1. Simple Stress and Strain in Civil Engineering

Simple Stress and Strain in Civil Engineering

Stress and strain are fundamental concepts in civil engineering, especially in Strength of Materials and Structural Analysis. They help engineers understand how materials behave under loads (forces) and ensure structures like beams, columns, bridges, and buildings are safe and do not fail.

1. What is Stress?

Stress (σ) is the internal resistance offered by a material when an external force is applied. It is the force per unit area.

Formula:

$$\sigma = \frac{P}{A}$$

Where:

• σ (sigma) = Stress (in N/mm², MPa, or kN/m²)
• P = Applied force/load (in Newtons or kN)
• A = Cross-sectional area perpendicular to the force (in mm² or m²)

Types of Simple Stress:

• Tensile Stress: When the force pulls the material apart (elongation).
  • Example: A steel rod in a truss bridge under tension.
• Compressive Stress: When the force pushes the material together (shortening).
  • Example: A concrete column supporting weight.
• Shear Stress: When forces tend to slide one part over another (not simple axial, but basic in rivets/bolts).

Units:

• Common: N/mm² (MPa) or kN/cm²
• 1 MPa = 1 N/mm²

2. What is Strain?

Strain (ε) is the deformation (change in shape or size) produced in the material due to stress. It is the change in length per unit original length.

Formula:

$$\varepsilon = \frac{\Delta L}{L}$$

Where:

• ε (epsilon) = Strain (dimensionless – no units)
• ΔL = Change in length (elongation or shortening)
• L = Original length

Types of Strain:

• Tensile Strain: Positive (elongation)
• Compressive Strain: Negative (shortening)
• Shear Strain: Angular distortion

Strain is usually very small (e.g., 0.001 or 0.1%).

3. Stress-Strain Relationship (Hooke's Law)

Within the elastic limit, stress is directly proportional to strain.

Hooke's Law:

$$\sigma = E \times \varepsilon$$

or

$$E = \frac{\sigma}{\varepsilon}$$

Where:

• E = Modulus of Elasticity (Young's Modulus) – measures stiffness of material
• Unit of E: Same as stress (N/mm² or GPa)

Meaning: Stiffer materials (high E) deform less under same stress.

Common Young's Modulus Values (approximate):

Material	Young's Modulus (E) in GPa
Steel	200–210
Concrete	20–40
Aluminium	70
Wood (along grain)	10–15

4. Stress-Strain Curve (Typical for Mild Steel)

The curve shows how material behaves under increasing tensile load:

1. Proportional Limit: Stress ∝ Strain (straight line – Hooke's Law applies)
2. Elastic Limit: Beyond this, permanent deformation starts
3. Yield Point: Material starts to deform plastically (large strain with little stress increase)
   • Upper and lower yield points in mild steel
4. Ultimate Stress Point: Maximum stress the material can withstand
5. Breaking Point: Material fractures

Key Points on Curve:

• Up to elastic limit: Material returns to original shape when load removed (elastic behavior)
• Beyond yield point: Permanent deformation (plastic behavior)

5. Important Terms

• Factor of Safety (FOS):
  $$FOS = \frac{\text{Ultimate Stress}}{\text{Allowable (Working) Stress}}$$
  Typical FOS: 1.5–3 for steel, higher for concrete.
• Allowable Stress = Ultimate Stress / FOS Used in design to keep structures safe.

6. Simple Example

A steel rod of length 2 m and cross-sectional area 100 mm² is subjected to a tensile force of 50 kN.

Calculate: a) Stress b) Strain (if E = 200 GPa) c) Elongation

Solution:

a) Stress:

$$\sigma = \frac{P}{A} = \frac{50,000}{100} = 500 \, \text{N/mm²} = 500 \, \text{MPa}$$

b) Strain:

$$\varepsilon = \frac{\sigma}{E} = \frac{500}{200,000} = 0.0025$$

c) Elongation (ΔL):

$$\Delta L = \varepsilon \times L = 0.0025 \times 2000 = 5 \, \text{mm}$$

The rod elongates by 5 mm.

Summary

• Stress: Force/Area → measures intensity of load
• Strain: Deformation/Original length → measures deformation
• Within elastic limit: σ = Eε (linear relationship)
• Engineers design so that stresses remain below allowable limits to prevent failure.

These concepts are the foundation for analyzing beams, columns, and all structural elements in civil engineering.