3. Strain Energy in Strength of Materials (SOM)

 

Strain Energy in Strength of Materials (SOM) - Civil Engineering Perspective

Strain energy is the internal potential energy stored in a material or structure due to elastic deformation when subjected to external loads. It equals the work done by the external forces during deformation (within the elastic limit). This concept is fundamental in Strength of Materials (SOM) for analyzing beams, shafts, columns, and frames in civil engineering, helping predict deflections, resilience, and failure under loads like axial, bending, shear, and torsion.

Basic Concept and Derivation

When a load is gradually applied, the work done is stored as strain energy $U$. For linear elastic behavior (Hooke's law: $\sigma = E \epsilon$):

• Strain energy density (energy per unit volume) $u = \int_0^\epsilon \sigma \, d\epsilon = \frac{1}{2} \sigma \epsilon = \frac{\sigma^2}{2E}$.
• Total strain energy $U = \int_V u \, dV = \frac{1}{2} P \delta$ (where $P$ is load, $\delta$ is deflection).

This is the area under the stress-strain curve up to the elastic limit.

Strain Energy for Different Loading Conditions

In civil engineering structures, strain energy expressions vary by load type:

1. Axial Load (tension/compression):
   $$U = \int_0^L \frac{N^2}{2AE} \, dx = \frac{P^2 L}{2AE}$$
   (For constant axial force $P$, length $L$, area $A$, modulus $E$.)
2. Bending (beams):
   $$U = \int_0^L \frac{M^2}{2EI} \, dx$$
   (Where $M$ is bending moment, $I$ is moment of inertia.)
3. Shear (transverse shear in beams):
   $$U = \int_0^L \frac{V^2}{2GA k} \, dx \quad \text{(often approximated with form factor \( k \), e.g., 1.2 for rectangular section)}$$
   ( $V$ is shear force, $G$ is shear modulus; shear energy is usually small compared to bending.)
4. Torsion (shafts):
   $$U = \int_0^L \frac{T^2}{2GJ} \, dx = \frac{T^2 L}{2GJ}$$
   ( $T$ is torque, $J$ is polar moment of inertia.)

Total strain energy in a structure is the sum of these components.

Resilience and Related Terms

• Resilience: Ability of a material to absorb and release energy elastically.
• Proof Resilience: Maximum strain energy stored up to the elastic limit (without permanent deformation).
  $$U_p = \frac{\sigma_y^2}{2E} \times V \quad (\sigma_y: \text{yield stress}, V: \text{volume})$$
• Modulus of Resilience: Proof resilience per unit volume (area under stress-strain curve up to yield).
  $$U_r = \frac{\sigma_y^2}{2E} \quad \text{(or } \frac{1}{2} \sigma_y \epsilon_y\text{)}$$
  Indicates energy absorption capacity before yielding (important for impact loads in civil structures).
• Modulus of Toughness: Energy absorbed per unit volume up to fracture (total area under stress-strain curve). Measures resistance to brittle failure.

Applications in Civil Engineering

• Castigliano's Theorem (Energy Method for Deflections):
  • Second Theorem: Deflection $\delta$ in the direction of a force $P$ is $\delta = \frac{\partial U}{\partial P}$.
  • Used for indeterminate structures (e.g., beams, trusses, frames) to find deflections/rotations without solving equilibrium equations directly.
  • Example: For a beam, $\delta = \int \frac{M}{EI} \frac{\partial M}{\partial P} \, dx$.
• Design of resilient structures (e.g., earthquake-resistant buildings absorb energy elastically).
• Failure theories (e.g., Maximum Strain Energy Theory for ductile materials).

Strain energy principles ensure safe, efficient designs in bridges, buildings, and dams by predicting deformations and energy absorption under static/dynamic loads. For complex cases, integrate over the structure or use software/tools.