Complex stresses
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Shivendra Nandan Galgotias University
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Complex stresses
- 1. INTRODUCTION 1 The state of stress on any plane in a strained body is said to be ‘Compound Stress’, if, both Normal and Shear stresses are acting on that plane. For, example, the state of stress on any vertical plane of a beam subjected to transverse loads will, in general, be a Compound Stress. In actual practice the state of Compound Stress is of more common occurrence than Simple state of stress. In a compound state of stress, the normal and shear stress may have a greater magnitude on some planes which are inclined (or, Oblique) to the given stress plane. Hence it is necessary, in a compound state of stresses, to find (i) The normal and shear stress on a plane which is inclined (or, Oblique) to the given stress plane; (ii) The inclination of max. and min. normal stress planes and the values of the normal stress (max. / min.) on them; (iii) The inclination of max. shear stress planes and the values of the shear stress (max.) on them.
- 2. 2 A E D C B σy τ τ σx σx θ σy (i) Normal & Shear stress on plane inclined (Oblique) to given stress plane: Consider an element ABCD subjected to a state of compound stress as shown in the Fig. Let: σx Normal Stress in x- direction σy Normal Stress in y- direction τ Shear Stresses in x & y – directions θ Angle made by inclined plane AE wrt vertical σn Normal Stress on inclined plane τn Shear Stress on inclined plane Normal Stress, σθ, and Shear stress, τθ, on inclined plane are given by: )1(2sin2cos 22 −−−+ − + + = θτθ σσσσ σθ yxyx )2(2cos2sin 2 −−−− − = θτθ σσ τθ yx
- 3. 2 (ii) The inclination of max. and min. normal stress planes and the values of the normal stress (max. / min.) on them Let, θP be the inclination of the plane of max. or min. normal stress and σP be the value of the max. or min. normal stress on that plane, then, from Eqn. (1): (1)---2sin2cos 22 PP yxyx P θτθ σσσσ σ + − + + = ( ) ( ) ( ) ( ) 0 02cos2sin 2 02cos22sin2 2 0 dθ dσ min.,ormax.betoσFor P P =⇒ =− − ⇒ =+− − ⇒ = P PP yx PP yx θτ θτθ σσ θτθ σσ Thus, the condition for max. or min. normal stress to occur on a plane is, shear stress on that plane should be zero. These planes on which shear stress is zero and the normal stress on them being either the max. or the min. are called ‘PRINCIPAL PLANES’.
- 4. 2 ( ) ( ) (3)--- 2 2tan02cos2sin 2 have,we,0, − =⇒=− − = yx PPP yx P From σσ τ θθτθ σσ τθ The above Eqn. (3), gives two values for θP, which differ by 900 . Thus, there are two mutually perpendicular Principal planes, on which there are only normal stresses, shear stress being zero on them. On one of them, the value of the normal stress is the max.; it is called the ‘Major Principal plane’, the max. normal stress on it is called the ‘Major Principal Stress’. On the other principal plane, the value of the normal stress is the min.; it is called the ‘Minor Principal plane’, the min. normal stress
- 5. ( ) 2 2 2 21 21 2 21 2 )( 2/ 2cos 2 )( 2sin get,we, 2 2tanFrom τ σσ σσ θ τ σσ τ θ σσ τ θ + − − ±= + − ±= − = P P yx P (σx-σy)/ 2 2θP τ ±√[(σ x-σ y/2]2 +τ2 Substituting for sin 2θP and cos 2θP in Eqn. (1), we get the equation for principal stresses as: ( ) )4( 22 2 2 −−−+ − ± + = τ σσσσ σ yxyx P The above equation (4) gives two values for principal stresses. The numerically max. of the two values (+ ve or − ve) is the Major Principal Stress, (σMajor or σMax) and the numerically min. (+ ve or − ve) is the Minor Principal Stress (σMinor or σMin) .
- 6. (iii) Inclination of max. shear stress planes, Max. shear stress Equation. Let, θS be the inclination of the plane of max. or min. shear stress and τS be the value of the max. or min. shear stress on that plane, then, from Eqn. (2): )2(2cos2sin 2 −−−− − = SS yx S θτθ σσ τ ( ) ( ) 12tan2thaveWe: (5)--- 2 2tan 02sin22cos2 2 0 dθ d min.,ormax.betoFor S S S −=× − −=⇒ =−− − ⇒ = SP yx S SS yx anNOTE θθ τ σσ θ θτθ σσ τ τ Eqn. (5) gives two values for θS, which differ by 900 . Thus, there are two mutually perpendicular planes, on which shear stress are max.; numerically equal but opposite in sense. The planes of Max. Shear stresses are inclined at 450 to the Principal planes.
