This session covers the basics that are required to analyse indeterminate trusses to maximum of two degree indeterminacy which includes,
strain energy stored due to axial loads and bending stresses,
maxwell's reciprocal deflection theorem,
Betti's law,
castigliano's theorems,
problems based on castigliano's theorems on beams and frames,
unit load method,
problems on trusses with unit load method,
lack of fit in trusses,
temperature effect on truss members.
This session covers the basics that are required to analyse indeterminate trusses to maximum of two degree indeterminacy which includes,
strain energy stored due to axial loads and bending stresses,
maxwell's reciprocal deflection theorem,
Betti's law,
castigliano's theorems,
problems based on castigliano's theorems on beams and frames,
unit load method,
problems on trusses with unit load method,
lack of fit in trusses,
temperature effect on truss members.
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Stiffness method of structural analysisKaran Patel
This method is a powerful tool for analyzing indeterminate structures. One of its advantages over the flexibility method is that it is conducive to computer programming.
Stiffness method the unknowns are the joint displacements in the structure, which are automatically specified.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
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Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
3. Objective(s)
• Familiarity with the definition of work
• Familiarity with the concept of virtual work by
• Axial forces
• Transverse shear forces
• Bending
• Torsion
• Familiarisation with unit load method
3
4. Comments
• Please post any comments either here or on BB:
https://padlet.com/damghani_mahdi/SDI
4
5. Introduction
• They are based on the concept of work and are
considered within the realm of “analytical mechanics”
• Energy methods are fit for complex problems such as
indeterminate structures
• They are essential for using Finite Element Analysis
(FEA)
• They provide approximates solutions not exact
• The Principle of Virtual Work (PVW) is the most
fundamental tool of analytical mechanics
5
7. Work
• Displacement of force times the quantity of force in the
direction of displacement gives a scalar value called work
cosFWF
2
1
a
FWF
2
2
a
FWF
21 FFF WWW MWF
7
8. Work on a particle
• Point A is virtually
displaced (imaginary
small displacement) to
point A’
• R is the resultant of
applied concurrent
forces on point A
• If particle is in
equilibrium?
R=0
WF=0
8
9. Principle of Virtual Work (PVW)
• If a particle is in equilibrium under the action of a
number of forces, the total work done by the forces
for a small arbitrary displacement of the particle is zero.
(Equivalent to Newton’s First Law)
• Can we say?
If a particle is not in equilibrium under the action of a
number of forces, the total work done by the forces for a
small arbitrary displacement of the particle is not zero.
R could make a 90 degree angle with
displacement
9
10. In other words
• The work done by a real force 𝑃 moving through an
arbitrary virtual displacement ≈ arbitrary test displacement ≈
arbitrary fictitious displacement 𝛿𝑢 is called the virtual work
𝛿𝑊. It is defined as; 𝛿𝑊 = 𝑃𝛿𝑢
• Note that The word arbitrary is easily understood: it simply
means that the displacements can be chosen in an
arbitrary manner without any restriction imposed on their
magnitudes or orientations. More difficult to understand are
the words virtual, test, or fictitious. All three imply that these
are not real, actual displacements. More importantly, these
fictitious displacements do not affect the forces acting on the
particle.
• Then we define PVW for both rigid bodies and deformable
bodies separately (see subsequent slides).
10
11. Note 1
• Note that, Δv is a small and purely imaginary
displacement and is not related in any way to the
possible displacement of the particle under the action
of the forces, F;
• Δv has been introduced purely as a device for setting
up the work-equilibrium relationship;
• The forces, F, therefore remain unchanged in
magnitude and direction during this imaginary
displacement;
• This would not be the case if the displacements were
real.
11
12. PVW for rigid bodies
• External forces (F1 ... Fr)
induce internal forces;
• Suppose the rigid body is
given virtual displacement;
• Internal and external forces
do virtual work;
• There are a lot of pairs like
A1 and A2 whose internal
forces would be equal and
opposite;
• We can regard the rigid body
as one particle.
21 A
i
A
i FF
eitotal WWW 0iW et WW
021
A
i
A
i WW
12
It does not undergo deformation
(change in length, area or shape)
under the action of forces.
Internal forces act and
react within the system
and external forces act
on the system
13. PVW for deformable bodies
• If a virtual displacement of Δ is applied, all particles do
not necessarily displace to the amount of Δ, i.e.
internal virtual work is done in the interior of the body.
• This principle is valid for;
• Small displacements.
• Rigid structures that cannot deform.
• Elastic or plastic deformable structures.
• Competent in solving statically indeterminate structures.
21 A
i
A
i FF 0 ietotal WWW
13
0iW
The distance between two
points changes under the
action of forces.
14. Work of internal axial force on
mechanical systems/structures
14
Isolate
Section
This truss element is working
under the action of axial load
only as a result of external
aerodynamic loading.
After imposing a virtual
displacement, the axial load
does virtual work on this truss
element.
