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Important_Relations_and_Results.pdf

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Appendix Important Relations and Results PL AE 2. Temperature stress in bar, s = a t E = a t s/e 3. Net strain in the direction of s1, e1 = s1/E – s2/E – s3/E 1. Elongation of a bar, ⌬ = 4. Relation between elastic constants, E = 2G(1 + ) = 3K(1 – 2 ) = 9KG 3K + G 5. Normal stress on an inclined plane = s cos2 u 6. Shear stress on an inclined plane = 7. Strain energy stored in a bar = 1 s 2 sin 2q P2L s 2 1 = 3 volume = 3 stress 3 strain 3 volume 2 AE 2 E 2 t2 2G 9. Maximum bending momen
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  Appendix ImportantRelations and Results  1. Elongation of a bar,   = PL AE   2. Temperature stress in bar, s  = a  t E  =  a  t   s / e  3. Net strain in the direction of s 1 ,  e 1  = s 1 /  E   – s 2 /  E   – s 3 /  E   4. Relation between elastic constants,  E   = 2 G (1 + ) = 3  K  (1 – 2 ) = 93 KGK G +  5. Normal stress on an inclined plane = s  cos 2 u  6. Shear stress on an inclined plane = 12  2 s q  sin  7. Strain energy stored in a bar = P L AE E  2 2 2 2 = s  3  volume = 12  3  stress 3  strain 3  volume 8. Shear strain energy per unit volume = t  2 2 G  9. Maximum bending moments in standard cases: In a cantilever with a point load at free end = Wl   In a cantilever with a uniformly distributed load = wl Wl 2 2 2 =  In a simply supported beam with a point load at mid-span = Wl 4  In a simply supported beam with a point load W   at a distance a  from one end of the span l   = Wa l al ( ) -  In a simply supported beam carrying a uniformly distributed load = wl Wl 2 8 8 =  10. The moment of inertia of a rectangular lamina of sides b  and d   about centroidal axis parallel to side b  = bd  3 12 SM RattanAppendix indd731 SM-RattanAppendix.indd 731  3/8/20114:44:35PM 3/8/2011 4:44:35 PM  732  Appendix  11. The moment of inertia of circular lamina = p  d  4 64  12. The relation governing the simple bending of beam is s   y M  I  E  R = =  13. The diameter of kern of a circular cross-section of diameter d   is d  /4. 14. Shear centre of a semi-circular arc is at 4 r  / p  15. The flexural rigidity of a beam is  EI   16. Deflection and slope of standard cases: Cantilevers: A point load at the free-end: Deflection of free end = Wl EI  3 3  (maximum)Slope of free end = Wl EI  2 2  (maximum) A point load at the mid-span, deflection of free end = 548 3 Wl EI   A uniformly distributed load, deflection of free end = wl EI  4 8  (maximum)Slope of free end = wl EI  3 6  (maximum) Simply supported beam:  Central point load, Deflection at mid-span = Wl EI  3 48  (maximum) Slope at mid-span = Zero Slope at the support = Wl EI  2 16  (maximum) Uniformly distributed load, Deflection at mid-span = 5384 4 Wl EI   (maximum) Slope at the supports = wl EI  3 24  Fixed beam: Central point load, Maximum deflection = Wl EI  3 192 Uniformly distributed load, Maximum deflection = wl EI  4 384  17. The relation governing the torsional torque in circular shafts is T  J r Gl = = t q   18. Torsional rigidity of a shaft = GJ   = Tl  / u  19. Maximum shear stress of a solid shaft is given by 16 3 T d  p  SM RattanAppendix indd732 SM-RattanAppendix.indd 732  3/8/20114:44:36PM 3/8/2011 4:44:36 PM   Appendix  733  20. Deflection of a closely coiled helical spring under axial load = 32 8 2434 WR lG d WD nGd  p  =  21. Shear stress in a closed-coiled helical spring under an axial load = 8 3 WDd  p   22. The angle of twist of a closely coiled helical spring = 64 4 TDn Ed   23. The equivalent stiffness of two springs joined in series, ss ss s =+ 1 21 2  24. The equivalent stiffness of two springs joined in parallel,  s =  s 1 =  s 2  25. The equivalent lengths of columns for different types of end conditions: both ends hinged, l  e  = l  one end fixed and the other free, l  e  = 2 l   both ends fixed, l  e  = l  /2one end fixed, other hinged, l l e  =  / 2  26. Euler crippling load for columns with both ends hinged = p  22  EI l  27. In a thin cylinder, hoop stress =  pd t  2 ; longitudinal stress =  pd t  4  28. In a thin spherical shell, hoop stress =  pd t  4  29. The volumetric strain in a thin spherical shell = 341  pd tE  ( ) - n   30. Hoop stress induced in a rotating ring, s r w r  q   = ◊ = ◊ r v 2 2 2  31. In a solid rotating disc, at the centre of the disc, s s  n rw  q  r   R = =+ 38 2 2  At the outer surface, s  n rw  q   =- 14 2 2  R ,  s r   = 0 32. In a hollow rotating disc, Radial stresses are zero at inside and outside radii. s r   is maximum at r R R i o =  and is 38 2 2 +- n rw   ( )  R R o i s u  is maximum at inner radius and is rw n n  22 2 41 3[( ) ( ) ] - + +  R R i o At outer radius, s  rw n n  q   = + - - 22 2 43 1[( ) ( ) ]  R R i o  33. In a long rotating solid cylinder, the radial stress at the centre = ( )( )3 28 1 2 2 -- n n rw   R  34. In plastic bending:Moment of resistance at first yield, M   y  = ( s  y / s w )  ?  M  w  Moment of resistance in fully plastic state, M   p  = SZ  s  y Load factor,  L  = W  c / W   = S  ( s  y / s c ) = Shape factor   Factor of safety SM RattanAppendix indd733 SM-RattanAppendix.indd 733  3/8/20114:44:36PM 3/8/2011 4:44:36 PM
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