-davidKWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY

SCHOOL OF ENGINEERING

DEPARTMENT OF MECHANICAL ENGINEERING

LABORATORY: STRENGTH OF MATERIALS

TITLE: DEFLECTION BY ENERGY METHODS

Name Index no.

Antwi Elijah Kwabena 3750009

Anti Eric Papa Kojo 3749909

Gyamfi Joseph 3752509

Opoku Anthony 3755809

Summary

The report describes an experiment to compare the theoretical and experimental deflection by energy method. The deflection is carried out on two conditions; offset end load, and lateral point load. The experiment was carried out successfully with each member of the group playing specific duties in the experiment process and the writing of the report finally. The contribution of each member is hereby stated below.

Joseph Gyamfi; making all the necessary sketches in the lab, helping to read values and typing the report.

Elijah .K. Antwi; reading values in the experiment lab, taking measurements ,tabulating experimental values and also did the theory.

Eric .P.K. Anti; reading of time during the experimental process, researching into the concept of the experiment.

Opoku Anthony; taking measurements, helping in reading the time and helping in tabulating experimental values.

EXECUTIVE SUMMARY

Energy methods are very useful for analysing structures, especially for those that are statically indeterminate. This experiment introduces the principle of virtual work and applies it to statically determinate and statically indeterminate frameworks. The experiment also shows how the method can be used for the plastic design of beams and rigid-jointed plane frames.

OBJECTIVETo compare theoretical and experimental deflections by the application of loads on a quadrant beam.

APPARATUSA bent quadrant beamCalipersMeter ruleDial gaugeApplied loads.

THEORY

2R2 = L2

R2 = L2/2 R = L/√2………………….(1)

When a tensile or compressive load P is applied to a bar there is a change in length x, which for an elastic material is proportional to the applied load. This at the proportional limit obeys Hook’s Law and hence work done is represented by

V = ½ Px……………………. (2)Let A be the cross-sectional area and L the length of a beam

P = A x = L/E

Where E = modulus of elasticityV = (2/2E) ×volume……… (3)

For a short distance say δs of the beam, as a distance y where the cross-sectional area is δA where y is distance from neutral axis, the strain energy due to the small length δs is given by

δV = f(2/2E)×volumeδV = δs f2 δA/2E

δV = δs/2E fM2y2δA/I2 where I = fy2.δA δV = (M2/2E I) δs

V = fM2δs /2EI………………….. (4)Where M is the bending moment applied to the beamFrom Castigliano’s theorem, the differential change in the total strain energy due to change in the external load applied produces the deflection in the direction of the load application.

δV = δN/δW………………….(5a)and δK =(δN/δP)p=0………….…(5b)where δV is the deflection in the vertical direction and δK is the deflection in the horizontal direction due to the forces W and P respectivelyFrom equations (4) and (5)

δV =1/2EIf2M(δM/δW).δs

δV =1/EIfM(δM/δW).δs

Moreover δK =1/2EIf[2M(δM/δW).δs]p=o

δK=1/EIf[M (δM/δW).δs]p=o

But M = WR sinθ + PR(1 – cosθ )= R [Wsinθ + 2Psin2θ/2]

δM/δW = Rsinθ δM/δP = 2sin2θ/2But ds = Rdθ

δV =1/EI of2π WR sinθ Rsinθ. Rdθ=WR3

of2πsin2θdθ=WπR3/4EI…………….. (6)

Also δK =1/EI of2π WR sinθ 2R sin2θ/2. Rdθ

=WR3ofπ/2sinθsin2θ/2dθ

=WR3/2EI…………….. (7)

PROCEDURELoads are hanged in increments of 1/10 lbf from 0 lbf to 1 lbf and in each case the corresponding deflections are measured for both loading and unloading for the quadrant beam.

RESULTSWEIGHTS (lbf) LOADING

DEFLECTION (mm)UNLOADING

DEFLECTION (mm)0 0.00 0.01

0.1 0.02 0.300.2 0.4 0.940.3 1.02 1.760.4 1.85 2.240.5 2.52 3.170.6 3.26 3.820.7 4.00 4.440.8 4.72 5.140.9 5.42 5.83

1 6.12 6.12

DERIVED RESULTS

Thickness of beam (t) = 5.3mm

Width of bar (b) = 19.8mmLength, L = 610mm

From R = L/√2 = 610/√2 = 431.34mmAlso I = bt3/12 = (19.8) (5.3)3/12 =245.65mm4 For mild steel, E = 202,000N/mm2

EXPERIMENTAL LOADING

WEIGHTS (N) LOADINGDEFLECTION (mm)

UNLOADINGDEFLECTION (mm)

0 0.0 0.010.445 0.02 0.300.89 0.40 0.941.34 1.02 1.761.78 1.85 2.242.23 2.52 3.172.67 3.26 3.823.12 4.00 4.443.56 4.72 5.144.01 5.42 5.834.45 6.12 6.12

THEORETICAL LOADINGFor the theoretical deflection using the equations (6) and (7)

δV =WR3/4EI at W = 0.445N = 1/10lb substituting in for E,R and I

δV = 0.445N× π ×431.343/4×202,000N × 245.65mm4

= 0.565mm

Also for δKδK =WR3/2

=0.445N × 431.343/2×202,000N × 245.65mm4

= 0.360mm

THEORETICAL LOADING

WEIGHTS (N) deflection δ(mm)

Δv δK0.0 0 0

0.445 0.565 0.3600.89 1.131 0.7201.34 1.702 1.0841.78 2.261 1.4402.23 2.834 1.8032.67 3.392 2.1603.12 3.963 2.5233.56 4.522 2.8794.01 5.094 3.2434.45 5.653 3.600

A GRAPH OF THEORITICAL DEFLECTION AGAINST WEIGHT

A GRAPH OF EXPERIMENTAL LOADING AND UNLOADING DEFLECTION AGAINST WEIGHT

CONCLUSIONIt can be seen from the experimental graphs that during loading a straight line graph was obtained that the beam was in proportional limits. The graph of the unloading deviated from the loading graph and this deviation is such that the area between the two graphs represents the energy that was stored in the beam after the loads had been removed.The horizontal deflection represents the increase at the edge of the beam, in the horizontal direction when the beam is further bowed.

PRECAUTIONDial gauge was set to zero before beginning the experiment.It was ensured that the setup was not disturbed by external forces such as shaking of the table on which setup is placed.The weights on the hanger were also checked that they were not oscillating during the experiment by placing the weights on the hanger carefully.

SOURCES OF ERROR

The beam may be damage due to continuous use of the beam hence it may affect the results of the experiment.Parallax of the pointer of the dial gauge when reading values.

ReferencesEngineering Mechanics By: Braja M. DasElements of Strength of MaterialBy: Timoshenko and YoungStrength of Material and StructuresBy: John Case and A.H Chilver