# 2009 14 CombinedLoad - Introduction to Solid Mechanics-Vm211 Introduction to Solid Mechanics-Vm211 Chapter 14 Combined Loadings CHAPTER OUTLINE 1

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1 SJTU Introduction to Solid Mechanics-Vm211 Chapter 14 Combined Loadings Combined Loadings SJTU Introduction to Solid Mechanics-Vm211 CHAPTER OUTLINE 1. Thin-Walled Pressure Vessels 2. State of Stress Caused by Combined Loadings SJTU Introduction to Solid Mechanics-Vm211 THIN-WALLED PRESSURE VESSELS ± Cylindrical or spherical vessels are commonly used in industry to serve as boilers or tanks ± When under pressure, the material which they are made of, is subjected to a loading from all directions ± However, we can simplify the analysis provided it has a thin wall SJTU Introduction to Solid Mechanics-Vm211 THIN-WALLED PRESSURE VESSELS ± “Thin wall” refers to a vessel having an inner-radius- thickness ratio of 10 or more ( r/t 10 ) ± When r / t = 10, results of a thin-wall analysis will predict a stress approximately 4% less than the actual maximum stress in the vessel SJTU Introduction to Solid Mechanics-Vm211 THIN-WALLED PRESSURE VESSELS ± Assumption taken before analysis is that the thickness of the pressure vessel is uniform or constant throughout ± The pressure in the vessel is understood to be the gauge pressure , since it measures the pressure above atmospheric pressure which is assumed to exist both inside and outside the vessel’s wall SJTU Introduction to Solid Mechanics-Vm211 THIN-WALLED PRESSURE VESSELS Cylindrical vessels ± A gauge pressure p is developed within the vessel by a contained gas or fluid, and assumed to have negligible weight

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2 SJTU Introduction to Solid Mechanics-Vm211 THIN-WALLED PRESSURE VESSELS Cylindrical vessels ± Due to uniformity of loading, an element of the vessel is subjected to normal stresses σ 1 in the circumferential or hoop direction and 2 in the longitudinal or axial direction SJTU Introduction to Solid Mechanics-Vm211 THIN-WALLED PRESSURE VESSELS Cylindrical vessels ± We use the method of sections and apply the equations of force equilibrium to get the magnitudes of the stress components ± For equilibrium in the x direction, we require Σ F x =0; 2 [ σ 1 ( tdy )] p ( 2rdy ) = 0 pr t 1 = SJTU Introduction to Solid Mechanics-Vm211 THIN-WALLED PRESSURE VESSELS Cylindrical vessels ± As shown, σ 2 acts uniformly throughout the wall, and p acts on the section of gas or fluid. Thus for equilibrium in the y direction, we require Σ F y =0; σ 2 (2 π rt ) p ( r 2 ) = 0 pr 2 t 2 = SJTU Introduction to Solid Mechanics-Vm211 THIN-WALLED PRESSURE VESSELS Cylindrical vessels ± For Eqns , 1 , 2 = normal stresses in the hoop and axial directions, respectively. Each is assumed to be constant throughout the wall of the cylinder, and each subjects the material to tension pr t 1 = pr 2 t 2 = SJTU Introduction to Solid Mechanics-Vm211 THIN-WALLED PRESSURE VESSELS Cylindrical vessels ± p
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## This note was uploaded on 08/09/2011 for the course EE 211 taught by Professor Liuxila during the Summer '09 term at Shanghai Jiao Tong University.

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2009 14 CombinedLoad - Introduction to Solid Mechanics-Vm211 Introduction to Solid Mechanics-Vm211 Chapter 14 Combined Loadings CHAPTER OUTLINE 1

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