1. COMBINED LOADING

2.  Analyze the stress developed in thin-walled pressure vessels  Review the stress analysis developed in previous chapters regarding axial load, torsion, bending and shear 2 Discuss the solution of problems where several of these internal loads occur simultaneously on a member’s x-section

3. 1. Thin-Walled Pressure Vessels 2. State of Stress Caused by Combined Loadings 3

4.  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  “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 actual maximum stress in the vessel 4

5.  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 5

6. Cylindrical vessels  A gauge pressure p is developed within the vessel by a contained gas or fluid, and assumed to have negligible weight  Due to uniformity of loading, an element of the vessel is subjected to normal stresses  1 in the circumferential or hoop direction and  2 in the longitudinal or axial direction 6

7. 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 7 pr t  1 = Equation 8-1

8. Cylindrical vessels  As shown,  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 8 pr 2t  2 = Equation 8-2

9. Cylindrical vessels  For Eqns 8-1 and 8-2, 9  1,  2 = normal stress in the hoop and longitudinal directions, respectively. Each is assumed to be constant throughout the wall of the cylinder, and each subjects the material to tension p = internal gauge pressure developed by the contained gas or fluid r = inner radius of the cylinder t = thickness of the wall ( r/t ≥ 10 )

10. Cylindrical vessels  Comparing Eqns 8-1 and 8-2, note that the hoop or circumferential stress is twice as large as the longitudinal or axial stress  When engineers fabricate cylindrical pressure vessels from rolled-form plates, the longitudinal joints must be designed to carry twice as much stress as the circumferential joints 10

11. Spherical vessels  The analysis for a spherical pressure vessel can be done in a similar manner  Like the cylinder, equilibrium in the y direction requires 11 pr 2t  2 = Equation 8-3

12. Spherical vessels  Note that Eqn 8-3 is similar to Eqn 8-2. Thus, this stress is the same regardless of the orientation of the hemispheric free-body diagram 12 An element of material taken from either a cylindrical or spherical pressure vessel is subjected to biaxial stress; normal stress existing in two directions

13. Spherical vessels  The material is also subjected to radial stress,  3. It has a value equal to pressure p at the interior wall and decreases to zero at exterior surface of the vessel 13 However, we ignore the radial stress component for thin-walled vessels, since the limiting assumption of r/t = 10, results in  3 and  2 being 5 and 10 times higher than maximum radial stress (  3 ) max = p

14. Cylindrical pressure vessel has an inner diameter of 1.2 m and thickness of 12 mm. Determine the maximum internal pressure it can sustain so that neither its circumferential nor its longitudinal stress component exceeds 140 MPa. Under the same conditions, what is the maximum internal pressure that a similar-size spherical vessel 14

15. Cylindrical pressure vessel Maximum stress occurs in the circumferential direction. From Eqn 8-1, we have 15  1 = ; pr t 140 N/mm 2 = p(600 mm) 12 mm p = 2.8 N/mm 2 Note that when pressure is reached, from Eqn 8-2, stress in the longitudinal direction will be  2 = 0.5(140 MPa) = 70 MPa.

16. Cylindrical pressure vessel Furthermore, maximum stress in the radial direction occurs on the material at the inner wall of the vessel and is (  3 ) max = p = 2.8 MPa. This value is 50 times smaller than the circumferential stress (140 MPa), and as stated earlier, its effects will be neglected. 16

17. Spherical pressure vessel Here, the maximum stress occurs in any two perpendicular directions on an element of the vessel. From Eqn 8-3, we have 17  2 = ; pr 2t 140 N/mm 2 = p(600 mm) 2(12 mm) p = 5.6 N/mm 2 Although it is more difficult to fabricate, the spherical pressure vessel will carry twice as much internal pressure as a cylindrical vessel.

18.  In previous chapters, we developed methods for determining the stress distributions in a member subjected to internal axial forces, shear forces, bending moments, or torsional moments.  Often, the x-section of a member is subjected to several of these type of loadings simultaneously  We can use the method of superposition to determine the resultant stress distribution caused by the loads 18

19. Application of the method of superposition 1. The stress distribution due to each loading is determined 2. These distributions are superimposed to determine the resultant stress distribution Conditions to satisfy  A linear relationship exists between the stress and the loads  Geometry of the member should not undergo significant change when the loads are applied 19

20.  This is necessary to ensure that the stress produced by one load is not related to the stress produced by any other load 20

21. Procedure for analysis Internal loading  Section the member perpendicular to its axis at the pt where the stress is to be determined  Obtain the resultant internal normal and shear force components and the bending and torsional moment components  Force components should act through the centroid of the x-section, and moment components should be computed about centroidal axes, which represent the principal axes of inertia for x-section 21

22. Procedure for analysis Average normal stress  Compute the stress component associated with each internal loading.  For each case, represent the effect either as a distribution of stress acting over the entire x- sectional area, or show the stress on an element of the material located at a specified pt on the x- section 22

23. Procedure for analysis Normal force  Internal normal force is developed by a uniform normal-stress distribution by  = P/A Shear force  Internal shear force in member subjected to bending is developed from shear-stress distribution determined from the shear formula,  = VQ/It. Special care must be exercised as highlighted in section 7.3. 23

24. Procedure for analysis Bending moment  For straight members, the internal bending moment is developed by a normal-stress distribution that varies linearly from zero at the neutral axis to a maximum at outer boundary of the member.  Stress distribution obtained from flexure formula,  = –My/I.  For curved member, stress distribution is nonlinear and determined from  = My /[ Ae ( R – y )] 24

25. Procedure for analysis Torsional moment  For circular shafts and tubes, internal torsional moment is developed by a shear-stress distribution that varies linearly from the central axis of shaft to a maximum at shaft ’ s outer boundary  Shear-stress distribution is determined from the torsional formula,  = T  / J.  If member is a closed thin-walled tube, use  = T /2 A m t 25

26. Procedure for analysis Thin-walled pressure vessels  If vessel is a thin-walled cylinder, internal pressure p will cause a biaxial state of stress in the material such that the hoop or circumferential stress component is  1 = pr / t and longitudinal  2 = pr / 2t.  If vessel is a thin-walled sphere, then biaxial state of stress is represented by two equivalent components, each having a magnitude of  2 = pr / 2t 26

27. Procedure for analysis Superposition  Once normal and shear stress components for each loading have been calculated, use the principle of superposition and determine the resultant normal and shear stress components  Represent the results on an element of material located at the pt, or show the results as a distribution of stress acting over the member ’ s x-sectional area 27