Deflection of loaded steel beams having different end supports

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Medway Campus – School of Engineering

Engineering Analysis and

Applications IICIVI 0008 - Level 6

Coursework Assignment I

Deflection for steel beams having different end supports Analysis MEng Civil Engineering

Baris Evran 000696354

Sagar Sohan

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Deflection of loaded steel beams havingdifferent end supports

Session: 2013 - 2014

MEng in Civil Engineering Year 3Tutor: Dr. Morteza Aboutalebi


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Scientific report Paper on “DEFLECTION OF LOADED STEEL BEAMS HAVINGDIFFERENTEND SUPPORT CONDITIONS ”

Baris Evran

The University of Greenwich

Submitted for the degreeof bachelors from the

University of Greenwich.

Content


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sMEng in Civil Engineering Year 3 Tutor: Dr. Morteza Aboutalebi.....1

1 Abstract.........................................................8

2 Introduction.....................................................8

2.1 Aim..........................................................8

3 Theory...........................................................9

3.1 Boundary Conditions:.........................................9

3.1.1 Simply supported Beam:...................................93.1.2 Pin supported beams.....................................10

3.3 Solving Methods for Differential Equations..................103.3.1 Rayleigh-Ritz Method....................................10

3.3.2 Direct Integration......................................113.3.3 Central Finite Difference Method........................12

4 Apparatus and Methodology.......................................15

4.1 Equipment...................................................15

4.2 Methodology.................................................174.3 Health & Safety.............................................17

5 Calculations....................................................18

5.1 Governing Differential Equation Experiment 1................18

5.2 Finite Difference Method Experiment 1.......................225.3 Rayleigh-Ritz Method Experiment 1...........................24

5.4 Finite Difference Method Experiment 2.......................275.5 Rayleigh Ritz Method Experiment 2...........................30

6. Results........................................................35

7 Graphs..........................................................36

8 Discussions and Conclusions.....................................37

8.1 Errors......................................................37

8.1.1 Human Errors............................................378.1.2 Equipment Errors........................................37


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8.1.3 Error due to Assumptions................................378.1.4 Actual Error............................................37

8.1.5 How can we improve on the error?........................378.2 Discussion of Results.......................................37

8.2.1 Experimental results....................................378.3 Final Conclusion............................................38

9 Bibliography....................................................38


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LIST OF FIGURES

Figure 1: Simply supported beam

Figure 2: Graphical representation of derivative

Figure 3:The picture represents the equipment which was used

for experiment.

Figure 4: This picture shows; the gauge which provides the

values of deflections, extremely sensitive and checks

millimetres.

Figure 5: This picture illustrates the hangers and other

experiment materials.

Figure 6: The materials used as free weights.


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Figure 7: Comparison of Theoretical Deflection to Experimental Deflection in 1st

Experiment

Figure 8: Comparison of Theoretical Deflection vs Experimental Deflection in 2nd

Experiment.


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1 Abstract

This report will discuss an experiment that was conducted in

which the structural behaviour of two steel beams is studied.

The beams are the same in every aspect apart from supporting

conditions, one beam is supported via pin and roller the other

beam is restrained horizontally via two pin supports. The beam

with two pins means that there will be horizontal relations

present due to beam elongation; this will cause a hogging

moment which counteracts the sagging moment. The purpose of

this report is to conduct the two experiments then conduct

theoretical calculations upon the proposed beam conditions

using different methods of calculation, the methods used are;

Rayleigh-Ritz method, Direct Integration method and Central

Finite Difference method. It is found within this report that

the best method for use for calculations is direct method in

most cases due to its moderate ease of use and moderate

accuracy. The Rayleigh-Ritz method is most accurate and is the

easiest to use for the beam with roller and pin, however for Deflection for steel beams having different end supports Analysis MEng Civil Engineering

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the pin-pin supported beam the calculation became too complex

and error within calculations grew exponentially. Finite

difference method is the hardest to use and conduct the

calculations; however it provides the least amount of error.

2 Introduction

The properties of steel make it a very versatile material with

a variety of different applications in the civil engineering

industry. The steel beams that are applied in the structures

are a main concept for many of the theoretical concepts this

report will cover. The ability of the beam is to sustain a

load carrying capacity into the inelastic range which is

essential for seismic design. The extent of this ability is

identified in its property known as ductility which can be

either that of the material itself or the actual structure.

However, it is widely known that steel is stronger in tension

but less stronger in compression when applied to a structure.

