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 well-posed elasticit problems (e.g. for finite element analses Solve simple elastic problems b the inverse and semiinverse method Concepts # quation 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 X, Y, Z are bod lement forces per unit volume 3 4
Complementar Principal of Shear M ( ( ( ( ( ( ( ( V V V V V V As and,, etc. 5 3D quations X, Y, Z are bod forces per unit volume F ( ( ( ( ( ( ( ( ( X( X Y Z 6 D Plane Stress D Reduction X Y Z X Y 7 D Plane Strain D Reduction const, X Y Z X Y Z 8
Components on an Inclined Plane 9 3D Scale Up Boundar Conditions F X s X As, d d X ds ds X l m Y l m Unit thickness X l m n Y l m n Z l m n cos m cos Strain Compatibilit # v u v u w v v u u w Strain Compatibilit # v u u w ( ( u v w u ( ( ( ( ( 3 l
Strain Compatibilit #3 ( ( ( 3 D Strain Compatibilit D Reduction or const, ( ( ( 4 Problem D lastic Problems 3D Set Up # Governing equations X Y Z Problem D lastic Problems 3D Set Up # Constitutive equations ( ( ( G G G u v w u v v w w u 5 6 4
Problem D lastic Problems 3D Set Up #3 ( ( ( Compatibilit condition 7 Problem D lastic Problems Plane Stress Set Up # Governing equations X Y Constitutive equations ( ( ( G X X X Y Y Y 8 Problem D lastic Problems Plane Stress Set Up # X Y Compatibilit condition ( ( ( ( X ( ( Y 9 Problem D lastic Problems Plane Stress Set Up #3 X Y ( ( X Y ( ( X Y ( ( X Y ( ( X Y ( ( 5
Problem D lastic Problems Plane Stress Set Up #4 Compatibilit condition X Y ( ( X Y ( ( ( ( Problem D lastic Problems Plane Stress Set Up #5 ( ( Assume a function,, Neglect the bod force ( ( ( ( 4 4 4 4 4 Air Stress Function Air Stress Function Neglect the bod force ( ( To solve the equations, use stress functions that satisf relationships,, Substitute into the compatibilit condition 4 4 Inverse Method Procedure Select the stress function Check compatibilit Fit in the boundar condition Advantage Simplified assumptions Disadvantage Find problems that fit solution 3 4 6
ample Timoshenko # For an elastic plate subjected to static loads shown, determine the stress function. The A, B and C are constants. Domain & Boundar conditions ample Timoshenko # Describe stress distributions within the plate A B C,, 4 4 4 4,, C A B 5 6 ample Timoshenko #3 A B C, C, A, B Let A, B, C 3 6 MPa, MPa, MPa 7 ample Timoshenko #4 6 MPa, MPa, MPa Plane stress, MPa 3 6.83 MPa.7 MPa MPa T 3 6.83 MPa v v ( ( 6.3 MPa 8 7
ample Timoshenko # 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, For top & bottom edges:, ample Timoshenko # Describe stress distributions A B C D 6 6,, 4 4 4 4 3 3 Satisf the condition if For left & right edges C D C D D For top & bottom edges A B A B For all edges B C B C 9 3 ample Timoshenko #3 D,, Let D MP MPa/m, a m, b m,, ample Timoshenko 3 # 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 8
ample Timoshenko 3 # A B C D 6 6 4 3 3 4 A, C, 4 4 4 4 A4C Thus, ( C A C D C D ( C A A B C B D C ample Timoshenko 3 #3 Describe stress distributions C D ( C A A B B C C B D C Satisf the condition if C D (C C A A B C B D C 33 34 ample Timoshenko 3 #4 Let A C D, B MPa MPa, MPa, MPa MPa MPa ample Timoshenko 3 #5 Let A MPa, B MPa, C 3 MPa, D 4 MPa 3 4 7 MPa, 3 MPa, 6 MPa 35 36 9
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 ample # For an elastic plate subjected to static loads shown, determine the stress distribution and displacements. Assume unit thickness. State the domain & boundar conditions 37 38 ample # A B 3 6, 4 4 4 4, B B A ample #3,, B B A Top and bottom edges BC: at A Bh A Bh 4 8,, Bh B B 8 Right edge BC: P da at 3 h / Bh B Bwh P ( wd BI B P/ I h / 8 P P B 3 P,, ( 4 ( 4 h h I wh 8 wh h 39 4
ample #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 end, are free to distort. Then, find displacement for plane stress problems ample #5 ( P I ( P I ( G u v w u v v w w u 4 4 ample #6 P u P P u d f( I I I P v P P v d g( I I I P u v ( h 4 8 GI P P P ( h 4 f( g( 8 GI I I Ph P P f ( P g ( 8GI GI I I Ph P g ( P f ( P F ( G ( 8GI I I GI ample #7 Ph P g ( P f ( P 8GI I I GI Ph G ( F ( G F 8 GI Ph As const, F( Fconst and G( Gconst. 8 GI P g ( g ( P G G I I P g ( ( Gd I 3 P g ( G C 6 I 43 44
ample #8 Ph P g ( P f ( P 8 GI I I GI Ph G ( F ( G F 8 GI Ph As F ( Fconst and G( Gconst. 8 GI P f ( P f ( P P F F I GI GI I f( P P GI I F d ( f ( P 3 3 P F C 6 GI 6 I ample #9 3 3 P P P P u f( F C I I 6GI 6I 3 P P P v g( G C I I 6I Boundar condition at the built-in end: L,, u v C 3 PL C GL 6 I 3 dv PL PL L,, G, C d I 3I Ph Ph PL Ph GF F G 8 G I 8GI I 8GI 45 46 ample # 3 3 P P P PL Ph u ( I 6GI 6I I 8GI 3 3 P P PL PL v Ans I 6I I 3I The v relationship at can be predicted b simple beam. These are displacements without shear of neutral plane. ample # P h Ph ( 4 at NA 8GI 8GI Ph L Deflections of NA due to shear strain ( 8 GI 3 3 P PL PL Ph v ( L Ans 6I I 3I 8GI 47 48
ample # Let L 5 m, h m, w.3 m, P. MN, GPa,.3, G ( P P,, ( h 4 8 I 49 3