For a beam, EI(d³y/dx³) represents:
Try it first: pick an option, then read the explanation.
Correct answer: D. shear force
Asked in SSC JE 2010
Detailed Explanation
Each derivative of the elastic curve has a physical meaning. Since M = EI d²y/dx² and V = dM/dx, one more derivative gives the shear force.
\[ EI\frac{d^3y}{dx^3} = \frac{d}{dx}\left(EI\frac{d^2y}{dx^2}\right) = \frac{dM}{dx} = V \]
Why “shear force” Is Correct
EI d³y/dx³ is the shear force; EI d⁴y/dx⁴ is the load intensity.
| Quantity | Expression | Physical meaning |
|---|---|---|
| Deflection | y | Vertical movement of the beam axis |
| Slope | dy/dx | Rotation of the tangent to the elastic curve |
| Bending moment | M = EI d²y/dx² | Curvature × flexural rigidity |
| Shear force | V = EI d³y/dx³ | Rate of change of bending moment |
| Load intensity | w = EI d⁴y/dx⁴ | Rate of change of shear force |
Successive derivatives of the elastic curve y(x).
Why the Other Options Are Wrong
- A. deflection: Deflection is y itself.
- B. slope: Slope is dy/dx.
- C. bending moment: Bending moment is EI d²y/dx².
Key Point
y → slope y' → M = EIy'' → V = EIy''' → w = EIy''''.
Key Concepts for Students
- dM/dx = V and dV/dx = −w (sign depends on convention).
- Shear force is zero where M is maximum.
- Four integrations of EIy'''' = w need four boundary conditions.
Practice more: this is question 35 in Slope and Deflection MCQs – Part 3. Try the whole set and check every answer instantly.