PRINCIPLE

Statically determinate body, the equilibrium equation of statics are sufficient to determine all unknown forces or other unknowns that appears in equilibrium equation. Whereas statically indeterminate body the equilibrium equations of statics are not sufficient to determine all unknown forces or other unknown.

The principle load- carrying portion of most structures, however lie in a single plane and since the loads are also coplanar, the above requirements for equilibrium reduce to :

∑ Fx = 0 : The sum of horizontal forces acting on the structure is equals zero.

∑ Fy = 0 : The sum of vertical forces acting on the structure is equals zero.

∑ Mo = 0 : The sum of moment of these force components about an axis perpendicular to the x-y plane (the z axis) and passing through point O is equals zero.

A simple rule of thumb to help ascertain whether an object is statically determinate or indeterminate is to compare the number of unknowns to the number of equilibrium equations.

If n < 3 The body is statically determinate and it can have partial or no fixity.

If n = 3 The body is statically determinate if it has full fixity but indeterminate if it has partial fixity.

If n > 3 The body is statically indeterminate and it have full fixity or partial fixity.

As a general rule, a structure can be identified as being either statically determinate or statically indeterminate by drawing free-body diagrams of all its members. For a coplanar structure there are at most three equilibrium equation for each part.

UNSTABLE STABLE STATICALLYDETERMINATE

STABLE STATICALLY INDETERMINATE

BEAM r < 3n r = 3n r > 3n

r : reaction.n : number of members.

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The equipment is used to investigate two beam arrangements as shown in Figure 1 below :

Figure 1: The theoretical Formula for Fixed and Propped Cantilever Beam

OBJECTIVE To find out the fix moment value between the theoretical with the experimental value for fixed and propped cantilever beam.

APPARATUS

I. The backboard unit set up in the rest frame.II. The knife - edge and encastre fixing, load cell support.

III. A digital forcemeter with leads.IV. The thin " flexible" beam.V. A set of weights, weight hanger and a knife-edge hanger.

VI. 4.9 N load.

Figure1: Show the equipment.

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PROCEDURE

1. The load cell support was put onto the test frame and slide to 400mm position2. The top clamp plate was removed from the load cell and the two screws on the front were

fix securely.3. The screws were loose, by used the hole at one end and had secured the beam to the

moment chuck on the backboard.4. The beam fully rested on the load cell by undo the moment arm locking screw.5. The beam was clamped evenly by two screws and clamping plate. Thus, the chuck screw

tighted and also the moment arm locking screw.6. The moment arm connected to input 1 and the load cell to input 2.7. Each reading was selected in turn and the relevant control used to zero the readings.8. The equipment was set up as a fixed beam. The left and right-hand end were setted to

measure the fixing moment and support reaction.9. Weight of 4.9N was applied to the beam, 40mm from the left-hand end.10. The readings were taken and recorded in the table.11. Procedure was repeated in 40mm increments across the span beam. The moment MA

was calculated by multiplied the force and length of the moment arm.12. The moment arm clamp screw was released and undo. Both clamp plates were removed

from the load cell support.13. The beam was rested back to the knife-edge and the moment arm clamp screw tighted.14. The moment arm and the load cell support was setted to zero.15. The propped cantilever equipment was setted.16. The experimental procedure was repeated as the fixed beam and results were recorded.17. The theoretical values of all the moments and reactions were calculated.

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Result of Experiment For Propped Cantilever Beam

Distance

A (m)

Distance

B (m)

Load

W (N)

Moment

Arm

Force (N)

Experimental

MA (Nm)

Theory

MA

(Nm)

Experimental

RB (N)

Theory

RB (N)

0.040 0.360 4.900 3.500 0.175 0.168 0.100 0.071

0.080 0.320 4.900 5.600 0.280 0.282 0.300 0.274

0.120 0.280 4.900 6.800 0.340 0.350 0.600 0.595

0.160 0.240 4.900 7.500 0.375 0.376 1.100 1.019

0.200 0.200 4.900 7.300 0.365 0.368 1.600 1.531

0.240 0.160 4.900 6.400 0.320 0.330 2.200 2.117

0.280 0.120 4.900 5.300 0.265 0.268 2.900 2.761

0.320 0.080 4.900 3.500 0.175 0.188 3.600 3.450

0.360 0.040 4.900 1.800 0.090 0.097 4.300 4.166

Table 1

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Result of Experiment For Fixed Beam

