Geometric Properties of Beams

Akshay Samker, SMKAKS001

Geometric Properties ExperimentLecturer: Matongo Kabani

SAMKER4/16/2014

Summary

The purpose of this experiment is to substantiate the

relationship between bending moments and sectional properties

of symmetrical cross sections, to measure beam displacement

using dial gauges and to process collected data using

statistical methods.

The linear relationship between load and deflection has been

verified and the Young’s Modulus (modulus of elasticity) has

been calculated using the relationship between bending moment

and curvature. A linear chart showing the difference between

theoretical deflection and experimental deflection has also

been computed.

The purposes of this report have been met. The linear

relationship between load and deflection has been proven.

Young’s Modulus has been calculated. There is a 4.4%

difference between the theoretical and experimental

deflection. The principle of virtual work is suited for roof

trusses.

Introduction

In this experiment we hope to substantiate the relationship

between bending moments and section properties of symmetrical

cross sections of Aluminium cantilever beams with the

following arrangement.

Experiment setup

(Fig. 1)

We define coordinate x across the beam length, y vertically

and z across the beam width. This system sets the top of the

beam to be in tension and the bottom of the beam to be in

compression. For symmetric cross sections with the line of

action of the force at the centroid of the cross section we

define the bending moment as

Mz=−P (Lx−x )(1)

Where P is the load and L is the span of the beam and x=0 at

the support.

If the beam cross section is symmetrical about the z or y axis

then the bending moment is defined by

Mz=EIzzkz

(2)

Where E is the Modulus of Elasticity and k is the curvature of

the beam.

We will be using 5 systems for this experiment in different

orientations and 3 beam types.

RH

S Major Axis

(Fig. 2)

RHS

Minor Axis

(Fig. 3)

T-Beam

(Fig. 4)

Channel

(Fig. 5)

The theoretical deflection can be calculated using a

manipulated version of the formulas

dt=PL3

3EI(3)

Methodology

Use Vernier callipers to measure the cross sectional

dimensions of each beam and a measuring tape to record the

span of the beam.

For each cantilever:

1. Ensure that it is fixed to the frame and that the beam is

in a horizontal direction.

2. Mount the dial gauges at the tip of the cantilever to

measure vertical deflections.

3. Place the hanger at the tip of the cantilever and zero

the gauge.

4. Place weights on the hanger and record the deflections at

each weight.

5. Remove weights in the same intervals and record the

deflections ant unloading

Results

RHS Minor Axis

Table 1: Average DimensionsTable 2: Average experimental displacement vs theoretical

t (mm) 2.34Load(N)

Average displacement (mm*0.01)

h (mm)25.4

0 0

w (mm)49.1

9 10

Span(cm)42.3

3 2030405060

Table 3: Bending moment vs CurvatureBending Moment (N.m)

Curvature(1/m)

Area Moment of Inertia (mm

0.00 0.00-4.23 0.00-8.47 -0.01-12.70 -0.01-16.93 -0.02-21.16 -0.02-25.40 -0.02

0.00 50.00 100.00 150.00010203040506070

f(x) = 0.527417230007633 xf(x) = 0.431567077673976 x

Experimental VS Theoretical Displacement to verify linear

relationship

ExperimentalLinear (Experimental)Theoretical Linear (Theoretical )

Displacement (mm*0.01)

Load (N)

RHS Major Axis


t (mm) 2.05Load(N)


Theoretical

h (mm)25.2

4 0 0.19 0

w (mm)50.1

1 10 7.34 6.62

Span(cm)42.5

8 20 14.85 13.2430 23.63 19.8640 31.89 26.4950 41.98 33.1160 49.68 39.73


Curvature(1/m)

Area Moment of Inertia(m4)

0.00 0.00 99673-4.26 0.00-8.52 0.00-12.78 0.00-17.03 -0.01-21.29 -0.01-25.55 -0.01

0.00 50.00 100.00 150.00010203040506070

f(x) = 0.527417230007633 xf(x) = 0.431567077673976 x


relationship

ExperimentalLinear (Experimental)Theoretical Linear (Theoretical )


Load (N)

0.00 10.00 20.00 30.00 40.00 50.00 60.000

10

20

30

40

50

60

70

f(x) = 1.51023464742347 xf(x) = 1.22289871233049 x


relationship

ExperimentalLinear (Experimental)TheoreticalLinear (Theoretical)


Load(N)

T-Beam


t (mm) 3.33Load(N)


Theoretical

h (mm)25.6

9 0 0.56 0.00

w (mm)25.7

4 10 65.90 68.37

Span(cm)42.7

3 20 131.89 136.7330 197.69 205.1040 262.06 273.4750 327.50 341.8360 391.38 410.20


Curvature(1/m)

Area Moment of Inertia (mm4)

0.00 0.00 9756-4.27 -0.01-8.55 -0.02

-12.82 -0.03-17.09 -0.04-21.37 -0.05-25.64 -0.06

0.00 10.00 20.00 30.00 40.00 50.00 60.000

10

20

30

40

50

60

70

f(x) = 1.51023464742347 xf(x) = 1.22289871233049 x


relationship



Load(N)

0.00 100.00200.00300.00400.00500.000

10

20

30

40

50

60

70

f(x) = 0.14627070074808 xf(x) = 0.152766166086125 x


relationship

ExperimentalLinear (Experimental)Linear (Experimental)TheoreticalLinear (Theoretical)


Load(N)

Channel


t (mm) 3.61Load(N)


h (mm)28.3

9 0 0.00

w (mm)51.2

8 10 32.64

Span(cm)40.7

8 20 63.6430 93.8040 127.0150 156.0160 186.33

Table 3: Bending moment vs CurvatureBending Moment(N.m)

Curvature(1/m)


0.00 0.00 27689-4.08 -0.01-8.16 -0.01-12.23 -0.02-16.31 -0.02-20.39 -0.03-24.47 -0.03

0.00 100.00200.00300.00400.00500.000

10

20

30

40

50

60

70

f(x) = 0.14627070074808 xf(x) = 0.152766166086125 x


relationship

ExperimentalLinear (Experimental)Linear (Experimental)TheoreticalLinear (Theoretical)


Load(N)

RHS 2 Minor Axis

Table 13: Average Dimensions

Table 14: Average experimental displacement vs theoretical

t (mm)2.59

Load(N)


h (mm)37.35 0 1.73

w (mm)75.76 10 15.85

Span(cm)45.63 20 27.00

30 38.0640 50.7450 67.9060 83.45


Curvature(1/m)


0.00 0.00 133132

0.00 50.00 100.00 150.00 200.000

10

20

30

40

50

60

70

Experimental VS Theoretical Displacement



Load(N)

-4.56 0.00-9.13 0.00-13.69 0.00-18.25 0.00-22.82 0.00-27.38 -0.01

0.00

20.00

40.00

60.00

80.00

100.00

120.00

0

10

20

30

40

50

60

70

Experimental Vs Theoretical Displacement to verify linearity



Load (N)

-0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

f(x) = 4501.27702994143 x − 1.03620292530137f(x) = 740.571980135535 x + 0.451274628288916f(x) = 398.906742809867 x + 0.0710147853694423f(x) = 3063.42880860839 x − 0.499661315415159f(x) = 1098.04487651599 x + 0.1226008363008

CURVATURE VS BENDING MOMENT

RHS MINOR Linear (RHS MINOR )Linear (RHS MINOR )RHS MAJOR Linear (RHS MAJOR )

CURVATURE

BENDING MOMENT(N.mm)

Table 16 : Experimental Young's Modulus

Beam TypeYoung's Modulus

RHS Minor Axis 32.1E+9RHS Major Axis 30.7E+9T-Beam 40.9E+9Channel 26.7E+9RHS 2 Minor 33.8E+9

Discussion

The factors that affect the bending of the beam rely heavily

on the physically properties of the beam. The type of support

plays a role in the bending of the beam. This cantilever beam

had a point support.

