DECLARATION AND CERTIFICATION

Except where mentioned, we verify that the experimental work, results, analysis

and conclusions are set out in this project is entirely our own efforts.

MFO 04/2012 large deflections of beams. Laboratory validation of proposed

project.

ACHOLA KEVIN : F18/1887/2007

ONUNGA ERICK: F18/1857/2007

..

KIRUGUMI DANIEL: F18/20991/2007

The above named students have submitted this report to the department of

Mechanical and manufacturing Engineering, University of Nairobi with my

approval as the supervisor (s)

Professor Oduori F.M Engineer Munyasi D.M

.

ii | P a g e

DEDICATION

We dedicate this project to human prosperity and enlightenment.

iii | P a g e

Abstract The deflection of a large cantilever beam made of linear elastic aluminum under the

action of a vertical concentrated load applied at mid span was analyzed .this was

done experimentally and numerically using Professor Oduoris1 theory of large

beam deflections.

The experiment was done using aluminum beam of dimension length 47cm by 5cm

and thickness of 1.7cm

From first principles, we derived the equation for the determination of large

deflection of beams. We set up tests in the laboratory in order to validate the theory;

we then compared theoretical and experimental results. They were in good

agreement.

1Professor department of mechanical and manufacturing engineering university of Nairobi

iv | P a g e

Acknowledgements

We take this opportunity to express our thanks to Professor Oduori and Engineer

Munyasi for their inspiration, support and the appreciation for their work under

taken in this project .we also express sincere gratitude to the staff at the

mechanical engineering workshop. We also appreciate the moral and general

support accorded to us by our friends.

v | P a g e

Nomenclature

b -Breath of the specimen

d Thickness of the specimen (depth)

E Youngs modulus

F Force causing deflection of the beam

I Second moment of area elasticity of aluminum

L, M Coordinates perpendicular to, and parallel to the force causing deflection of

aluminum

M Bending moment

P Axial force

q Uniformly distributed load

R Radius of curvature

S Length along the deflected cantilever beam

w, y -deflection

X, Y Cartesian coordinates in the plane respectively

- Angular deformation of the deflected cantilever beam

b Bending stress

vi | P a g e

Table of contents

Declaration and certification .......i

Dedication ii

Abstract iii

Acknowledgements ....iv

Nomenclature ..v

Chapter One 1.0 Introduction .1

1.1Assumptions.....3

1.2Objective ..4

Chapter Two 2.0 Literature review5

2.1Theoretical Analysis 6

2.2 Formulation of the model ..21

2.3 Model Validation and adaptation for laboratory experimentation .28

Chapter Three 3.0 Description of apparatus used 32

3.1 Hand tools .....32

3.1.1 Tape measure 32

3.1.2 Steel rule 32

3.1.3 Vernier calipers 32

3.1.4 Vernier Height gauge ..32

3.1.5 Try-square ..33

3.1.6 Dial gauge ..33

3.1.7 Rough and smooth file 33

3.1.8 Spirit level ..34

vii | P a g e

3.1.9 Scriber .34

3.2 Machines .35

3.2.1 Power hacksaw ....................35

3.2.2 Lathe machines 36

3.2.3 Milling machine 37

3.2.4 Planar machine .38

3.2.5 TIC machine ..38

Chapter four

4.0 laboratory data acquisition ..40

4.1 specimen preparation .40

4.1.1 Procedure of preparation 40

4.2 Test Rig preparation 42

4.2.1 Preparation of the support roller bearings..43

4.3 Test of specimen 45

Chapter five

5.0 Analysis of experimental results..51

5.1 Results ....52

5.2 Data analysis .56

5.2.1 Sample calculation.56

Chapter Six 6.0 Discussion, conclusion and recommendations .66

6.1 Discussions 66

6.2 Conclusion .....................................................67

6.3 Recommendations for further work .68

7.0 References ..69

viii | P a g e

Table of figures

Figure page

Fig 1 crop stem deflected by the reel2

Fig 2 diagrammatic representation of deflected beam .6

Fig 3 bending of Euler-Bernoulli beam 8

Fig 4 elastic curve derivation .11

Fig 5 triangle CDQ .11

Fig 6 first area principle (semi-graphic form) .........................................................................................15

Fig 7 large deflections of buckled bars (the elastica) ...16

Fig 8 model of deflection ....21

Fig 9 transformed model of deflected stem ....23

Fig 10 triangle ABC ..26

Fig 11 the model adopted for laboratory validation ..28

Fig 12 transformed model adopted for laboratory validation .29

Fig 13 force diagram .30

Fig 14 loading arrangement 50

Fig 15 deflections in progress 50

Fig 16 triangle adapted for data analysis ..56

Fig 17 beam cross section 57

ix | P a g e

Table of pictures

Unless otherwise stated all pictures were taken at the department of mechanical and manufacturing

Engineering workshop at the university of Nairobi.

