FAKULTI KEJURUTERAAN MEKANIKALUNIVERSITI TEKNIKAL MALAYSIA MELAKA
TUTORIAL 4SOLID MECHANICS 2 (BMCS 2333)
SEMESTER 2 SESSI 2005/2006Statically Indeterminate Beam (Castigliano,s Theorem) & Strut
Buckling
1. A curved beam is in the form of a semicircle as shown in Figure 1. The free end is pinned but is restrained to move in the horizontal direction only. If a horizontal force, F is applied at the free end, show that the vertical restraining force on the end of the beam is
given
by .
Figure 1Solution
,
Vertical deflection at right support,
2. Determine the reactions for the fixed-end beam AB with a concentrated load P shown in Figure 2 by using Castigliano,s
Theorem. Also determine the maximum deflection of the beam.
Solution
Unknown: Ra, Rb, Ma, Mb
Static equation:
Since it is symmetry, …
(1)
Portion CB:
,
Deflection at B,
Portion AC:
F
2L
2L
P
A B
Figure 2
F
R
x
Mx
Q
Rsin
R-Rcos
2L
2L
P
A BMa Mb
RbRa
C
1x
BMb
Rb
1xM
2x 2L
P
BMb
Rb
C2xM
,
Deflection at B,
Total deflection at B,
Since ,
Maximum deflection occurred at center point.
Deflection at center,
3. Determine the reaction for fixed-ed AB subjected to a uniform load of intensity q as shown in Figure 3. (Use Castigliano,s Theorem)
Solution
Unknown: Ra, Rb, Ma, Mb
Static equation:
Since it is symmetry, …
(1)
Portion AB:
,
Deflection at B,
Replace into above equation
Since ,
4. A horizontal bar is supported by a pinned-end column as shown in the figure. The column is a steel bar of square cross section having length and side dimensions . Calculate the allowable load if the factor of safety with respect to buckling of the column is .
d 2d
b L
AC
B
D
Q
Figure 4
Figure 3L
q
A B
2L
2L
qA B
Ma Mb
RbRa
x
BMb
Rb
xM2x
qx
x
B3
16PL
2P
xM
FAKULTI KEJURUTERAAN MEKANIKALUNIVERSITI TEKNIKAL MALAYSIA MELAKA
TUTORIAL 4SOLID MECHANICS 2 (BMCS 2333)
SEMESTER 2 SESSI 2005/2006Statically Indeterminate Beam (Castigliano,s Theorem) & Strut
Buckling
Solution
5. The steel rod is attached to the rigid bar and to the fixed support at . Knowing that , determine.i. the diameter, of rod for
which the critical load of the system is 350N.
ii. the critical load for the system when .
Solution
+ :
……..(1)
a) Data;
From (1),
b) Data;
From (1),
6. From the Figure 6, determine.i. The critical load for the aluminum
strut.ii. The dimension for which the
steel strut will have the same critical load.
iii. Express the weight of the aluminum strut as a percentage of the weight of the steel strut.
d
A
B
C
P
375mm
500mm
Figure 5
B
crP
BT
BR
375
h
mm
Bh
B
BC
B
BT500L mm
iv. The dimension so that the aluminum and steel strut will have the same weight and compute the critical load for each strut
Solutiona) For aluminum, ,
=6.94kN
b) For steel strut,
c) Weight,
d) For aluminum, same as solution (a),
For steel strut, since ;
=14.7mm
=2.37kN
Figure 6
A
B
P
d1.8mm d
C
D
P
25mm 1.8m
m25mm
Aluminum
Steel