Project concerning modern theory

Project in Partial di¤erential equations -
Assignment 2
given by Dag Lukkassen
The compulsory assignment must be formed as a scienti…c report, with table
of contents, summary, references, pictures, page numbers, etc. max. 10 pages
with 12 pt. text size. The problem will be dependent of a parameter s which is
determined by the student as follows:
s =
(Your mothers age) (Your age)
:
20
Note that each subtask task must be performed by each student individually
without using any parts done by others. Recall that the …nal report must be
attached to the examination paper. For the Ansys task below, a print of the
log…le from Ansys must be attached to the report.
1 Torsion of unbrako fastener
An allen-wrench (also called unbrako fastener) is usually de…ned as a L-shaped
bar with a hexagonal head, used to turn screws with hexagonal sockets (see
Figure 1). However, for simplicity we will consider the case of an allen-wrench
with quadratic head , of which we are going to calculate the torsional rigidity.
In order to do this we have to solve the following partial di¤erential equation:
8
<
:
@2u @x2 +
1
@2u @x2 2
= 2 x = (x1; x2) 2 ;
u = 0 on @ ;
where is the quadrat = [ s; s] 2 (see Figure 2):
a) Find the corresponding weak formulation of (1), and explain why this formulation
has a unique solution.
Figure 1: Allen-wrench with a hexagonal head
1
(1)

Figure 2: Allen-wrench with a quadratic head
b) Find a numerical solution of the above problem by using the …nite element
method presented for the 2-dimensional case in the compendium with N = 3
(i.e. use the triangulation with 3 3 interior nodes and h = s=2): Note that in
order to …nd the numerical solution you have to solve a problem of the type
A = b:
This problem can easily be solved (e.g. by using the program Scienti…c Workplace)
after you have found the sti¤ness matrix A and the vector b. The matrix
A is found exactly as in the lectures. For the computation of b, note that
each component bi can be found by …rst calculating the volume of the pyramid
bounded by the x1; x2 plane and the shape of the basis-function vi.
c) When the allen-wrench is subject to a torque T; the cross section parallel
with the x1; x2 plane and at distance x3 from the x1; x2 plane will be rotated at
an angle x3: The parameter is called the relative twist. The torsional rigidity
is de…ned by
It is possible to show that
= T :
Z
= 2G
u dx;
where G is the shear modulus of the material and u is the solution of (1). Find
an approximate value of for the case when G = 1 by using the numerical
solution found in b).
2

d) The problem (1) is of the same type as the heat conductivity problem for
which the conductivity = 1 and u denotes the temperature. The thermal
energy E is given by
E = 1
2
Explain why the torsional rigidity
Z
jgrad uj 2 dx = 1
Z
2
= 2GE;
jgrad uj 2 dx:
e) Use the FEM-program Ansys to solve the above problem more accurately
than above by using a much larger …nite dimensional function space consisting
of quadratic polynomials (8 node elements called plane77 with element edge
length 0.07). In particular, …nd the torsional rigidity for the case when G = 1
by calculating the thermal energy E and then multiply with 2 (as shown in
the previous subtask). Compare with the value you found in b). A print of
the log…le from Ansys must be attached to the report. Since you may not be
familiar with Ansys we add some step by step guidelines in the Appendix below:
f) The shear stresses 13 and 23 of the allen-wrench near its head are given by
13 = G @u
; 23 = G
@x2
@u
:
@x1
Illustrate 13 and 23 by plotting the gradient of u (the ”temperature”) in x1
and x2 direction. Where in is it most likely that the material fail?
2 Appendix (step by step guidelines for the Ansys
task)
Before you start Ansys, turn o¤ Norman Virus Control (stopp sanntidssøkning).
Start Ansys launcer, choose to save the …les on C:n
Preferences: thermal, OK
Preprocessor: Element Type: Add, Add, Thermal solid, 8 node 77 (=plane77),
ok, close
Preprocessor: Material Props: Material Models: material model number 1:
Thermal; Conductivity, isotropic, conductivity KXX=1, ok.
Preprocessor: Modelling: create, areas rectangle, by dimensions (sidelengths=4
! x1 = s, x2 = s, y1 = s; y2 = s), ok.
Preprocessor: Meshing, size cntrls, manual size: global, size, SIZE Element
edge length 0.07. ok.
File: write db log …le: test.lgw, ok. File save as jobname.db. ok
Preprocessor: Mesh, areas, free, click on area with the mouse, ok.
Save db.
Solution, De…ne loads, apply, thermal, temperature, on lines, click on all four
boundary lines with the mouse, OK, Lab2=Temp, Value Load TEMP value= 0:
3

Solution, De…ne loads, apply, Heat Generat, On areas, click on the area with
the mouse, iok, VALUE Load HGEN value = 2,ok.
Solution, solve, current LS, OK, Solution is done!close.
General Postproc, Read results, Last set.
General Postproc, Element table, De…ne table, add, Lab=energy, Item,
Comp Results data item = energy, ok. Close.
General Postproc, Element table, Sum og each item, Energy= .......
General Postproc, Plot results, Contour plot, nodal solu, Nodal solution,
Thermal Gradient, x-component of thermal gradient, ok. (example).
General Postproc, Plot results, Contour plot, nodal solu, Nodal solution,
DOF solution, Temperature OK. (example).
File: write db log …le: test.lgw, ok. File save as jobname.db. ok
File exit.
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