The integration constant C₁ used in Macaulay's method of analysis is obtained by applying ______ to the first integral (slope equation) of the differential equation of the elastic curve.
Try it first: pick an option, then read the explanation.
Asked in SSC JE 07/06/2024 (9 am–11 am)
Detailed Explanation
The elastic curve satisfies EI d²y/dx² = M. Integrating once gives the slope equation, which contains one unknown constant C₁. Integrating again gives the deflection equation, with a second constant C₂.
\[ EI\frac{dy}{dx} = \int M\,dx + C_1 \quad\text{(slope)} \] \[ EIy = \iint M\,dx\,dx + C_1x + C_2 \quad\text{(deflection)} \]
Why “boundary condition for slope” Is Correct
C₁ appears first in the slope equation, so it is found from a point where the slope is known — zero slope at a fixed support, or zero slope at the centre of a symmetrically loaded beam. C₂ is then found from a point of known deflection (y = 0 at a support). When no slope is known directly (e.g. an unsymmetric simply supported beam), both constants come from the two deflection conditions.
| Constant | Appears in | Typical condition used |
|---|---|---|
| C₁ | Slope equation (1st integral) | dy/dx = 0 at a fixed end or at mid-span of a symmetric beam |
| C₂ | Deflection equation (2nd integral) | y = 0 at a support |
Why the Other Options Are Wrong
- A. load factor: A load factor is a design (limit state) concept, not an integration condition.
- B. partial factor of safety: Partial safety factors belong to design codes, not to deflection analysis.
- C. boundary condition for deflection: The deflection condition is used for C₂ in the second integral.
Key Point
C₁ → slope boundary condition (1st integral); C₂ → deflection boundary condition (2nd integral).
Key Concepts for Students
- Fixed end: y = 0 and dy/dx = 0.
- Simple support: y = 0 (slope not zero).
- Symmetric loading: dy/dx = 0 at the line of symmetry.
Practice more: this is question 2 in Slope and Deflection MCQs – Part 1. Try the whole set and check every answer instantly.