Application Solutions of Plane Elasticity

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Presentation transcript:

Application Solutions of Plane Elasticity Professor M. H. Sadd

Solutions to Plane Problems Cartesian Coordinates Airy Representation Biharmonic Governing Equation R S Traction Boundary Conditions x y

Uniaxial Tension of a Beam y T 2l 2c

Note Integrated Boundary Conditions Pure Bending of a Beam x y M 2l 2c Note Integrated Boundary Conditions

Bending of a Beam by Uniform Transverse Loading x y w 2c 2l wl x/w - Elasticity x/w - Strength of Materials l/c = 2 l/c = 4 l/c = 3 Dimensionless Distance, y/c

Bending of a Beam by Uniform Transverse Loading x y w 2c 2l wl Note that according to theory of elasticity, plane sections do not remain plane For long beams l >>c, elasticity and strength of materials deflections will be approximately the same

Cantilever Beam Problem x y N P L 2c Stress Field Displacement Field

Cantilever Tapered Beam x y L p  A B Stress Field x = L x = L

Solutions to Plane Problems Polar Coordinates Airy Representation Biharmonic Governing Equation R S Traction Boundary Conditions x y  r 

General Solutions in Polar Coordinates

Thick-Walled Cylinder Under Uniform Boundary Pressure r1/r2 = 0.5 r/r2 r /p  /p Dimensionless Distance, r/r2 Internal Pressure Case

Stress Free Hole in an Infinite Medium Under Uniform Uniaxial Loading at Infinity r/a

Stress Concentrations for Other Loading Cases Biaxial Loading T Biaxial Loading T Unaxial Loading K=3 K=2 K=4

Stress Concentration Around Elliptical Hole x y b a ()max/S Circular Case (K=3)

Half-Space Under Concentrated Surface Force System (Flamant Problem) x y Y X r  C Normal Loading Case (X=0) Dimensionless Distance, x/a y/(Y/a) xy/(Y/a) y = a

Notch-Crack Problems Contours of Maximum Shear Stress y r    x  = 2 -  r  x Contours of Maximum Shear Stress

Two-Dimensional FEA Code MATLAB PDE Toolbox - Simple Application Package For Two-Dimensional Analysis Initiated by Typing “pdetool” in Main MATLAB Window Includes a Graphical User Interface (GUI) to: - Select Problem Type - Select Material Constants - Draw Geometry - Input Boundary Conditions - Mesh Domain Under Study - Solve Problem - Output Selected Results

FEA Notch-Crack Problem (vonMises Stress Contours)

Curved Beam Problem P  r a b  = /2 b/a = 4 a/P Dimensionless Distance, r/a a/P Theory of Elasticity Strength of Materials  = /2 b/a = 4

Disk Under Diametrical Compression = P Flamant Solution (1) + + Flamant Solution (2) Radial Tension Solution (3)

Disk Under Diametrical Compression + + = P 2 y x 1 r1 r2

Disk Results Theoretical, Experimental, Numerical Theoretical Contours of Maximum Shear Stress Finite Element Model (Distributed Loading) Photoelastic Contours (Courtesy of Dynamic Photomechanics Laboratory, University of Rhode Island)