A cylindrical pressure tank of mean diameter 1.6 m, is fabricated by butt welding a 8 mm thick plate along a helix which forms an angle of 20° with the transverse plane. The end caps are spherical having wall thickness 10 mm. If the internal gauge pressure is 0.8 MPa, determine,

(a) stresses in the direction and perpendicular to the weld,
(b) normal stress and maximum shear stress in the spherical cap.

Question

A cylindrical pressure tank of mean diameter 1.6 m, is fabricated by butt welding a 8 mm thick plate along a helix which forms an angle of 20° with the transverse plane. The end caps are spherical having wall thickness 10 mm. If the internal gauge pressure is 0.8 MPa, determine,

(a) stresses in the direction and perpendicular to the weld,
(b) normal stress and maximum shear stress in the spherical cap.

Accepted Answer

(a) Cylindrical tank:

First consider the equilibrium of forces at a transverse cut for evaluating longitudinal stress [latex]\sigma _1[/latex] as shown in Fig. 2.20.

Inward force, [latex]p imes \pi r^2[/latex] = Outward force, [latex]\sigma_1 imes \pi r imes t[/latex]

or, [latex]\sigma_1=\frac{p r}{2 t}=\frac{0.8 imes 800}{2 imes 8}=40 MPa[/latex]

Similarly consider the section of a unit length of tank as shown in Fig. 2.20 for circumferential stress [latex]\sigma_2[/latex],

[latex]2 imes \sigma_2 imes 1 imes t=p imes 1 imes 2 r [/latex]

or, [latex]\sigma_2=\frac{p r}{t}=\frac{0.8 imes 800}{8}=80 MPa[/latex]
Let stresses across and along the weld be [latex]\sigma_{x^{\prime}} and au_{x^{\prime}y^{\prime}} ext{Angle between} \sigma_{x^{\prime}} and \sigma_1=20^{\circ}. [/latex] Using Eqs (2.1) and (2.2),

[latex]\sigma _{x^{\prime}}=\frac{\sigma _x+\sigma _y}{2}+\frac{\sigma _x-\sigma _y}{2}\cos 2 heta - au _{xy}\sin 2 heta[/latex] (2.1b)

[latex] au _{x^{\prime}y^{\prime}}=\frac{\sigma_x- \sigma_y}{2}\sin 2 heta + au _{xy}\cos 2 heta[/latex] (2.2)

[latex]\sigma_{x^{\prime}}=\frac{40+80}{2}+\frac{40-80}{2} \cos 40^{\circ} [/latex] = 44.68 MPa (tension)

[latex] au_{x^{\prime}y^{\prime}} =\frac{40-80}{2} \sin 40^{\circ} [/latex] = -12.86 MPa (Anticlockwise)

(b) Spherical cap:

The stress in spherical cap is same in all directions due to symmetry. A section of a sphere along any diameter will be similar to the transverse section of the cylinder in Fig. 2.20.

∴ [latex] \sigma_1=\sigma_2=\frac{p r}{2 t}=\frac{0.8 imes 800}{2 imes 10}=32 MPa [/latex]

Shear stress in the plane of the sphere is zero. The shear stress on the walls of the vessel is greater if third dimension is also considered.

Question 2.13: A cylindrical pressure tank of mean diameter 1.6 m, is fabri...

Related Answered Questions

Figure 2.30(a) shows an element under axial stress 20 MPa and 50 MPa along with shear stress 40 MPa in x-y plane. Determine the state of stress in x’ – y’ plane at 30° counter-clockwise to x-y plane. ...

Verified Answer:

Verified Answer:

Tensile stresses of 100 MPa and 50 MPa are acting on two perpendicular planes in a body. Determine the resultant stress inclined at 35° to the major principal plane using stress ellipsoid. ...

Verified Answer:

Figure 2.28 shows the state of stress at a point. Determine the stress tensor on x’y’z’ reference frame rotated such that x’-axis makes equal angle with x, y, z directions in first quadrant and y’-axis makes 60° with x-axis. ...

Verified Answer:

Figure 2.28 shows the state of stress at a point. Determine the principal normal and shear stresses and the orientation at which they act. Also determine the octahedral normal and shear stresses. ...

Verified Answer:

Solve Example 2.16 if the rotation of xyz reference is 30° clockwise. ...

Verified Answer:

he state of stress at a point in xyz reference is as follows. τ = |80 -60 0 -60 -40 0 0 0 40|MPa Determine the stress tensor in the direction x’y’z’ by rotating xyz through 30° anticlockwise about z-direction as shown in Fig. 2.26. ...

Verified Answer:

Verified Answer:

A horizontal force 1 kN is applied at the end of a cantilever bent of diameter 40 mm as shown in Fig. 2.22. Determine maximum principal normal stresses 200 mm above the base. ...

Verified Answer:

solve example 2.11 using Mohr’s circle. ...

Verified Answer: