 2=0
( )
1 6 0 2 d
= = F 2=F1 1 =153lb
8 x d2
5
Truss Analysis
Electronics
AND Truth Table OR Truth Table NOT Truth Table NAND Truth Table NOR Truth Table
A B Y A B Y A B A B Y A B Y
0 0 0 0 0 0 1 0 0 0 1 0 0 1
0 1 0 0 1 1 0 1 0 1 1 0 1 0
1 0 0 1 0 1 1 0 1 1 0 0
1 1 1 1 1 1 1 1 0 1 1 0
NAND to NOT NAND to OR NAND TO AND
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