Rigid Body vs. Deformable Body
Rigid Body

Retains its shape/form when a load is applied (external or internal).

Example: Step on a stone; it keeps its shape.

Idealization: In Statics, bodies are assumed rigid for simplicity.
Deformable Body

Changes shape/form when a load is applied.

Example: Press modeling clay; it deforms.

Reality: All bodies deform under load. Even a stone will break under a heavy enough force.
➡️ In this subject, you're dealing with Deformable Bodies, not rigid ones.
Forces That Cause Deformation
1. Normal Force (P)

Acts perpendicular to the cross-sectional area.

Example: Force applied along the axis of a cylinder.
2. Shear Force (V)

Acts parallel to the cross-sectional area.

Tends to cause sliding between two adjacent sections.
3. Torsion / Twisting Moment (T)

A moment causing twisting around the longitudinal axis.

Example: Wrench twisting a bolt.
4. Bending Moment (M)

Causes the body (beam/cylinder) to bend.

Creates tension and compression on opposite sides.
Centroidal (Axial) Loading

Load passes through the centroid of the cross-section.

Applies to bars: 3D objects with consistent cross-sectional shape.

Uniform loading → Resultant force passes through the centroid.
Normal Stress (σ)
Formula:
σ=PAσ = \dfrac{P}{A}

σ (sigma): Normal stress

P: Axial force

A: Cross-sectional area perpendicular to P
Units of Normal Stress
Quantity
SI Units
English Units
Normal Stress (σ) Pa, kPa, MPa psi, ksi
Normal Force (P)
N, kN
lbs, kips
Area (A)
m², mm²
in², ft²
➡️ Conversion:
1 N/mm2=1 MPa1 \text{ N/mm}^2 = 1 \text{ MPa}
Steps in Solving Normal Force Problems
1. Equilibrium Analysis

Draw the Free-Body Diagram (FBD).

Determine all external reactions.

Use Method of Sections:
o
Make an imaginary cut.
o
Isolate and analyze internal forces.
2. Computation of Stress

After finding P, apply
σ=PAσ = \dfrac{P}{A}

Calculate the average normal stress.
Actual Stress vs. Allowable Stress
Type
Description
Actual Stress
Result from external loads and equilibrium.
Allowable Stress Based on material properties (working/design).
➡️ Design Rule:
σactual≤σallowableσ_{actual} \le σ_{allowable}
Example (Structural Steel):

Allowable stress ≈ 36,000 psi
Shear Stress (τ)
Formula:
τ=VAτ = \dfrac{V}{A}

V: Shear force

A: Area parallel to V
Types of Shear
1. Single Shear
o
Failure through one plane
o
Example: Lap joint with a single bolt
2. Double Shear
o
Failure through two planes
o
Example: Gusset plate connections
3. Punching Shear
o
Failure around a perimeter
o
Example: Bolt hole tearing through a plate
Bearing Stress (σb)
Occurs When:
Two bodies are in contact (compressive).
Examples:

Pressure between rivet and hole

Soil pressure under a footing
Formula:
𝝈𝒃 =
𝒑
𝒕𝒅

P: Bearing load (same as V in shear)

A_b: Projected area = thickness (t) × diameter (d)
Thin-Walled Pressure Vessels
Types:

Cylindrical

Spherical
➡️ Thin-walled condition:
rt≥10\dfrac{r}{t} \ge 10
Types of Stress:
1. Tangential (Hoop) Stress (σt)
o
Acts around the circumference
2. Longitudinal Stress (σl)
o
Acts along the length
Summary Formulas
Stress Type
Normal Stress
Shear Stress
Formula
𝝈=
𝑷
𝑨
𝝉=
𝑽
𝑨
Bearing Stress
𝝈𝒃 =
𝑷
𝒕𝒅
STRAIN
Definition:
Strain measures deformation of a body under force.
Types of Strain
1. Normal Strain

Change in length/dimensions

Elongation (tension) or Shortening (compression)
2. Shear Strain

Change in angles

Example: A square distorts into a rhombus under shear force.
Relationship Between Stress and Strain

Stress causes strain.

They exist together.
Average Axial Strain Formula
𝝐=

ε: Strain

δ: Deformation (change in length)