Picture page

Picture 1 vernier height gauge .33

Picture 2 power hacksaw .35

Picture 3 and 4 lathe machines 36

Picture 4 milling machine .37

Picture 5 planar machine .38

Picture 6 TIC machine .39

Picture 7 the unprepared aluminium beam 40

Picture 8 aluminium bar on the planar workstation .41

Picture 9 aluminium beam after preparation 42

Picture 10 cast iron block with roller bearing supports .43

Picture 11 roller bearings 44

Picture 12 knife-edge .45

Picture 13 loading tip (knife-edge) 2 ..46

Picture 14 roller bearings on each groove .47

Picture 15 loading arrangement 48

Picture 16 the general arrangement on a TIC machine ..49

x | P a g e

Table of tables

Table page

Table 1: experimental raw data 52

Table 2: load and respective deflection (Experimental) .53

Table 3: table of analyzed values ...............................................................................................58

Table 4: load and calculated deflection ..61

Table 5: comparisons between experimental and calculated values of deflections ..62

Table 6: load and percentage deviations 65

xi | P a g e

Table of graphs

Graph page

Graph 1: of load against deflection 54

Graph 2: for load against deflection for the elastic region..55

Graph 3: of

..59

Graph 4: of calculated deflection against experimental deflection ..................................................63

1 | P a g e

Chapter one

1.0 Introduction Beams are common elements of many architectural, civil and mechanical engineering

structures and the study of the bending of straight beams forms an important and

essential part of the study of the broad field of mechanics of materials and structural

mechanics. All undergraduate courses on these topics include the analysis of the

bending of beam but only small deflections of the beam are usually considered. In such

a case, the differential equation that governs the behavior of the beam is linear and can

easily be solved. Here we consider large deflections in cantilever beams.

By definition beams are structural members capable of sustaining loads normal to their

axis, a cantilever beam is a beam that is fixed at one end, while the other end is

unsupported but suspended.

A beam in application may be strong enough to resist safely the bending moments due

to applied load yet not be suitable because its deflection is too great. Excessive

deflection may impair the strength and stability of the structure giving rise to minor

troubles such as cracking as well as affecting the functional needs and aesthetic

requirements. Thus, there is always a need to consider deflection when designing

beams.

In much of the study and practice of mechanical and structural engineering, the

equations used for determination of beam deflections, are derived with assumption of

small deflections .This is appropriate because, in most mechanical and structural

engineering applications, small deflections are a functional requirement. However, there

arise cases in agricultural machinery engineering for instance, where beam deflections

can no longer be assumed small. Then, it becomes necessary to develop and use

equations other than those commonly found in mechanical and structural1 engineering

documents, which are largely based on small deflections. Timoshenko and Gere

derived solutions to large deflection problems, which led to an elliptical integral. Elliptical

integral problems can only be evaluated numerically, which is tedious and long. Hence,

there is sufficient reason therefore to seek analytical solutions to problems of large

deflections .such an equation is developed and evaluated in this presentation.

An example of an application that would involve large crop stem (beam) deflections, in

the design and operation of the combine harvester reel as illustrated in fig 1.10

2 | P a g e

Combine harvester reel

Reel tine

Deflected crop stem

Ground

Fig 1 crop stem deflected under the effect of a reel

3 | P a g e

1.1 Assumptions The assumptions made in formulating a model of the deflected crop stems include

A bunch of deflected crop stems deflected by the reel shall be considered by the

reel shall be considered to behave like a single, initially vertical cantilever that is

fixed at the base.

At its point of action, the deflecting force shall be considered directed normal to

the curvature of the cantilever beam.

The stress-strain relationship for the deflected stems shall be tentatively

assumed linearly elastic.

We do not make the assumption of small deflections commonly made in

mechanical and structural engineering applications.

4 | P a g e

1.2 Objective The purpose of this work is to validate a new theory for the determination of large beam

deflections of a class of cantilever beams under the action of a concentrated force or

load. This is done both experimentally and theoretically then the results compared to

ascertain the validity of the theory and hence formula presented for use in deriving

solutions analytically to large beam deflections.

5 | P a g e

Chapter two

2.0 Literature review In the literature, large deflections behavior of beams continues to be the subject of

intensive research. Numerous researchers have studied the problem under different

conditions and using different conditions and methods of solutions

Jong-Dar Yau developed closed-form solutions of large deflections for a guyed

cantilever column pulled by an inclined cable. He used elliptical integral method in

deriving analytical solution for tip displacement of the guyed column. His theory

can be useful in cable-stayed bridges, radio masts and cable supported roofs all of

which involve large deflections.

N. Tolou and J.L. Herder developed a semi analytical approach to large deflections

in compliant beams under point load. In their work, they successfully investigated

the feasibility of ADM (Adomian decomposition method) in analyzing compliant

mechanisms.

A.Kimiaeifar, G. Domairry, S.R. Mohebpour, A.R. Sohouli and A.G Davodi4

developed analytical solutions for large deflection of a cantilever beam under a non-

conservative load based on homotopy analysis method (HAM)

Stephen P. Timoshenko and James M. Gere1 analyzed large deflections by

considering the case of buckled bars (the elastica)

6 | P a g e

2.1 Theoretical analysis Beams are essentially one-dimensional systems. Figure 2.0 shows a cantilever beam of

length L with a concentrated force P applied at the mid span. In the evaluation of the

various parameters that come into play when the beam deflects with the application of

load, the assumption made in formulating the model of the deflected crop stems in

chapter one are made here as well, that is:

The stress-strain relationship for the deflected beam is to be linearly elastic. The

material of the beam is homogeneous and its stress-strain curve is linear, i.e. the

stress is proportional to strain by Hookes law.

It will also be assumed that the cross section of the beam remains constant

across the length of the beam, meaning that the effect of poisons ratio, or the

ratio of axial elongation to lateral contraction can be neglected.

