Civil Engineering is the Branch of Engineering.The Civil engineering field requires an understanding of core areas including Mechanics of Solids, Structural Mechanics - I, Building Construction Materials, Surveying - I, Geology and Geotechnical Engineering, Structural Mechanics, Building Construction, Water Resources and Irrigation, Environmental Engineering, Transportation Engineering, Construction and Project Management. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
In Engineering Mechanics the static problems are classified as two types: Concurrent and Non-Concurrent force systems. The presentation discloses a methodology to solve the problems of Concurrent and Non-Concurrent force systems.
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
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
In Engineering Mechanics the static problems are classified as two types: Concurrent and Non-Concurrent force systems. The presentation discloses a methodology to solve the problems of Concurrent and Non-Concurrent force systems.
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
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Bending Stresses are important in the design of beams from strength point of view. The present source gives an idea on theory and problems in bending stresses.
This document gives the class notes of Unit 6: Bending and shear Stresses in beams. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
If a body is subjected to many forces on its plane at a single point then they are called as Coplanar Concurrent Forces. The effect of the forces acting on the body is unknown. It is necessary to determine the resultant force of the coplanar forces to know this effect. Copy the link given below and paste it in new browser window to get more information on Law Of Polygon:-
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Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visi tus: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Coplanar forces res & comp of forces - for mergeEkeeda
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Bending Stresses are important in the design of beams from strength point of view. The present source gives an idea on theory and problems in bending stresses.
This document gives the class notes of Unit 6: Bending and shear Stresses in beams. Subject: Mechanics of materials.
Syllabus contest is as per VTU, Belagavi, India.
Notes Compiled By: Hareesha N Gowda, Assistant Professor, DSCE, Bengaluru-78.
If a body is subjected to many forces on its plane at a single point then they are called as Coplanar Concurrent Forces. The effect of the forces acting on the body is unknown. It is necessary to determine the resultant force of the coplanar forces to know this effect. Copy the link given below and paste it in new browser window to get more information on Law Of Polygon:-
http://www.transtutors.com/homework-help/mechanical-engineering/force-systems-and-analysis/law-of-polygon.aspx
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Coplanar forces res & comp of forces - for mergeEkeeda
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
Definition of Force
Definition of a Vector
Vector Diagram
Graphical Representation of a Vector
Parallelogram of Forces
Resolving and inclined force
Resultant of two forces acting at a point Force
Resultant of a system of forces acting at a point
Mechanical Engineering is the Branch of Engineering.The mechanical engineering field requires an understanding of core areas including mechanics, dynamics, thermodynamics, materials science and structural analysis,Fluid Mechanics, Metrology and Instrumentation, Dynamics of Machinery- II, Manufacturing Processes II, Industrial Drafting and Machine Design, Engineering Graphics, Power Plant Engineering. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus. Visit us: https://ekeeda.com/streamdetails/stream/mechanical-engineering
Ekeeda Provides Online Video Lectures for Mechanical Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/mechanical-engineering
Ekeeda Provides Online Video Lectures for Mechanical Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/mechanical-engineering
Civil Engineering is the Branch of Engineering.The Civil engineering field requires an understanding of core areas including Mechanics of Solids, Structural Mechanics - I, Building Construction Materials, Surveying - I, Geology and Geotechnical Engineering, Structural Mechanics, Building Construction, Water Resources and Irrigation, Environmental Engineering, Transportation Engineering, Construction and Project Management. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Kinetics of particles impulse momentum methodEkeeda
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Learn Online Courses of Subject Introduction to Civil Engineering and Engineering Mechanics. Clear the Concepts of Introduction to Civil Engineering and Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/introduction-to-civil-engineering-and-engineering-mechanics
Learn Online Courses of Subject Introduction to Civil Engineering and Engineering Mechanics. Clear the Concepts of Introduction to Civil Engineering and Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/introduction-to-civil-engineering-and-engineering-mechanics
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us: http://www.infomaticaacademy.com/
Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us: http://www.infomaticaacademy.com/
Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us/l http://www.infomaticaacademy.com/
Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us: http://www.infomaticaacademy.com/
Infomatica, as it stands today, is a manifestation of our values, toil, and dedication towards imparting knowledge to the pupils of the society. Visit us: http://www.infomaticaacademy.com/
Ekeeda Provides Online Engineering Subjects Video Lectures and Tutorials of Mumbai University (MU) Courses. Visit us: https://ekeeda.com/streamdetails/University/Mumbai-University
Ekeeda Provides Online Engineering Subjects Video Lectures and Tutorials of Mumbai University (MU) Courses. Visit us: https://ekeeda.com/streamdetails/University/Mumbai-University
Ekeeda Provides Online Engineering Subjects Video Lectures and Tutorials of Mumbai University (MU) Courses. Visit us: https://ekeeda.com/streamdetails/University/Mumbai-University
Discover the innovative and creative projects that highlight my journey throu...dylandmeas
Discover the innovative and creative projects that highlight my journey through Full Sail University. Below, you’ll find a collection of my work showcasing my skills and expertise in digital marketing, event planning, and media production.
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s DholeraAvirahi City Dholera
The Tata Group, a titan of Indian industry, is making waves with its advanced talks with Taiwanese chipmakers Powerchip Semiconductor Manufacturing Corporation (PSMC) and UMC Group. The goal? Establishing a cutting-edge semiconductor fabrication unit (fab) in Dholera, Gujarat. This isn’t just any project; it’s a potential game changer for India’s chipmaking aspirations and a boon for investors seeking promising residential projects in dholera sir.
