Q.54 In the double integration method for a simply supported beam subjected to UDL over its entire span, the slope and deflection calculation will require:
- Aonly the symmetry condition
- Bonly the boundary conditionCorrect
- Cthe boundary condition as well as the symmetry condition
- Dthe boundary condition as well as the continuity condition
Solution
Concept
In double integration, the two integration constants are found from the support (boundary) conditions.
Formula
EI d2y/dx2 = Mx = wLx/2 - wx2/2
Step-by-step solution
1. Bending moment Mx = wLx/2 - wx2/2 is one expression for the full span. 2. Integrate once: EI dy/dx = wLx2/4 - wx3/6 + C1 3. Integrate again: EI y = wLx3/12 - wx4/24 + C1 x + C2 4. Boundary condition y = 0 at x = 0 gives C2 = 0 5. Boundary condition y = 0 at x = L gives C1 = -wL3/24 6. Both constants are found from boundary conditions only.
Explanation
The moment is one single expression for the whole span, so there are only two constants. The two support conditions (zero deflection at both ends) are enough to find them. The symmetry condition (zero slope at mid-span) is an optional shortcut, not a requirement.
Why the other options are wrong
A: Symmetry alone gives only one condition, but there are two constants. C: Symmetry can be used, but it is not required. D: Continuity is needed only when the moment has different expressions in different parts, such as for point loads or partial UDL.
Shortcut
One moment expression means two constants, and two supports give two boundary conditions. That is enough.
Common mistake
Thinking that the zero-slope-at-mid-span condition is compulsory. It is only a convenient alternative.
Correct answer: B) only the boundary condition