Mechanics of Materials II UET, Taxila Lecture No. (8&9)

Cylindrical vessel with hemispherical ends

hemispherical ends

Consider now the vessel shown in next Figure in which the wall thickness of the cylindrical and hemispherical portions may be different.

Cross-section of a thin cylinder with hemispherical ends. ts tc

(this is sometimes necessary since the hoop stress in the cylinder is twice that in a sphere of the same radius and wall thickness).

For the purpose of the calculation the internal diameter of both portions is assumed equal. From the preceding sections the following formula are known to apply:

a- For the cylindrical portion For the cylindrical portion hoop or circumferential stress =

After substitution: circumferential strain for cylindrical part

b- For the hemispherical ends

After substitution

Equilibrium Thus equating the two strains in order that there shall be no distortion of the junction, Then:

With the normally accepted value of Poisson’s ratio for general steel work of 0.3, the thickness ratio becomes: = 0.41

i. e. the thickness of the cylinder walls must be approximately 2 i.e. the thickness of the cylinder walls must be approximately 2.4 times that of the hemispherical ends for no distortion of the junction to occur.

In these circumstances, because of the reduced wall thickness of the ends, the maximum stress will occur in the ends.

For equal maximum stresses in the two portions the thickness of the cylinder walls must be twice that in the ends but some distortion at the junction will then occur.

Effects of end plates and joints

The preceding sections have all assumed uniform material properties throughout the components and have neglected the effects of endplates and joints which are necessary requirements for their production.

In general, the strength of the components will be reduced by the presence of, for example, riveted joints or welding, and this should be taken into account by the introduction of a joint eficiency factor “  ” into the equations previously derived.

Welded joints are not as strong as the parent plate unless welds are thoroughly inspected and, if flawed, repaired during manufacture - all of which is expensive.

This strength reduction is characterised by the: weld or joint efficiency η = joint strength / parent strength

It varies from 100% for a perfect weld (ie It varies from 100% for a perfect weld (ie. virtually seamless) through 75-85% for a tolerably good weld and much less than that for bad weld.

Thus, for thin cylinders: Hoop stress acting on longitudinal Junction : where L is the efficiency of the longitudinal joints,

Why L with hoop stress and c with longitudinal stress

longitudinal stress acting on circumferential Junction : where c is the efficiency of the circumferential joints.

Pressure Vessels

For thin spheres: Hoop stress acting on the Junction

Normally the joint efficiency is stated in percentage form and this must be converted into equivalent decimal form before substitution into the above equations.