-0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

f(x) = 4501.27702994143 x − 1.03620292530137f(x) = 740.571980135535 x + 0.451274628288916f(x) = 398.906742809867 x + 0.0710147853694423f(x) = 3063.42880860839 x − 0.499661315415159f(x) = 1098.04487651599 x + 0.1226008363008

CURVATURE VS BENDING MOMENT

RHS MINOR Linear (RHS MINOR )Linear (RHS MINOR )RHS MAJOR Linear (RHS MAJOR )

CURVATURE BENDING MOMENT(N.mm)

The moment of inertia and the orientation of the beam play a

vital role in the amount of deflection in the beam. The moment

of inertia was computed using the cross sectional area. The

dimensions of the cross section was obtained using a Vernier

calliper, which has an associated uncertainty, using the

uncertainty in the mean of the dimensions and calculating

using the range method, of 0.12mm.

The span of the beam is also factor in the bending moment

which plays a role in the deflection.

The dial gauge calibration and manner of loading and unloading

affect the reading. Gently loading the beam will offer less

“shock” to the gauge and hence give a more accurate reading.

The noise factor in the data is obtained from the initial

reading of the deflection in the beam, i.e.: when no load is

applied. These values need to be subtracted from the

subsequent readings in order to give the correct reading.

According to the dial gauge resolution the uncertainty in the

deflection is 0.05×0.01mm. Using the root sum method the

uncertainty in the vertical deflection is 14.4nmm. The noise

factors and other factors of error are constituent of the

variation in the between the experimental and the theoretical

deflection.

Conclusion

The purposes of this lab have been met. The relationship

between load and vertical displacement is linear and the

Young’s modulus for each type of beam has been calculated

experimentally taking into account the uncertainty with the

given data.

Uncertainty in the measurement of deflections and dimensions

were accounted for using the range method and instrument

resolution. The percentage difference between the theoretical

and experimental deflection in the T-Beam was 4.4% which shows

the most accuracy of all the beams analysed. The Young’s

modulus calculated was 40.9±2.3GPa. 39GPa does fall in thisrange.

The principle of virtual work is suited for estimating the

deflection in roof trusses. The uncertainty is low and hence

accurate estimations can be made, given that the roof truss is

made of an isotropic material.

Recommendations

During this experiment, the readings from the dial gauge were

not clear. The sensitivity of the gauge was not given. A

digital gauge is recommended as the readings would be quicker

to establish and would also be correctly calibrated.

The hollow of the beams were covered making it difficult to

obtain precise readings using the Vernier calliper. A sample

piece of each beam is recommended for the purpose of measuring

dimensions.

The level of friction in the dial gauge has not been given.

This could affect the readings if the gauge is not installed

in a perfectly vertical arrangement. In this case a digital

gauge is recommended.

The use of strain gauges instead of a dial gauge would have

been a quicker method to achieving the objectives of this

experiment, i.e. the use of engineering stress and engineering

strain.

References

1. Kabani,M. CIV2037F Course Notes

2. Gere, J.M.; Timoshenko, S.P. Mechanics of Materials:Forth

edition, Nelson Engineering, (1996),

3. Norris, Charles Head; John Benson Wilbur, Senol Utku,

Elementary Structural Analysis (3rd ed.). McGraw-Hill.

Pp, (1976). 313–326

Appendices

Sample Calculations

Area Moment of Inertia of RHS Minor (all other moment of

inertia is calculated using the same principle)

Ixx= ∫−h/2

h /2

∫−w /2

w/2

y2dxdy

Ixx= ∫−h/2

h /2

y2wdy

Ixx=h3w12

Iyy=hw3

12

Ixx=25.4349.19

12+20.72344.51

12

Ixx=34179mm4

Uncertainty in the T-Beam

Dimensions (error in the mean), using the Range Method

∆h∧∆b=3.4−3.2

√3=0.12mm×0.01

Bending Moment

Uncertainty in Bending Moment = Uncertainty in span

∆L=42.75−42.7

√3=0.028cm =

Uncertainty in Moment of Inertia using Root Sum Method

∆I=√(h312 ∆w)

2+(3h2w12 ∆h)

2

∆I=223.74mm4

Uncertainty in Theoretical Deflection using Root Sum Method

∆ ∆y=√(L2PEI ∆L)

2+(

−L3P3EI ∆I)

2=14.4nmm

Uncertainty in Curvature using the Root Sum Method

∆ky=√(3∆∆y)2+(−6∆LL3 )

2

∆ky=39.32×10−6 1m

Uncertainty in the Young’s Modulus using the Root Sum Method

∆E=√(∆M)2+(−∆II2

)2+(

−∆kk2 )