It is assumed that the Bernoulli Eurler theorem is valid, which states that the

curvature of the beam is proportional to the bending moment.

The beam has the same modulus of elasticity in compression as in tension

The deflection due to the weight of the beam is neglected

R load

y Neutral axis

X

A

X B

Fig 2 diagrammatic representation of deflected beam

7 | P a g e

Several beam theories are in use to calculate and analyze deflections of beams under

the action of different kinds of loading .The most common ones are Eurler-Bernoulli

beam theory and Timoshenko beam theory.

Euler-Bernoulli beam theory which is also known as engineers beam theory or

classical beam theory is a simplification of the linear theory of elasticity which provides a

means of calculating the load carrying and deflection characteristics of beams .It

covers the case for small deflections of the beam which is subjected to lateral loads

only.

Timoshenko beam theory covers beams under both lateral and axial loading.

8 | P a g e

Static beam equation of Euler-Bernoulli beam

Fig 3 bending of an Euler Bernoulli beam

9 | P a g e

The Euler-Bernoulli equation describes the relationship between the beams deflection

and the applied load. Each cross-section of the beam is at 90 degree to the neutral axis

The curve describes the deflection of the beam at some position considering

the beam model is a one-dimensional object. q is a distributed load. It is force per unit

length analogous to pressure, that is force per unit area. It may be a function of or

other variables.

Note that E is the elastic modulus and that I is second moment of area I must be

calculated with respect to the centroidal axis perpendicular to the applied loading. For a

Eurler-Bernoulli beam not under any axial loading, this axis is axial loading.

Often, and EI is a constant so that

This equation describing the deflection of a uniform, static beam is used widely in

engineering practice. For more situations that are complicated, the deflection can be

determined by solving the Eurler-Bernoulli equation using techniques such as slope,

moment distribution method, moment area method, conjugate beam method, the

principle of virtual work, direct integration, Castiglianos method, Macaulays method or

the direct stiffness method and elastic curve method, the most commonly used methods

include.

1. Area moment method

2. The elastic curve method

The elastic method requires the use of calculus since it is based on the solution of the

differential equation for the elastic curve by double integration. In its analysis, we get the

equation of the elastic curve.

From fig.2, it is evident that the magnitude of the bending stresses varies directly with

the distance from the neutral axis, that is;

Where R is the radius of curvature of the elastic curve. Taking the general flexure

expression;

10 | P a g e

Then

Now consider unloaded beam AB as shown in fig 4, the beam is deflects to A1B1 under

a load q.

11 | P a g e

Y O

d

R B1

D

C dy

Q

A1

dx

A B

Fig. 4 elastic curve derivation

From the figure above, the following can be deduced from the triangle CDQ.

D

ds dy

C dx Q

Fig. 5 Triangle CDQ

12 | P a g e

CD=

For small angle of

The curved line A1B1 in fig.3 represents the neutral surface of a beam after bending .It is

the elastic surface of a deflected beam of indefinite length. C and D are points on this

elastic surface and are separated by a small distance

When two lines are constructed perpendicular to the elastic curve at points C and D,

their extensions will intersect at the centre of curvature O, a distance R from the

curvature and will form a small angle,

The curvature is actually very slight; therefore, CQ can replace the horizontal length

Then from geometry

However, since triangle CDQ is small

And

Therefore,

13 | P a g e

But

Substituting and evaluating, we get the differential equation of elastic curve as shown

below

From the general flexural equation

Hence,

Where

EI-is the flexural stiffness

- is the curvature of the neutral axis

In the above derivation of the elementary theory, the assumption of small deflections

has been made. Hence, this equation cannot be used in analyzing large beam

deflections problems. The elementary theory also neglects the square of the first

derivative in the curvature formula and provides no correction for the shortening of the

moment arm as the loaded end of the beam deflects.

From the area moment principle, the neutral surface of a homogeneous beam is

considered to be on a continuous plane, which passes through the centre of gravity of

14 | P a g e

each right section. When the beam deflects, the neutral surface becomes a continuous

curved surface. Deflections are measured from the original position of the neutral

surface to the elastic surface.

The derivation of the area moment equations is shown in the semi-graphic form in fig 6

15 | P a g e

O x

d D

C

A E

B1

d y

A1

M

EI

dx

X

Fig 6 first area principle (semi-graphic form)

16 | P a g e

From fig.6, the angle between the two tangents AA1 and BB1 is equal to . And the

summation of all elemental areas of the

diagram between the two tangent points (the

cross hatched area shown) is presented in equation form as .

This is the first area moment. it considers the assumptions of small deflections and thus

cannot be used to determine large deflections .

Timoshenko and Gere analyzed large deflections by considering the case of buckled

bars (the elastic) as shown below.

O

s

L P

m xa

n

ya

Fig 7 large deflections of buckled bars (the elastica)

17 | P a g e

At the critical loading, the bar can have any value of deflection, provided the deflection

remains small, the differential equations used to calculate the critical loads are based on

the approximate expression

for the curvature of the buckled bar. If the exact

expression for the curvature is used, there will be no indefiniteness in the value of

deflection. The shape of the elastic curve, when found from the exact differential

equation is called the elastica.

Consider the slender rod shown in fig.7, which is fixed at the base and free at the upper

end. If the load P is taken somewhat larger than the critical value, a large deflection of

the bar is produced. Taking the axis as shown in the figure and measuring the distance

s along the axis of the bar from the origin O, we find that the exact expression for the

radius of the curvature of the bar is

Since the bending moment in the bar is equal to the product of flexural rigidity the

curvature, the exact differential equation of deflection is;

The change in length of the column due to compression is neglected.

Differentiating equation (i) with respect to s and using the relation

We obtain,

Multiplying through by and integrating gives

Where

Hence

18 | P a g e

Therefore

Boundary conditions

At the upper end of the bar;

And also,

Substituting the boundary conditions and solving for constant c, we get ,

Therefore, we have,

Or

Solving for ds and noting that

is negative we have,

In addition, for the total length of the bar, after the limits of integration are interchanged

we have

19 | P a g e

Introducing and letting

such that,

Differentiating equation

Substituting equation (iii) and noting that

We obtain

As the value of decrease, the integral K and the load P increase

In calculating the deflection of the bar, we note that

The total deflection of the top of the top of the bar in the horizontal direction becomes

From equation (iv) we have;

And therefore,

By using the relation,

20 | P a g e

We find that;

Substituting expression (v)(vi) and (ix) into equation (viii) and changing the limits

accordingly, we obtain;

21 | P a g e

2.2 Formulation of the model However, the integral appearing in equation (vii) is known as a complete elliptical

integral of the first kind and is designated by K (p). Elliptical integrals can only be

evaluated numerically which is both tedious and does not give very accurate results. In

contrast, the introduction of L as the variable of integration instead of s greatly simplifies

the problem.

Hence, analytical solutions are sought to problems of large deflections. In this case, a

cantilever beam is considered fixed horizontally and a point load applied near the free

end such that the beam deflects upwards.

Ym

Fcos m

dy Y

Beam dx

ds

r + dr

d r X Xm

Fig 8 model of deflection

22 | P a g e

The assumptions of small deflections commonly made in mechanical and structural

engineering are not made here.

Referring to fig.8 according to the elementary theory of elastic bending at an arbitrary

point denoted (X, Y) along the length of the deflected cantilever, we may write;

Where r is the radius of curvature of the elastic curve

Taking the general flexure expression

We have

Therefore

Then

From fig 2.15, the bending is

The following figure 9 illustrates the deflection model transformed in a manner similar to

the transformation of rectangular Cartesian coordinates, common in the study of

kinematics

23 | P a g e

Ym

Nm

m

m

Y

dy

dL dx

N X Xm

Lm L

Ground

Fig 9 transformed model of deflected stem

24 | P a g e

Accordingly,

Since L and N are both zero when X and Y are both zero, then

The transformation matrix of equations (3a) and (3b) is symmetric and therefore equal

to its own transpose i.e.

It is also orthogonal and therefore its transpose is equal to its inverse, considering the

context of the plane transformation geometry, this matrix has the form of reflection

matrix, which implies that;

Lm = X m and Nm = Y m .(3c)

From equation (3b)

From equation (3c), it follows that

And,

The relations in (3c) above imply that;

Dividing equation (8) by (9) gives,

25 | P a g e

But Lm = X m and Nm = Ym

Therefore

From trigonometry

Thus

Therefore

The maximum deflection Ym can be developed from equation (10), that is

Equation (11) and (12) are based entirely on the geometry of deflection .they do not

involve material properties of the deflected cantilever.

26 | P a g e

From equation 3a, it follows that the bending moment at L is;

And,

dL

A B

ds

C

Fig 10 triangle ABC

27 | P a g e

Combining equations (1)(13)and (14), it follows that,

But

Hence

Rearranging gives,

Cos (

The introduction of L as the variable of integration instead of s, greatly simplifies the

problem. Equation (15) can be integrated if the relationship between EI and L is known.

The following assumptions may be made as a matter of investigations.

The product EI does not vary with L since =0 when L=0.

By integrating equation (15) we get;

28 | P a g e

2.3 Model validation

Adaptation of the model for laboratory validation

With the equipment available at the university of Nairobi mechanical engineering

workshop it was easier and more convenient to use an initially horizontal, simply

supported beam of uniform cross-section rather than an initially vertical cantilever. The

maximum load at the mid span of the beam is p. for easy analysis; each half of such a

beam is equivalent to an initially horizontal cantilever, with a span that is half of that of a

simply supported beam as illustrated in fig 10

P

Rh Rh

Rv Xm Rv

Fsin m

Ym

0.5 Xm F cos m

Fig 11 the model adapted for laboratory validation

29 | P a g e

From the figure Rh is the horizontal component of the reaction and Rv is the vertical

component of the reaction

From statics

Therefore

Where p is the load at mid-span

L- axis

Lm

Nm c

m

m F

O A B

m

N-axis

Fig 12 the transformed model adopted for laboratory validation

30 | P a g e

From fig 12 we can draw triangle ABC as shown below

C

m

F

A B

Fig 13 force diagram

The force diagram can be analyzed as;

From equation (14) of chapter two

Substituting for F in equation (17)

From trigonometry;

31 | P a g e

Substituting this into the equation (19), we get

Now, in accordance with equation (3c) of chapter two, it follows from fig 10 and 11 that

And

Substituting equation (21), for Lm into equation (20), we get;

The experimental results were also in agreement with the predictions of the theory from

equation (12) below

Therefore, it follows from equation (22)

Dividing through by 4 gives

Substituting the value of

from equation (23) into equation (12) gives

The above equation was used to validate the model.

32 | P a g e

CHAPTER THREE

3.0 A DESCRIPTION OF APPARATUS USED.

3.1 HAND TOOLS

3.1.1 TAPE MEASURE

A tape measure was used to measure the dimensions of the aluminium specimen.

3.1.2 STEEL RULE

The steel rule was used in measuring the dimensions of the aluminium bar, aiding in

drawing straight lines on the bar and as a straight guide for marking with the scriber.

3.1.3 VERNIER CALIPERS

The vernier calipers was used to measure the thickness of the beam specimen, the

diameter of the rollers and the measurement of the length x.

3.1.4 VERNIER HEIGHT GAUGE

This is tool is used in marking out lines with particular interest in accuracy. With the

underside as a datum, using the sharpened pointer (carbide tip) as a scriber, markings

were made all round the bar where the machining would take place

33 | P a g e

Picture 1 vernier height gauge

3.1.5 TRY-SQUARE

The try square was for marking and measuring the aluminium work piece. It was also

used to check the straightness to the adjoining surface.

3.1.6 DIAL GAUGE

The dial gauge was used to measure the deflection of the aluminium specimen when a

load was applied. The mounting of the dial gauge was done in such a way that the dial

gauge sensor was perpendicular to the specimen axis.

3.1.7 ROUGH AND SMOOTH FILE

The hand files were to make smooth the edge of the aluminium specimen after

machining.

3.1.8 SPIRIT LEVEL

The spirit level was used on the specimen to check whether it was level.

34 | P a g e

3.1.9 SCRIBER

A scriber was used in conjunction with the rule and the try square to obtain thin semi-

permanent lines where they were required for machining of the specimen.

3.2 MACHINES

3.2.1 POWER HACKSAW The power hacksaw shown in figure below was for cutting the aluminium specimen.

Picture 2 power hacksaw

35 | P a g e

3.2.2 LATHE MACHINE

It consists of the tailstock that supports the work piece, the headstock and the tool post.

It was for facing and turning of the roller bar for fitting into the roller bearing.

Picture 3 and 4 lathe machines

36 | P a g e

3.2.3 MILLING MACHINE

A milling machine is a tool used to machine/cut metal solid bars. The spindle is

stationary spindle with the table moving perpendicular and parallel to the spindle axis to

accomplish cutting. The aluminium bar was clamped on the bed.

Picture 4 milling machine

37 | P a g e

3.2.4 PLANAR MACHINE

The milling machine was also used in planing of the machined aluminium bar.

Picture 5 planar machine

3.2.5 T.I.C MACHINE

The T.I.C machine is for subjecting a specimen to tension by applying a load by means

of a hydraulic lever operated manually. Reading off a gauge on the machine gives the

load. The loaded specimen undergoes a deflection. The pre set dial gauge gives the

value of deflection.

38 | P a g e

Picture 6 TIC machine

39 | P a g e

CHAPTER FOUR

4.0 LABORATORY DATA ACQUISITION

4.1 PREPARATION OF SPECIMEN

4.1.1 PROCEDURE OF PREPARATION

1. An aluminium beam of 50mm by 50mm was obtained from the workshop store.

2. Using a tape measure the length of the beam was measured and was found to

be 470mm. The beam was as shown below.

Picture 7 aluminium beam

40 | P a g e

The bar was then marked using a vernier height gauge to produce a cross-section

of 17mm thickness by 50mm width. A 2mm allowance was put into consideration

for the cutting disc and for the planning.

3. The bar was then carefully set and clamped on the bed of the milling machine.

A vernier caliper was used to measure accurately the length from the edge of

the bed to ensure the bar lied parallel to its axis.

4. The milling machine was used to split the bar lengthwise to produce a bar of the

cross-section 17mm by 50mm. Paraffin coolant was used to keep the work

piece at a stable low temperature and reduce tip welding of the cutter.

5. The bar was then carefully mounted on the bed of the planar with the flat side

facing downwards. The machine was turned on and the work was given a

reciprocating movement while the tool was fed crosswise. A thickness of 17mm

was achieved.

Picture 8 aluminium bar on the planar workstation

41 | P a g e

6. We filled the machined surface to smoothen the edge. A spirit level was used to

check for the flatness of the bar indicated that it was level.

7. The edges were verified to be perpendicular by means of a try-square. The

prepped beam is as shown below.

Picture 9 aluminium beam after preparation

4.2 TEST RIG PREPARATION. Requirements:

A block of cast iron with two grooves for holding the support roller

bearings as shown in picture 10

Four roller bearings each of inside diameter 25mm and outside diameter

of 50mm are required.

Two pieces of smoothened mild steel bar of length 96mm and diameter

of 25.4 are required.

The cast iron block of length 520mm, and cross section 98mm by 70mm

with two grooves on upper surface, which had a diameter of 30mm and

depth of 10mm each was made available. The grooves that were 300mm

apart were thoroughly smoothened.

