INSTRUCTIONAL MANUAL
OF
CURVED MEMBER
APPARATUS
By:
ENGINEERING MODELS & EQUIPMENT
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2
CURVED MEMBER APPARATUS
CONTENTS:
Page No.
1.0
Theory
03
2.0
Objective
05
3.0
Apparatus
05
4.0
Suggested experimental work
05
5.0
Results & Discussions
05
6.0
Sample Data Sheet
06
7.0
Comments
07
8.0
Precautions
07
3
CURVED MEMBER APPARATUS
1.0
THEORY:
Castigliano's first theorem is used to find the elastic displacements of curved members.
Theorem states" Partial derivative of the total strain energy of a structure with respect to
any force gives the displacement of the point of its application in the direction of the
force".
The total strain energy of any structure is determined in terms of all the load with their
actual values and a fictitious load P applied at the point at which the deflection is required
and it is acting in the same direction in which the deflection is required. In case no
external load is acting at the joint in the direction desired, a fictitious load is applied in
that direction and forces in all members are worked out. After partial differentiation with
respect to P, zero is substituted for fictious load P (or if P is not fictitious its actual value
is substituted). Thus the result is the required deflection.
In all cases the horizontal BH and vertical deflection BV due to vertical load W are to be
determined. These deflections are obtained by using Castigliano's first theorem where
strain energy due to bending only is taken into account. The results obtained for the four
curved members shall be as follows:
(a)
Quadrant of a circle
Fixed at A and free at B (radius R) and subjected to a concentrated load W at free
end.
B
W
R

A
(a)
Vertical displacement of load point = BV 
WR 3
4 EI
WR 3
Horizontal displacement of load point B = BH 
2 EI
4
(b)
Quadrant with a straight leg
From A to B, quadrant of a circle of radius R, from B to C straight length of y.
A
W
R

B
y
C
(b)
Vertical displacement of load point A = AV =
WR 3
4 EI
Horizontal displacement of load point A == AH =
(c)

WR 2 y
EI
WR
( R  y) 2
2 EI
Semicircle with straight arm
From A to C semi circle of radius R, A to B straight length of y
W
A
B

y
R
C
(C)
Vertical displacement of loaded point B = BV =
W
 2 y 3  3R (2y 2  8 y R  R 2 ) 
6 EI
WR 2
(y  2 R)
Horizontal displacement of loaded point B == BH =
EI
5
(d)
Circle of radius R
W
B

X
R
X
(D)
Vertical displacement of loaded point B = BV 
2.0
WR 3
( 2  8)
4EI
OBJECTIVE:
To determine the elastic displacement of the curved members experimentally and
verification of the same by analytical methods.
3.0
APPARATUS:
Apparatus consists of a steel bar which, is used to make the different curved members
Viz. Circle, semicircle with straight arm, a quadrant of a circle and quadrant of a circle
with straight arm. The bottom ends of the members are fixed to the base. Under the
application of load at free end, its horizontal and vertical deflection is measured with
the help of a dial gauge.
4.0
SUGGESTED EXPERIMENAL WORK:
Step1: Place a load on the hanger to activate the member and treat this as the initial position
for measuring deflections.
Step2: Fix the dial gauges for measuring horizontal and vertical deflections.
Step3: Place the additional loads at the steps mentioned in the table below for each case and
tabulate the values of dial gauge reading against the applied loads.
5.0
1.
RESULTS AND DISCUSSIONS:
Plot the graph load Vs deflection for each case to show that the structure remains
within the elastic limit.
6
2.
6.0
Measure the value of R and straight length in each case. Find width and depth of steel
section and calculate the value of I as bd3/12
SAMPLE DATA SHEET:
Name of the experiment:
Deflection of Curved Member apparatus
Name of student:
Semester
Width of section (mm) b
Depth of section (mm) d
=
=
Least moment of inertia
=
Batch
Session
bd 3
12
= 2 x 106
E (kg/cm2)
(a)
Quadrant of a circle
Sl.
No.
Additional
load (kg)
Dial gauge reading (mm)
Deflection (mm)
Horizontal
direction
Horizontal
direction
(b)
Quadrant with Straight leg
Sl.
No.
Additional
load (kg)
Vertical
direction
Dial gauge reading (mm)
Horizontal
direction
(c)
Semi-circle with straight leg
Sl.
No.
Additional
load (kg)
Vertical
direction
Dial gauge reading (mm)
Horizontal
direction
Vertical
direction
Vertical
direction
Deflection (mm)
Horizontal
direction
Vertical
direction
Deflection (mm)
Horizontal
direction
Vertical
direction
7
(d)
Circle
Sl.
No.
Additional
load (kg)
Dial gauge reading (mm)
Horizontal
direction
Vertical
direction
7.0
COMMENTS:
8.0
PRECAUTIONS:


Apply the loads gently.
Measure the displacements very accurately.
Deflection (mm)
Horizontal
direction
Vertical
direction