1 ) Par t A) Set up but do not solve the integr al for m of
the ar ea moment of iner tia about the y-axis for the
shaded r egion. M ake sur e to include pr oper limits
of integr ation. (8 pts)
Any of the four solutions shown below are valid.
2
Iy
x
2
1
0
x2
dx
4
1 2 y
x 2 dxdy
Iy
0
0
2 1
x 2 dydx
Iy
0 x2
4
1
Iy
0
2 3/ 2
y dy
3
1
2 y 3/ 2 dy
0
2
Par t B) The centr oid for the channel’s
cr oss-sectional ar ea is located at
= 2 in.
Deter mine the ar ea moment of iner tia, Ix’,
about the x’-axis. For maximum par tial
cr edit, show your calculations in the
tabular for mat below. (1 2 pts)
1
3
Part
y˜
I˜x
A
dy
Ix'
1
3
(1/12)(2)(6)3
(2)(6)
-1
48
2
1
(1/12)(12)(2)3
(2)(12)
1
32
3
3
(1/12)(2)(6)3
(2)(6)
-1
48
Ix’ channel = _____1 2 8 in4 ______________
/ 20
2 ) Par t A) Deter mine the pr oduct of iner tia for the flanged shape shown above. For maximum par tial
cr edit, show your calculations in the tabular for mat below. (1 4 pts)
Part
Ix'y'
dx
dy
A
Ixy
(1)
0
1.1
–5.9
32
–207.68
(2)
0
–1.9
–0.4
18
13.68
(3)
0
0.1
5.1
12
6.12
–187.88 in.4
Par t B) Compute the pr oduct of iner tia for the r ight
tr iangle with r espect to the x and y axes shown in the
sketch. (6 pts)
A
Ix' y'
I xy
1
bh
2
1
(75 mm)(45 mm) 1, 687.50 mm 2
2
1 2 2
bh
72
1
(75 mm) 2 (45 mm) 2 158, 203.1 mm 4
72
Ix' y' dxd y A
158.203.1 mm 4 (80 mm)( 50 mm)(1,687.50 mm 2 )
/ 20
6,591, 796.9 mm
4
4 ) Par t A) For the L1 5 2 x 1 0 2 x 1 2 .7 r olled steel
angle shape shown to the r ight, the moments and
pr oduct of iner tia with r espect to the x and y axes
have been computed and ar e
Ix
7.24 106 mm 4 Iy
2.61 106 mm 4 Ixy
2.54 106 mm 4
Deter mine the following:
i) the or ientation of the pr incipal axes of the section
about O. Sketch the pr incipal axes and label the
or ientation on the dr awing shown at r ight
ii) the values of the pr incipal moments of iner tia of
the section
iii) the moments and pr oduct of iner tia of the section
with r espect to the x’ and y’ axes for ming an angle of 6 0 °
with the x and y axes. (1 8 pts)
Par t B) Select the most appr opr iate answer . For a given ar ea with a set of r ectangular x and y axes
located at a point O, M ohr ’s cir cle is a method that can be used to calculate _____________. (4 pts)
(a)
the or ientation of the pr incipal axes for an axis system with origin at point O
(b)
the pr incipal moments of inertia for an axis system with or igin at point O
(c)
the moments of inertia with respect to any other pair of r ectangular axes passing
thr ough O
(d)
the pr oduct of iner tia with respect to any other pair of rectangular axes passing
thr ough O
(e)
the moments and pr oducts of inertia with r espect to a pair of r ectangular axes
passing thr ough O at an angle of 4 5 ° counter clockwise fr om the x and y axes
(f)
all of the above
_______________________________________________________________________________________
The remainder of this page may be used as extra space. Be sure to reference this work
space if used!!
/ 22
4 ) A 3 0 0 -kN for ce is applied at point E of the r od
assembly and is applied in a dir ection that is par allel to the x-axis. The suppor t at A is a balland-socket joint. The suppor t at D is a smooth
jour nal bear ing for which the couple moment
r eactions shall be neglected. Suppor t r od BC is
par allel to the x-axis and should be tr eated as a
two-for ce member .
SET UP (BUT DO NOT SOLVE) the equilibr ium
equations to find all suppor t r eactions. W r ite
the moment equilibr ium equations so that m oments ar e summed about a set of axes passing
thr ough the or igin. (2 4 pts)
/ 24