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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
1/23
Analyse Static Engineering
SystemsEngineering ScienceBy Brendan Burr
8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
2/23
Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Table of Contents
TABLE OF CONTENTS ........................................................... 2
TASK 1 ................................................................................ 3
Solve Shear Force and Bending Moment problems ................................ 3Solution:- ..............................................................................................................3Shear Force Diagram:- ..........................................................................................7Bending Moment Diagram:- ..................................................................................9Check:- ...............................................................................................................10
TASK 2 .............................................................................. 12
Select standard rolled steels sections for beams ................................... 12Solution:- ............................................................................................................12Shear Force Diagram:- ........................................................................................15Bending Moment Diagram:- ................................................................................16Check:- ...............................................................................................................17
Shear Stress and the Angular Deflection due to Torsion in Circular Shafts......................................................................................................... 19
Solution:- ............................................................................................................19Check:- ...............................................................................................................21
EVALUATION ..................................................................... 22
CONCLUSION ..................................................................... 22
Books ................................................................................................ 23
Catalogues ......................................................................................... 23
Websites ............................................................................................ 23
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Task 1
Solve Shear Force and Bending Moment problems
Horizontal Beam ABCD, 13m long, is simply supported at B and D.AB = 3m, BC = 2m, and CD = 8m.Concentrated load of 4kN and 7kN act at A and C respectively, and auniformly distributed load of 1kN/m extends from C to D.Neglecting the effect of gravity on the mass of the beam, sketch theShearing Force and Bending Moment Diagrams for the beam, insertingprincipal values, and determine the position and magnitude of themaximum bending moment.
Solution:-
Not To Scale
Anticlockwise Moments = Clockwise MomentsTaking Moments about D:
( ) ( ) ( )
B
B
B
=
=++=++
10
140
10525632
104137884
kNB 14=
Upwards Forces = Downwards Forces
( )
14874
817414
++=++=+
D
D
kND 5=
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
5/23
Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Bending Moment Calculations:
( )
( )
( )( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) kNBM
kNBM
kNBM
kNBM
kNBM
kNBM
kNBM
kNBM
kNBM
kNBM
kNBMkNBM
kNBM
kNBM
m
m
m
m
m
m
m
m
m
m
m
m
m
m
02
81871014134
5.42
7177914124
82
6
167814114
5.102
5157714104
122
414761494
5.122
313751484
122
212741474
5.102
111731464
82
010721454
211444
1201434824
414
004
2
13
2
12
2
11
2
10
2
9
2
8
2
7
2
6
2
5
4
3
2
1
0
=
+++
+=
+++
+=
+++
+=
+++
+=
+++
+=
+++
+=
+++
+=
+++
+=
+++
=+=+
=
==
Therefore the maximum bending moment is positioned at 8 meters and has a
magnitude of +12.5 kN/m.
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Bending Moment from 5 meters to 13 meters:-
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) kNmx
kNmx
kNmx
kNmx
kNmx
kNmx
kNmx
kNmx
kNmx
xxBM
088385.0
5.487375.0
886365.0
5.1085355.0
1284345.0
5.1283335.0
1282325.0
5.1081315.0
880305.0
835.0
2
8
2
7
2
6
2
5
2
4
2
3
2
2
2
1
2
0
2
=++
+=++
+=++
+=++
+=++
+=++
+=++
+=+++=++
++=
Differentiate:
1=
=
n
n
naxdx
dy
axy
x
x
xxdx
BM
=
+=
+=
1
3
310
315.02 1112
3=x Meters
( ) ( ) 83335.0 2 ++=MAXBM
5.12=MAX
BM kNm
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Shear Force Diagram:-
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
9/23
Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Bending Moment Diagram:-
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Check:-
For the check I entered appropriate values into an online calculator for simplysupported beams.
Location of left support,
xS1m
Location of right
support, xS2
m
Point load #1, P1 N
Location of pointload #1,
xP1
m
Point load #2, P2 N
Location of load #2, xP2 m
Couple load #1, C1 N.m
Location of couple #1,
xC1
m
Couple load #2, C2 N.m
Location of couple #2,
xC2
m
Distributed load #1, left-
hand-side, W1l
N/m
Left hand side locationof load #1, xW1l
m
Distributed load #1,
right-hand-side, W1r
N/m
Right hand side location
of load #1, xW1r
m
Distributed load #2, left-
hand-side, W2l
N/m
Left hand side location
of load #2, xW2l
m
Distributed load #2,
right-hand-side, W1r
N/m
Right hand side location
of load #2, xW1r
m
This then returns values which correspond to the calculations I made.