To obtain the amount of virtual
work, we obtain the work on
the section and then
throughout the length (next
slide).
15. Work of internal axial force
A
A
N
AN
• Work done by small axial force due to
small virtual axial strain for an
element of a member:
xNxdA
A
N
w v
A
vNi ,
• Work done by small axial force due to
small virtual axial strain for a member:
L
vNi dxNw ,
• Work done by small axial force due to
small virtual axial strain for a structure
having r members:
rm
m
vmmNi dxNw
1
,
15
x
xl
l
vA
A
vv
:reminder
16. Work of internal axial force for
linearly elastic material
• Based on Hook’s law (subscript v denotes virtual);
• Therefore, we have (subscript m denotes member m);
EA
N
E
vv
v
...
21 22
22
11
11
1
,
L
v
L
v
rm
m L mm
vmm
Ni dx
AE
NN
dx
AE
NN
dx
AE
NN
w
m
16
rm
m
vmmNi dxNw
1
,
17. Work of internal shear force
AS
• Work done by small shear force due to
small virtual shear strain for an element
of a member (β is form factor):
xSxdA
A
S
xdAw vv
A
vSi ,
• Work done by small shear force due to
small virtual shear strain for a member
of length L:
L
vSi dxSw ,
δS
• Work done by small shear force due to
small virtual shear strain for a structure
having r members:
rm
m L
vmmmSi dxSw
1
,
17
18. Work of internal shear force for
linearly elastic material
• Based on Hook’s law (subscript v denotes virtual);
• Therefore, we have (subscript m denotes member m);
GA
S
G
vv
v
...
21 22
22
2
11
11
1
1
,
L
v
L
v
rm
m L mm
vmm
mSi dx
AG
SS
dx
AG
SS
dx
AG
SS
w
m
18
19. Work of internal bending moment
• Work done by small bending due to
small virtual axial strain for an
element of a member:
x
R
M
x
R
y
dAw
vA v
Mi ,
• Work done by small bending due to
small virtual axial strain for a member:
L v
Mi dx
R
M
w ,
• Work done by small bending due to
small virtual axial strain for a structure
having r members:
rm
m vm
m
Mi dx
R
M
w
1
,
A
vMi xdAw ,
19
Radius of curvature due
to virtual displacement
v
v
EI
My
v R
y
IE
My
EI
M
R
v
,1
20. Work of internal bending moment for
linearly elastic material
• We have (subscript m denotes member m);
1 v
v
M
R EI
...
21 22
22
11
11
1
,
L
v
L
v
rm
m L mm
vmm
Mi dx
IE
MM
dx
IE
MM
dx
IE
MM
w
m
20
21. Work of internal torsion
• See chapter 2 of Reference 1,
chapter 15 of Reference 2 or chapter
9 of Reference 3 for details of this
• Following similar approach as
previous slides for a member of
length L we have;
L
v
Ti dx
GJ
TT
w ,
• For a structure having several
members of various length we have;
...
21 22
22
11
11
1
,
L
v
L
v
rm
m L mm
vmm
Ti dx
JG
TT
dx
JG
TT
dx
JG
TT
w
m
21
22. Virtual work due to external force
system
• If you have various
forces acting on
your structure at
the same time;
L
yvvvxvyve dxxwTMPWW ,,, )(
L L L
vAvAvA
L
vA
i dx
GJ
TT
dx
EI
MM
dx
GA
SS
dx
EA
NN
W
0 ie WW
22
23. Note
• So far virtual work has been produced by actual forces
in equilibrium moving through imposed virtual
displacements;
• Base on PVW, we can alternatively assume a set of
virtual forces in equilibrium moving through actual
displacements;
• Application of this principle, gives a very powerful
method to analyze indeterminate structures;
23
24. Example 1
• Determine the bending moment at point B in the
simply supported beam ABC.
24
25. Solution
• We must impose a small virtual displacement which
will relate the internal moment at B to the applied
load;
• Assumed displacement should be in a way to exclude
unknown external forces such as the support
reactions, and unknown internal force systems such
as the bending moment distribution along the length
of the beam.
25
26. Solution
• Using conventional equations of equilibrium method;
26
RA RC0CM
A
A
R L Wb
b
R W
L
B A
ab
M R a W
L
27. Solution
• Let’s give point B a virtual displacement;
27
β
b
a
baBv ,
b
L
B
BvBBie WMWW ,
L
Wab
MWa
b
L
M BB
Rigid
Rigid
28. Example 2
• Using the principle of virtual work, derive a formula in
terms of a, b, and W for the magnitude P of the force
required for equilibrium of the structure below, i.e. ABC
(you may disregard the effects weight).
