Structures Laboratory II [AE39003]
Experiment - 1
Unsymmetrical Bending
Submitted by:
19AE10038 Shrey Sharma
19AE10039 Piyush Prakash Nashani
19AE30001 Chetan Gowda
19AE30010 Marisha Bhatti
19AE30017 Pranjal Verma
19AE30023 Sunidhi Tiwari
19AE30025 Mudit Kumar Bhugari
Submitted To: Prof. D.K. Maiti | Course Instructor
Aim: To study unsymmetrical bending of a Z-beam
Apparatus:
● Z Beam
● Strain Gauge
● Travelling Microscope
● Known Weight
Theory:
Symmetrical Bending: In the elementary theory of bending it is assumed that
the plane of loading on the beam is parallel to or contains a principal inertial
axis of the beam cross-section. This also means that the applied bending
moment acts about one or the other of the principal axes of the cross section.
Unsymmetrical Bending: Unsymmetrical bending will occur when the bending
is caused by loads that are inclined to the principal planes as shown in the
figure below. In other words, the bending moment is inclined with respect to
the principal axes as shown in the following figure. In this case, the neutral
axis is in general not perpendicular to the axial plane in which the bending
moments are acting.
Fig 1: Unsymmetrical bending due to loads inclined to the principal planes
Fig 2: Unsymmetrical bending due to B.M. inclined to principal axes
The neutral axis is the axis about which the beam bends. The neutral axis is
the axis of zero stress. The bending moment can be resolved into Mx and My
such that Mx = Msinθ and My = Mcosθ.
Fig 3: Developed Bending Moment is resolved into two components Mx and My
ξ
Strain=ε = ρ
𝑧
where ξ=xsinα+ ycosα: displacement w.r.t. neutral axis
and ρ=radius of curvature
ξ
Stress, σ = ε 𝐸 = 𝐸 ρ = E(xsinα + ycosα)/ρ by hooke’s law.
𝑧 𝑧
Mx = ∫Aσ ydA ; My = ∫Aσ xdA
𝑧 𝑧
Ixx = ∫Ay2dA ; Iyy = ∫Ax2dA ; Ixy = ∫AxydA
𝑀𝑦𝐼𝑥𝑥−𝑀𝑥𝐼𝑥𝑦 𝑀𝑥𝐼𝑦𝑦−𝑀𝑦𝐼𝑥𝑦
We know that: σ = 2 𝑥− 2 𝑦
𝑧 𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦 𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦
' '
𝑀𝑦 𝑀𝑥
σ𝑧 = 𝐼𝑦𝑦
𝑥+ 𝐼𝑥𝑥
𝑦;
' '
𝑀𝑦 𝑀𝑦𝐼𝑥𝑥−𝑀𝑥𝐼𝑥𝑦 𝑀𝑥 𝑀𝑥𝐼𝑦𝑦−𝑀𝑦𝐼𝑥𝑦
𝑤ℎ𝑒𝑟𝑒 𝐼𝑦𝑦
= 2 𝑎𝑛𝑑 𝐼𝑥𝑥
= 2
𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦 𝐼𝑥𝑥𝐼𝑦𝑦−𝐼𝑥𝑦
Deflection due to bending (ξ) normal axis can be calculated as follows:
𝑢 =− ξ𝑠𝑖𝑛 α ; 𝑣 =− ξ𝑐𝑜𝑠 α
2 2 2
1 ∂ξ ∂𝑢 𝑠𝑖𝑛 α ∂𝑣 𝑐𝑜𝑠 α
ρ
= 2 ; 2 =− ρ
; 2 =− ρ
∂𝑧 ∂𝑧 ∂𝑧
' '
∂𝑢
2 𝑀𝑦 2
∂𝑣 𝑀𝑥
2 =− 𝐸𝐼𝑦𝑦
; 2 =− 𝐸𝐼𝑥𝑥
∂𝑧 ∂𝑧
∂𝑀'𝑥 '
Considering a cantilever beam, My=0 implies ∂𝑧 = 𝑆𝑦
'
𝑆𝑦 (𝑧−𝐿)
3
𝑧𝐿
2 3
𝐿
Solving, we get: 𝑣 =− 𝐸𝐼𝑥𝑥
[ 6
− 2
+ 6
]
' 3 ' 3
𝑆𝑦𝐿 𝑆𝑥𝐿
Tip deflection: 𝑣𝑇 = 3𝐸𝐼𝑥𝑥
; 𝑢𝑇 = 3𝐸𝐼𝑦𝑦
Shear forces: Sy=-P and Sx=0;
𝐼
𝑃 𝐼𝑥𝑦
' 𝑃 '
we get: 𝑆𝑦 =− 2
𝐼𝑥𝑦
; 𝑆𝑥 = 𝐼𝑥𝑦
𝑥𝑥
2
1− 𝐼 𝐼
1− 𝐼 𝐼
𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦
Fig 4: Dimensions of unsymmetrical section under bending
Set Up:
A cantilevered Z section beam and a travelling microscope is set to observe the
vertical and horizontal deflections of the free end section. An arrangement is
also made to introduce vertical load at the free end.