42 | P a g e

Picture 10 cast iron block with roller bearing supports

4.2.0 PROCEDURE FOR THE PREPARATION OF THE SUPPORT ROLLER

BEARINGS

a. A length of 98mm was measured along the smoothened mild steel of

length 570mm and diameter 25.4mm and that point marked by use of a

scriber.

b. The bar was properly fixed and tightened onto the work piece holder on

the power hacksaw and cut along the mark.

c. The procedure as illustrated in (a) and (b) were repeated to get the

second identical piece.

d. The two pieces were then mounted on a lathe, each at a time and their

ends faced to a final length of 96mm while at the same time turning to a

diameter of 25mm from the edge to a length of 16mm on either side of

43 | P a g e

the roller bar. Sand paper was used on the surface to obtain a smooth

finish.

e. The pressing machine was then used to press fit the roller bearing onto

each either side of the mild steel roller bar.

f. The above step (e) was repeated for the other mild steel roller bar and

the resulting support roller bearing were as shown in picture 11

Picture 11 roller bearings

44 | P a g e

4.2.1Test of specimen

i. The test rig was placed at the base of the T. I. C such that the centre of

the cast iron block coincided with the centre of the lower tip of the loading

point of the machine. ( Knife edge )

Picture 12 knife-edge

45 | P a g e

Picture 13 loading tip (knife-edge) 2

46 | P a g e

I. The two roller bearings were placed on each groove.

Picture 14 roller bearings on each groove

ii. The beam to be tested was then placed on the two roller bearing such

that its centre at 235mm coincided with the centre of the test rigs well as

the centre of the loading point. In this way, the beam specimen would be

a simply supported member. This ensured that a span of 30mm of the

beam was between the two roller bearings.

iii. The dial gauge was mounted onto the setup as shown in the figure

below.

iv. The load was lowered such that the loading point is just touching the

specimen. The reading of the T. I. C machine and the dial gauge was

adjusted such that the readings of the load applied and deflection

respectively, were zero. The picture next illustrates the loading

arrangement.

47 | P a g e

Picture 15 loading arrangement

v. The length of the overhanging span on one side of the beam was noted

and denoted by the letter x. the heights of the beam before loading were

also noted from the base of the test rig and noted by h1 and h2 as shown

in figure X below

vi. The two measurements of height h1 coincided with that at the roller while

h2 coincided with the end of the beam span (test specimen).

vii. A load of 454.8N (0.41.137KN) was then applied slowly and carefully.

The dial gauge was let to stabilize before that reading corresponding to

the load was taken.

viii. The values of x, h1, and h2 were taken and recorded as well.

ix. With an increment of 0.41.137KN steps (viii) and (ix) were repeated

each time noting the readings while also making sure the loading was still

within the elastic limit

48 | P a g e

Picture 16 the general arrangement on TIC machine

49 | P a g e

Fig 14 loading arrangement

Fig 15 deflections in progress

50 | P a g e

Chapter five Analysis of experimental results

The test was carried out using T.I.C machine found in the strength of materials

laboratory of the University of Nairobi, department of mechanical and manufacturing

engineering workshop. After a series of preliminary tests on the specimens, finally

stable results were recorded in the tables that follow.

We recorded the exact values of load and deflections as shown in table 2.

51 | P a g e

Table 1: Experimental raw data

Reading of load (*1.137KN)

Deflection (y) (mm)

h1 (mm) h2 (mm) X (mm)

0.0 0 37.50 37.50 54.00

0.4 0.28 37.57 37.68 53.90

0.8 0.52 37.65 37.86 53.81

1.2 0.79 37.75 38.05 53.74

1.6 1.04 37.85 38.25 53.67

2.0 1.27 37.93 38.42 53.56

2.4 1.52 37.99 38.58 53.44

2.8 1.75 38.07 38.75 53.38

3.2 1.99 38.16 38.94 53.26

3.6 2.22 38.24 39.11 53.17

4.0 2.46 38.31 39.27 53.11

4.4 2.72 38.36 39.42 53.00

4.8 2.97 38.45 39.60 52.90

5.2 3.2 38.56 39.80 52.83

5.6 3.46 38.62 39.95 52.74

6.0 3.69 38.70 40.13 52.66

6.4 4.01 38.79 40.31 52.55

6.8 4.24 38.85 40.46 52.49

7.2 4.49 39.10 40.82 52.37

7.6 4.86 39.24 41.97 52.27

8.0 5.16 39.41 42.13 52.13

8.4 5.62 39.60 42.34 51.97

8.8 6.06 39.82 42.60 51.78

9.2 6.52 40.08 42.91 51.55

9.6 7.08 40.43 43.30 52.35

10.0 7.76 40.85 43.75 51.12

10.4 8.78 41.34 44.43 50.88

10.8 9.92 41.86 45.22 50.65

11.2 10.88 42.38 46.13 50.41

11.6 12.2 42.74 47.04 50.12

52 | P a g e

Table 2: Load and respective deflection (Experimental) P (N) Deflection (mm)

0 0

454.8 0.28

909.6 0.52

1364.4 0.79

1819.2 1.04

2274 1.27

2728.2 1.52

3183.6 1.75

3638.4 1.99

4093.2 2.22

4548 2.46

5002.8 2.72

5457.6 2.97

5912.4 3.2

6367.2 3.46

6822 3.69

7276.8 4.01

7731.6 4.24

8186.4 4.49

8641.2 4.86

9096.0 5.16

9550.8 5.62

10005.6 6.06

10460.4 6.52

10915.2 7.08

11370.0 7.76

11824.8 8.78

12279.6 9.92

12734.4 10.88

13189.2 12.2

53 | P a g e

From table 2, we plotted a graph of load against deflection as shown below graph 1.

