Thin And Thick Cylinders 333b3k

Thin Cylinders:  Hoop or circumferential stress-(Tensile stress):  Longitudinal stress-(Tensile stress) :  Stress along the radial direction-(compressive) = p  Greatest stress:  Circumferential (hoop) strain:  Longitudinal strain:  Change in volume/Original volume=elong+2*ecirc Thin Spherical shells:  Hoop or circumferential stress: d=diameter, p= pressure intensity, t= wall thickness, E = Young’s modulus, 1/m=Poisson’s ratio. Thick cylinder:  longitudinal strain is constant  Radial pressure px and hoop stress fx at any radius x are: pouter=0 Thick Spherical shells:  Radial pressure px and hoop stress fx at any radius x are: pouter=0

Only internal pressure

Only External Pressure

 Shrinking a hoop over an inner cylinder-Self- hooping or Autofrettage. This allows the cylinder to operate at higher fluid pressure  if ηl is the efficiency of a t in the longitudinal direction, influencing the hoop stress, then the stress will be given as

 if ηh is the efficiency of a t in the circumferential direction, influencing the longitudinal stress, then the stress will be given as

Concept Problems: 1) A cylindrical boiler is 2.5 m in diameter and 20 mm in thickness and it carries steam at a pressure of 1.0 N/mm2. Find the stresses in the shell. Ans: Longitudinal stress=31.25 N/mm2 ; Hoop stress= 62.5 N/mm2 2) A thin cylindrical vessel of 2 m diameter and 4 m length contains a particular gas at a pressure of 1.65 N/mm2. If the permissible tensile stress of the material of the shell is 150 N/mm2, find the maximum thickness required. Ans: thickness=11mm 3) A cylindrical compressed air drum is 2 m in diameter with plates 12.5 mm thick. The efficiencies of the longitudinal (ηl) and circumferential (ηc) ts are 85% and 45% respectively. If the tensile stress in the plating is to be limited to 100 MN/m2, find the maximum safe air pressure. Ans: maximum safe air pressure = 1.063 N/mm2 4) A cylindrical shell, 0.8 m in a diameter and 3 m long is having 10 mm wall thickness. If the shell is subjected to an internal pressure of 2.5 N/mm2, determine (a) Change in diameter, (b) Change in length, and (c) Change in volume. Take E = 200 GPa and Poisson’s ratio = 0.25.

Ans: Change in diameter=0.35mm, Change in length=0.375mm, Change in volume=1507e3 mm3 5) The internal and external diameters of a thick hollow cylinder are 80 mm and 120 mm respectively. It is subjected to an external pressure of 40 N/mm2 and an internal pressure of 120 N/mm2. Calculate the circumferential stress at the external and internal surfaces and determine the radial and circumferential stresses at the mean radius Ans: Circumferential stresses=168N/mm2, 88N/mm2; Radial 2 2 stress=68.16N/mm , Circumferential stress=116.16N/mm 6) A thick cylinder of 0.5 m external diameter and 0.4 m internal diameter is subjected simultaneously to internal and external pressures. If the internal pressure is 25 MN/m2 and the hoop stress at the inside of the cylinder is 45 MN/m2 (tensile), determine the intensity of the external pressure. Ans: external pressure=12.4 MN/m2

Previously asked problems 1) A thin walled spherical shell is subjected to an internal pressure. If the radius of the shell is increased by 1% and the thickness is reduced by 1%, with the internal pressure remaining the same, the percentage change in the circumferential (hoop) stress is(Gate: ME 2012) (A) 0 (B) 1 (C) 1.08 (D) 2.02 2) A thin cylinder of inner radius 500 mm and thickness 10 mm is subjected to an internal pressure of 5 MPa. The average circumferential (hoop) stress in MPa is (Gate: ME 2011) (A) 100 (B) 250 (C) 500 (D) 1000 Common Data for Q. 3 and 4: A cylindrical container of radius R = 1 m, wall thickness 1 mm is filled with water up to a depth of 2 m and suspended along its upper rim. The density of water is 1000 kg/m3 and acceleration due to gravity is 10 m/s2. The self weight of the cylinder is negligible. The formula for hoop stress in a thin

