Theories of Failure 2.1 RANKINE S THEORY OR MAXIMUM NORMAL STRESS THEORY

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1 CHPTER Theories of ailure n the previous chapter, we have en that a member is subjected to any of the simple stress tensile, compressive, shear or bending stress then it is easy to predict the failure of the member. But in practice machine members will be subjected to more than one type of stress simultaneously and hence it will be difficult to predict the failure of such machine members using the simple stress theories. n order to predict the failure of such members subjected to combined stress, the following theories of failure are being suggested by different people: (i)rankine s theory or maximum normal stress theory. (ii) Guest s theory or maximum shear stress theory. (iii)hencky-von-mis theory or distortion energy theory or shear energy theory. (iv) Saint Venant theory or maximum strain theory. BX STRESSES WTH SHER STRESS y.1 RNKNE S THERY R MXMUM NRM STRESS THERY x x igure.1 shows an element subjected to stress, s x action along x-direction (tensile or compressive), s y acting along y-direction ^ lr to x (tensile or ig..1 compressive) combined with shear stress, t xy. ccording to the maximum normal stress theory or Rankine s theory of failure, equivalent stress s e 1 ( s s ) ( s s ) x + y + x - y + t 5 0 xy... P 58. y

2 THERES URE 55 PRBEMS Problem 1: machine element is subjected to the following stress s x 0 MPa, s y 5 MPa, t xy 0 MPa. ind the factor of safety if it is made of C5 steal having yield stress as 5 MPa, using the following theories of failure. (i)maximum principal stress theory, (ii) Maximum shear stress theory, (iii)shear energy theory, and (iv) Maximum strain theory taking Poisson ratio as 0. Given data: s x 0 MPa, s y 5 MPa, t xy 0 MPa yield stress, s ys 5 MPa Poisson ratio v 0.. (i)ccording to maximum principal stress or Rankine s theory of equivalent stress \ S s ys s e 1 ( s + s ) + ( s ) + t...(5-0) s e 1 ( ) + ( 0-5 ) + ( 0 ) (ii) ccording to max. shear stress theory or Guest s theory equivalent stress 8. MPa t e 1 ( s ) + t...(5-1) or s e ( s ) + t Q t e ( 0-5) + ( 0) 1.85 MPa K J \ S s ys 5/ s e (iii)ccording to shear energy theory or Hencky-Von-Mis theory, equivalent stress \ S s ys s e s + s s + t...(5-) s e MPa (iv) ccording to Max-Strain theory or Saint-Venant theory. Equivalent stress

3 5 DESGN MCHNE EEMENTS s e 1 ( 1 - v )( s + s ) + ( 1 + v ) ( s ) + t...(5-) ` s e 1 ( 1-0. )( ) + ( ) ( 0-5 ) + ( 0 ) \ S s ys 7.95 MPa Problem : M.S. shaft having yield stress as MPa is subjected to the following stress. s x 10 MPa, s y 0 MPa and t xy MPa. ind the factor of safety using: (i)rankine s theory of failure, (ii) Guest s theory of failure and (iii)von-mis theory of failure. Given data: Yield stress, s ys MPa s x 10 MPa, s y 0 MPa and t xy MPa. ccording to Rankine s theory or maximum normal stress theory of failure s e 1 ( s + s ) + ( s ) + t s e 1 ( 10-0 ) + [ 10 - ( - 0 )] + ( ) S s ys (ii) ccording to Guest s theory or max shear stress theory of failure t e 1 ( s ) + t or s e ( s ) + t [ 10 -(- 0)] + ( ) s e MPa \ S s ys (ii) ccording to Hencky-Von-Mis theory or shear energy theory of failure s e s + s s + t 1.9 MPa 10 + (-0) - 10 (- 0) + ( ) MPa \ S s ys

4 THERES URE 57 Problem : machine member is subjected to the following stress s x 150 MPa, t xy MPa. ind the equivalent stress as per the following theories of failure. (i)shear stress theory, (ii)normal stress theory, (iii)von-mis theory. Given data: s x 150 MPa, t xy MPa (s y Not given)(s y 0, Not given) (i) ccording to maximum shear stress theory, equivalent stress s e ( s ) + t s e MPa (ii) ccording to maximum normal stress theory, equivalent stress s e 1 ( s + s ) + ( s ) + t s e ( ) MPa (iii) ccording to Von-Mis theory, equivalent stress s e s + s s + t s e ( ) MPa. Problem : ind the diameter of a rod subjected to a bending moment of knm and a twisting moment of 1.8 knm according to the following theories of failure, taking normal yield stress as 0 MPa and factor of safety as. (i)normal stress theory, (ii)shear stress theory. Given data: Bending moment, M b knm 10 N-mm Twisting moment, M t 1.8 knm N-mm Yield stress, s ys 0 MPa S \ llowable stress, s s e s ys S 0 10 MPa Bending stress, s M b C 10 d/ ( pd / ) d s d s x Shear stress, t Mr t J d/ ( pd / ) d t xy

