Understanding Beam Derivatives and Deflection

This question relates the deflection curve of a prismatic beam, denoted as $y(x)$, to its corresponding internal forces (shear force and bending moment), the applied load intensity, and the slope. The relationships are derived from the fundamental equations of beam theory.

Key Relationships in Beam Theory

For a prismatic beam where the flexural rigidity $EI$ is constant, the following differential relationships hold:

• The slope of the deflection curve is the first derivative of the deflection:

  Slope $= \frac{dy}{dx}$

• The bending moment ($M$) is related to the second derivative of the deflection:

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

• The shear force ($V$) is related to the third derivative of the deflection:

  $V(x) = \frac{dM}{dx} = EI \frac{d^3y}{dx^3}$

• The loading intensity ($w$) is related to the fourth derivative of the deflection (with a negative sign depending on convention):

  $w(x) = -\frac{dV}{dx} = -EI \frac{d^4y}{dx^4}$

Matching Beam Properties with Derivatives

Based on these relationships, we can match the given terms with their corresponding derivatives:

• A. Shear force: Corresponds to the third derivative of deflection, $\frac{d^3y}{dx^3}$.
• B. Loading intensity: Corresponds to the fourth derivative of deflection, $\frac{d^4y}{dx^4}$.
• C. Bending moment: Corresponds to the second derivative of deflection, $\frac{d^2y}{dx^2}$.
• D. Slope: Corresponds to the first derivative of deflection, $\frac{dy}{dx}$.

Correct Combination

Now, let's match these findings with the Roman numeral notations provided:

• i. $\frac{d^2y}{dx^2}$ matches C. Bending moment.
• ii. $\frac{d^3y}{dx^3}$ matches A. Shear force.
• iii. $\frac{d^4y}{dx^4}$ matches B. Loading intensity.
• iv. $\frac{dy}{dx}$ matches D. Slope.

Therefore, the correct combination is:

A - ii (Shear force)

B - iii (Loading intensity)

C - i (Bending moment)

D - iv (Slope)

Final Answer Verification

Comparing this derived combination with the given options, Option 2 provides the correct matches:

A-ii, B-iii, C-i, D-iv