Deflection Of Beams

The deflection is measured from original neutral surface of the beam to neutral surface of deformed beam. The configuration of neutral surface of deformed beam is known as elastic curve of beam.

Differential Equation of Elastic Curve for the loaded beam

Assumptions:

• Stress is proportional to strain i.e. hooks law applies.
• Small deflection
• pure bending or Any deflection resulting from the shear deformation of the material or shear stresses is neglected.

from analytic geometry, curvature of line is given by,

\[k = \frac{1}{R} = \frac{\frac{d^2y}{dx^2}}{\Big[1 + \frac{d^2y}{dx^2}\Big]^{3/2}} \approx \frac{d^2y}{dx^2}\]

from simple bending theory equation,

\[\frac{σ}{y} = \frac{M}{I} = \frac{E}{R} \quad or \quad \frac{1}{R} = \frac{M}{EI}\]

So basic differential equation governing deflection of beam is

\[\boxed{M = EI \frac{d^2y}{dx^2}}\]

since shear force is \(\frac{dM}{dx}\) thus,

\[\boxed{Slope = \frac{dy}{dx}}\]

\[\boxed{Bending \quad Moment = EI\frac{d^2y}{dx^2}}\]

\[\boxed{Shear \quad Force = EI\frac{d^3y}{dx^3}}\]

\[\boxed{Load \quad Distribution = EI\frac{d^4y}{dx^4}}\]

Deflection for Common Loadings

1. Moment load at the free end of cantilever beam

• Maximum Bending Moment       M = -M   
• Slop at end     \(θ = \frac{ML}{EI}\)
• Maximum deflection (at end)     \(δ = \frac{ML^2}{2EI}\)
• Deflection Equation    \(EI y = -\frac{Mx^2}{2}\)

2. Concentrated load at the free end of cantilever beam

• Maximum Bending Moment       M = -PL   
• Slop at end     \(θ = \frac{PL^2}{2EI}\)
• Maximum deflection (at end)     \(δ = \frac{PL^3}{3EI}\)
• Deflection Equation    \(EI y = -\frac{Px^2}{6}(3L-x)\)

3. Cantilever Beam subjected to a Uniformly distributed load

• Maximum Bending Moment       \(M = -\frac{qL^2}{2}\)   
• Slop at end     \(θ = \frac{qL^3}{6EI}\)
• Maximum deflection (at end)     \(δ = \frac{qL^4}{8EI}\)
• Deflection Equation    \(EI y = -\frac{qx^2}{120L}(6L^2 -4Lx + x^2)\)

4. Cantilever Beam subjected to a Uniformly Triangular distributed load

• Maximum Bending Moment       \(M = -\frac{q_oL^2}{6}\)   
• Slop at end     \(θ = \frac{q_oL^3}{24EI}\)
• Maximum deflection (at end)     \(δ = \frac{q_oL^4}{30EI}\)
• Deflection Equation    \(EI y = -\frac{q_ox^2}{120L}(10L^3 -10L^2x + 5Lx^2 - x^3)\)

5. Concentrated load at the mid-span of simply supported beam

• Maximum Bending Moment       \(M = -\frac{PL}{4}\)   
• Slop at end     \(θ_A = θ_B = θ = \frac{PL^32}{16EI}\)
• Maximum deflection (at midspan)     \(δ = \frac{PL^3}{48EI}\)
• Deflection Equation (for 0 < x < L/2) \(EI y = -\frac{Px}{48}(3L^2 - 4x^2)\)

6. Uniformly distributed load of simply supported beam

• Maximum Bending Moment       \(M = -\frac{qL^2}{8}\)   
• Slop at end     \(θ_A = θ_B = θ = \frac{qL^3}{24EI}\)
• Maximum deflection (at mid span)     \(δ = \frac{5qL^4}{384EI}\)
• Deflection Equation    \(EI y = -\frac{qx}{24}(L^3 -2Lx^2 + x^3)\)

Strain Energy Method (Castigliano’s Theorem)

Castigliano’s Theorem states that the partial derivative of the strain energy with respect to an applied force is equal to the displacement of the force along its line of action.

\[\boxed{δ = \frac{\partial U}{\partial P}}\]

The strain energy is given by

\[U = \int\limits_{0}^{L} \frac{M^2}{2EI} dx\]

\[\boxed{δ = \int\limits_{0}^{L} \frac{\partial M}{\partial P}\frac{M}{EI} dx}\]

Example : Take a cantilever beam subjected to concentrated load P at free end

M = P(L - x)

\(δ = \int\limits_{0}^{L} \frac{\partial M}{\partial P}\frac{M}{EI} dx\)

\(δ = \int\limits_{0}^{L} (L - x)\frac{P(L-x)}{EI} dx\)

\(δ = \frac{PL^3}{3EI}\)