5 58 DESGN MCHNE EEMENTS (i)ccording to maximum normal stress theory, (Here s y 0, no stress in ^lr direction) s e 1 ( s + s + ( s ) + t 10 1 \ d 1.8 mm (ii) ccording to maximum shear stress theory d d d s e ( s ) + t d d \ d.7 mm \ Recommended diameter, d.7 ~ mm. (Take bigger one always). Problem 5: bolt is subjected to a tensile load of 18 kn and a shear load of 1 kn. The material has an yield stress of 8. MPa. Taking factor of safety as.5, determine the core diameter of bolt according to the following theories of failure. (i)rankine s theory, (ii) Shear stress theory, (iii)shear energy theory and (iv) Saint Venant s theory. Take Possion ratio 0.98 Given data: Tensile load, T 18 kn N Shear load, s 1 kn 1 10 N Yield stress, s ys 8. MPa S.5 \ llowable stress, s e s ys S Tensile stress, s T Shear stress, t s (s y 0, not given) 11. MPa. s x t xy

6 THERES URE 59 (i)ccording to Rankine s theory of failure x x xy s e 1 s + s + t pd c \ Core dia, dc 15.5 mm (ii) ccording to maximum shear stress theory, s e s + t x xy \ 8. pd c \ Core dia, dc mm (iii)ccording to Von-Mis theory of failure s e s + t 11. x xy pd c \ Core dia, dc 1. mm (iv) ccording to Saint Venant s theory of failure M s e 1 ( 1 - v )( s ) + ( 1 + v ) s + t x x xy ( ) ( ) M pd c \ Core dia, dc mm.

7 0 DESGN MCHNE EEMENTS Problem : SE 105 steel rod (s ys 09.9 MPa) of 80 mm diameter is subjected to a bending moment of knm and torque T. Taking actor of safety as.5, find the maximum value of torque T that can be safely carried by rod according to: (i)maximum normal stress theory, (ii)maximum shear stress theory. Given data: Material SE 105. Yield stress, s ys 09.9 MPa S.5 diameter d 80 mm \ llowable stress, s e s ys S MPa. 5 Bending moment, M b knm 10 N-mm. \ Bending stress, s M b C 10 ( 80/ ) ( p/ 80 ) 59.8 MPa s x Torque, M t T \ Shear stress, t M t r J T ( 80/ ) ( p/ 80) ( ) MPa \ t t xy ( ) T (s y 0, not given) (i)ccording to maximum normal stress theory x x xy s e 1 s + s + t T (. 10 ) \ Torque, T N-mm knm (ii) ccording to maximum shear stress theory x xy t e 1 s + t ssuming, t e 0.5 s e MPa T (. 10 ) Torque, T N-mm 5. knm.

8 THERES URE 1 Problem 7: stresd element is loaded as shown in ig... Determine the following: (i)von-mis stress, (ii) Maximum shear stress, (iii)maximum normal stress, (iv)ctahedral shear stress. Given data: rranging in descending order 150 ³ 150 > 100 \ s MPa, s 150 MPa and s 100 MPa (compressive) ig.. (i) Von-Mis stress 100 MPa 150 MPa 150 MPa t e ( s1 ) + ( s ) + ( s-s1) ( ) + ( ) + ( ) (ii)maximum shear stress t 1 s t s 150 -(-100) 15 MPa t 1 s 1- s 150 -(-100) 15 MPa \ t max 15 MPa (max of the values) (iii) Maximum normal stress s 1 > s > s 50 MPa then s max s MPa. (iv) ctahedral shear stress t e 1 ( s ) + ( s ) + ( s ) ( ) + ( ) + ( ) MPa. Problem 8: material has a yield strength of 00 MPa. Compute the factor of safety for each of the failure theories for the each of the following stress: (i) s 1 0 MPa, s 10 MPa, s 0, (ii) s 1 0 MPa, s 180 MPa, s 0,

9 THERES URE (a)von-mis theory, s e - ( s1 ) + ( s ) + ( s-s1) s e ( 0-180) + ( 180-0) + ( 0-0).97 MPa \ S t ys t e (b)max. normal stress theory, s e s 1 0 MPa \ S s ys (c)max. shear stress theory t 1 s MPa t s MPa t 1 s MPa \ t max 10 MPa t e s ys \ S t max (iii) s 1 0, s 180 MPa, s 0 MPa (a) Von-Mis theory, s e ( ) + ( ) + ( 0+ 0).9 \ S s ys (b)max. normal stress theory, s e s 1 0 \ S 00 µ But, in compression S

10 DESGN MCHNE EEMENTS (c)max. shear stress theory t 1 s t s t 1 s S 1 1 s ys t max MPa MPa MPa \ t max 10 MPa 1.8. Problem 9: hot rolled bar has yield stress of 90 MPa. Compute the factor of safety for the following theories of failure: (i)maximum normal stress theory, (ii) Maximum shear stress theory and (iii)distortion energy theory for the following states of stress. (a) s 1 5 MPa, s 5 MPa, s 0 (b) s 1 5 MPa, s 10 MPa, s 0 (c) s 1 5 MPa, s 0, s 10 MPa. Given data: Yield stress, s ys 90 MPa S s ys (a) s 1 5 MPa, s 5 MPa, s 0 s 1 > s > s (i) Maximum normal stress theory, s e s 1 5 MPa \ S (ii) Maximum shear stress theory t 1 s t s MPa t 1 s 1- s MPa \ t e t max 11.5 MPa and S s ys t max