walled cylinder can be used at all points along the height of the cylindrical container. 3) The axial and circumference stress (σd,σc ) experienced by the cylinder wall at mid-depth (1 m as shown) are(Gate: ME 2008) (A) (10, 10)MPa (B) (5, 10)MPa (C) (10, 5)MPa (D) (5, 5)MPa 4) If the Young’s modulus and Poisson’s ratio of the container material are 100 GPa and 0.3, respectively, the axial strain in the cylinder wall at mid-depth is(Gate: ME 2008) (A) 2 × 10-5 (B) 6 × 10-5 (C) 7 × 10-5 (D) 1.2 × 10-5 5) A thin walled cylindrical vessel of wall thickness t and diameter d is filled with gas to a gauge pressure of p. the maximum shear stress on the vessel wall will then be(Gate: ME 1999) (A) (B) 6)

7)

8)

9)

(C) (D) A thick cylinder is subjected to an internal pressure of 60 MPa. If the hoop stress on the outer surface is 150 MPa, then the hoop stress on the internal surface is(Gate: ME 1996) (A) 105 MPa (B) 180 MPa (C) 210 MPa (D) 135 MPa A thin walled cylindrical pressure vessel having a radius of 0.5m and wall thickness of 25mm is subjected to an internal pressure of 700kPa. The hoop stress developed is(Gate: CE 2009) (A) 14MPa (B) 1.4MPa (C) 0.14MPa (D) 0.014MPa Where does the maximum hoop stress in a thick cylinder under external pressure occur? (IES: ME 2008) (A) At the outer surface (B) At the inner surface (C) At the mid-thickness (D) At the 2/3rd outer radius A thin cylindrical shell of diameter d length l and thickness t is subjected to an internal pressure p. What is the ratio of longitudinal strain to hoop strain in of Poisson’s ratio (1/m)? (IES: ME 2004) (A) (B) (C)

(D)

10) A thick cylinder of internal radius a and external radius b is subjected to internal pressure p as well as external pressure p Which One of the following statements is correct? (IES: ME 2004) The magnitude of circumferential stress developed is (A) Maximum at radius r = a (B) maximum at radius r=b (C) Maximum at radius (D) constant √ 11) A thick cylinder with internal diameter d and outside diameter 2d is subjected to internal pressure p. Then the maximum hoop stress developed in the cylinder is(IES: ME 2003) (A) p (B) (2/3)p (C) (5/3)p (D) 2p 12) The volumetric strain in case of a thin cylindrical shell of diameter d, thickness t, subjected to internal pressure p, is (IES: ME 2003) (A) (B) (C) (D) 13) A thin cylinder of radius r and thickness t when subjected to an internal hydrostatic pressure P causes a radial displacement u, then the tangential strain caused is (IES: ME 2002) (A) du/dr (B) 1/r.du/dr (C) u/r (D) 2u/r 14) For the same internal diameter, wall thickness, material and internal pressure, the ratio of maximum stress, induced in a thin cylindrical and in a thin spherical pressure vessel will be(IES: ME 2001) (A) 2 (B) 1/2 (C) 4 (D) 1/4 15) A thin cylinder contains fluid at a pressure of 500 N/m2, the internal diameter of the shell is 0.6 m and the tensile stress in the material is to be limited to 9000 N/m2. The shell must have a minimum wall thickness of nearly(IES: ME 2000) (A) 9mm (B) 11mm (C) 17mm (D) 21 mm 16) Consider the following statements at given point in the case of thick cylinder subjected to fluid pressure : 1. Radial stress is compressive. 2. Hoop stress is tensile. 3. Hoop stress is compressive.

4. Longitudinal stress is tensile and it varies along the length. 5. Longitudinal stress is tensile and remains constant along the length of the cylinder. Which of the statements given above are correct? (IES: ME 2006) (A) Only 1, 2 and 4 (B) Only 3 and 4 (C) Only 1, 2 and 5 (D) Only 1, 3 and 5