Variables Values Units Maximum Shear force, Vmax 10000.0N
Location of Vmax 5.0m
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Maximum Bending moment, Mmax 12500.0N.m
Location of Mmax 8m
Reaction force, R1 14000.0N
Reaction force, R2 5000.0N
The Shear Force and Bending Moment Diagrams are also the same as thosecreated from my calculations above.
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Task 2
Select standard rolled steels sections for beams
A beam, 5m long, simply supported at its ends, and carries a uniformlydistributed load of 15kN/m over its entire length, together with aconcentrated load of 30kN at a point 1m from the left-hand support. Ifthe maximum stress allowable in the material is 100MN/m, select astandard rolled steel section from the tables.
Not To Scale
Solution:-
Anticlockwise Moments = Clockwise Moments
Taking Moments about A:
( ) ( )
C
C
C
=
=+=+
5
5.217
55.18730
55.275130
kNC 5.43=
Upwards Forces = Downwards Forces
( )
5.437530
515305.43
+=+=+
A
A
kNA 5.61=
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Shear Force Calculations:-
kNSF
kNSF
kNSF
kNSF
kNSFkNSF
kNSF
kNSF
m
m
m
m
m
m
m
m
05.434.43
5.43155.28
5.28155.13
5.13155.1
5.1155.165.16305.46
5.46155.61
5.615.610
5
99.4
4
3
2
1
99.0
0
=+=
==+=+=
+=+=+
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Bending Moment Calculations:-
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) kNmx
kNmx
kNmx
kNmx
kNmx
xxBM
055.755.3130
3645.745.3130
5735.735.3130
6325.725.3130
5415.715.3130
5.75.3130
2
5
2
4
2
3
2
2
2
1
2
=+
+=+
+=+
+=+
+=+
+=
Differentiate:
1=
=
n
n
naxdx
dy
axy
x
x
xxdx
BM
=
+=
+=
15
5.31
5.31150
5.315.72 1112
1.2=x Meters
( ) ( ) 301.25.311.25.7 2 ++=MAXBM
075.63=MAX
BM kNm
36
6
3
1075.630
10100
10075.63
cm
M
y
IM
=
=
Therefore from the BS5950-1:2000 Table I have selected Serial Size:
305 x 165
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Shear Force Diagram:-
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Bending Moment Diagram :-
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
This then returns values which correspond to the calculations I made.
Variables Values Units
Maximum Shear force, Vmax 61500.0N
Location of Vmax 0.0m
Maximum Bending moment, Mmax 63000.0N.m
Location of Mmax 2.1m
Reaction force, R1 61500.0N
Reaction force, R2 43500.0N
The Shear Force and Bending Moment Diagrams are also the same as thosecreated from my calculations above.
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Task 3
Shear Stress and the Angular Deflection due to Torsion in Circular Shafts
A hollow shaft has an external diameter of 150mm and an internaldiameter of 100mm. If the maximum shear stress is not to exceed50MN/m, determine the torque, the maximum power that could betransmitted by the shaft when rotating at 180 rev/min, and the angle oftwist (in degrees) over a 5m length when transmitting this power. Themodulus of the material is 80GN/m.
