4.3 Theory of Flexure and Columns

4.3 Theory of Flexure and Columns

Introduction to Structural Deformation

• When subjected to transverse loads, slender members primarily experience bending deformation. This flexural behavior is governed by the relationship between applied moments, internal stresses, and resulting curvatures and deflections.

• Simultaneously, members subjected to axial compressive loads are susceptible to buckling, a failure mode characterized by sudden lateral deflection.

• This section synthesizes the fundamental theories for analyzing bending stress, beam deflection, and the stability of columns, forming the cornerstone of structural design for beams and compression members.

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1. Flexure: Co-planar and Pure Bending

1.1 Co-planar Bending

• Definition: Bending that occurs in a single plane when transverse loads are applied perpendicular to the longitudinal axis of the member and act within a plane containing one of the principal axes of the cross-section.

• Condition: The plane of loading must coincide with a plane of symmetry of the cross-section, or the member must be constrained to bend only in that plane. This ensures bending occurs without twisting.

• Result: The beam deflects in the same plane as the applied loads.

1.2 Pure Bending

• Definition: A special case of bending where a beam segment is subjected to a constant bending moment and zero shear force.

• Condition: Achieved by applying equal and opposite couples at the ends of a beam segment. No transverse loads act on the segment.

  $\boldsymbol{V = 0 \quad \text{and} \quad M = \text{constant}}$

• Significance: Under pure bending, the beam deforms into a circular arc, which simplifies the derivation of the fundamental flexure formula. It represents the most elementary stress state for developing bending theory.

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2. Kinematics of Bending: Deformation Geometry

• To derive stress relationships, we first analyze the geometry of deformation under the assumption of plane sections remaining plane.

2.1 Elastic Curve and Angle of Rotation

• Elastic Curve: The deflected shape of the longitudinal neutral axis of the beam under load.

• Angle of Rotation (Slope - $\boldsymbol{\theta}$): The angle, in radians, through which the cross-section rotates relative to its original position. For small deflections, it is approximately equal to the first derivative (slope) of the deflection curve.

  $\boldsymbol{\theta(x) \approx \tan \theta = \frac{dy}{dx}}$

  where, $\boldsymbol{y(x)}$ is the transverse deflection and $\boldsymbol{\theta(x)}$ is the slope/rotation.

2.2 Radius of Curvature and Its Relationship

• Radius of Curvature ($\boldsymbol{\rho}$): The radius of the osculating circle that best fits the elastic curve at a given point. It quantifies how sharply the beam is bent.

• Differential Relationship: For the elastic curve $\boldsymbol{y(x)}$, the exact expression for curvature is:

  $\boldsymbol{\kappa = \frac{1}{\rho} = \frac{\frac{d^2y}{dx^2}}{\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2}}}$

• Small Deflection Simplification: When slopes are very small ($\boldsymbol{(dy/dx)^2 \ll 1}$), the relationship simplifies to a fundamental equation linking geometry and moment:

  $\boldsymbol{\frac{1}{\rho} \approx \frac{d^2y}{dx^2}}$

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3. Flexural Stiffness and Constitutive Relation

• Combining deformation geometry with material behavior yields the governing equation for beam bending.

3.1 Flexural Stiffness (EI)

• Definition: The product of the material's Modulus of Elasticity (E) and the cross-section's Second Moment of Area (I).

• Physical Interpretation: A measure of a beam's resistance to bending. A higher $\boldsymbol{EI}$ value means the beam is stiffer and will deflect less under the same applied moment.

  $\boldsymbol{EI}$

• Second Moment of Area (I): A geometric property that depends solely on the shape and dimensions of the cross-section about the neutral axis. It quantifies the distribution of area relative to the bending axis.

  • For a rectangle of width $\boldsymbol{b}$ and depth $\boldsymbol{d}$: $\boldsymbol{I = \frac{b d^3}{12}}$

  • For a solid circle of diameter $\boldsymbol{D}$: $\boldsymbol{I = \frac{\pi D^4}{64}}$

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4. Stress and Deflection Analysis

4.1 Bending Stress (Flexure Formula)

• Core Formula: Derived from equilibrium, compatibility, and Hooke's Law, this formula gives the normal stress due to bending at any point in the cross-section.

  $\boldsymbol{\sigma = -\frac{M y}{I}}$

  where, $\boldsymbol{\sigma}$ is the bending stress, $\boldsymbol{M}$ is the bending moment at the section, $\boldsymbol{y}$ is the distance from the neutral axis (positive in the direction of beam curvature), and $\boldsymbol{I}$ is the second moment of area about the neutral axis.