The graph illustrates that the elastic limit of the beam is reached when a load of 8186.4 KN is applied all

the values after this are insignificant to this project.

Graph 1

0

2000

4000

6000

8000

10000

12000

14000

0 2 4 6 8 10 12 14

Load

(N)

Deflection (mm)

load against deflection

54 | P a g e

The graph for load against deflection for the elastic region lo

Graph 2

y = 1844.6x - 44.532

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 1 2 3 4 5

Load

(N)

Deflection(mm)

Elastic region

Elastic limit Graph

Linear (Elastic limit Graph)

55 | P a g e

Data analysis

Sample calculations From figure 15

h2-h1

x

Fig 16 triangle used for data analysis

We can say that from fig 16

For a load of 2.274 KN, deflection of 1.27, h1 of 37.93 mm, h2 of 38.42mm and x of 53.56 mm

And

Therefore

56 | P a g e

Hence

We calculated the value of the right hand term of equation (22) using the above values as;

For P=2.274 KN

E = 70 109 NM-2

Where

E is youngs modulus of aluminium

I is second moment for a rectangular section and is given by

b = width of the beam -0.017m

d = the beam thickness 0.05m

The cross section of the beam in this case was as shown below

d

b

Fig 17 beam cross section

57 | P a g e

Therefore;

Then

Thus for the load of 2.274 KN mentioned above;

=0.017853

The other values of m , sin 2m and

were calculated and the results recorded as shown in

table3

We then plotted a graph of against

Table 3: Analyzed values Load (N) h1 (mm) h2 (mm) X (mm) (degrees)

0 37.50 37.5 54.00 0 0 0 0

454.8 37.68 37.57 53.90 0.002041 0.23388 0.0040816 0.003571

909.6 37.86 37.65 53.81 0.003903 0.44725 0.0078059 0.007141

1364.4 38.05 37.75 53.74 0.005582 0.63964 0.0111635 0.010712

1819.2 38.25 37.85 53.67 0.007468 0.85575 0.014935 0.014282

2274 38.42 37.93 53.56 0.009169 1.05066 0.018336 0.017853

2728.2 38.58 37.99 53.44 0.011052 1.26641 0.022101 0.021424

3183.6 38.75 38.07 53.38 0.012767 1.46291 0.025529 0.024994

3638.4 38.94 38.16 53.26 0.014669 1.68082 0.029331 0.028565

4093.2 39.11 38.24 53.17 0.016381 1.87695 0.032753 0.032135

4548 39.27 38.31 53.11 0.018113 2.07536 0.036213 0.035707

5002.8 39.42 38.36 53.00 0.020038 2.29587 0.040059 0.039276

5457.6 39.60 38.45 52.90 0.021768 2.49403 0.043515 0.042847

5912.4 39.80 38.56 52.83 0.023511 2.69366 0.046995 0.046418

6367.2 39.95 38.62 52.74 0.025218 2.88915 0.050403 0.049988

6822 40.13 38.70 52.66 0.027345 3.13272 0.054649 0.053560

7276.8 40.31 38.79 52.55 0.029305 3.35714 0.058559 0.057129

7731.6 40.46 38.85 52.49 0.030673 3.51376 0.061288 0.060700

8186.4 40.82 39.10 52.37 0.032843 3.76217 0.065615 0.064271

58 | P a g e

A graph of

Graph 3

y = 1.0105x + 0.0003

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

sin

2

m

pX2m /8EI

59 | P a g e

More sample calculations

Theoretical values of deflection Based on the deflection formula the following sample calculations were obtained.

Sample 1

For a load of 2.274 KN

=1.3391mm

Sample 2

For a load of 4.0932

=2.4106mm

Values of Ym for corresponding loads in the elastic region were calculated and tabulated as shown in

table 4 below.

60 | P a g e

Table 4: Load and calculated deflection

Load P(N) Deflection y (mm)

0 0

454.8 0.2678

909.6 0.5355

1364.4 0.8034

1819.2 1.0712

2274 1.3391

2728.2 1.6069

3183.6 1.8748

3638.4 2.1427

4093.2 2.4106

4548 2.6787

5002.8 2.9466

5457.6 3.2146

5912.4 3.4828

6367.2 3.7509

6822 4.0192

7276.8 4.2873

7731.6 4.5557

8186.4 4.8240

61 | P a g e

Table 5: comparison between experimental and calculated

values of deflections

Load P (N) Experimental Calculated

0 0 0

454.8 0.28 0.2678

909.6 0.52 0.5355

1364.4 0.79 0.8034

1819.2 1.04 1.0712

2274.0 1.27 1.3391

2728.2 1.52 1.6069

3183.6 1.75 1.8748

3638.4 1.99 2.1427

4093.2 2.22 2.4106

4548.0 2.46 2.6787

5002.8 2.72 2.9466

5457.6 2.97 3.2146

5912.4 3.20 3.4828

6367.2 3.46 3.7509

6822.0 3.69 4.0192

7276.8 4.01 4.2873

7731.6 4.24 4.5557

8186.4 4.49 4.8240

62 | P a g e

A graph of calculated deflection against experimental deflection

Graph 4

y = 1.0869x - 0.0269

-1

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

the

ori

tica

l

experimental

theoretical vs experimental deflection

63 | P a g e

Percentage deviations Sample calculations

For a load of 5.9124 KN

=8.8375

For a load of 6.3672 KN

=8.4075

The percentage deviations representing relative errors for the corresponding loads were calculated and

tabulated as below.