Solution:-
L
G
RJ
T ==
( )( )
mmGN
mmmMN
dD
Torque5/80
75/50
32
22
44
==
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )
075.0
994175024.1
075.0
10501025.40632
075.0
10501010010150
32
1075
/105010015032
36
34343
3
2344
=
=
=
=
T
T
T
m
mNmmmm
T
58900032.26=T Nm
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
J
T
L
G=
( )
( )
6
6
6
6
1080
56665.666666
6665.6666665
1080
1088350049.39
58900032.26
5
1080
=
=
=
60416666666.0= Radians
18060416666666.0
180
=
=
= 387324146.2
sRAD
revrev
/6
623sec/3min/180
=
=
TPower =
658900032.26 =Power
1908484.501=Power Watts
Therefore we see that:
Total Torque (T) = 26.58900032 NmPolar Second Moment Of Area (J) = 39.88350049 x 10-6 m4Shear Stress ( ) = 50 x 103 N/m2
Radius (R) = 75 x 10-3 mModulus Rigidity (G) = 80 x 106 N/m2
Angle Of Twist ( ) = 2.387324146 o OR 0.04166666666 RadiansRadial Line Of Length (L) = 5 mTransmitted Power (P) = 501.1908484 Watts
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Check:-
( )6666.666666
5
60416666666.01080
6667.6666661075
1050
6665.6666661088350049.39
58900032.26
6
3
3
6
=
=
=
=
=
=
L
G
R
J
T
From this we can see that all values must be correct due to the consistency ofthe answers derived from the Engineers Theory of Torsion Formulae.
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8/2/2019 BTEC HNC - Science - Analyse Static Engineering Systems
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Evaluation
I can honestly say that I found this assignment difficult to grasp. Now I knowhow to calculate the Bending Moments I can easily work out the maximum
Bending Moment and the position of it.In Task 1, I could work out the Reactant Forces without any difficulty, so wasable to move straight on to working out the Bending Moments. This is where Ibegan to have trouble, with the first 5 meters I didnt have any problems, itwas when I had to work out the uniformly distributed load for the last 8 meters.I was unable to find any notes that I could put into practice, because of writingit down incorrectly. This meant I had to do research into it, which took hoursafter I had managed to decipher all of the relevant information.After attaining all of the appropriate points I created a Shear Force Diagramand a Bending Moment Diagram, by using Graphmatica. This is a very goodpiece of software once you know how to use it, and by simply typing in theBending Moment equation you can create the required parabolic curve whichis formed under a UDL.To check my answer I found an online Simply Supported Beam Calculator.By entering the values taken from the question I could compare my answer tosee if there was any deviation. Fortunately for me there wasnt whichstrengthens conclusion that Task 1 is correct.Task 2 is a similar question to Task 1, however it involved a bit of furtherinvestigation into the calculation of the Engineers Theory of Bending Formula.I had already been given the Maximum Stress value in the question, all I needto do was work out the Maximum Bending Moment and then search for the
appropriate beam using the BS5950-1:2000 Table. This question has taughtme how engineers calculate which materials are suitable for the requiredapplication.I checked this answer in the same way as Task 1.Task 3 was mainly a test on ensuring the correct values are entered into thecalculation. There were values in various forms such as Meters, Centimetres,Millimetres, Giga-Newtons, Mega-Newtons and Kilo-Newtons. This gave wayfor the potential of radical inaccuracies. My check involved entering thevalues attained into the Engineers Theory of Torsion, whilst it had beenbroken down. Each of the three fractions had to equal the same value, whichthey did.
Conclusion
I feel this assignment pushed me to work the answers out for myself. It hasbeen a difficult challenge and involved a lot of research and investigation.I am very pleased with the outcome, and a lot of time and effort has gone intothe presentation of this assignment.I will be taking thorough notes in class from now on to ensure the nextassignment flows much more smoothly. I plan to work just as hard on thenext assignment as I have done on this.
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Brendan Burr BTEC Higher National Certificate in ElectronicsAnalyse Static Engineering Systems
Bibliography
Through guidance from my lecturer, the following text books, catalogues andwebsites I was able to complete this assignment:
BooksHigher Engineering Mathematics (John Bird)ISBN: 0-7506-8152-7
Higher National Engineering (Mike Tooley & Lloyd Dingle)ISBN: 978-0-7506-6177-5
Catalogues
N/A
Websites
http://www.mechengcalculations.com/jmm/beam28_process.jsp
http://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-405-shear-and-moment-diagrams
http://www.mechengcalculations.com/jmm/beam28_process.jsphttp://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-405-shear-and-moment-diagramshttp://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-405-shear-and-moment-diagramshttp://www.mechengcalculations.com/jmm/beam28_process.jsphttp://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-405-shear-and-moment-diagramshttp://www.mathalino.com/reviewer/mechanics-and-strength-of-materials/solution-to-problem-405-shear-and-moment-diagrams