Strength of Materials- Quick Review

Strength of Materials Quick review  Babak Jamshidi

Stress and Strain Strain:

Stress:

Intensity of force per unit area.

Mathematically,

Elastically

σ  = lim

σ  =

∆ A→0

 P   A

 P 

 A dimensionless quantity which relates to deformation of a physical body under the action of applied forces.

∆ A Elastically

and

ε  =

∆ L  L

σ  = E ε 

σ: stress

ε: strain

P: Applied force

E: Module of Elasticity

 A: Area

L: Initial Length  ∆L: Deformed length

Bending Beam Theory  dθ ρ

Beam Axis x

y

Undeformed

dx

Deformed

Fundamental assumption: Plane sections through a beam taken normal to its axis remain plane after the beam is subjected to bending.

ds =  ρ dθ    d θ  ds

=

1  ρ 

= κ  (curvature)

σ  x =  E ε  x = − E κ  y

Bending Beam Theory  ∫

 M  z  =  E κ   y 2 dA

From Equilibrium Equations

 A

∫

 I  z  =  y 2 dA

Second Moment of the Area

 A

- For rectangle: z

 I  z  =

h

bh 3 12

b

σ  x = −

 M  z   I  z 

 y

therefore;

σ max =

 Mc  I 

c =  y max

Beam Analysis Notes •

To derive the deflection, internal forces the following information

should be provided: 1. Applied Forces and/or displacement 2. Boundary Conditions 3. Material Properties •

Forces drawn in a free body diagram should be in equilibrium.

 M  z −left 

 F  y −left 

 

 F  y − right 

 M  z − right   F  x − right 

 F  x −left  General Free body diagram of a beam

Beam Analysis Notes •

Beam deflection

1  ρ 

= κ  = −

d 2 y dx

2

=

 M   EI 

ε 

1

 y

 ρ 

=

 M   EI 

From Bending Beam Theory

Governing Equation

Typical Approach: - Calculate the moment equation as a function of length (x) - Plug in the governing equation M, E and I as a function of length. - Integrate twice - Apply the boundary conditions

References For further study the following references are recommended: 1. Egor P. Popov,”Engineering Mechanics of Solids”, Prentice Hall 2. Stephen Timoshenko, “Theory of Elasticity”, McGraw Hill 3. Any basic structural or mechanical analysis and design book …