## Transcribed Text

1. (15 pt) Let the stress tensor be given by
in the (x,y,z) coordinate system. Consider a plane through a point in the (x,y,z) system
having unit normal [n] = [n x,ny, n, f.
Let the same stress tensor be given in the eigenvector system (x1, -~\, x3 ) as
0
and consider the same plane through the same point in the ( x1, x2 , x3 ) system
having the same unit normal now expressed as [nf = [n:, n;, n; f.
a. (7 pt) Express the stress vectors [Tn] and [T n] • associated with each coordinate system.
b. (4 pt) Express the normal component of the stress vector in each coordinate system: N = Tn • n and
N*=[r]*·[nf.
c. (4 pt) If [Q] is the transformation from the original (x, y,z) to the eigenvector system (xpx2 , x3 ) , how
would we get [G]* and [n] • from [G] and [n]? (. ,,,..-/ ,'ri,1 t-,-,μ / _
2. (25 pt) Consider the Airy stress function </J = f3y2 x + axy + C with zero body forces, where ex, /J
and C are constants. (a) (5 pt) Verify that it satisfies the biharmonic equation. (b) (5 pt) Determine the
in-plane stresses <Y x, <Y Y, r ;y . (c) (6 pt) Determine the tractions on the two rectangular boundaries
x = b, y = c (see figure). (d) (9 pt) As a plane strain solution, determine r xz , r yz, <Y, and all of the
strain components.
A e}
(O,c) (b,c)
e" .x
(0,0) (b,O)
3. (10 pt) Thermoelastic problems.
a. (3 pt) What is meant by the strains and temperature being uncoupled in the energy equation? How
does it help us? Do we need to know the strains to determine the temperature in the
uncoupled case?
b. (3 pt) The "mechanical" and thermal strains in an isotropic thermoelastic solid are given by:
M l+v v
eij = E (J" ij - E (J" kk sij
e; = a(T-T0 )Sij
Write the corresponding expression for the total strain eij .
Let v ,£,a > 0.
i . (2 pt) When <J"ij = 0 and T > T0 , what is the state of total strain eij? (What are the nonzero
total strain components? Expansion or contraction?)
ii. (2 pt) When O'ij = 0 and T < T0 , what is the state of total strain? (What are the nonzero
total strain components? Expansion or contraction?)
4. (25 pt) The assumed bending stress for a beam in flexure is
<J", = (Bx+ Cy)(l - z)
where l is the unsupported length of the beam, z is the length coordinate, Band Care constants, and the loads are
applied at the end z=l.
It is also assumed that
a. (1 pt) Why is the relation CJ', =(Bx+ Cy)(l - z) assumed?
b. (1 pt) Why are the relations <J"x = O'Y = 'r xy = 0 assumed?
c. (1 pt) What kind of General Solution Strategy is this - Direct, Inverse, or Semi-Inverse?
d. (1 pt) The boundary conditions at the free end are not given point-wise, but only in terms of resultant
forces and moments. What is the name of this common type of boundary condition?
e. (4 pt) Write the the general z-direction equilibrium equation. Next obtain from it the z-equilibrium
equation with a , = (Bx+ Cy )(l - z) .
f. (9 pt) Show that the z-equilibrium equation of part e can be written in the form:
o 1 2 o 1 2 - [ -r - - Bx ] + - [-r - - Cy ] = 0
ox xz 2 oy yz 2
Show (rewrite the equation) and explain why the stress function F (Eq. (9.8.3)) is assumed in a form
,,. __ oF +_!_Bx2 . ,,. ___ oF +_!_Cy2 including its derivatives . Note: • • (9 8 3)
xz oy 2 ' yz ox 2 . . .
g. (6 pt) Show (derive) how both compatibility equations expressed at Eq. (9.8.4) are needed to obtain the
governing equation (9.8.5).
o 2 vB
(Note: -(V F)+--=0 (9.8.4a),
oy l+v
o 2 ve --(V F)+-=0 (9.8.4b),
ox l+v
2 V V F=-(Cx-By)+k (9.8.S))
l+v
h. (2 pt) In the decomposition of the stress function
F(x, y) = ¢(x, y) + lf(x, y),
The governing equation is
2 V V F= - (Cx - By)-2μa
l+v
How do we know that the torsional part of the governing equation is
5. {25 pt) Monoclinic material under torsion.
For a monoclinic material (with the x-y plane the plane of symmetry), The following matrix equations express
Hooke's law, where Cij are stiffness moduli and Sij are compliance moduli:
<Y, ell -C12 Cn 0 0 Cl6 e, e, S11 S12 S13 0 0 s l6 <Y,
(J"y C12 C22 C23 0 0 c 26 eY eY S12 S22 S23 0 0 s26 {jy
<Y, C13 c 23 C33 = 0 0 c 36 e, e, Sn = S23 S33 0 0 s 36 <Y,
Ty, 0 0 0 C44 C45 0 2eYZ 2ey, 0 0 0 S44 S45 0 Ty,
Tzx 0 0 0 C45 C 55 0 2ezx 2ezx 0 0 0 S45 S 55 0 '"zx
rxy Cl6 c 26 c 36 0 0 c66 2exy 2exy Sl6 s 26 s36 0 0 s66 ~
a. (2 pt) Write equations for the four nonzero stress components caused by a non-zero ey (with all
other strain components zero.)
b. (2 pt) Write equations for the four nonzero strain components caused by a non-zero r xy (with
all other stress components zero.)
c. Consider a stress formulation for torsion of a monoclinic bar, with ax = <YY = r xy = 0 . The
governing PDE is (11.4.9):
s a2,y - 2s a2,y + s a2vr = -2
44 dX2 45 dXdY 55 dy2
The stress function is given by
If/'= J(Px2 + Q.xy + Ry2 -S)
i. (7 pt) Show the stress function satisfies the governing PDE by using the governing PDE to solve
for J.
ii. (7 pt) Obtain the stresses r,,, 'fy, .
iii. (7 pt) Write the three equilibrium equations. What can you conclude about the non-zero stress
components? (Which are a function only of (x,y), and which may be a function of (x,y,z)?)

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