To obtain the symbolic expressions for the stress components in the local (x, y) reference frame, we can consider the equilibrium of the small element of the cylindrical wall.
In the (x, y) reference frame, the stress component along the x-axis (σx) will be equal to the internal pressure p, as there is no deformation in the hoop direction:
σx = p
The stress component along the y-axis (σy) will arise due to the deformation resulting from the internal pressure. Since the cylindrical wall is thin, we can use the thin-wall approximation and consider the cylindrical wall as a membrane subjected to an internal pressure. For a cylindrical membrane, the stress in the y-direction is given by:
σy = (p * R) / (2 * t)
The shear stress component in the xy-plane (τxy) will be negligibly small, considering the thin-wall assumption and the fact that the tank is subject to internal pressure only.
Now, let's move on to the stress components acting on the seam in the local (x1, y1) reference frame.
The stress component along the y1-axis (σy1) will be equal to the stress component along the y-axis (σy) in the (x, y) reference frame:
σy1 = σy
The shear stress component in the x1y1-plane (τx1y1) will be equal to the stress component along the x-axis (σx) in the (x, y) reference frame:
τx1y1 = σx = p
Therefore, the symbolic expressions for the stress components in the local (x, y) reference frame and the stress components on the seam are:
σx = p
σy = (p * R) / (2 * t)
τxy = 0
σy1 = (p * R) / (2 * t)
τx1y1 = p
Now, let's determine the magnitude of the load W required to achieve the desired compressive stress on the seam.
The compressive stress magnitude σo is given by:
σo = pR / (8t)
Since the load W is evenly distributed on the transverse cross-section of the cylindrical wall, the force per unit area acting on the cylindrical wall will be:
F/A = W / 2πRt = σo
Rearranging the equation, we can solve for W:
W = σo * 2πRt
Substituting the expression for σo, we get:
W = (pR / (8t)) * 2πRt
W = (π/4) * pR^2
Therefore, the symbolic expression for the magnitude of the load W required to achieve the desired compressive stress on the seam is:
W = (π/4) * pR^2