Presentation on theme: "Mechanics of Materials II"— Presentation transcript:
1Mechanics of Materials II UET, TaxilaLecture No. (6)
2Cylinders & Pressure vessels Cylindrical or spherical pressure vessels are commonly used in industry to carry both liquids and gases under pressure.
3Classification of applications Cylinders find many applications, two of the most common categories being :a- fluid containers such as :pressure vessels, hydraulic cylinders, gun barrels, pipes, boilers and tanks.b- interference-fitted bearing bushes, sleeves and the like.
4Other applicationsCylinders can act as beams or shafts eg. ( load building blocks) but in the present chapter cylinders are loaded primarily by internal and external pressures due to adjacent fluids or to contacting cylindrical surfaces.
5Pressure LoadingWhen the pressure vessel is exposed to this pressure, the material comprising the vessel is subjected to pressure loading, and hence stresses, from alldirections.
6Factors that affect stresses The normal stresses resulting from this pressure are function of :1- the radius of the element under consideration,2- the shape of the pressure vessel (i.e., open ended cylinder, closed end cylinder, or sphere)3- the applied pressure.
7Two types of analysis are commonly applied to pressure vessels. The most common method is based on a simple mechanics approach and is applicable to “thin wall” pressure vessels which by definition have a ratio of inner radius (r), to wall thickness (t) of r/t ≥ 10.
8The second method is based on elasticity solution and is always applicable regardless of the r/t ratio and can be referred to as the solution for “thick wall” pressure vessels.
10Thin-Walled Pressure Assumptions Several assumptions are made in this method.1) Plane sections remain plane2) r/t ≥ 10 with t being uniform and constant3) The applied pressure, p, is the gauge pressure (where p is the difference between the absolute pressure and the atmospheric pressure)
114) Material is linear-elastic, isotropic and homogeneous. 5) Stress distributions throughout the wall thickness will not vary6) Element of interest is remote from the end of the cylinder and other geometric discontinuities.7) Working fluid has negligible weight.
14Classifications of Cylinders Cylinders are classed as being either :open - in which there is no axial component of wall stress, orclosed - in which an axial stress must exist to equilibrate the fluid pressure.
16When a thin-walled cylinder is subjected to internal pressure, three mutually perpendicular principal stresses will be set up in the cylinder material.
171- The circumferential or hoop stress. 2- The longitudinal stress. Types of stressesNamely:1- The circumferential or hoop stress.2- The longitudinal stress.3- The radial stress.
18Provided that the ratio of thickness to inside diameter of the cylinder is less than 1/20, it is reasonably accurate to assume that the hoop and longitudinal stresses are constant across the wall thickness.
19Also, the magnitude of the radial stress set up is so small in comparison with the hoop and longitudinal stresses that it can be neglected.
20This is obviously an approximation since, in practice, it will vary from zero at the outside surface to a value equal to the internal pressure at the inside surface.
21For the purpose of the initial derivation of stress formulae it is also assumed that the ends of the cylinder and any riveted joints present have no effect on the stresses produced; in practice they will have an effect and this will be discussed later.
241- Hoop or circumferential stress This is the stress which is set up in resisting the bursting effect of the applied pressure and can be most conveniently treated by considering the equilibrium of half of the cylinder.
25It is required to calculate the hoop stress in terms of: Pressure (p)Inner diameter (d)Thickness (t)
26Half of a thin cylinder subjected to internal pressure showing the hoop and longitudinal stresses acting on any element in the cylinder surface.
27Consider the equilibrium of forces in the x-direction acting on the sectioned cylinder shown in figure 2. It is assumed that the circumferential stress H (or θ( is constant through the thickness of the cylinder.
39Force owing to internal pressure Total force on the end of the cylinder owing to internal pressureForce on cylinder end :Force =
40For equilibrium of forces we need to calculate the End section area
41End Section AreaThe cross-sectional area of the cylinder wall is characterized by the product of its wall thickness and the mean circumferenceFor the thin-wall pressure vessels whereD >> tthe cylindrical cross-section area may be approximated by pDt.
47And change in length = longitudinal strain x original length Then change in length =
48(b) Change in diameterAs above, the change in diameter may be determined from the strain on a diameter, i.e. the diametral strain.
49Now the change in diameter may be found from a consideration of the circumferential change. The stress acting around a circumference is the hoop or circumferential stress Hgiving rise to the circumferential strain H.