Homework SourseA circular beam is loaded as shown in the following figure T
A circular beam is loaded as shown in the following figure. The dimensions a, b and d, the applied forces F_1 and F_2, the reaction force R_0 and moment M_0 at the wall, and the elasticity E, yield strength S_y and ultimate strength S_ut are known values. You are given the moment as a function of distance down the length of the beam to be: M(x) = M_0 degree + R_0 ^1 - F_1 ^1 - F_2 ^1 (Refer to section 4-6 if you have forgotten singularity function notation. Section 4-3 reviews the relation between moment and deflection.) Can you simplify the moment expression, assuming values of x on [0, b] are of interest? Determine the deflection at the end (x = b) of the beam Find the conservative factor of safety for static failure if the beam is made from a brittle material. Assume that bending normal stress at the top fiber of the beam is the stress of concern. Specify which failure criteria you used and why.
Solution
The free-body diagram of the beam’s right segment sectioned through an arbitrarypoint shown in Fig.
a
will be used to write the shear and moment equations of the beam.
*6–4.
Draw the shear and moment diagrams for the canti-lever beam.
2 kN
/
m6 kN
m2 m
A
‚(1)
a
‚(2)
+©
M
=
0;
-
M
-
2(2
-
x
)
c
12 (2
-
x
)
d
-
6
=
0
M
=
{
-
x
2
+
4
x
-
10}kN
#
m
+c©
F
y
=
0;
V
-
2(2
-
x
)
=
0
V
=
{4
-
2
x
} kNThe shear and moment diagrams shown in Figs.
b
and
c
are plotted using Eqs
