The roof beam of a building is subjected to the loading as shown below. The vertical displacement of point A should not exceed [tex]\frac{1}{2} \text{ in}[/tex] to avoid damage to the ceiling and roofing materials. Using the double integration method, calculate the minimum required moment of inertia of the beam.

Given:
- Modulus of Elasticity, [tex]E = 29000 \text{ ksi}[/tex]

Loading:
- 4 kips/ft
- 2 kips/ft

Supports:
- Roller at B
- Pin support at C

Length:
- 5 ft and 10 ft

Calculate the minimum required moment of inertia.

To solve this problem, we're going to use beam theory to determine the minimum required moment of inertia for the beam, ensuring that the vertical displacement of point A does not exceed 1/2 inch.

Assumptions and Given Data:

The beam length from B to A is 5 ft and from A to C is 10 ft.
The beam is subjected to a uniformly distributed load of 4 Kips/ft over the 5 ft span and a point load of 2 Kips point load at A.
Modulus of Elasticity [tex]E = 29000 \text{ ksi}[/tex].
Maximum allowable deflection at A [tex]\delta_{max} = \frac{1}{2} \text{ inch}[/tex].

Calculation Steps:

1. Determine Support Reactions:

Before applying the double integration method, you need to find the reactions at the supports.

Applying equilibrium equations:

Sum of moments about C:
[tex]5 \times 4 \times 5/2 + 2 \times 10 - R_B \times 15 = 0[/tex]
Simplify to find [tex]R_B[/tex]:
[tex]R_B = \frac{100 + 20}{15} = 8 \text{ Kips}[/tex]
Sum of vertical forces:
[tex]R_B + R_C = 20 + 2[/tex]
[tex]R_C = 22 - 8 = 14 \text{ Kips}[/tex]

2. Use the Double Integration Method:

The elasticity and beam deflection relationship is given by:

[tex]EI \frac{d^2v}{dx^2} = M(x)[/tex]

Use known beam theory cases or integration to find the deflection at point A:

Differential equation of the elastic curve is established using the moment equation.
Integrate to find slope and deflection equations, using boundary conditions.

3. Calculate Moment of Inertia:

Given that [tex]\delta \leq \frac{1}{2} \text{ inch}[/tex].

Use the deflection equation that results from integrating the moment equation:
[tex]\frac{W L^3}{384 EI} + \frac{P a^2 b^2}{3 EI L} \leq \frac{1}{2} \text{ inch}[/tex]

4. Substitute Terms and Solve for

[tex]I[/tex]

:

With calculated reactions and known loadings, solve the inequality:

[tex]I \geq \frac{L^3}{384 E} \left( \frac{1}{2} \text{ inch in converted units} \right)[/tex]

Ensure consistent units for calculations.

Conclusion:

By substituting the given values and solving this equation using the given modulus of elasticity, solve for the minimum moment of inertia [tex]I[/tex]. This ensures the deflection does not exceed the specified limit of 1/2 inch.

This process involves detailed application of static equilibrium and beam deflection theories, where accuracy in calculations and unit conversions is crucial.