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