28
A
C
B
W
P a
b
29. Solution
• We assume that AB and AC are rigid and therefore
internal work done by them is zero
• Apply a very small virtual displacement to our system
29
Just to confirm the answer,
you would get the same
result if you took moment
about B, i.e. 𝑀 𝐵 = 0
Virtual movements
C
A
r b
r a
Virtual work
0
0
0
/
A C
U
P r W r
Pa Wb
P bW a
A
C
B
W
P a
b
Cr
Ar
31. Solution
• This structure has 1 degree of
indeterminacy, i.e. 4 reaction (support)
forces, unknowns, and 3 equations of
equilibrium
• Let’s apply an infinitesimally small virtual
displacement where we intend to get the
force
• Equating work done by external force to
that of internal force gives
31
BvCv
CvBv
,,
,,
3
4
43
)tan(
kNFF ABBvABCv 4030 ,,
32. Example 4
• We would like to obtain slope for the portal frame
below;
32
P
2rk1rk
1M 2M
h
a
33. Solution
33
1 1 2 2
e
i
W P
W M M
1 2 / h
1 2
1 2
/ /i
i
W M h M h
W M M
h
1 2 1 2
1 2
0
1 1
0 0
/
i i i
e i
M k
r r
W W
P M M P M M
h h
Ph k k
P
2rk1rk
1M 2M
h
a
34. Note
• The amount of virtual displacement can be any
arbitrary value;
• For convenience lets give it a unit value, for example
in the previous example lets say Δv,B=1;
• In this case the method could be called unit load
method.
34
35. Note
• If you need to obtain force in a member, you should
apply a virtual displacement at the location where force
is intended;
• If you need to obtain displacement in a member, you
should apply a virtual force at the location
displacement is intended.
35
37. Solution
• Apply a virtual unit load in the direction of displacement to be calculated
37
LxxMv )( 2
22
222
)( xL
wwL
wLx
wx
xM
• Work done by virtual unit load
Be vw 1
• Work done by internal loads
L
L
v
Mi xL
EI
w
dx
EI
MM
w
0
3
,
2
• Equating external work with internal
EI
wL
vxL
EI
w
v B
L
B
82
1
4
0
3
Virtual system
Real system
38. Example 6
• Using unit load method determine slope and deflection
at point B.
38
AC B D
5kN/m
IAB=4x106 mm4
IBC=8x106 mm4
8kN
2m 0.5m 0.5m
E=200 kN/mm2
39. Solution
• For deflection we apply a unit virtual load at point B in the
direction of the intended displacement;
39
Virtual system
Real system
Segment Interval I (mm4) M v (kN.m) M (kN.m)
AD 0<x<0.5 4x106 0 8x
DB 0.5<x<1 4x106 0 8x-2.5(x-0.5)2
BC 1<x<3 8x106 -x+1 8x-2.5(x-0.5)2
...
21 22
22
11
11
1
,
L
v
L
v
rm
m L mm
vmm
Mi dx
IE
MM
dx
IE
MM
dx
IE
MM
w
m
40. Solution
40
12B mm
• For slope we apply a unit virtual moment at point B
2 2
0.5 1 3
6 6 6
0 0.5 1
0 8 2.5 0.5 1 8 2.5 0.50 8
1
200 4 10 200 4 10 200 8 10
v
B
L
x x x x xM M x
dx dx dx dx
EI
1kN.m
...
21 22
22
11
11
1
,
L
v
L
v
rm
m L mm
vmm
Mi dx
IE
MM
dx
IE
MM
dx
IE
MM
w
m
41. Solution
41
Segment Interval I (mm4) M v (kN.m) M (kN.m)
AD 0<x<0.5 4x106 0 8x
DB 0.5<x<1 4x106 0 8x-2.5(x-0.5)2
BC 1<x<3 8x106 1 8x-2.5(x-0.5)2
3
1
6
21
5.0
6
25.0
0
6
108200
5.05.281
104200
5.05.280
104200
80
1 dx
xx
dx
xx
dx
x
dx
EI
MM
L
v
B
radB 0119.0
42. Q1
• Use the principle of virtual work to determine the
support reactions in the beam ABCD.
42
43. Q2
• Find the support reactions in the beam ABC using the
principle of virtual work.
43
44. Q3
• Find the bending moment at the three-quarter-span
point in the beam. Use the principle of virtual work.
44
45. Q4
• Use the unit load method to calculate the deflection at
the free end of the cantilever beam ABC.
45
46. Q5
• Calculate the deflection of the free end C of the
cantilever beam ABC using the unit load method.
46
47. Q6
• Use the unit load method to find the magnitude and
direction of the deflection of the joint C in the truss. All
members have a cross-sectional area of 500mm2 and
a Young’s modulus of 200,000 N/mm2.
47
48. Q7
• Calculate the forces in the members FG, GD, and CD
of the truss using the principle of virtual work. All
horizontal and vertical members are 1m long.
48
Editor's Notes
https://padlet.com/mahdi_damghani/SDIAero
The total virtual work done by all forces acting on a rigid body in static equilibrium is zero for small and admissible virtual displacements from the equilibrium state. In equation form it is written as above.