Distance A (m)

Distance B (m)

Load w (N)

Moment Arm

Force (N)

Experimental MA (Nm)

Theory MA

(Nm)

Experimental RB (N)

Theory RB (N)

0.040 0.360 4.900 3.400 0.170 0.159 0.200 0.137

0.080 0.320 4.900 5.200 0.260 0.251 0.600 0.510

0.120 0.280 4.900 5.900 0.295 0.288 1.100 1.058

0.160 0.240 4.900 5.800 0.290 0.282 1.800 1.725

0.200 0.200 4.900 5.100 0.255 0.245 2.400 2.450

0.240 0.160 4.900 4.100 0.205 0.188 3.200 3.175

0.280 0.120 4.900 2.800 0.140 0.123 3.800 3.842

0.320 0.080 4.900 1.500 0.075 0.063 4.300 4.390

0.360 0.040 4.900 0.500 0.025 0.018 4.700 4.763

Table 2

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Example the calculation of table 1(Fixed Beam)

W= 4.9 N A= 40mm B=(L-A)mm L=400mm

WAB2

L2 = 4.9(40)(360)2

4002 ÷ 1000 W A2(L+2B)

L3 = 4.9(40)2(400+2 (360 ))4003

MA = 0.16 N RB = 0.14 N

Example the calculation of table 2(Propped Cantilever)

W= 4.9 N A= 40mm B=(L-A)mm L=400mm

WAB(L+B)2L2 =

4.9×40×360(360+400)2(400)2 ÷ 1000

MA = 0.17 N

WAL - ML = 4.9(40)400 – 0.17(1000)

400 RB= 0.07 N

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DISCUSSION

1. Relationship between experimental and theoretical.

Moment Arm

Force (N)

Experimental MA (Nm)

Theory MA

(Nm)

Error(%)

Experimental RB (N)

Theory RB (N)

Error(%)

3.500 0.175 0.168 4.17 0.100 0.071 40.85

5.600 0.280 0.282 -0.71 0.300 0.274 9.49

6.800 0.340 0.350 -2.86 0.600 0.595 0.84

7.500 0.375 0.376 -0.03 1.100 1.019 7.95

7.300 0.365 0.368 -0.82 1.600 1.531 4.51

6.400 0.320 0.329 -2.74 2.200 2.117 3.92

5.300 0.265 0.268 -0.11 2.900 2.761 5.03

3.500 0.175 0.188 -6.91 3.600 3.450 4.35

1.800 0.090 0.097 -7.22 4.300 4.166 3.22

Table 3 : Propped Cantilever Beam

Moment Experimental Theory Error Experimental RB Theory Error

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Arm Force

(N)MA (Nm)

MA

(Nm)(%) (N) RB (N) (%)

3.400 0.170 0.159 6.92 0.200 0.137 45.99

5.200 0.260 0.251 3.59 0.600 0.510 17.65

5.900 0.295 0.288 2.43 1.100 1.058 3.97

5.800 0.290 0.282 2.84 1.800 1.725 4.35

5.100 0.255 0.245 4.08 2.400 2.450 -2.04

4.100 0.205 0.188 9.04 3.200 3.175 0.79

1.800 0.140 0.123 13.82 3.800 3.842 -1.09

1.500 0.075 0.063 19.05 4.300 4.390 -2.05

0.500 0.025 0.018 38.89 4.700 4.763 -1.32

Table 4 : Fixed Beam

DISCUSSION

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From the result obtained by the experiment, we has achieved the objective of the experiment that is determined the fix moment value for the fixed and propped cantilever beam.

For the fixed beam and propped cantilever beam, the result showed that the maximum value for moment at point A can be occurs when the distance of load from point A is 160mm and for the reaction at point B maximum value can be occurs when the distance of load from point A is 360mm. It showed the maximum moment can be occurs when load is applied at the a few middle of the fixed beam, but for the maximum value for reaction at point B can be occurs when load is applied at a little end of the fixed beam.

