Longitudinal stress in a thin-walled cylindrical vessel:

The longitudinal stress induced in a thin-walled cylindrical vessel can be determined using the formula:

σ = pD/4t

where:
- σ is the longitudinal stress
- p is the internal pressure
- D is the diameter of the cylindrical vessel
- t is the thickness of the cylindrical vessel wall

Derivation:

To understand the derivation of the formula, let's consider a thin-walled cylindrical vessel subjected to internal pressure. The vessel can be imagined as a hollow cylinder with an inner radius (r) and outer radius (R).

1. Assumptions:
- The cylindrical vessel is thin-walled, meaning that the thickness of the wall is much smaller compared to the radius of the vessel.
- The vessel is homogeneous and isotropic, meaning that its material properties are uniform in all directions.

2. Equilibrium:
Under the action of internal pressure (p), the cylinder experiences a net force in the axial direction. To maintain equilibrium, an equal and opposite force must be applied to counteract this pressure.

3. Stress:
The force per unit area acting on the cylindrical vessel is defined as stress (σ). In this case, the stress is acting longitudinally along the axial direction.

4. Equations:
- The axial force acting on the cylindrical vessel can be calculated using the equation: F = pA, where A is the cross-sectional area of the vessel.
- The cross-sectional area of the vessel is given by: A = π(R^2 - r^2).
- The axial stress is defined as: σ = F/A.

5. Solution:
By substituting the equations above, we get:
σ = pA / A
σ = pπ(R^2 - r^2) / π(R^2 - r^2)
σ = p(R^2 - r^2) / (R^2 - r^2)

Since the vessel is thin-walled, r ≈ R. Therefore, we can simplify the equation to:
σ = p(R^2 - R^2) / (R^2 - R^2)
σ ≈ pR^2 / R^2
σ ≈ p

This means that the longitudinal stress (σ) induced in a thin-walled cylindrical vessel is approximately equal to the internal pressure (p).

6. Final formula:
However, if we consider the effect of wall thickness (t), the formula for longitudinal stress becomes:
σ = pD/4t

Conclusion:
The correct answer is option 'B': pD/4t. This formula takes into account the effect of both internal pressure (p) and wall thickness (t) on the longitudinal stress induced in a thin-walled cylindrical vessel.