This experimental analysis will identify two steel beams. The

first beam is pin support while the other is roller support.

The second steel beam was pin supported at both ends of the

beam. Load was applied in exact conditions for both


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experiments. The deflection was then studied in during this

both experimentally and theoretically using different

analytical methods.

2.1 Aim

The aim of the experiment is to:

Study the behaviour of two beams with identical cross

sections shown subjected to point loads at quarter

distance from both supports.

To be able to use ordinary differential equations to

solve engineering problems using different methods

To compare the experimental behaviour to the expected

behaviour of two beams with identical cross sections but

one simply supported and other with two pin ends


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3 Theory

In engineering applications the problems will arise which

will be converted into a mathematical model and mathematical

problem which is in this experiment obtaining governing

ordinary differential equation. There are various different

methods which can be used to solve different types of ordinary

differential equation which are Analytical Methods, Numerical

Methods, Variational Methods and Weighted Residual Methods.

There are various methods which can be used to solve

differential equations in engineering applications. Some of

them will be listed below.

Analytical Methods

1. Bernoulli’s Equation

2. Separable Variables

3. Homogenous 1st Order Equations

4. Linear 1st Order Equations

Numerical Method.

1. Numerical Integration( Simpson’s Rule)

2. Runge-Kutta Method with different orders

3. Finite Difference Method


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4. Euler’s Method

5. Modified Euler’s Method

6. Mid-point Method

Variational Approach Methods

1. Rayleigh –Ritz Method

(This method uses the principle of Virtual Work with

assumed behaviour of the element)

2. Finite Element Method using Variational Approach

Weighted Residual Method.

1. Galerkin Method.

2. Least Square method.

3.1 Boundary Conditions:

Boundary conditions are the conditions which will be input to

model the behaviour of structural elements when they are

subjected to different loading conditions. These include

restraints at the supports of the elements, slope and

curvature at different locations at the elements of the

structure


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3.1.1 Simply supported Beam:

Simply supported beams are the beams which have hinge support

at one end and roller support at other end of the beam. The

figure of simply supported beam and boundary conditions will

be listed below

Boundary conditions for the above simply supported beam states

that:

At x=0, y(0) = 0; because the beam is pinned to its

support therefore is vertically and horizontally

restrained from moving

At x=0, y’’ (0) =0. Because there is no curvature right

at the point of the support but at the vicinity it is not

zero

At x=L, y(L) =0; This is because the beam is restrained

from displacing in vertical direction but it is not

restrained from moving horizontal direction


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At x=L, y” (L) =0. This is because there is no curvature

like at the location where x=0

At x = 0.5L = y’(0.5L) = 0 as there is no slope at the

point of maximum deflection

3.1.2 Pin supported beams

Pin supported beams have pin joints at both ends. The

behaviour of this

type of beam will

be slightly

different to simply

supported beams.

Figure 1: Simply supported beam

Boundary Conditions


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At x=0 and x = L y(0) = 0; because the beam is pinned

to its supports therefore is vertically and horizontally

restrained from moving

At x=0 and x = L y’’ (0) =0. Because there is no

curvature right at the point of the supports but at the

vicinity it is not zero

At x = 0.5L = y’ (0.5L) = 0 as there is no slope at the

point of maximum deflection.

3.3 Solving Methods for Differential Equations

In this report three methods will be used for solving

governing differential equation of the behaviour of simply

supported beam subjected to quarter point loads which are

Direct Integration, Rayleigh-Ritz Method and Central Finite

Difference Method

Two methods will be used for solving differential equations

for the behaviour for pin ended beams subjected to quarter

point loads which are Rayleigh-Ritz and Central Finite

Difference Method


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3.3.1 Rayleigh-Ritz Method

Rayleigh-Ritz method is one of the variational methods which

are used to solve the governing differential equation which

uses tentative function y=C1ϕ(x) which models the approximate

behaviour of the beams with behavioural factor C1.

This is variational approach because this method is based on

determining the stationary value of the integral of the

function over the whole element which models the assumed

behaviour of the structure. This function will include the

constant

I = ¿∫0

L

F(x,y,y'(x),y''(x)).dx

Where

x = distance from the start of the beam to specified location

y = Assumed Shape function

y’ = dydx

y’’ = d2yd2x


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In Rayleigh-Ritz Method we obtain C1 as the stationary value of

the integral of the function and substitute it into the

tentative solution y

To obtain C1 we first differentiate the obtained integral

which satisfies this condition

dIdC1

= 0

Then we obtain C1 from the differentiated function the equation

above and substitute into assumed shape function which will

produce an approximate model of the behaviour of the

structure. In Rayleigh-Ritz Method the function which we use

to integrate is developed using theorem of virtual work on

structural elements subjected to different static loading

conditions.

