lab span deflection - copy

SPAN DEFLECTION (DOUBLE INTEGRATION METHOD)GROUP 3 SECTION 3

1.0 OBJECTIVE

To determine the relationship between span and deflection.

2.0 INTRODUCTION

A beam must possess sufficient stiffness so that excessive deflections do not

have an adverse effect on adjacent structural members. In many cases, maximum

allowable deflections are specified by Code of Practice in terms of the dimensions of

the beam, particularly the span. The actual deflections of a beam must be limited to the

elastic range of the beam, otherwise permanent distortion result. Thus in determining

the deflections of beam under load, elastic theory is used. In this experiment double

integrations method is used to give the complete deflected shape of the beam.

3.0 THEORY

M x-x = EI d²y = P L dx² 2 2


v x-x = EI dy = PLx _ Px² + Adx 4 4

y x-x = EIy = PLx² _ Px³ + Ax + B 8 12

When x = 0 ; dy = 0 hence A = 0

When x = L / 2 ; y = 0; hence 0 = PL³ _ PL³ + B32 96

B = - PL³48

When x = 0 ; Y mak = - PL³ ( mid span ; c )48EI

x = L / 2 ; v mak = PL² ( at support )16 EI

Where E can be obtained from the backboard

I = bd³12

b

d

4.0 APPARATUS


1. Deflection of Beam and Cantilever Appratus

2. Digital Dail Test Indicator

3. Hanger and Masses


4. Specimen Beam

5.0 PROCEDURES


1. The moveable knife-edge supports were positioned so that they are

400mm apart. The chosen beam was place on the support.

3. The hanger and the digital dial test indicator placed at mid span. The

digital reading zeroed.

4. An incremental load was apply and the deflection recorded for each


increment in the table below.

5. Repeated the above using span of 300mm and 500mm and another beam.

6.0 RESULT


Experiment 1: Span = 500mm

No. Mass (N) Deflection

(experimental)

Theoretical Def.

(Ymax)

% Different

1 2.453 - 2.01 - 0.00374 98.14 %

2 2.453 - 0.68 - 0.00072 99.89 %

3 2.453 - 1.49 - 0.0015 99.90 %

Table 1



(experimental)

Theoretical Def.

(Ymax)

% Different

1 2.453 - 1.02 - 0.00192 99.81 %

2 2.453 - 0.37 - 0.00031 99.92 %

3 2.453 - 0.80 - 0.00077 99.90 %

Table 2



(experimental)

Theoretical Def.

(Ymax)

% Different

1 2.453 - 0.44 - 0.00082 99.81 %

2 2.453 - 0.16 - 0.00016 99.90 %

3 2.453 - 0.34 - 0.00032 99.91 %

Table 3

7.0 DATA ANALYSIS


Experiment 1, number 1;

When x = 0; Y mak = - PL³ (mid span ; c )

48EI

I aluminium = bd³12

b = 19mmd = 2.5mm

I = (19) ( 2.5 ) 3 12

= 24.740 mm4

= 2.474 x 10-11m4

E = 207GNm-2

Ymak = - ( 2.453 )(0.5) 3 48(69G)(2.474 x10-11) = - 0.00374 mm

% different = theo – exp x 100% exp = - 2.01 – (- 0.00374 ) x 100%

- 2.01 = 98.14 %

When x = 0;


Y mak = - PL³ (mid span ; c )48EI

I tembaga = bd³12

b = 18mmd = 3mm

I = (1 8 ) ( 3.0 ) 3 12

= 40.50 mm4

= 4.05 x 10-11m4

E = 105GNm-2

Ymak = - ( 2.453 )(0. 5 ) 3 48(105G)(4.05 x10-11) = - 0.0015 mm


- 1.49 = 99.90 %

When x = 0;


Y mak = - PL³ (mid span ; c )48EI

I steel = bd³12

b = 19mmd = 3.0mm

I = (19) ( 3.0 ) 3 12

= 42.75 mm4

= 4.275 x 10-11m4

E = 207GNm-2

Ymak = - ( 2.453 )(0. 5 ) 3 48(207G)(2.474 x10-11) = - 0.00072 mm


- 0.68 = 99.89 %

8.0 DISCUSSION


Comment on the different between the theoretical and experimental result.

The range of percentage of the different between the theoretical and experimental

result was 99.14% to 99.92%. Therefore, there might have some errors occur during

this experiment. The surrounding environment might be effect the reading of the result

of deflection. The errors also can occur when the hanger was hang at a wrong place or

the masses was not accurate.

9.0 EXTRA QUESTIONS

1. Calculate the deflection when x = L / 3 (experiment 1, no 3). Check the result by

placing the digital dial at this position.

y x-x = EIy = PLx² _ Px³ + - PL³ 8 12 48EI

P = 2.453N

x = 0.5/3 = 0.167m

L =0.5m

E = 105GNm-2

I = 40.50 mm4

= 4.050 x 10-11m4

- PL³ = - ( 2.453 )(0.5) 3 48EI 48(105G)(4.05 x10-11) = - 0.0015 mm

y x-x = ( 2.453 x 0.5 x 0.167 2 ) – ( 2.453 x 0.167 3 ) – 0.0015 x 10-3

8 12 = 3.32 x 10-3 m

= 3.32 mm

2. Calculate V max in experiment 2, no. 2.

x = L / 2 ;


v mak = -PL² ( at support )16 EI

P = 2.453N

L =0.4m

E = 207GNm-2

I = 42.75 mm4

= 4.275 x 10-11m4

v mak = ( 2.453 )(0.4) 2 16(207G)( 4.275 x 10-11)

= 1.109 x 10-3

10.0 CONCLUSION

Macaulay’s method (the double integration method) is a technique used in

structural analysis to determine the deflection of Euler-Bernoulli beams. Use of

Macaulay’s technique is very convenient for cases of discontinuous and/or discrete

loading. Typically partial uniformly distributed loads (u.d.l.) and uniformly varying

loads (u.v.l.) over the span and a number of concentrated loads are conveniently

handled using this technique.

The first English language description of the method was by Macaulay. The

actual approach appears to have been developed by Clebsh in 1862 Macaulay's method

has been generalized for Euler-Bernoulli beams with axial compression, to

Timoshenko beams, to elastic foundations, and to problems in which the bending and

shear stiffness changes discontinuously in a beam.

From this experiment, we can concluded that we were achieve the objective that

to determine the relationship between span and deflection. We also can conclude that

the deflection will be increase when the mass or forces are increase. Beside that, the

span or the length of the beam will also affect the value of deflection. The shorter beam

http://en.wikipedia.org/w/index.php?title=Elastic_foundation&action=edit&redlink=1

http://en.wikipedia.org/wiki/Timoshenko_beam_theory

http://en.wikipedia.org/wiki/Beam_theory

http://en.wikipedia.org/wiki/Deflection

http://en.wikipedia.org/wiki/Structural_analysis


will have a smaller value of deflection

11.0 REFERENCES

Labsheet Lab Structure Span Defelection (Double Intergration Method)

www.wikipedia.com.my

Mechanics Of Materials Second Edition Universiti Tun Hussein Onn

Malaysia.

CONTENT

http://www.wikipedia.com.my/


OBJECTIVE…………………… 1

INTRODUCTION………........... 1

THEORY………………………. 1

APPARATUS………………….. 3

PROCEDURE………………..... 5

RESULT……………………...... 7

DATA ANALYSIS……………... 8

DISCUSSION………………….. 11

EXTRA QUESTION…………... 11

CONCLUSION………………... 12

REFERENCES………………… 13

lab span deflection - copy

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