L: Original length
Conditions for Using Axial Strain Formula
1. Constant Cross-Section (e.g., cylinders)
𝜹
𝑳
2. Homogeneous Material (uniform composition)
3. Axial Load passes through the centroidal axis
Stress-Strain Curve (Tension Test)
Test Machine: Universal Testing Machine (UTM)
Graph: Stress vs. Strain Curve
Regions of the Stress-Strain Curve
1. Proportional Region (Hooke’s Law) 𝝈 =
o
𝑬
𝝐𝝈
Valid up to Proportional Limit
2. Elastic Region
o
Returns to original shape if unloaded before Elastic Limit
3. Yield Region
o
Deforms significantly at Yield Stress
4. Ultimate Strength
o
Maximum stress the material can withstand
5. Rupture (Failure Point)
o
Material fails (necking and fracture)
Types of Rupture Strength Calculations
Type
Nominal Rupture
True Rupture
Formula
𝝈𝒏𝒐𝒎𝒊𝒏𝒂𝒍 =
𝝈𝒕𝒓𝒖𝒆 =
𝑭𝑨𝑰𝑳𝑼𝑹𝑬 𝑳𝑶𝑨𝑫
𝑶𝑹𝑰𝑮𝑰𝑵𝑨𝑳 𝑨𝑹𝑬𝑨
𝑭𝑨𝑰𝑳𝑼𝑹𝑬 𝑳𝑶𝑨𝑫
𝑹𝑬𝑫𝑼𝑪𝑬𝑫 𝑨𝑹𝑬𝑨
➡️ True rupture strength is higher because the area is smaller at necking.
Allowable / Working Stress

Steel: Based on Yield Stress

Other materials: Based on Ultimate Strength
Formulas for Deformation (δ)
If ALL 3 Conditions Are Met (Axial Load, Constant Section, Homogeneous Material):
𝜹=

P: Axial Load

L: Length

A: Cross-sectional Area

E: Modulus of Elasticity
𝑷𝑳
𝑨𝑬
Key Takeaways

Strain measures deformation

Stress and strain occur together

Use simplified formulas only if all conditions are met

The Stress-Strain Curve shows how materials behave:
Elastic → Yield → Plastic → Rupture
Deformation Analyses
1. The magnitude of internal force P is determined from equilibrium analysis.
2. A positive force (tension) causes elongation; a negative force (compression) causes shortening.
3. Consistent units are critical. The modulus of elasticity, E, typically uses GPa in the SI system.
Statically Indeterminate Members

These members cannot be analyzed by equilibrium alone.

Additional equations are required—called compatibility equations, which come from strain
equations.
Equations to Use
1. Equilibrium Equations
2. Deformation Equations (Compatibility)
Steps
1. Draw the Free Body Diagram (FBD).
2. Derive the compatibility equations.
3. Use Hooke’s Law to express strain in terms of force.
4. Solve all equations simultaneously.
Thermal Stresses
Thermal Strain Formula
ΔL = α * ΔT * L
Where:

ΔL = Change in length

α = Coefficient of thermal expansion

ΔT = Temperature change (Final - Initial)

L = Original length
➡️ Thermal stresses arise when deformation is restrained.
Torsion of Circular Shafts
Assumptions of Deformation
1. Circular sections remain circular; plane sections remain plane and perpendicular to the axis.
2. No deformation occurs in the plane of the cross-section.
3. Distances between cross-sections do not change (no axial strain).
4. Radial lines stay straight in projection.
5. Shaft is loaded by twisting couples in planes perpendicular to its axis.
6. Stresses do not exceed the proportional limit.
Practical Notes

Two equal and opposite forces create torsion.

Circular shafts stay circular (no warping or distortion).

No elongation or shortening occurs.
Angle of Twist (θ)
Formula:
𝜃=
𝑇𝐿
𝐽𝐺
𝝉=
𝑻𝒓
𝑱
Where:

θ = Angle of twist (radians)

T = Torque

L = Length

J = Polar moment of inertia

G = Modulus of rigidity
➡️ Similar to PL/AE, but for torsion.
Shear Stress in Torsion
Formula:
Where:

τ = Shear stress

T = Torque

ρ = Radial distance from center

J = Polar moment of inertia
➡️ Max shear stress (τmax) occurs at ρ = r (outer radius).
Polar Moment of Inertia (J)
Solid Shaft
𝑱=(
𝝅𝒓𝟒
𝝅𝑫𝟒
)
)=(
𝟐
𝟑𝟐
Hollow Shaft
𝝅
𝝅
𝑱 = ( (𝑹𝟒 − 𝒓𝟒 ) = ( (𝑫𝟒 − 𝒅𝟒 )
𝟐
𝟑𝟐

D = Outer diameter

d = Inner diameter
➡️ Solid shafts have no hole, hollow shafts have an inner hole.
Key Takeaways

Use consistent units.

Hooke’s Law applies within elastic limits.

Know your basic torsion formulas: 𝜃 = 𝐽𝐺 & 𝜏 = 𝑇𝑟/𝑗

Identify solid vs hollow shafts for J.

Thermal stress happens when expansion/contraction is restrained.
𝑇𝐿
Power Transmission

Power (P) is directly proportional to Torque (T).
➡️ As torque increases, power increases.
Key Relationships
1. P = ω * T
o
ω (omega) = angular speed (radians per second)
2. ω = 2π * f
o
f = frequency (Hertz or revolutions per second)
3. Substituting ω into the first equation:
P = 2π * f * T
Rearranging for Torque (T):
T = P / (2π * f)
Variables:

P = Power (Watts or Joules/second)

T = Torque (Newton-meters)

ω = Angular speed (radians/second)

f = Frequency (Hertz or revolutions/second)
Application:

These equations are used to design shafts that transmit power under specified speed and
torque conditions.