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# bentbar_moving

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# bentbar_moving   { BENTBAR_MOVING.PDE

This problem is a moving mesh variant of BENTBAR.PDE

}

title "Timoshenko's Bar with end load"

variables

U(1e-6)           { X-displacement }

V (1e-6)         { Y-displacement }

Xm = move(x)

Ym = move(y)

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=GLOBALMAX(abs(v)) { for grid-plot scaling }

mag=1/(amplitude+1e-6)

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))

{ Timoshenko's analytic solution:  }

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

initial values

U = 0

V = 0

equations             { the displacement equations }

! force the displacements to evolve in pseudo-time, to allow a smooth deformation of the mesh.

! the time scale of these equations is arbitrary

U:  dx(Sx) + dy(Txy) = dt(U)

V:  dx(Txy) + dy(Sy) = dt(v)

! the mesh surrogate variables.  They move at the same rate as the material deformation

Xm: dt(Xm) = dt(U)

Ym: dt(Ym) = dt(V)

boundaries

region 1

start (0,-hW)

load(U)=0         { free boundary on bottom, no normal stress }

line to (L,-hW)

value(U) = Uexact { clamp the right end }

mesh_spacing=hW/10

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

line to (L,hW)

load(U)=0         { free boundary on top, no normal stress }

mesh_spacing=10

line to (0,hW)

line to close

time 0 to 1e-8 !by 1e-10

plots

for cycle=1

! x and y have already been moved by u and v, but this is small compared to mag*u, etc.

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

end "JqARWPEgsA1HmFOIXNmCNfb+qWfQGus/TmzjXhVHKZ2Q6NnpPJOCREi6wK8g8g+Pnvy7DriMhUjFAl1qcBgg5yF+MkmYxGJoetheissilVHa1rvSKiA1A0QLkuYpmTaJG2+/v5g68e+flbisgxXY2m4KBHCsFJGbJsy2FFnplf6"