Procedure:
1. Measure the dimensions of the given section.
2. Focus the travelling microscope on a location of the beam section and
take the initial readings.
3. Load the free end in steps of 200 grams gradually from 0 grams to 2000
grams for loading.
4. Carefully take readings of the vertical tip deflection and horizontal tip
deflection by adjusting the travelling microscope.
5. Repeat the same procedure for unloading.
Observations:
Dimensions of the beam before deflection:
● Length of z-section = 96 cm
● Average dimension of b1 : 32.41 mm
Observations 32.22 32.37 32.54 32.52
(mm)
● Average dimension of b2 : 33.24 mm
Observations 32.56 33.45 33.66 33.29
(mm)
● Average dimension of h : 51.67 mm
Observations 51.05 50.33 52.33 52.96
(mm)
● Average thickness t : 2.21 mm
Observations 2.35 2.19 2.15 2.15 2.21 2.18
(mm)
Table 1: Deflection during Loading and Unloading of varied weights
Loading Unloading Average + Normalise
Load x - axis y - axis x - axis y - axis x - axis y - axis
(kg) deflection deflection deflection deflection deflection deflection
(mm) (mm) (mm) (mm) (mm) (mm)
0.0 11.62 14.29 11.64 14.39 0 0
0.2 11.88 14.59 11.89 14.68 0.255 0.295
0.4 12.14 14.88 12.19 14.92 0.535 0.560
0.6 12.33 15.15 12.43 15.18 0.750 0.825
0.8 12.61 15.42 12.68 15.48 1.015 1.110
1.0 12.89 15.75 12.93 15.75 1.280 1.410
1.2 13.16 15.95 13.18 15.99 1.540 1.630
1.4 13.39 16.29 13.50 16.35 1.815 1.980
1.6 13.75 16.59 13.78 16.59 2.135 2.250
1.8 13.99 16.86 14.04 16.87 2.385 2.525
2.0 14.25 17.17 - - 2.620 2.830
Graph 1: x-axis and y-axis Deflection vs Load
Calculations:
We first calculate the second moments of inertia using the dimensions of the Z
section beam:
Ixx= b1d13/12 + A1(Y’ - Y1)2 + b2d23/12 + A2(Y’ - Y2)2 + b3d33/12 + A3(Y’ - Y3)2
Ixx= 127000 mm4
Iyy= b1d13/12 + A1(X’ - X1)2 + b2d23/12 + A2(X’ - X2)2 + b3d33/12 + A3(X’ - X3)2
Iyy= 42400 mm4
Ixy= A1(X’ - X1)(Y’ - Y1) + A2(X’ - X2)(Y’ - Y2) + A3(X’ - X3)(Y’ - Y3)
Ixy= -55800 mm4
Conclusion:
Experimental Slopes:
X - deflection: 1.310 (mm/Kg); Y - deflection: 1.415 (mm/Kg)
Since the graph gives a straight line. Hence, Hooke’s law is well applicable.
Discussion:
Unsymmetrical Bending occurs in the following cases:
● Section is symmetrical but the load line is inclined.
● Section is unsymmetrical and the load line is along or parallel to axes.
Following assumptions were made during the experiment:
● While deriving the theoretical expressions an assumption was made
that the cross section remains parallel to the plane perpendicular to the
z plane even after deflection.
● The stresses are assumed to be small so that they are within the elastic
range and hooke’s law is applicable.
● The beam is considered to be uniform and of homogeneous material.
● There is no net internal axial force acting on the beam.
After taking the readings and averaging and normalising (taking deflections at
0 grams load to be zero and extending the new change in deflection to all the
readings) them, the x axis and y axis deflections for both loading and
unloading of varied weights the best fit graph was made to minimise the error.
The slope was averaged from the loading and the unloading curves.
Comments/Suggestion for Improvement:
● Increasing the number of data points by reducing the step size in which
the weight is applied can improve accuracy.
● The error can be due to parallax, oscillations of the beam and certain
imperfections pertaining to the material properties.
● The hanger on which the weights were placed also moved which can add
to the sources of errors.
