Stress Functions. First Semester, Academic Year 2012 Department of Mechanical Engineering Chulalongkorn University

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1 Lecture Note 3 Stress Functions First Semester, Academic Year 01 Department of Mechanical Engineering Chulalongkorn Universit 1

2 Objectives #1 Describe the equation of equilibrium using D/3D stress elements Set well-posed elasticit problems (e.g. for finite element analses) Solve simple elastic problems b the inverse and semiinverse method

3 Concepts #1 Equation of equilibrium is the governing equation. Well-posed problems can be formulated b substituting appropriate constitutive relationships, loads, initial conditions and boundar conditions into the governing equations With appropriate conditions from constitutive relations, the stress distribution can be found b fitting the boundar conditions into the appropriate stress functions 3 3

4 Equilibrium Equilibrium Element X, Y, Z are bod forces per unit volume 4 4

5 Equilibrium Equilibrium Complementar Principal of Shear M z 0 ( )( z) ( )( z) ( )( z ) ( )( z ) 0 V V V V V V 0 As 0 and 0,, etc

6 Equilibrium Equilibrium 3D Equations F 0 ( )( z) ( z) ( )( z) ( z) X, Y, Z are bod forces per unit volume ( z z z)( ) z ( ) X( z) 0 z z X 0 z z Y 0 z z z z Z 0 z 6 6

7 Equilibrium Equilibrium D Plane Stress D Reduction z z z 0 z z X 0 z z Y 0 z z z z Z 0 X 0 Y 0 7 7

8 Equilibrium Equilibrium D Plane Strain D Reduction z const, z z 0 z z z z z z X Y 0 0 z Z 0 z X 0 z z Y 0 Z 0 8 8

9 Equilibrium Equilibrium Components on an Inclined Plane 9 9

10 Equilibrium Equilibrium Boundar Conditions 3D Scale Up Unit thickness F 0 1 Xs X 0 As 0, X d ds d ds X l m n z Y l mzn X l m Z zl zmzn Y l m l cos m cos 10 10

11 Equilibrium Equilibrium Strain Compatibilit #1 z z v u v u w v v u z u z w z z z z z z z z 11 11

12 Equilibrium Equilibrium Strain Compatibilit # z v u u w ( ) ( ) z z z z u v w u ( ) ( ) z z z z z z ( ) z z z ( z ) z z z z z ( ) z z 1 1

13 Equilibrium Equilibrium Strain Compatibilit #3 z z z z z z z z z z ( ) z z z ( z ) z z z ( z z ) z 13 13

14 Equilibrium Equilibrium D Strain Compatibilit D Reduction 0 or const, 0 z z z z z z z z z z z z z z z z z ( ( z ( z z z z ) z z ) z z ) 14 14

15 Problem D Elastic Problems 3D Set Up #1 Governing equations z X 0 z z Y 0 z z z z Z 0 z 15 15

16 Problem D Elastic Problems 3D Set Up # Constitutive equations 1 ( z) E 1 ( z ) E 1 ( ) z z E G z z G z z G z z z u v w z u v v w z w u z 16 16

17 Problem D Elastic Problems 3D Set Up #3 z z z z z z z z z z ( ) z z z z ( ) z z z ( z z ) z Compatibilit condition 17 17

18 Problem D Elastic Problems Plane Stress Set Up #1 Governing equations Constitutive equations 1 X 0 ( ) E 1 Y 0 ( ) E 1 (1 ) G E 0 X X X Y 0 Y Y 18 18

19 Problem D Elastic Problems Plane Stress Set Up # X Y Compatibilit condition (1 ) ( ) ( ) (1 ) X Y (1 )( ) 19 19

20 Problem D Elastic Problems Plane Stress Set Up #3 X Y (1 )( ) X Y (1 )( ) X Y (1 )( ) X Y (1 )( ) X Y (1 )( ) 0 0

21 Problem D Elastic Problems Plane Stress Set Up #4 Compatibilit condition X Y (1 )( ) X Y ( )( (1 ) )( ) 1 1

22 Problem D Elastic Problems Plane Stress Set Up #5 Neglect the bod force ( )( ) 0 0 ( )( ) 0 ( )( ) Assume a function 0 4 4,,

23 Air Stress Function Air Stress Function Neglect the bod force ( )( ) 0 To solve the equations, use stress functions that satisf relationships,, Substitute into the compatibilit condition

24 Plate Inverse Method Procedure Select the stress function Check compatibilit Fit in the boundar condition Advantage Simplified assumptions Disadvantage Find problems that fit solution 4 4

25 Plate Eample Timoshenko 1 #1 For an elastic plate subjected to static loads shown, determine the stress function. The A, B and C are constants. Domain & Boundar conditions 5 5

26 Plate Eample Timoshenko 1 # Describe stress distributions within the plate A B C , 0, C,, A B 6 6

27 Plate Eample Timoshenko 1 #3 A B C, C, A, B Let A 1, B, C 3 6 MPa, MPa, MPa 7 7

Plate Eample Timoshenko 1 #4 6 MPa, MPa, MPa Plane stress, z 0 MPa 1 3

28 Plate Eample Timoshenko 1 #4 6 MPa, MPa, MPa Plane stress, z 0 MPa MP 6.83 MPa 1.17 MPa 0MPa 6.83 MPa T v v 1 3 ( ) ( ) MPa 8 8

29 Plate Eample Timoshenko #1 For an elastic plate subjected to static loads shown, determine the stress function. The D is a constant. Boundar conditions For left & right edges: D, 0 For top & bottom edges: 0, 0 9 9

30 Plate Eample Timoshenko # A B C D , 0, Describe stress distributions Satisf the condition if For left & right edges C D C D D For top & bottom edges A B A B 0 For all edges B C B C

31 Plate Eample Timoshenko #3 D, 0, 0 Let D 1 MPa/m, a m, b 1 m 1, 0,

32 Plate Eample Timoshenko 3 #1 For an elastic plate subjected to static loads shown, determine the stress function. The B is a constant. State the domain & boundar conditions 3 3

33 Plate Eample Timoshenko 3 # A B C D E A, C, E A 4 C E 0 C D E Thus, E ( C A) C D ( C A) A B C B D C 33 33

34 Plate Eample Timoshenko 3 #3 Describe stress distributions C D ( C A) A B C B D C Satisf the condition if C D (C A B C B C D C A ) 34 34

35 Plate Eample Timoshenko 3 #4 Let A C D 0, B MPa 0 MPa, MPa, MPa MPa MPa 35 35

36 Plate Eample Timoshenko 3 #5 Let A 1 MPa, B MPa, C 3 MPa, D 4 MPa MPa, 3 MPa, 6 MPa 36 36

37 Beam Semi-Inverse Method Procedure Select the stress function b assuming reasonable assumptions Check compatibilit Fit in the boundar condition Advantage More versatile stress functions Disadvantage Inaccuracies in regions which assume the St. Venant s principle 37 37

38 Beam Eample Beam 1#1 For an elastic plate subjected to static loads shown, determine the stress distribution and displacements. Assume unit thickness. State the domain & boundar conditions 38 38

39 Beam Eample Beam 1 # B 3 A , 0, B B, 0, A 39 39

40 Beam Eample Beam 1 #3 B B, 0, A h Top and bottom edges BC: 0 at Bh Bh A 0 A 4 8 Bh B B, 0, 8 Right edge BC: P da at 0 3 h / ( Bh B ) Bwh P wd BI B P/ I h / 8 1 P 1P B 3P, 0, ( 4 ) ( 4 ) h h I wh 8 wh 40 40

41 Beam Eample Beam 1 #4 The shear force over the free end as the integration of shear stress No resultant normal force across the sections All sections, including the built-in in end, are free to distort. Then, find displacement for plane stress problems 41 41

42 Beam Eample Beam 1 #5 1 ( ) P E E EI u u 1 ( ) P E E EI v ( ) z E w z 1 z G u v v w z z w u z z 4 4

43 Beam Eample Beam 1 #6 P u P P u d f( ) EI EI EI P v P P v d g( ) EI EI EI P u v ( h 4 ) 8GI P P P ( h 4 ) f( ) g( ) 8GI EI EI Ph P P f ( ) P 8GI GI EI EI g ( ) Ph P g ( ) P f ( ) P F( ) G( ) 8GI EI EI GI 43 43

44 Beam Eample Beam 1 #7 Ph P g( ) P f ( ) P 8GI EI EI GI Ph G ( ) F ( ) GF 8 GI Ph As const, F( ) Fconst and G( ) Gconst. 8GI P g( ) g( ) P G G EI EI P g ( ) ( Gd ) EI 3 P g ( ) G C 1 6EI 44 44

45 Beam Eample Beam 1 #8 Ph P g ( ) P f ( ) P 8GI EI EI GI Ph G ( ) F ( ) G F 8GI Ph As const, F( ) Fconst and G( ) Gconst. 8 GI P f ( ) P f ( ) P P F F EI GI GI EI P P f( ) ( F) d GI EI 3 3 P P f ( ) F C 6GI 6EI 45 45

46 Beam Eample Beam 1 #9 3 3 P P P P u f ( ) F C EI EI 6GI 6EI 3 P P P v g( ) G C1 EI EI 6 EI Boundar condition at the built-in end: L, 0, u v 0 C 0 3 PL C 1 GL 6EI 3 dv PL PL L, 0, 0 G, C1 d EI 3EI Ph Ph PL Ph G F F G 8G I 8GI EI 8GI 46 46

47 Beam Eample Beam 1 #10 u v 3 3 P P P PL Ph ( ) EI 6 GI 6 EI EI 8 GI 3 3 P P PL PL EI 6EI EI 3EI Ans The v relationship at 0 can be predicted b simple beam. These are displacements without shear of neutral plane

48 Beam Eample Beam 1 #11 P Ph ( h 4 ) at NA 8GI 8GI Ph Deflections of NA due to shear strain ( L ) 8GI 3 3 P PL PL Ph v 0 ( L ) Ans 6 EI EI 3 EI 8 GI 48 48

49 Beam Eample Beam 1 #1 Let L 5 m, h 1 m, w 0.3 m, P 0. MN, E 00 GPa, 0.3, G E (1 ) P P, 0, ( h 4 ) I

With appropriate conditions from constitutive relations, the stress distribution can be found by fitting the boundary

With appropriate conditions from constitutive relations, the stress distribution can be found by fitting the boundary Lecture Note 3 Stress Functions First Semester, Academic Year Department of Mechanical ngineering Chulalongkorn Universit Objectives # Describe the equation of equilibrium using D/3D stress elements Set