- 7. (σx-σy)/ 2 2θS τ ±√[(σ x−σ y)/2]2 +τ2 ( ) 2 2 2 21 21 2 21 2 )( 2/ 2sin 2 )( 2cos get,we, 2 2tanFrom τ σσ σσ θ τ σσ τ θ τ σσ θ + − − ±= + − ±= − −= S s yx S Substituting for sin 2θS and cos 2θS in Eqn. (2), we get the equation for Max. shear stresses as: )6(2 .max 2 2 .max −−− −±=⇒ + − ±= MinorMajor yx σστ τ σσ τ The above equation (6) gives two values for Max. shear stresses, which are numerically equal but opposite in sense.
- 8. 2EQUATINS, NOTATIONS & SIGN CONVENTIONS : Normal Stress, σθ, and Shear stress, τθ, on inclined plane are given by: )1(2sin2cos 22 −−−+ − + + = θτθ σσσσ σθ yxyx )2(2cos2sin 2 −−−− − = θτθ σσ τθ yx ( ) )4( 22 2 2 −−−+ − ± + = τ σσσσ σ yxyx P The equation for Principal stresses are: Inclination of Principal planes θP . There are two mutually perpendicular Principal planes (3)---- 2 2tan − = yx P σσ τ θ )6(2 .max 2 2 .max −−− −±=⇒ + − ±= MinorMajor yx σστ τ σσ τ Inclination of Max. shear stress planes θS = θP + 450 . Also, given by Eqn (5). The two mutually perpendicular Max. shear stress planes have equal & opposite shear stresses. (5)---- 2 2tan τ σσ θ − −= yx S
- 9. SIGN CONVENTIONS : A E D C B σy τ τ σx σx θ σy σx Normal Stress in x- direction σy Normal Stress in y- direction τ Shear Stresses in x & y – directions θ Angle made by inclined plane wrt vertical σθ Normal Stress on inclined plane AE τθ Shear Stress on inclined plane AE σθ τθ All the parameters are shown in their +ve sense in the Fig. Normal stresses, σ Tensile stresses +ve. Shear Stresses, τ, in x – direction & Inclined Plane Clockwise +ve. Shear Stresses, τ, in y – direction Anti-Clockwise +ve. Angle, θ measured wrt vertical, Anti-Clockwise +ve.
- 10. 1]. The state of stress at a point in a strained material is shown in Fig. (i) Locate the principal planes and find the principal stresses, (ii) Locate the Max. shear stress planes and find the max. shear stress. AD C B 32 30 30 80 80 32 Stresses are in MPa Inclination of Principal planes θP wrt Vertical : .104.09&09.14or,18.282 2 )32(80 30 2 2tan 000 ==⇒ −− = − = PP yx P θθ σσ τ θ Principal stresses, σP : ( ) ( ) (C)MPa39.53-(T);87.53MPa 30 2 )32(80 2 )32(80 22 2 2 2 2 +⇒ + −− ± −+ ⇒ + − ± + = τ σσσσ σ yxyx P
- 11. AD C B 32 30 30 80 80 32 Inclination of Max. shear stress planes θS = θP + 450 . θS = 14.090 + 450 = 59.090 &149.090 . (wrt Vertical) Max. shear stress τmax : MPa.53.63)53.39(53.87 .max ±=−−±⇒ −±= MinorMajor σστ 2]. The state of stress at a point in an elastic material is shown in Fig. Find the resultant stress on a plane AE inclined at 550 to horizontal. A E D C B 40 550 40 60 200 60 200 Stresses are in MPa Shear stress τ = 60 sin 20 = 20.52 Mpa Normal stress σx = 60 cos 20 = 56.38 Mpa Inclination of AE wrt Vertical θ = 90 – 55 = 350
- 12. A E D C B 40 20.52 20.52 56.38 56.38350 40 Stresses are in MPa Normal Stress, σθ, and Shear stress, τθ, on inclined plane AE: (T)MPa95.43 )352sin(52.20)352cos( 2 )40(38.56 2 4038.56 2sin2cos 22 +⇒ ×+× −− + − ⇒ + − + + = θτθ σσσσ σθ yxyx )(ClockwiseMPa26.38 )352cos(52.20)352sin( 2 )40(38.56 2cos2sin 2 +⇒ ×−× −− ⇒ − − = θτθ σσ τθ yx ( ) ( ) MPa.27.58 26.3843.95 2222 =⇒ +=+= θ θθθ τσ f f 0 11 04.41 95.43 26.38 tantan =⇒ = = −− α σ τ α θ θ Resultant stress, fθ and its inclination α :
- 13. 3]. The principal stresses at a point in a strained material are 80 MPa and 40 MPa both tensile. Find the normal, tangential and resultant stress on a plane inclined at 500 to the major principal plane. AD C B 40 80 80 40 Stresses are in MPa Since σx and σy are principal stresses, the shear stress, i.e, τ, on their planes is zero. Normal Stress, σθ, and Shear stress, τθ, on plane inclined at θ = 500 to plane of σx : 500 (T)MPa53.56 0)502cos( 2 4080 2 4080 2sin2cos 22 +⇒ +× − + + ⇒ + − + + = θτθ σσσσ σθ yxyx )(ClockwiseMPa68.190)502sin( 2 4080 2cos2sin 2 +⇒−× − ⇒ − − = θτθ σσ τθ yx
