UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

Conical Shell filled with liquid:

Solution: Let consider a conical shell subjected to load P in the direction of the axis of cone as

shown in figure. Let the semi apex angle of the cone be α. Therefore at point where radius of

parallel circle is ro

....................................... (1)

If the load P is applied in the direction of the axis of cone, then the stress distribution is

symmetrical.

From the equation of equilibrium along axial direction we know,

.................................... (2)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

Since the curvature of the meridian in the case of a cone is zero, r1 = ∞

.................................. (3) { ro = r2 Sin θ}

For the above case of loading z = 0

............................... (4)

Now, Let us consider a conical tank to be filled with a liquid of specific weight ɣ shown in

Figure. Measuring the distances y from the bottom of the tank and denoting by the total depth of

liquid in the tank, the pressure at any parallel circle mn is

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

P = -z

∴ P = ɣ (d -y)

∴ z = -ɣ (d -y)........................................ (5)

Also for such tank :

............................................. (6)

And

........................................ (7)

Where, α semi-cone angle of the tank. Substituting value of φ from equation 6 in the equation 3

we obtain

Put the values of –z and r0 from equations 5 and 7 respectively

................................ (8)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

......................... (9)

(Since r0 = y tan α)

UNIT IV: - Membrane theory of thin cylindrical shells, Spherical and conical domes/shells

Prof. Shaikh Zahoor Prof. Ansari Fatima M.E (Civil- Structure). M.E (Civil- Structure)

.................................. (10)