In the theorem of virtual work the virtual displacement which

is used will be obtained from the assumed shape function in

terms of C1

3.3.2 Direct Integration

The governing differential equation of the beam subjected to

certain loading conditions is need to be developed before you Deflection for steel beams having different end supports Analysis MEng Civil Engineering

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start solving them to determine the displacement along the

beam using direct methods, numerical methods and other types

of approximate methods.

We must develop the following equations:

−EI d2y

dx2=Mx=bendingmomentatpointxalongthebeam

The equation above is the bending moment equation at point x

along the beam is equal to the second order governing

differential equation satisfying the boundary conditions

within the considered range of the beam

If we integrate the equation of bending moment over the

certain portions of the beam we will obtain the slope

−EIdydx

=θx=slopeatpointxalongthebeam

The equation above is used to determine the slope of the

tangent of the curve at point x along the beam is equal to the

first order governing differential equation satisfying

boundary conditions for that part of the beam. If we integrate

the slope equation over the whole length of the beam

−EIy=δx=deflectionatpointxalongthebeamDeflection for steel beams having different end supports Analysis MEng Civil Engineering

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When we develop differential equation for beam in bending

subjected to quarter point loads and with length L we have to

develop three differential equations because each of them will

only cover a certain location of the beam. These ranges are.

0<x<L4

L4

<x<3L4

3L4<x<L

After we obtain and integrate differential equations we equate

the slope equations and deflection equations to determine the

constants which we have developed throughout the process of

integration. This will be provided in the calculations that we

have done.

After we obtain these constants they will be inputted into the

deflection equation of the beam subjected to the loading

condition.


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3.3.3 Central Finite Difference Method

Central Finite Difference Method is a numerical method which

is used to solve linear differential equations of any order.

In this method we convert differential equations into a set of

linear equation which can be solved using various different

methods such as inverse matrix, gauss elimination, gauss

Jordan, augmented matrix etc.

Finite difference equations are derived from Taylor’s Series

function. We will remove all the series from second order

onwards as they produce very small values. This is called

truncation process and this will cause truncation error.

3.3.3.1 Development of finite difference equations

Figure 2.: Graphical representation of derivative


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The Taylor series expansion:

y (xo+h )=y (xo)+y'(xo)

1!h+

y''(xo)

2!h2+…+

yn(x0)n!

hn

Where

n = the order of each term

Truncation process will lead to the following equation

f (xo+h)=f (xo )+f' (xo)1! h

Develop backward difference and forward difference equations

Forward difference

Backward difference

yi+1=yi+Δx(dydx )iyi=yi−1+Δx(dydx )i−1

Solving for dy/dx:

(dydx )i=yi+1−yi

Δx (dydx )i−1=yi−yi−1

Δx

Differentiating once:


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(d2ydx2 )i=(dydx )i+1

−(dydx )iΔx (d2y

dx2 )i−1=

(dydx )i−(dydx )i−1

Δx

Equations

(d2ydx2 )i=yi+2−2yi+1+yi

(Δx )2 (d2ydx2 )i−1

=yi−2yi−1+yi−2

(Δx )2

We will obtain central finite difference equtios by averaging

the derivatives of forward differentials and backward

differentials.

Central Finite difference equations

(dydx )i=yi+1−yi−1

2Δx (d2ydx2 )i=yi+1−2yi+yi−1

(Δx)2


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4 Apparatus and Methodology

4.1 Equipment

Figure :3 The picture represents the equipment which was used

for experiment.


Roller support:has got only vertical

Steel beam: elastically deflected under

Pinned support: has got 2 reactions

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Figure 4: This picture shows; the gauge which provides the

values of deflections, extremely sensitive and checks

millimetres.

Figure 5: This picture illustrates the hangers and other

experiment materials.


The hangers were used to apply free weights for showing the deflection in the

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Figure 6: The materials used as free weights.

4.2 Methodology

Felt-tip pen

Hand weights

Pad and pen

Steel beams

Steel measuring tape


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Camera

Micrometer

50 mm dial gauge

Firstly Start with a steel beam with pin-roller support.

Dial gauge is then set to zero, making sure the pointer

is at 0.

