Question

5 a. A hollow circular cylinder is made of cast iron and has an outside diameter of 120mm and inside diameter of 95mm. The length is 960mm. It is subjected to an axial tensile load of 160kN. Determine the normal stress and extension of the cylinder if the Modulus of elasticity is 230GPa.

b. A hollow bar of rectangular cross section has an external cross section of 30cm by 55cm with a thickness of 10cm. It is subjected to and internal pressure of 2.3MPa, bending force of 8kN acting over a distance of 3m and a torsional load of 35kNm. Determine the maximum tensile and shear stresses. Using a factor of safety of 6, whatis the maximum allowable combined stress in the bar.

Answer 1

5 a. Data given:

Outside diameter of the hollow circular cylinder; D = 120 mm,

Inside diameter of the hollow circular cylinder; d = 95 mm,

Length of the hollow circular cylinder; L = 960 mm,

Axial tensile load; P = 160 kN, and

Modulus of elasticity of cast iron; E = 230 GPa.

We have to determine the normal stress () and extension of the cylinder ().

Assumptions: Following assumptions can be made to simplifying the analysis:

(i) The material is isotropic and homogenious, hence elastic properties are constant.

(ii) No other forces and/or couple are acting except a single axial tensile load.

Analysis: Normal stress is determined by the relation given below:

(i)

Substituting the given values in the above relation and on simplifying, we obtained;

= 37901.1 kPa = 37.9 MPa (ii) Ans(a)

Further, extension of the cylinder is determined by the formula given below:

         (iii)

Substituting the given values in the above relation and on simplifying, we obtained;

= 37.9*960/{230*(10^3)} = 0.16 mm    (iv) Ans(a)

5 b. Data given:

External cross section (rectangular) of a hollow bar; b*d = 30 cm by 55 cm,

Thickness, t = 10 cm,

Internal pressure, Pi = 2.3 MPa,

Bending force, F = 8 kN; (acting over a) distance, l = 3m,

Torsional load, T = 35 kNm, and

Factor of safety, S = 6,

We have to determine the maximum tensile () and shear stresses(). Also the maximum allowable combined stress (_Combined) in the bar.

Assumptions: Following assumptions can be made to simplifying the analysis:

(i) The material is isotropic and homogenious, hence elastic properties are constant.

Analysis: For the axial (longitudinal) stress(_l) calculation, balancing the forrces along the length of the hollow rectangular bar:

Pi*(inside area) = _l*(area of action)

or; Pi * (b-2t)*(d-2t) = _l*{2t(b+d-2t)} (i)

Substituting the known values in the above relation and on simplifying, we obtained;

_l = 0.62 MPa (ii)

Similarly, we can find out hoop stress and can easily compare both for the maximum value.

For Bending Stess = Bending moment/ section modulus

and Torsion stress = Torsion load/ section modulus

(Note: For circular cross section we use polar modulus)

And finally we can evaluate combined stress using all the three stresses.

Ultimately; with the help of factor of safety we can determine the maximum allowable combined stress (_Combined) in the bar.

Thanks students. Kindly approach the calculation as per given direction.