Key Assumptions in Beam Bending Theory

• The material of the beam is homogeneous and isotropic.
• The beam has a constant cross-sectional shape along its length.
• Plane sections remain plane and perpendicular to the neutral axis after bending.
• The beam obeys Hooke's Law (linear elastic behavior).

Types of Beams

• Simply Supported Beam: Supported at both ends, free to rotate.
• Cantilever Beam: Fixed at one end, free at the other.
• Overhanging Beam: Extends beyond its support on one or both sides.
• Fixed Beam: Both ends are fixed, resisting rotation.
• Continuous Beam: Spans over multiple supports.

Neutral Axis and Moment of Inertia

The neutral axis is the line within the cross-section of a beam where the fiber experiences zero stress. The moment of inertia (I) quantifies the beam's resistance to bending and depends on the cross-sectional geometry.

Governing Equations of Beam Bending

Bending Stress (σ):

The bending stress is given by:

σ = (M * y) / I

Flexure Formula:

The flexure formula is:

M / I = σ / y = E / ρ

• M: Bending moment (Nm)
• y: Distance from the neutral axis (m)
• I: Moment of inertia (m⁴)
• E: Modulus of elasticity (Young's Modulus) (Pa)
• ρ: Radius of curvature (m)

Beam Deflection (v):

The deflection of a beam is calculated using:

d²v/dx² = M / EI

Relationship Between Load, Shear, and Moment:

• V = dM/dx
• q = dV/dx

Worked Example

Problem:

A simply supported beam of length L = 6 m carries a uniformly distributed load q = 2 kN/m. Calculate:

1. Maximum bending moment.
2. Maximum deflection.

Solution:

Step 1: Reactions at Supports

Using equilibrium:

R_A + R_B = qL

R_A = R_B = (qL) / 2 = 6 kN

Step 2: Bending Moment

The bending moment at any point x from the left support is:

M(x) = R_Ax - (qx²) / 2

At the midpoint (x = L/2):

M_max = 9 kNm

Step 3: Deflection

The deflection equation is:

v_max = (5qL⁴) / (384EI)

Substituting values:

v_max = 33.6 mm

Applications of Beam Bending

• Structural Engineering: Design of beams in bridges, buildings, and other civil structures.
• Mechanical Engineering: Shafts, levers, and machine components.
• Aerospace Engineering: Wing and fuselage design.
• Automotive Engineering: Chassis and suspension systems.

Advanced Topics in Beam Bending

• Nonlinear Beam Theory
• Composite Beams
• Beam Buckling
• Dynamic Bending
• Thermal Stresses in Bending
• Timoshenko Beam Theory
• Elastic Foundation Beams
• Plastic Bending

References

1. Gere, J. M., & Goodno, B. J. (2012). Mechanics of Materials. Cengage Learning.
2. Hibbeler, R. C. (2020). Mechanics of Materials. Pearson Education.
3. Timoshenko, S., & Gere, J. M. (1961). Theory of Elastic Stability. McGraw-Hill.
4. Beer, F. P., Johnston, E. R., & DeWolf, J. T. (2017). Mechanics of Materials. McGraw-Hill.

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