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,

Question

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=