Static analysis with external distributed mass

1

I am importing a cantilever beam bdf file from commercial package( Ansys ) to Gmsh and converting it to XDMF using meshio and importing that XDMF file into fenics.I have to incorporate distributed point mass of 2kg at the free end of this beam.
image (5)

I am unable to implement this in coding.
The geometry and xdmf file is attached here
The MWE without point mass is attached as-

from dolfin import *
mesh = Mesh()
with XDMFFile("mesh/tetra.xdmf") as infile:
    infile.read(mesh)

def eps(v):
    return sym(grad(v))

E, nu = 200e9, 0.3  
rho = 7850
mu = E/2/(1+nu)
lmbda = E*nu/(1+nu)/(1-2*nu)
def sigma(v):
    return lmbda*tr(eps(v))*Identity(3) + 2.0*mu*eps(v)
rho_g=7850*9.8066     
f = Constant((0, -rho_g, 0))
V = VectorFunctionSpace(mesh, 'Lagrange', degree=2)
du = TrialFunction(V)
u_ = TestFunction(V)
a = inner(sigma(du), eps(u_))*dx
l = inner(f, u_)*dx
support = CompiledSubDomain("near(x[0], 0, tol) && on_boundary", tol=1e-14) 
bc = DirichletBC(V, Constant((0.,0.,0.)), support)
u = Function(V, name="Displacement")
solve(a == l, u, bc)
file_results = XDMFFile("elasticity_results.xdmf")
file_results.write(u)

Can you please help in incorporating distributed point mass of 2kg at the free end of this beam.

2

Hi, you should consider a distributed surface load q of intensity q=mg/S where w=2 kg and S is your beam cross-section. You should also tag the corresponding end-boundary either using gmsh or manually, for instance, assuming a beam of length L in the x direction:

def end(x, on_boundary):
    return near(x[0], L) and on_boundary
facets = MeshFunction("size_t", mesh, 2)
AutoSubDomain(end).mark(facets, 1)

then use a ds measure with the appropriate facets markers and add the corresponding load:

ds = Measure("ds", domain=mesh, subdomain_data=facets)
S = assemble(1*ds(1))
q = Constant((0, 0, -m*g/S))
l = dot(q, u)*ds(1)