- 14. Resultant stress, fθ and its inclination α : AD C B 40 80 80 40 Stresses are in MPa 500 ( ) ( ) MPa.86.59 68.1956.53 2222 =⇒ +=+= θ θθθ τσ f f 0 11 19.19 53.56 68.19 tantan =⇒ = = −− α σ τ α θ θ 4]. At a certain point in a shaft, the normal stresses across two planes at right angles to each other are 60 MPa (C) and 40 MPa (T). The major principal stress is known to be 150 MPa (C). Find the shear stresses on the two said planes. Stresses are in MPa AD C B 40 60 60 40 τ τ ( ) ( ) MPa.130.77 2 4060 2 4060 150 22 2 2 2 2 ±=∴ + −− − +− =−⇒ + − ± + = τ τ τ σσσσ σ yxyx major
- 15. 5]. A 50mm×100mm tie member of a timber truss has a glued joint (shown in Fig.) at an inclination of 400 to the longitudinal axis. If it is subjected to an axial force of 200 kN, check whether there is a risk of failure. The permissible normal and shear stress for the joint are 25 MPa and 16 MPa respectively. P P A E 40 400 40 Stresses are in MPa Normal Stress σx = (200×103)/(50 ×100) σx = 40 MPa (T) Normal Stress, σθ, and Shear stress, τθ, on plane AE inclined at θ = 500 to vert. plane : (T)MPa5.160)502cos( 2 40 2 40 2sin2cos 22 +⇒+× + ⇒ + − + + = θτθ σσσσ σθ yxyx )(ClockwiseMPa68.190)502sin( 2 40 2cos2sin 2 +⇒−× ⇒ − − = θτθ σσ τθ yx τθ > 16 MPa, There is a risk of failure. ∴∴∴
- 16. 6]. Determine the max. safe uni-axial force that a strut of C/S area 60 ×103 mm2 can carry safely, if, permissible normal and shear stress are 25 MPa and 12.5 MPa respectively, on a critical plane (shown in Fig.) inclined at 300 to the vertical. A E σx 300 σx Let P be the max. safe load and σx be the stress under the load P. Normal Stress, σθ, and Shear stress, τθ, on plane AE inclined at θ = 300 to vert. : (1)----MPa33.33 0)302cos( 22 25 2sin2cos 22 +=⇒ +−× − + − =−⇒ + − + + = x xx yxyx σ σσ θτθ σσσσ σθ P P -(2)-MPa8.8720)302sin( 2 5.12 2cos2sin 2 =⇒−−× − = − − = x x yx σ σ θτθ σσ τθ From (1) and (2), max. safe σx = 28.87 MPa P = σx × a => 28.87 × 60 ×103 P = 1732.2 ×103 N = 1732.2 kN
- 17. 7]. At a transverse section of a beam of rectangular C/S, 100mm × 240mm, the BM is 76.8 kNm (Hogging) and SF is 96 kN. Find at that section, 40mm above the N.A. (i) Principal Stresses and (ii) Max. shear stress. 8040 80 120 100 120 C/S AD C B 5.33 26.67 26.67 5.33 State of stress At 40mm above N.A. (1) Bending stress σ and shear stress τ : ( ) ( ) MPa33.5 100102.115 80100801096 Ib yaF MPa(T)67.2640 102.115 108.76 y I M 6 3 y 6 6 = ×× ×××× == =× × × =×= τ σ (2) Principal stress σP and max. shear stress τmax : ( ) ( ) (C)MPa1.03-(T);MPa69.72 33.5 2 067.26 2 067.26 22 2 2 2 2 +⇒ + − ± + ⇒ + − ± + = τ σσσσ σ yxyx P M.I. → I = 100×2403 = 115.2×106 mm4
- 18. (2) Principal stress σP and max. shear stress τmax : AD C B 5.33 26.67 26.67 5.33 State of stress MPa.4.361)03.1(69.27 .max ±=−−±⇒ −±= MinorMajor σστ 8]. A propeller shaft of 200mm external diameter and 100mm internal diameter is subjected to a torque of 116.14 kNm, BM 10 kNm (sagging) and a thrust of 100 kN. Find the Max. and Min. principal stresses and inclinations of their planes. (1) Geometrical Propertise of the C/S : ( ) ( ) ( ) 4644 4644 2322 mm1026.147100200 32 JM.I.Polar mm1063.73100200 64 IM.I.Plane mm1056.23100200 4 AareaC/S ×=−×=→ ×=−×=→ ×=−×=→ π π π (2) Stresses at outermost layer at the top : (C)MPa82.1758.1324.4stress,NormalNet MPa43.109001 1026.147 1014.161 r J T Torque,toDue (C)MPa58.13001 1063.73 1010 r I M BM,toDue (C)MPa24.4 1056.23 10100 A P Thrust,toDue x 6 6 e 6 6 eb 3 3 d =+=+= =× × × =×= =× × × =×= = × × == bd σσσ τ σ σ
