Name: Apoorv Agarwal (Group 6)

Roll no.: Y9124

Course Name: Experiments in structures (AE 331)

Date of Experiment: 25-08-2011

AE 331 LABORATORY REPORT - 4

Determination of Shear Centre of a beam of a given cross-section

Objective: To determine the position of the shear centre for the given metal beam of

C-shaped cross-section by using Dial Gages and by balancing the torque acted upon

beam by the weights.

Introduction & Theory:

Shear centre is an imaginary point on a section, where a shear force can be applied without inducing any torsion. EXPERIMENTALLY: WA & WB are the loads applied on the rod that passes through the centre of the beam and coincides with the Axis of symmetry of the beam.

A

lB lA

V WA

WB

e

LA & LB are distances from the center of the beam where the respective loads are applied and let ‘e‘ be the distance of shear center from the center of the beam. The Torque ‘ T ‘ experienced at the center of the beam due the load applied on the rod passing through the center of the beam taking anti-clockwise sense of direction as positive is

T = WA LA - WB LB Kg cm Assuming that the shear center lies at a distance of ‘ e ‘ towards WA from the center of the beam, Torque acted at the center of the beam by the load (WA + WB) acted upon shear center is

T = e ( WA + WB ) Kg cm Equating both torques, we get

e = (WALA - WBLB) / (WA + WB)

THEORETICALLY:

b F1 MP =

t

h F2 F1 = F3 =

e P F3 Under No Twisting condition, F*e = MP

F which gives, e =

Equipments:

Clamps, standard weights of (5g, 10g, 20g, 50g, 100g, 200g, 500g, 1kg and 2kg), metre scale, Vernier Calliper, C section beam, 2 Load pans of equal weight, 2 Dial gauges. Procedure & Measurements:

• Fix the aluminium C-section beam and the dial gauges at the appropriate position.

• Measure the dimensions of the beam using the Vernier calliper. • Measure LA & LB on the rod passing through the centre of the beam using

Vernier calliper and mark various distances using marker. • LA is maintained at 10cm throughout the experiment while LB is varied as

6cm, 10cm and 14cm respectively. Initially it is set to 6cm and marks are made along the horizontal rod for clarity.

• The readings in both the dial gauges are set to zero. • A 1 kg weight is put on pan A. Loading is done very slowly to avoid any errors

in reading due to drastic changes. • Weights are varied on pan B such that the readings in two dial gauges are the

same. This is done for all three different values of LB. • The above procedure is repeated for 2 kg and 3 kg weights in pan A and the 9

readings for weight in pan B are recorded as such.

Results & discussion:

a) Sample Calculations:

Beam dimensions and settings: 49.88 mm

Weight of pan: 52 gm Least count of dial gauge: 0.01 mm

Theoretically,

e =

Now, b = 49.88 mm, h = 76.33 mm Therefore eth = 19.87 mm

Also, Experimental e = (WALA - WBLB) / (WA + WB) b) Data Presentation:

IA (in cm) WA ( in Kg ) IB ( in cm ) WB ( in Kg ) e ( in mm)

10 1 6 1.535 3.11637

10 1 10 0.902 5.15247

10 1 14 0.63 7.239

10 2 6 2.9 5.30612

10 2 10 1.855 3.76135

10 2 14 1.295 5.67527

10 3 6 4.305 5.70842

10 3 10 2.8 3.44828

10 3 14 2.08 1.73228

From the above Table we can obtain the mean value for ‘e’ as 4.5711 mm

1.15 mm

76

.33

mm

c) Discussion &Error Analysis:

Experimental value of shear centre is found to be 4.5711 mm which is not close to the calculated theoretical value of 19.87 mm.

Percentage error =[|(Experimental – Theoretical Values)|/Theoretical value ] x 100%= [15.2989/19.87]x100 = 76.99%

Standard Deviation = ((∑ (ei - emean)2)/9)1/2 = 1.585 mm

The deviation from the expected value is due to the errors involved in the experiment. The possible causes for errors are: 1) Human errors and rounding off values. 2) Residual Strains in the dial gauges. 3) Beam is not made up of pure material. 4) Friction between the moving components of the apparatus can cause

errors in the readings. 5) Weights used may not be exactly accurate. 6) Vibration of beam affects the accuracy of measurements.

Conclusion :

Shear centre is located at a distance of 4.5711 mm from the centre of the beam and the % error in locating the shear centre came out to be 76.99 % with a standard deviation of 1.585 mm.