• Sign Convention: A negative sign is often included so that a positive moment (sagging) produces compressive stress (negative) at positive $\boldsymbol{y}$ (top fibers) and tensile stress (positive) at negative $\boldsymbol{y}$ (bottom fibers).

• Maximum Bending Stress: Occurs at the fibers farthest from the neutral axis ($\boldsymbol{y = y_{max} = c}$).

  $\boldsymbol{\sigma_{max} = \frac{M c}{I} = \frac{M}{S}}$

  where, $\boldsymbol{S = I/c}$ is the section modulus, a direct geometric measure of bending strength.

4.2 Beam Deflection Equation

• Governing Differential Equation: The fundamental equation that relates deflection to applied moment, stiffness, and load.

  $\boldsymbol{EI \frac{d^2y}{dx^2} = M(x)}$

  This is the most commonly used form. Higher-order derivatives relate to shear and load:

  $\boldsymbol{EI \frac{d^3y}{dx^3} = V(x) \quad \text{and} \quad EI \frac{d^4y}{dx^4} = w(x)}$

• Solution Methods: Deflection $\boldsymbol{y(x)}$ is found by integrating these equations, applying appropriate boundary conditions (e.g., deflection and slope at supports) and continuity conditions.

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5. Column Buckling: Euler's Theory

5.1 Concept of Buckling

• Definition: Buckling is a stability failure mode for slender compression members (columns). It is characterized by a sudden, large lateral deflection under an axial compressive load, often occurring at stresses well below the material's yield strength.

• Critical Load ($\boldsymbol{P_{cr}}$): The maximum axial load a column can carry before it becomes unstable and buckles. It is a function of geometry, material stiffness, and end support conditions.

5.2 Euler's Formula for Long Columns

• Assumptions:

  1. Column is perfectly straight, homogeneous, and isotropic.

  2. Load is perfectly axial (no eccentricity).

  3. Material obeys Hooke's Law (elastic behavior).

  4. Deflections are small.

  5. Cross-section is uniform.

• General Formula: The critical buckling load is given by:

  $\boldsymbol{P_{cr} = \frac{\pi^2 E I_{min}}{(K L)^2}}$

  where,

  • $\boldsymbol{E}$ is the Modulus of Elasticity.

  • $\boldsymbol{I_{min}}$ is the minimum second moment of area (buckling occurs about the axis with least stiffness).

  • $\boldsymbol{L}$ is the actual unsupported length of the column.

  • $\boldsymbol{K}$ is the effective length factor.

• Effective Length ($\boldsymbol{K L}$): The length of an equivalent pinned-pinned column that buckles at the same load. It accounts for end restraint conditions.

  • Pinned-Pinned: $\boldsymbol{K = 1.0}$

  • Fixed-Fixed: $\boldsymbol{K = 0.5}$

  • Fixed-Pinned: $\boldsymbol{K \approx 0.7}$

  • Fixed-Free: $\boldsymbol{K = 2.0}$

• Slenderness Ratio: A key non-dimensional parameter governing buckling.

  $\boldsymbol{\lambda = \frac{K L}{r}}$

  where, $\boldsymbol{r = \sqrt{I/A}}$ is the radius of gyration. Euler's formula is valid for columns with a slenderness ratio above a certain threshold (long columns).

• Limitations: Euler's theory overpredicts strength for short and intermediate columns, where inelastic buckling or yielding governs failure. Other formulas (e.g., Johnson's parabolic formula, Rankine-Gordon) are used in these regimes.

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Summary

• Pure bending (constant moment, zero shear) leads to circular arc deflection and is the basis for deriving the flexure formula ($\boldsymbol{\sigma = -M y / I}$).

• The elastic curve is described by the differential equation $\boldsymbol{EI \frac{d^2y}{dx^2} = M(x)}$, linking flexural stiffness ($\boldsymbol{EI}$), moment, and deflection.

• Bending stress is linearly distributed across the section, maximum at the outer fibers, and proportional to the moment and distance from the neutral axis.

• Column buckling is a stability problem. Euler's formula ($\boldsymbol{P_{cr} = \pi^2 E I / (KL)^2}$) predicts the critical elastic buckling load for long, slender columns, which depends crucially on end conditions via the effective length factor (K).

• The slenderness ratio ($\boldsymbol{KL/r}$) is the primary parameter classifying columns and determining their failure mode (elastic buckling vs. inelastic failure/yielding).