64 | P a g e

Table 6: Load and percentage deviations Load P (N) percentage deviations (relative errors)

0 0

454.8 4.3750

909.6 2.9808

1364.4 1.6962

1819.2 3.0000

2274 5.4409

2728.2 5.7171

3183.6 7.1314

3638.4 7.6734

4093.2 8.5853

4548 8.8902

5002.8 8.3309

5457.6 8.2397

5912.4 8.8375

6367.2 8.4075

6822 8.9214

7276.8 6.9152

7731.6 7.4458

8186.4 7.4388

65 | P a g e

Chapter six

Discussions, conclusions and recommendations

Discussion The results as tabulated were extensively analyzed as shown previously in the previous chapter. The

analysis was restricted to the elastic region before yielding as we had made a prior assumption of linear

elasticity in the project formulation.

To determine the accuracy and the validity of the method of solution proposed in this study an in depth

analysis of experimental and theoretical values was done. We first obtained the results experimentally

of the large deflection of a simply supported beam under the action of a concentrated load P. secondly;

the numerical results are calculated using the proposed theory. Comparison of the two set of values was

then established. The agreement between the values obtained using our numerical method and those

obtained experimentally was good.

The distributed load due to the weight of the beam was considered negligible and therefore was not

used in the analysis. The numerically calculated deflections were obtained using the Oduoris proposed

method, which is given by the following equation:

Where is the deflection, P is load, E is youngs modulus, I is the second moment of area and is

the mean distance between the roller supports.

Using equation 22,

from which the final large deflections formula for above was

derived, a graph of

was plotted, from the equation, the plotted graph should be

having an expected gradient of 1 in our case it was 1.0105.

Table 1 shows the deflection as a function of the applied load P whereas the table 4 represents the

numerically calculated values with the aid of the numerical equation. We also included the relative error

(percentage) in the values of calculated numerically as compared with the values measured

experimentally. To further the validation, a comparison was done between theoretical and the

experimental deflection then a graph was plotted of calculated against theoretical deflection. The

expected gradient in case of absolute agreement was 1 however the gradient obtained was 1.0869.

66 | P a g e

The comparisons above show there are some errors that were incurred during the experiment that

caused the deviations from theoretical values. These errors could include:

Calculations did not include deflections due to shearing

Possible residual stresses introduced in the aluminium beam during machining while preparing

the specimen. These stresses have an effect of decreasing the buckling load and are caused by

heat and incompatible internal strains

Errors incurred when taking experimental data ( ), taking the beam dimensions

There was some play encountered from the dial gauge that was used to obtain deflection

The T.I.C machine could not handle heavy tensions and started leaking at high values of load

Youngs modulus for Aluminium varies from 65 GPa to 75 GPa, for the purposes of evaluation 70

GPa was used as the average which may have been inaccurate.

Conclusion The study was a success. The laboratory validation of the new theory (Oduoris formulae) for

determination of large deflections for a class of cantilever beams by laboratory experimentation was

carried out to satisfaction and found to hold.

The theoretical analysis resulted in the following deflection equation corresponding to the general case

of large deflections.

And as shown in chapter 2 this equation is directly derived from equation 22 as quoted below.

From values of

a linear graph through the origin with a gradient of

1.662 against the expected gradient of 1. The variation occurred because of the experimental errors

It can be concluded that the theory as proposed by professor Oduori is valid for linear analysis of large

deflections of cantilever beams

67 | P a g e

Recommendations for further work We recommend further work to prove this theory based on better experimentation methods and

equipment. There should be extra vigilance to avoid the errors mentioned before. The following should

be taken into consideration

Shear deflections should be incorporated.

The measurement of should be done with extra caution. Using a vernier as we did

introduces errors and it is extremely hard to obtain exact values.

The beam should be prepared using standard preparation methods and its properties like

youngs modulus should be gauged properly. Dimensional in accuracies should also be avoided.

The experiment could be done using a more reliable loading system other than the TIC

machines.

68 | P a g e

Chapter 7 References

B. Shvartsman, large deflections of a cantilever beam subjected to a follower force, J. Sound and

Vib.304, pp.969-973 (2007) .

F.V. Rhode, ` Large deflections of cantilever beams with uniformly distributed load, Q. Appl. Math. ,

11, pp. 337-338 (1953)

J. M Gere, S.P .Timoshenko, `Mechanics of materials, second edition, books Engineering Division,

California (1984)

M.F Oduori, J. Sakai, E. Inoue, `A paper on Modeling of crop stem Deflection in the context of the

combine harvester reel design and operation.

J. Case, C. Lord, and Carl T.F.R., `Strength of materials and structures , fourth edition chapter 13,

deflections of beams,

S.W. Crawley, R.M. Dillon steel buildings analysis and design, second edition, chapter 4, beam

deflections

Wikipedia: http://en.wikipedia.org/wiki/Timoshenko beam theory

Wikipedia: http://en.wikipedia.org/wiki/Euler-Bernoulli beam theory

Wikipedia: http://en.wikipedia.org/wiki/Moment area method beam theory