Theory of Elasticity, Article 18, Generic 2D Stress Distributions

Transcription

1 Theory of Elasticity, Article 18, Generic 2D Stress Distributions One approach to solve 2D continuum problems analytically is to use stress functions. Article 18 of the third edition of Theory of Elasticity by Timoshenko & Goodier, published in 1969 by McGraw- Hill, contains several versions of polynomial stress functions, Φ, all satisfying the governing equation 4 Φ x Φ x 2 y + 4 Φ 2 y = 4 although some with certain conditions on the constant coefficients in Φ. In the following the stress functions are visualized to understand the stress pattern implied by each of them. In this document the notation and axis directions are somewhat different than in the book. Each section below displays the following items: $ Stress function, Φ $ Axials stress in x-direction, σ xx $ Axial stress in y-direction, σ yy $ Shear stress, τ xy $ Sometimes a plot to visualize the stresses Stress function x 2 (Constant σ yy ) Φ = C x2 x 2 ; 2 C x2 Examples Updated January 7, 218 Page 1

2 Stress function y 2 (Constant σ xx ) Φ = C y2 y 2 ; 2 C y2 Stress function xy (Constant τ xy ) Φ = C xy x y; -C xy Stress function x 3 (Linearly varying σ yy along x-axis) Φ = C x3 x 3 ; 6 x C x3 Examples Updated January 7, 218 Page 2

3 Stress function x 2 y (Linearly varying σ yy and τ xy ) Φ = C x2y x 2 y; 2 y C x2y -2 x C x2y plotvalues = {C x2y 1}; σxxplot = Plot3D[σ xx /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σxx]; σyyplot = Plot3D[σ yy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σyy]; σxyplot = Plot3D[τ xy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel τxy]; Stress function xy 2 (Linearly varying σ xx and τ xy ) Φ = C xy2 x y 2 ; Examples Updated January 7, 218 Page 3

4 2 x C xy2-2 y C xy2 plotvalues = {C xy2 1}; σxxplot = Plot3D[σ xx /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σxx]; σyyplot = Plot3D[σ yy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σyy]; σxyplot = Plot3D[τ xy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel τxy]; Stress function y 3 (Linearly varying σ xx along y-axis) Φ = C y3 y 3 ; 6 y C y3 Examples Updated January 7, 218 Page 4

5 Stress function with x 4 and x 2 y 2 When using stress functions with fourth-order terms it is no longer guaranteed that the differentiation equation is satisfied. For example, by itself the term x 4 by itself does not work. This problem is mitigated by adding other fouth-order terms. For example: Φ = C x4 x 4 + C x2y2 x 2 y 2 ; This stress function satisfies the differential equation under this condition: solution = Solve[ x x x x Φ + 2 x x y y Φ + y y y y Φ ] C x4 - C x2y2 3 When that condition is satisfied the stresses are: /. solution[[1]] 2 x 2 C x2y2 /. solution[[1]] -4 x 2 C x2y2 + 2 y 2 C x2y2 /. solution[[1]] -4 x y C x2y2 Those expressions and the following plots shows that, e.g., σ xx varies parabolically with x. Examples Updated January 7, 218 Page 5

6 plotvalues = {C x2y2 1}; σxxplot = Plot3D[σ xx /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σxx]; σyyplot = Plot3D[σ yy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σyy]; σxyplot = Plot3D[τ xy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel τxy]; Stress function x 3 y Φ = C x3y x 3 y; 6 x y C x3y -3 x 2 C x3y Examples Updated January 7, 218 Page 6

7 plotvalues = {C x3y 1}; σxxplot = Plot3D[σ xx /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σxx]; σyyplot = Plot3D[σ yy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σyy]; σxyplot = Plot3D[τ xy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel τxy]; Stress function xy 3 Φ = C xy3 x y 3 ; 6 x y C xy3-3 y 2 C xy3 Examples Updated January 7, 218 Page 7

8 plotvalues = {C xy3 1}; σxxplot = Plot3D[σ xx /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σxx]; σyyplot = Plot3D[σ yy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel σyy]; σxyplot = Plot3D[τ xy /. plotvalues, {x,, 1}, {y,, 1}, AxesLabel {x, y}, Ticks {{}, {}, {}}, PlotLabel τxy]; Examples Updated January 7, 218 Page 8