{ BENTBAR.PDE
*****
This is a test problem from Timoshenko: Theory of Elasticity, pp41-46
A cantilever is loaded by a distributed shearing force on the free end,
while a point at the center of the mounted end is fixed.
The solution is compared to Timoshenko's analytic solution.
*****
}
title "Timoshenko's Bar with end load"
select
cubic { Use Cubic Basis }
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=GLOBALMAX(abs(v)) { for grid-plot scaling }
mag=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
initial values
U = 0
V = 0
equations { define the displacement equations }
U: dx[Sx] + dy[Txy] = 0
V: dx[Txy] + dy[Sy] = 0
boundaries
region 1
start (0,-hW)
load(U)=0 { free boundary on bottom, no normal stress }
load(V)=0
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 }
load(V)=0
mesh_spacing=10
line to (0,hW)
load(U) = 0
load(V) = dist { apply distributed load to Y-displacement equation }
line to close
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(Txy,Txyexact) from(0,0) to (L,0) as "Center Shear Stress"
end