Lame's Equation.pptx

Lame's Theory facilitates the calculation of stresses in Thick Cylinders

1. Lame’s equation Class Notes By Dr. Sewa Singh Professor, CTIEMT , Jalandhar

2. Lame’s Theory Consider a thick cylinder Let length of cylinder = l Internal radius = r1 External radius = r2 Uniformly distributed internal pressure intensity = p1 Uniformly distributed external pressure intensity = p2 Assumptions: The cylinder material is linear, homogeneous and isotropic. Plane transverse section remaining plain under the pressure. (As a result of this assumption the longitudinal strain is constant at all the points in the cylinder wall, i.e. independent of radius)

3. Lame’s Theory Let: (all assumed tensile) Therefore, longitudinal Strain: + )] , E, , are all constant Therefore, + ………Eq. 1  

4. Lame’s Theory Consider an annular ring of cylinder between radii r and r+ δ r as shown in fig. (a) Let: = Internal Radial Stress = External Radial Stress (as shown in fig. (b))   Considering the equilibrium of half the ring: Simplifying and neglecting small terms, we get: . +2 .r  

5. Lame’s Theory For equilibrium of the ring Resisting Force = Bursting Force i.e. In limiting case, we get Substituting in eq. 1 we get  

6. Lame’s Theory Or Integrating, we get  

7. Lame’s Theory Taking Anti log on both sides (Where B is another constant)  

8. Lame’s Theory Substituting Eq.3 in Eq.1 Eq. 3 & Eq4 are known as Lame’s Equations Value of the constants A and B can be calculated from the boundary conditions at and At , At ,  

9. Lame’s Theory Substituting in Eq.3 we get and Therefore so  

10. Lame’s Theory And  

11. Lame’s Theory Therefore And  

12. Lame’s Theory (Special Cases) Internal Pressure only , i.e. , Therefore And  

13. Lame’s Theory (Special Cases) External Pressure only , i.e. , Therefore And  

14. Lame’s Theory (Special Cases) Solid Circular Shaft having external radial pressure e only , i.e. and Therefore Thus the radial and hoop stresses are equal and constant throughout the shaft and are of same nature.  

15. Lame’s Theory (Special Cases) Longitudinal Stress Consider the cross-section of the thick cylinder with closed ends subjected to internal and external pressures (usual notations). Therefore, for horizontal equilibrium: i.e. Note that eq.13 gives the value same as that of constant A, i.e. Longitudinal stress is constant throughout the length of cylinder  

16. Lame’s Theory (Special Cases) Maximum Shear Stress Shear stress will be maximum at, Therefore,