Question

The expression $EI \frac{d^2y}{dx^2}$ at a section of a beam represents

• A. Rate of loading
• B. Deflection
• C. Shear force
• D. Bending moment

Hint:

Memorize the hierarchy of beam equations:
- $y(x)$ = Deflection
- $\frac{dy}{dx}$ = Slope
- $EI \frac{d^2y}{dx^2}$ = Bending Moment ($M$)
- $EI \frac{d^3y}{dx^3}$ = Shear Force ($V$)
- $EI \frac{d^4y}{dx^4}$ = - (Load Intensity, $w$)

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks to identify the physical quantity represented by the mathematical expression $EI \frac{d^2y}{dx^2}$ in beam theory.

Step 2: Key Formula or Approach:
This expression comes from the Euler-Bernoulli beam theory, which establishes the relationship between the beam's deflection and the internal forces. The fundamental differential equation of the elastic curve (deflection curve) of a beam is:
\[ \frac{d^2y}{dx^2} = \frac{M}{EI} \] where:
$y$ is the deflection of the beam at a position $x$ along its length.
$M$ is the internal bending moment at that section.
$E$ is the modulus of elasticity of the beam material.
$I$ is the moment of inertia of the beam's cross-section.
$EI$ is the flexural rigidity of the beam.

Step 3: Detailed Explanation:
By rearranging the differential equation, we get:
\[ M = EI \frac{d^2y}{dx^2} \] This shows that the expression $EI \frac{d^2y}{dx^2}$ is exactly equal to the bending moment ($M$) at that section of the beam.
The other quantities are related to higher derivatives:
- Shear Force ($V$) = $\frac{dM}{dx} = EI \frac{d^3y}{dx^3}$
- Rate of loading ($w$) = $-\frac{dV}{dx} = -EI \frac{d^4y}{dx^4}$
- Deflection is simply $y$.

Step 4: Final Answer:
The expression $EI \frac{d^2y}{dx^2}$ at a section of a beam represents the bending moment.