A thin-wall cylindrical tank has radius R, wall thickness t≪R and is subjected to an internal pressure p. The cylindrical wall of the tank is constructed by welding steel plates along 45∘ seams,

Consider a small element of the cylindrical wall, and define local coordinate axes (x,y) aligned, as indicated, with the y axis parallel to the axis of the cylindrical tank, and the x axis along the hoop direction.

Consider also, on this small element, the local rotated coordinate frame (x1,y1) defined, as indicated in the figure, with the coordinate axis y1 normal to the line of the of the seam.

Obtain symbolic expressions for the Cartesian components of the stress acting on the small element of cylindrical wall of the tank in the local (x,y) reference frame, as well as the normal and shear stress components acting on the seam, σy1, and τx1y1, respectively. Express these quantities in terms of R, t, and p.

σx=

σy=

τxy=

σy1=

τx1y1=

In service, the tank is observed to leak at the welded seams.

A quick temporary fix is decided in which a load W is to be applied to the top cap of the tank, axially compressing the tank and changing the stress state in the cylindrical wall of the tank.

Assume that the the load W is applied to the center of the cap so that it is evenly distributed on the transverse cross section of the cylindrical wall. Since t≪R, take the thin-wall approximation for the transverse cross sectional area of the cylindrical wall of the tank (A=2πRt).

Also assume that, to prevent further leaks, we want to have a compressive stress of magnitude σo=pR8t acting normal to the seam .

Obtain a symbolic expression for the magnitude of the load W that, when combined with the effects of the internal pressure p, will result in the desired level of normal stress on the seam. Express your answer in terms of R and p. There will be a factor of π in your answer: enter this as pi.

W=

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