2 ,∆M=∆L

∆E=2.3GPa

Percentage Difference in gradient of Theoretical Deflection

and Experimental Deflection

Theoretical Deflection gradient = 0.1528

Experimental Deflection gradient = 0.1463

0.1528−0.14630.1463

×100=4.4%

Raw Data

1.0 RHS, minoraxis

t (mm) 2.14 2.26 2.01 2.94 2.34

h (mm)25.0

0 26.2025.1

1 25.28 25.40

w (mm)48.2

0 50.4048.0

3 50.14 49.19Span(cm

)42.8

0 43.2041.5

0 41.80 42.33

Deflection (mm) x 0.01Load(N)

loading

unloading

loading

unloading

loading

unloading

loading

unloading

0 0.0 2 0 1 0 0.1 0 210 20.5 29 20 28 20 28 21 2720 41.1 56 41 53 41 53 42 5030 62.0 81 61 76 61 76 65 7340 85.0 102 85 97 83 97 88 95

50112.5 123 109 118 109 118 114 119

60141.5 143.5 138 138 138 138 141 141

2.0 RHS, majoraxis

t (mm) 2.20 2.00 1.95 2.05

h (mm)25.2

1 25.5424.9

825.2

4

w (mm)50.1

4 50.1850.0

050.1

1

Span(cm)

43.20 42.75

41.80

42.58


loading

unloading

loading

unloading

loading

unloading

loading

unloading

0 0.0 -0.5 0 0.5 0 -1.5 0 310 4.2 5.1 8 8.5 7.5 6.5 8 1120 12.0 12.8 16 11.5 15.5 15 17 1930 20.8 21.2 24 25 23 23 25 2740 28.7 29.8 32.5 34 30.1 31 34 3550 38.2 40.1 43 43.5 40 41 45 4560 47.2 47.2 51.5 51.5 48 48 52 52

3.0 T-beam

t (mm) 3.20 3.40 3.40 3.33

h (mm)25.4

6 25.6026.0

025.6

9

w (mm)25.6

1 26.0025.6

025.7

4Span(cm)

42.70 42.75

42.75

42.73


loading

unloading

loading

unloading

loading

unloading

loading

unloading

0 0.0 1.5 0 1 0 1 0 110 64.0 67 65.1 67 65 67 65.1 67

20130.1 133 131 133 131 133 131 133

30196.5 197 197 199 197 199 197 199

40260.0 263.5 262 263 262 263 260 263

50326.0 329 327 328 327 328 327 328

60390.0 393 391 392 391 391 391 392

4.0 Channel

t (mm) 3.40 3.40 4.22 3.40 3.61

h (mm)25.6

0 25.6226.7

1 35.62 28.39

w (mm)50.8

0 50.8052.7

0 50.80 51.28Span(cm)

41.00 41.00

40.10 41.00 40.78


loading

unloading

loading

unloading

loading

unloading

loading

unloading

0 0.0 3 0 3 0 0.8 0 310 32.0 34 32 34 31.3 31.8 32 3420 63.5 65 63.5 65 61.3 62.3 63.5 6530 93.5 95 93.5 95 92.1 92.8 93.5 95

40130.0 126 130 126

123.8 124.3 130 126

50156.0 157 156 157

154.3 154.8 156 157

60186.5 186.5

186.5 187.5

184.8 184.8

186.5 187.5

5.0 RHS 2 minor axis bending

t (mm)2.16

0 3.4202.20

02.59

3

h (mm)38.820 36.600

36.630

37.350

w (mm)76.340 76.600

74.350

75.763

Span(cm)

44.200 49.500

43.200

45.633

Deflection (mm) x 0.01Load load unload load unload load unload load unload

(N) ing ing ing ing ing ing ing ing0 0.0 1.5 0 6 0 4.5 0 1.810 11.2 18 16 26 13.9 21 8 12.720 23.5 29 27 38 24 34.1 17.5 22.930 34.0 40.5 39 49 34.5 45.1 27.1 35.340 48.5 42.5 57 62 50.5 59 39 47.450 65.0 68.5 75 78 66.9 76 54.2 59.660 83.5 83.5 94 94 83 91 69.3 69.3

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Geometric Properties of Beams