REVIEW : STRESS TRANFORMATIONS II SAMPLE PROBLEMS INTRODUCION TO MOHR S CIRCLE

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1 LECTURE #16 : 3.11 MECHANICS OF MATERIALS F03 INSTRUCTOR : Professor Christine Ortiz OFFICE : PHONE : WWW : REVIEW : STRESS TRANFORMATIONS II SAMPLE PROBLEMS INTRODUCION TO MOHR S CIRCLE

2 Review Lecture #14 : Stress Transformations I 1. Plane Stress Rotations : Given a state of plane stress, what is the equivalent stress state on an element rotated b an arbitrar angle, θ + CCW defined relative to -ais original coordinate ais : /, stresses in original coordinate ais : σ, σ,,θ new coordinate ais : /, stresses in new coordinate ais : σ, σ,,θ σ O σ σ θ O θ σ Consider a free bod diagram of a wedge where a cut is made along an inclined plane where the transformed stresses are desired : σ A o A o θ A o tanθ A o /cosθ O σ A o tanθ θ σ A o /cosθ stresses var continuousl as the ais is rotated : Equations of Static Equilibrium, Geometr, Trig σ + σ ( σ σ )cos( θ) σ ' + + sin( θ) σ + σ ( σ σ )cos( θ) σ ' θ σ σ θ ( )sin( ) '' + θ cos( ) (sin( ) STRESS TRANSFORMATION EQUATIONS :

3 Principal and Maimum Shear : Principal Stresses: σ 1, + σ σ σ σ ± + σ 1 ma imum normal stress σ min imum normal stress σ 1 σ σ ma θ p1 σ σ σ min Principal Angles / Planes : tan( θ p) σ σ θ p1 σ1, θ p σ O θ p Maimum Shear Stresses: ma,min ma min σ σ ± + ma imum ( + ) shear stress min imum ( ) shear stress σ ma σ Planes / Angles of Maimum Shear : σ σ tan( θ s) θ, θ ( ) s1 ma s min O θ s1 Relationships Between Principal and Shear : θ s θ p ± 45 θ p1 ma and θ θs1 and θ o o p 90 apart (mutuall perpendicular planes) o s 90 apart (mutuall perpendicular planes) ( σ1 σ )

4 Derivation of General Equation for Principal Stresses σ + σ ( σ σ )cos( θ) σ ' + + sin( θ) ( )cos( ) θ σ σ σ + σ σ σ θ σ ' (sin( θ) tan( p)

5 Sample Problem #1 : Stresses on an Inclined Element An element in plane stress is subjected to : σ -16,000 psi σ 6000 psi 4000 psi Find and draw the stresses acting on an element inclined at an angle θ45 o. O z-face or / plane STRESS TRANSFORMATION EQUATIONS : σ + σ ( σ σ )cos( θ) σ ' + + sin( θ) σ + σ ( σ σ )cos( θ) σ ' θ σ σ θ ( )sin( ) '' + θ cos( ) (sin( )

6 Sample Problem #1 : Stresses on an Inclined Element An element in plane stress is subjected to : σ -16,000 psi σ 6000 psi 4000 psi Find and draw the stresses acting on an element inclined at an angle θ45 o. STRESS TRANSFORMATION EQUATIONS : σ + σ ( σ σ )cos( θ) σ ' + + sin( θ) σ + σ ( σ σ )cos( θ) σ ' θ σ σ θ ( )sin( ) '' + θ cos( ) (sin( )

7 Sample Problem #1 : Stresses on an Inclined Element O

8 Sample Problem : Principal and Maimum Shear Stresses An element in plane stress is subjected to : σ 1300 psi σ -400 psi psi (a) Determine the principal stresses and show them on a sketch of a properl oriented element (b) Determine the maimum shear stresses and show them on a sketch of a properl oriented element O z-face or / plane

9 Sample Problem : Principal and Maimum Shear Stresses An element in plane stress is subjected to : σ 1300 psi σ -400 psi psi (a) Determine the principal stresses and show them on a sketch of a properl oriented element (b) Determine the maimum shear stresses and show them on a sketch of a properl oriented element σ + σ ( σ σ )cos( θ) σ ' + + sin( θ)

10 Sample Problem : Principal Stresses O

11 Sample Problem : Principal and Maimum Shear Stresses An element in plane stress is subjected to : σ 1300 psi σ -400 psi psi (a) Determine the principal stresses and show them on a sketch of a properl oriented element (b) Determine the maimum shear stresses and show them on a sketch of a properl oriented element tan( θ s) ma, min ( σ σ ) σ σ ± +

12 Sample Problem : Maimum Shear Stresses O σ + σ ( σ σ )cos( θ) σ ' + + sin( θ) σ + σ ( σ σ )cos( θ) σ ' θ σ σ θ ( )sin( ) '' + θ cos( ) (sin( )

13 MOHRS CIRCLE Stress Transformation Equations : σ + σ ( σ σ )cos( θ) σ ' + + sin( θ) ( )cos( ) σ + σ σ σ θ σ ' (sin( θ) ( σ σ )sin( θ) '' + cos( θ) (3) (1) ()

14 MOHRS CIRCLE ( σ ' σ ave) + '' R σ R ave ( σ + σ ) σ σ +

15 CONSTRUCTION OF MOHR S CIRCLE ( σ ' σ ave) + '' R σ ave ( σ + σ ) R Given stress state : σ σ σ + O C O θ z-face or / plane σ

MECHANICS OF MATERIALS

CHAPTER 7 Transformations MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech Universit of Stress and Strain 006 The McGraw-Hill Companies,