- 19. AD C B 109.43 17.82 17.82 109.43 State of stress PROBLEM (8) CONTINUED : (3) Principal stresses and Inclination of their planes : Principal stresses, σP : ( ) ( ) (T)MPa9.001(C);MPa7.181 43.109 2 082.17 2 082.17 22 2 2 2 2 +−⇒ + −− ± +− ⇒ + − ± + = τ σσσσ σ yxyx P Inclination of Principal planes θP wrt Vertical : .132.67&67.42or,34.852 2 082.17 109.43 2 2tan 000 ==⇒ − = − = PP yx P θθ σσ τ θ
- 20. MOHR’S CIRCLE : DEFINITION : Mohr’s circle is a graphical method for solving problems in compound stresses. It is defined as a circle drawn such a way that, the X and Y – coordinates of a point on the circumference of the circle representes, respectively, the normal and shear stresses, on any plane at a point in a material subjected to a Two-Dimensional stress system. Sign Conventions: Shear stresses on all the planes, (Hz., Vert., Inclined), are taken as +ve, if, anti-clockwise. All other sign conventions remain the same as for analytical method. Consider an element ABCD subjected to a state of compound stress as shown in the Fig. Let: A E D C B σy τ τ σx σxθ σy σx Normal Stress in x- direction σy Normal Stress in y- direction τ Shear Stresses in x & y – directions θ Angle made by inclined plane AE wrt vertical σθ Normal Stress on inclined plane τθ Shear Stress on inclined plane
- 21. MOHR’S CIRCLE : (I) CONSTRUCTION OF MOHR’S CIRCLE : 1]. Select a convenient point as the origine and draw the X and Y – axes. Choose convenient scale to plot the stresses. 2]. Plot the stresses on the vertical plane, taking σx as x-coordinate and τ as y-coordinate, to get the point ‘V’. 3]. Plot the stresses on the horizontal plane, taking σy as x-coordinate and τ as y-coordinate, to get the point ‘H’. 4]. Join the points ‘V’ and ‘H’ by a straight lne and mark its point of intersection with X-axis as ‘C’, to get the center of the Mohr’s Circle. 5]. With c as center and CV/CH as radius draw a circle to pass through H and V, to get the Mohr’s Circle.
- 22. MOHR’S CIRCLE : (II) TO FIND, PRINCIPAL STRESSES, THEIR PLANES AND MAX. SHEAR STRESSES, THEIR PLANES : 1]. Mark the points A and B at which the Mohr’s Circle cuts the X-axis. As y-coordinates of these points are zero, they represent the Principal stresses. 2]. Measure the lengths OA and OB, the max. of the two represents, to scale, the Major Principal stress and the min. represents the Minor Principal stress. 3]. Half of the angle VCA and half of the angle VCB represent the angles of the respective Principal planes wrt to the vertical. 4]. Mark the points D and E at the ends of the vertical diameter of the Mohr’s Circle. As y-coordinates of these points are the max., they represent the Max. shear stresses. 5]. Measure the lengths CD and CE, they represent, to scale, the Max. shear stress. 6]. Half of the angle VCD and half of the angle VCE represent the angles of the respective Max. shear stress planes wrt to the vertical.
- 23. MOHR’S CIRCLE : (III) TO FIND NORMAL STRESS, SHEAR STRESS AND RESULTANT STRESS ON ANY GIVEN INCLINED PLANE ‘AE’ : 1]. Draw the radial line CP in the Mohr’s Circle at an angle ‘2θ’ to CV. 2]. The x-coordinate of ‘P’ represents the Normal stress and the y-coordinate of ‘P’ represents the Shear stress. 3]. The length ‘OP’ represents the resultant stress on given plane ‘AE’. 4]. The angle POA represents the angle of the resultant stress on the given plane ‘AE’ wrt to the normal of the plane.