The hanger is weighted and then placed on the beam. The

deflection due to hanger is noted by reading the dial

gauge.

1 kg weight is placed on each hanger and the centre

deflection is taken into consideration by reading the

dial gauge.

Keep adding the 1 kg weight and measuring the central

deflection until 4kg interval is obtained on each hanger.

Then 4 plates of 1 kg are removed and replaced by 5 kg

plate on each hanger. The central deflection is noted

again by reading the dial gauge.

Repeat the process by adding 1 kg weight and noting the

central deflection at each increment until 10 kg. At the


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end remove all loads and hanger and read the dial gauge.

Note if there is any residual deflection in the beam or

not.

Repeat the procedure for the second experiment, a steel

beam with pin-pin support.

4.3 Health & Safety

Health and safety is essential in when conducting laboratory

experiments. It ensures that no one gets injured and the

experiment is carried out effectively and safely. All PPE was

worn when conducting this experiment. All Health and Safety

rules laid by the university should be followed when in the

laboratory.


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5 Calculations

5.1 Governing Differential Equation Experiment 1


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5.2 Finite Difference Method Experiment 1


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5.3 Rayleigh-Ritz Method Experiment 1


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5.4 Finite Difference Method Experiment 2


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5.5 Rayleigh Ritz Method Experiment 2


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6. Results

Experiment 1

Load

Case

No:

W1(N) W2(N)Experiment

al

y(mm)

Direct

Integration

y(mm)

Central

Finite

Difference

y(mm)

Rayleigh-

Ritz

y(mm)

1 4.905 4.905 0.78 1.2368776 1.349321 1.25375

2 14.715 4.905 1.52 2.8110855 3.035972 2.507499

3 22.525 4.905 2.24 4.064354 4.37876 3.505642

4 34.335 4.905 3.2 5.9595013 6.409275 5.014998

5 44.145 4.905 4.15 7.5337092 8.095926 6.268748

6 53.955 4.905 5.1 9.1079171 9.782578 7.522497

7 53.955 14.715 6.51 10.007465 10.79457 8.776247

8 53.955 24.525 7.55 10.907012 11.80656 10.03

9 53.955 34.335 8.51 11.806559 12.81855 11.28375

10 53.955 44.145 9.59 12.706107 13.83054 12.5375

11 53.955 53.955 10.49 13.605654 14.84253 13.79124


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Experiment 2

Load

Case

No:

W1(N) W2(N) Experimenta

l

y(mm)

Rayleigh-Ritz

Method

y(mm)

Central Finite

Difference Method

y(mm)

1 4.905 4.905 0.760.726554 0.796228

2 14.71

5

4.905

1.761.287854 1.640105

3 22.52

5

4.905

2.831.634172 2.194215

4 34.33

5

4.905

3.682.050559 2.882735

5 44.14

5

4.905

4.552.332446 3.352911

6 53.95

5

4.905

5.232.575292 3.753881


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7 53.95

5

14.71

5 5.732.789544 3.843786

8 53.95

5

24.52

5 6.372.981941 3.918128

9 53.95

5

34.33

5 6.73.157069 3.979599

10 53.95

5

44.14

5 7.153.318181 4.030213

11 53.95

5

53.95

5 7.543.467676 4.071535


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7 Graphs

0 2 4 6 8 10 120246810121416

Experiment 1

Central Finite DifferenceRayleigh-RitzExperimental

Load Case No:

Cent

ral De

flec

tion(m

m)

Figure 7: Comparison of Theoretical Deflection to Experimental Deflection in 1st

Experiment


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0 2 4 6 8 10 120

1

2

3

4

5

6

7

8

Experiment 2

Experimental ResultsCentral Finite DifferenceRayleigh-Ritz

Loac Case NoCentral Deflection(mm)

Figure 8: Comparison of Theoretical Deflection vs Experimental Deflection in 2nd

Experiment.


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8 Discussions and Conclusions

8.1 Errors

8.1.1 Human Errors

There were some human errors which have occurred in this

experiment especially in recording the results of central

deflection of the simply supported beams and pin ended beams

because our reading wasn’t precise which did affect our

experimental results. Errors can also come from measurements

of our cross sectional dimensions of the beams

8.1.2 Equipment Errors

Equipment errors are the errors which come the meter which

displays the deflection of the beams in this experiment. These

errors can have small effect on the readings of deflection

which we take for each loading therefore can affect our

results.

8.1.3 Error due to Assumptions

There were some assumptions we made when we developed

governing differential equations which one of them is that the

deflection is very small and plain sections remain plain.


