An overhang beam ABC is loaded with uniformly distributed load (UDL) throughout its entire span as per the figure below. The intensity of UDL is 12 kN/m. The bending moment at an intermediate point P (between A and B) is found to be zero. The length (in m) of the portion AP is _____________ (rounded off to two decimal places)
Step 1: Given data
UDL $w = 12 \text{ kN/m}$
$AB = 6 \text{ m}, \quad BC = 2 \text{ m}$
Total length $AC = 8 \text{ m}$
Step 2: Reactions at supports
Total load:
$ W = w \times 8 = 96 \text{ kN}$
Taking moments about A:
$ R_B \times 6 = 96 \times 4 $
$ R_B = \frac{384}{6} = 64 \text{ kN}$
Then,
$ R_A = 96 - 64 = 32 \text{ kN}$
Step 3: Bending moment at point P (distance $x$ from A)
For section between A and B:
$ M(x) = R_A x - \frac{w x^2}{2} $
$ M(x) = 32x - \frac{12x^2}{2} = 32x - 6x^2 $
Step 4: Condition for zero bending moment
$ 32x - 6x^2 = 0 $
$ x(32 - 6x) = 0 $
Non-zero solution:
$ x = \frac{32}{6} = 5.33 \text{ m}$
Final Answer:
$\boxed{5.33 \text{ m}}$
Important Questions from Strength of Materials
- A steel wire of $5.65\text{ mm}$ diameter and $50\text{ m}$ length is used for a hoisting crane. The wire is used to vertically lift a weight of $200\text{ kg}$ attached to its lowest end. Assume the Young's Modulus of Elasticity of Steel as $2 \times 10^5\text{ N/mm}^2$ and gravitational acceleration as $10\text{ m/sec}^2$. The elongation of the steel wire (in mm ) will be ______ [rounded off to two decimal places].
A rectangular beam section of size 300 mm (width) X 500 mm (depth) is loaded with a shear force of 600 kN. The maximum shear stress on the section in N/mm² is ___________
- A load of 30 kN is applied vertically downward at the free end of a cantilever of span 5 m. If the elastic modulus of the cantilever is 30 GPa and the section has a width of 0.3 m and a depth of 0.6 m, then, the elastic deflection (in mm) is ________
- A solid straight steel rod of diameter 100 mm is bent in single curvature into a circular arc by a moment of 50 kNm applied at its ends. If elastic modulus, E, for steel is $2 \times 10^5$ MPa, the radius of curvature (mm) of the arc assuming $\pi=3.14$ is ________
A simply supported RCC beam of cross section $0.4 \text{ m} \times 0.6 \text{ m}$ covers a span of $8 \text{ m}$. It is subjected to a uniformly distributed load of $30 \text{ kN/m}$. If the unit weight of concrete is $24 \text{ kN/m}^3$, the tensile stress (in $N/mm^2$, rounded off to two decimal places) at the bottom of the beam at mid-span is______