The relationship between theoretical value and experiment value show has a few different that is means it has some errors when the experiment conducted.

The value of theoretical and experiment must be equal. But there were some errors when the experiment was conducted so that the result obtained is not accurate. The errors are including:

The equipment was not set up in proper conditions. The screw at the beam for 2 cases that is fixed beam and propped cantilever beam not

screw for suitable cases. For the fixed beam cases the beam screw not equally tightened. For the propped cantilever beam cases the beam screw is loosed. Have some errors when calculating number of load that was applied at the beam. Parallax error while taking measured for the distance of the cell load from point A.

If compare with experiment value and theoretical value there has a few error that can show by average percentage errors value for the experiment.

The average percentage errors show in table 3:

Average Percentage Errors

Fixed Beam Propped Cantilever Beam

MA (Nm) RB (N) MA (Nm) RB (N)

12.04 9.88 11.67 2.15

Table 3: The Average Percentage Errors.

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The following precautions must be considered while conducting the experiment in order to obtain an accurate result.

Set up the equipment properly. Screwed the beam for the suitable cases that is fixed beam cases and propped cantilever

beam cases. Equally tighten screwed for the fixed beam . Loosed the screw for the propped cantilever beam. The distance of the cell load must be surely measured. Calculating the number of load equally according to the experiment procedure.

The value of moments and theoretical deflections for the propped cantilever and the fixed beams can be calculated using this calculation:

Example the calculation of table 1(Fixed Beam)

W= 4.9 N A= 40mm B=(L-A)mm L=400mm

WAB2

L2 = 4.9(40)(360)2

4002 ÷ 1000 W A2(L+2B)

L3 = 4.9(40)2(400+2 (360 ))4003

MA = 0.16 Nm RB = 0.14 N

Example the calculation of table 2(Propped Cantilever)

W= 4.9 N A= 40mm B=(L-A)mm L=400mm

WAB(L+B)2L2 =

4.9×40×360(360+400)2(400)2 ÷ 1000

MA = 0.17 Nm

WAL - ML = 4.9(40)400 – 0.17(1000)

400 RB= 0.07 N

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ADVANTAGES & DISADVANTAGES OF A SIMPLE BRIDGE

Advantages

Beam bridges are helpful for short spans. Long distances are normally covered by placing the beams on piers.

Disadvantages

Beam bridges may be costly even for rather short spans, since expensive steel is required as a construction material. Concrete is also used as beam material, and is cheaper. However, concrete is comparatively not that strong to withstand the high tensile forces acting on the beams. Therefore, the concrete beams are normally reinforced by using steel mesh.

When long spans are required to be covered, beam bridges are extremely expensive due to the piers required for holding the long beams. Building of the support piers may not always be possible due to the limitation of space.

Bridge beams are likely to droop between the piers, due to the different bridge loads acting downwards. The forces acting upwards at the pier supports also influence the drooping effect. The sagging tendency is increased when the bridge span or load is increased.

The advantages are that you reduce the moment in the beam thus also reducing the deflection.

The disadvantages are that you are causing moment at the top over supports thus you will need some reinforcing in the top of the beam.

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CONCLUSION

Statically determinate body, the equilibrium equations of statics are sufficient to determine all unknown forces or other unknowns that appears in equilibrium equations. Whereas statically indeterminate body the equilibrium equations of statics are not sufficient to determine all unknown forces or other unknown

This experiment is use for designed the bridge it consider all load that is moving such as a vehicle. It can know the value of the moment that has been occurs when moving load applied. It also can know the value of the support reaction at any end point.

The objective of the experiment is achieved but the value for the experiment and theory has a different value and has an error. It happened because of several errors that has when the experiment conducted. As the conclusion, the experiment is accepted because of the percentage errors is low but if want to get the accurate result the experiment should be repeated by considering all the precautions stated.

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APPENDIX

4.9 N loads. A digital forcemeter with leads

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