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Other assumption that we made is that we assumed a deflection

profile when we were calculating the deflection using

Rayleigh-Ritz Method. The errors can also come from assuming

the profile of the behaviour of the structure subjected to

certain loading conditions. Assumption errors can cause tiny

error in the theoretical results we obtained

8.1.4 Actual Error

The actual error is large errors which did significantly

affect our theoretical and experimental results. These errors

are large differences from theoretical and experimental

results which we have in our experiment. Experiment 2 has a

lot of error as there were some big differences between

theoretical results and experimental results. These might come

from either the load cases we did of due to the assumption of

the behaviour because the experimental results do not agree

completely with central finite difference method result and

Rayleigh-Ritz method. Difference in results between central

finite difference and Rayleigh-ritz method is small. In most

of our load cases the beam does not behave exactly like the

assumed shape function because of the skew.


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8.1.5 How can we improve on the error?

The errors can be minimised by taking the precise readings of

the results, using equipment and gauge of good quality which

is more advanced which gives precise readings. We can also put

use a beam which is purely homogenous throughout the cross

section and whole length of the beam. Taking precise readings

of the cross sections using gauge will also reduce the error

in our experiment.

8.2 Discussion of Results

8.2.1 Experimental results

According to the results that have been obtained and displayed

on the results section on one of the previous pages we have

determined that with the same loading conditions experiment 2

was only subjected to smaller deflection compared to the

central deflection of the beam in experiment 1 as we were

expected but most of the experimental results in experiment 2

does not agree with the theoretical results obtained from

Rayleigh-Ritz method except the results obtained from first

few set of load cases e.g. on 1st load case the experimental


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deflection was 0.76mm and almost same with Rayleigh-Ritz

method but on the 10th load case the experimental deflection is

7.54mm and Rayleigh-Ritz method gives 4.16mm which means that

there is an error

For Experiment 1 the results obtained from experiment can be

agreed with theoretical results obtained from direct

integration, Rayleigh-Ritz method and central finite

difference method as it can be seen on the graph and on the

results table

The central deflection obtained from experiment one is closer

to the central deflection obtained from 2nd experiment which is

what have been expected but experimental results for 2nd

experiment is not comparable with theoretical results whereas

for 1st experiment the results are comparable with theoretical

results.

Results obtained from Central Finite Difference Method in

experiment 1 can agree with the experimental results, direct

integration and Rayleigh-Ritz method but for experiment 2 they

are only be agreed on 1st two load cases but for all the other


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load cases they are not agreeable because the results are far

away from the experimental results.

8.3 Final Conclusion

From the experimental results it is observed that the

deflection within the pin-pin beam, in the middle of the beam,

is much lower than the other beam. The reason for this is the

fact that pin-pin supports will restrict the beam from moving

horizontally by introducing horizontal reactions on both ends.

Whereas a roller support allows horizontal movement. This

horizontal reaction on the beam will cause a hogging moment

within the beam itself; this hogging moment will counteract

the sagging moment that is making the beam deflect. Therefore

deflection will be lower with pin-pin support conditions.

When comparing theoretical calculations for the two different

load cases using the three different methods it was found, as

the graphs show, that for experiment 1, pin-roller, the

Rayleigh-Ritz method is the most accurate and easy to use, the

Direct integration method is the second most accurate and

Central difference method is the least accurate. It is found

that for this method Rayleigh-Ritz is the easiest and quickest


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method to complete therefore it is concluded that for

experiment 1 the best method for theoretical calculation is

Rayleigh-Ritz.

As the graph for experiment 2 shows the Rayleigh-Ritz method

is generally similar, at least for slope of graph, compared to

the experimental results. The Central Difference method is

also relatively accurate at the start, however as loading is

increased it becomes increasingly inaccurate. Therefore it

may be concluded that the Rayleigh Ritz method is the best

method to conduct theoretical calculations for both

experiments.

9 Bibliography

Rao, S. S., 1999. The Finite Eleement Method in Engineering ( Fourth Edition)

Pages(169-178). Woburn, MA: Butterworth-Heinemann.


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The Work done:

By Baris Evran:

-Abstract

- Introduction

-Apparatus

-Word template

-Methodology

-Health and safety

-Conclusion

-Calculations.

By Sagar Sohan

-Theory

-Discussion

- Results

- Calculations

-Errors

-Results.


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Duration: 96 hours.


Deflection of loaded steel beams having different end supports

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