Job # 6
Study the bending of Z section
Objective:
To confirm the theory that “Loading the beam along principle axis always produces
symmetrical bending.”
APPARATUS:
• Steel tape
• Vernier Calipers
• Loading pan and weights
• Dial gauges (two in no. having L.C=.001”)
• Cantilever beam (Z-section) with a rotating arrangement
RELATED THEORY:
I. Axis of Symmetry:-
If an axis divides a section in two parts in such a manner that the parts are the mirror
images of each other such an axis is called axis of symmetry.
II. Symmetrical Section:-
A section such that it has at least one axis of symmetry is called a
symmetrical section. e.g. W-section and T-section.
Rectangular
Section W- Section T- Section
1
III. Unsymmetrical Section:-
Such sections which do not have any axis of symmetry is called unsymmetrical section.
e.g. Z-section and L-section.
Z- Section L- Section
IV. Principle Axes:-
Principle axes are the set of rectangular axis such that the moments of inertia are maxima or
minima but the product moment of inertia is zero at the same axes.
Axes of symmetry are always principle axes, but converse is not necessarily true.
V. Orientation of Principle Axes:-
In case of a symmetrical section the axes of symmetry are the principle axes but in case
of unsymmetrical sections we have determine their position.
VI. Symmetrical Bending:-
When loading and deflection are parallel or along the principle axes,
the bending is sail to be symmetrical axes.
VII. Unsymmetrical Bending:-
When deflection occurs in more than one plane the bending is termed as unsymmetrical
bending.
Unsymmetrical bending may occur in symmetrical and unsymmetrical sections.
Principle Axis
Assumptions of Flexural Formula:-
i. The plane section of the beam remains plane.
ii. The material in the beam is homogeneous and obeys Hook’s law. iii. The
moduli of elasticity for tension and compression are equal.
iv. The beam is initially straight and of constant cross section.
v. The plane of loading must contain a principle axis of the beam cross section and the loads
must be perpendicular to the longitudinal axis of the beam.
y Plane of loading
Therefore it can be concluded that flexural formula may be applied only when the bending loads
act in a longitudinal plane parallel to or containing one of the principle axis of the section. These
are the principle planes of bending. Deflections are in one direction only.
Unsymmetrical Bending in Symmetrical Sections:-
Orientation of N.A:
PROCEDURE
1. Measure the dimensions (length, depth, thickness of flanges and web) of the cantilever Z
section using steel tape and Vernier Calipers.
2. Calculate the area, centroid and M.O.I. about horizontal and vertical axes i.e.
Izz , Iyy
3. Determine the inclination of principle axes. i.e. α and also determine the M.O.I. about
principle axes i.e. Iz1z1 and Iy1y1
4. Take the horizontal and vertical DGR from the deflection gauges attached to the section
at 0 load.
5. Apply 20N, 40N and 60N loads
respectively and note down DGR
from horizontal and vertical
deflection gauges.
6. Unload the section and take DGR
against 40N, 20N & 0 and take the
mean value of DGR for loading and
unloading conditions.
7. For setting-2 set the angle α to make
the loading plane along the principal
plane to make sure unidirectional
bending
8. Repeat the same procedure for
setting-2 as that of setting –1.
Description Observation (mm)
L 1060
b1 26.5
b2 26.5
tf1 1.9
tf2 1.9
tw 1.9
d 55.5
X 25.55 mm
Y 27.75 mm
Ixx 94236.9 mm4
Iyy 21157.5 mm4
Ixy 33194.7 mm4
alpha -21.12 Degree
Setting-I Setting-II
Ix1x1 107064 mm4
Iy1y1 8330.81 mm4
E 200000 Mpa
Orientation of Principal Axes (α) = -20.87 degree
I’xx = 109.01 x 103
I’yy = 8380.29 mm
OBSERVATIONS AND CALCULATIONS
Determination of Experimental Deflection
Setting Load 0 20 40 60
H.D.G.R 0 84 173
1.5 87 175.5 268
3 90 178
ΔH (mm) 0.00 2.17 4.42 6.77
1
V.D.G.R 0 64 129
-1.5 63 129 199
-3 62 129
ΔV (mm) 0.00 1.64 3.31 5.09
H.D.G.R 0 3.5 12
5 8.75 15 13
10 14 18
ΔH (mm) 0.00 0.10 0.25 0.20
2
V.D.G.R 0 32 68
0 32 68.5 100
0 32 69
ΔV (mm) 0.00 0.81 1.74 2.54
Determination of Theoretical Deflection
Load (p) ΔY’ ΔX’ ΔV ΔH ΔR
Setting
N (mm) (mm) (mm) (mm) (mm)
0 0.000 0.000 0.000 0.000 0.000
20 0.346 1.716 0.940 1.476 1.750
1
40 0.692 3.431 1.881 2.952 3.500
60 1.038 5.147 2.821 4.427 5.250
0 0.000 0.000 0.000 0.000 0.000
20 0.371 0.000 0.371 0.000 -0.371
2
40 0.742 0.000 0.742 0.000 -0.742
60 1.112 0.000 1.112 0.000 -1.112
Comparison Table
Horizontal Deflection Vertical Deflection
Setting Load (N)
Th. Def Exp. Def % Diff Th. Def Th. Def Exp. Def
0 0.000 0.00 0 0.000 0 0
20 1.476 2.17 31.99198 0.940 1.64 42.66717
1
40 2.952 4.42 33.2229 1.881 3.31 43.1868
60 4.427 6.77 34.60381 2.821 5.09 44.58202
0 0.000 0.00 0 0.000 0 0
20 0.000 0.10 - 0.371 0.81 54.22084
2
40 0.000 0.25 - 0.742 1.74 57.37802
60 0.000 0.20 - 1.112 2.54 56.2034
Plotting of Curves:
Load ~ Horizontal Deflection
8.000
7.000
6.000
5.000
Load (N)
4.000
3.000
2.000
1.000
0.000
0 10 20 30 40 50 60 70
Deflection (mm)
Hz Th Setting 1 Hz Exp Setting 1 Hz Th Setting 2 Hz Exp Setting 2
Load ~ Vertical Deflection
6.000
5.000
4.000
Load (N)
3.000
2.000
1.000
0.000
0 10 20 30 40 50 60 70
Deflection (mm)
Vt Th Setting 1 Vt Exp Setting 1 Vt Th Setting 2 Vt Exp Setting 2
COMMENTS:
