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# resolve

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# resolve

{ RESOLVE.PDE

This is a test problem from Timoshenko: Theory of Elasticity, p41

The RESOLVE statement has been added to force regridder to resolve the

shear stress.

}

title "RESOLVE shear stress in bent bar"

select

elevationgrid=500

cubic

variables

U   { X-displacement }

V   { Y-displacement }

definitions

L = 1               { Bar length }

hL = L/2

W = 0.1             { Bar thickness }

hW = W/2

eps = 0.01*L

I = 2*hW^3/3       { Moment of inertia }

nu = 0.3           { Poisson's Ratio }

E  = 2.0e11         { Young's Modulus for Steel (N/M^2) }

{ plane stress coefficients }

G  = E/(1-nu^2)

C11 = G

C12 = G*nu

C22 = G

C33 = G*(1-nu)/2

amplitude=1e-6     { a guess for grid-plot scaling }

mag=0.1/amplitude

force = 250         { total loading force in Newtons (~10 pound force) }

dist = 0.5*force*(hW^2-y^2)/I     { Distributed load }

Sx = (C11*dx(U) + C12*dy(V))       { Stresses }

Sy = (C12*dx(U) + C22*dy(V))

Txy = C33*(dy(U) + dx(V))

Vexact = (force/(6*E*I))*((L-x)^2*(2*L+x) + 3*nu*x*y^2)

Uexact = (force/(6*E*I))*(3*y*(L^2-x^2) +(2+nu)*y^3 -6*(1+nu)*hW^2*y)

Sxexact = -force*x*y/I

Txyexact = -0.5*force*(hW^2-y^2)/I

small = 1e-5

initial values

U = 0

V = 0

equations { define the displacement equations }

U:  dx(C11*dx(U) + C12*dy(V)) + dy(C33*(dy(U) + dx(V))) = 0

V:  dx(C33*(dy(U) + dx(V)))   + dy(C12*dx(U) + C22*dy(V)) = 0

{  force regridder to resolve the shear stress.

Avoid the ends, where the stress is extreme. }

resolve (Txy, 100*(x/L)*(1-x/L))

boundaries

region 1

start (0,-hW)

{ free boundary on bottom, no normal stress }

{ clamp the right end }

value(U) = Uexact     line to (L,0) point value(V) = 0

line to (L,hW)

{ free boundary on top, no normal stress }

{ apply distributed load to Y-displacement equation }

plots

grid(x+mag*U,y+mag*V)   as "deformation"   { show final deformed grid }

elevation(V,Vexact) from(0,0) to (L,0) as "Center Y-Displacement(M)"

elevation(V,Vexact) from(0,hW) to (L,hW) as "Top Y-Displacement(M)"

elevation(U,Uexact) from(0,hW) to (L,hW) as "Top X-Displacement(M)"

elevation(Sx,Sxexact) from(0,hW) to (L,hW) as "Top X-Stress"

elevation(Sx,Sxexact) from(0,0) to (L,0) as "Center X-Stress"

elevation(Txy,Txyexact) from(0,hW) to (L,hW) as "Top Shear Stress"

elevation(Txy,Txyexact) from(0,0) to (L,0) as "Center Shear Stress"

elevation(Txy,Txyexact) from(hL,-hW) to (hL,hW) as "Center Shear Stress"

end