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Unveiling the Secrets How Does Generative AI Work.pdfSam H
At its core, generative artificial intelligence relies on the concept of generative models, which serve as engines that churn out entirely new data resembling their training data. It is like a sculptor who has studied so many forms found in nature and then uses this knowledge to create sculptures from his imagination that have never been seen before anywhere else. If taken to cyberspace, gans work almost the same way.
Memorandum Of Association Constitution of Company.pptseri bangash
www.seribangash.com
A Memorandum of Association (MOA) is a legal document that outlines the fundamental principles and objectives upon which a company operates. It serves as the company's charter or constitution and defines the scope of its activities. Here's a detailed note on the MOA:
Contents of Memorandum of Association:
Name Clause: This clause states the name of the company, which should end with words like "Limited" or "Ltd." for a public limited company and "Private Limited" or "Pvt. Ltd." for a private limited company.
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Registered Office Clause: It specifies the location where the company's registered office is situated. This office is where all official communications and notices are sent.
Objective Clause: This clause delineates the main objectives for which the company is formed. It's important to define these objectives clearly, as the company cannot undertake activities beyond those mentioned in this clause.
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Liability Clause: It outlines the extent of liability of the company's members. In the case of companies limited by shares, the liability of members is limited to the amount unpaid on their shares. For companies limited by guarantee, members' liability is limited to the amount they undertake to contribute if the company is wound up.
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Capital Clause: This clause specifies the authorized capital of the company, i.e., the maximum amount of share capital the company is authorized to issue. It also mentions the division of this capital into shares and their respective nominal value.
Association Clause: It simply states that the subscribers wish to form a company and agree to become members of it, in accordance with the terms of the MOA.
Importance of Memorandum of Association:
Legal Requirement: The MOA is a legal requirement for the formation of a company. It must be filed with the Registrar of Companies during the incorporation process.
Constitutional Document: It serves as the company's constitutional document, defining its scope, powers, and limitations.
Protection of Members: It protects the interests of the company's members by clearly defining the objectives and limiting their liability.
External Communication: It provides clarity to external parties, such as investors, creditors, and regulatory authorities, regarding the company's objectives and powers.
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Binding Authority: The company and its members are bound by the provisions of the MOA. Any action taken beyond its scope may be considered ultra vires (beyond the powers) of the company and therefore void.
Amendment of MOA:
While the MOA lays down the company's fundamental principles, it is not entirely immutable. It can be amended, but only under specific circumstances and in compliance with legal procedures. Amendments typically require shareholder
Digital Transformation and IT Strategy Toolkit and TemplatesAurelien Domont, MBA
This Digital Transformation and IT Strategy Toolkit was created by ex-McKinsey, Deloitte and BCG Management Consultants, after more than 5,000 hours of work. It is considered the world's best & most comprehensive Digital Transformation and IT Strategy Toolkit. It includes all the Frameworks, Best Practices & Templates required to successfully undertake the Digital Transformation of your organization and define a robust IT Strategy.
Editable Toolkit to help you reuse our content: 700 Powerpoint slides | 35 Excel sheets | 84 minutes of Video training
This PowerPoint presentation is only a small preview of our Toolkits. For more details, visit www.domontconsulting.com
RMD24 | Retail media: hoe zet je dit in als je geen AH of Unilever bent? Heid...BBPMedia1
Grote partijen zijn al een tijdje onderweg met retail media. Ondertussen worden in dit domein ook de kansen zichtbaar voor andere spelers in de markt. Maar met die kansen ontstaan ook vragen: Zelf retail media worden of erop adverteren? In welke fase van de funnel past het en hoe integreer je het in een mediaplan? Wat is nu precies het verschil met marketplaces en Programmatic ads? In dit half uur beslechten we de dilemma's en krijg je antwoorden op wanneer het voor jou tijd is om de volgende stap te zetten.
Personal Brand Statement:
As an Army veteran dedicated to lifelong learning, I bring a disciplined, strategic mindset to my pursuits. I am constantly expanding my knowledge to innovate and lead effectively. My journey is driven by a commitment to excellence, and to make a meaningful impact in the world.
[Note: This is a partial preview. To download this presentation, visit:
https://www.oeconsulting.com.sg/training-presentations]
Sustainability has become an increasingly critical topic as the world recognizes the need to protect our planet and its resources for future generations. Sustainability means meeting our current needs without compromising the ability of future generations to meet theirs. It involves long-term planning and consideration of the consequences of our actions. The goal is to create strategies that ensure the long-term viability of People, Planet, and Profit.
Leading companies such as Nike, Toyota, and Siemens are prioritizing sustainable innovation in their business models, setting an example for others to follow. In this Sustainability training presentation, you will learn key concepts, principles, and practices of sustainability applicable across industries. This training aims to create awareness and educate employees, senior executives, consultants, and other key stakeholders, including investors, policymakers, and supply chain partners, on the importance and implementation of sustainability.
LEARNING OBJECTIVES
1. Develop a comprehensive understanding of the fundamental principles and concepts that form the foundation of sustainability within corporate environments.
2. Explore the sustainability implementation model, focusing on effective measures and reporting strategies to track and communicate sustainability efforts.
3. Identify and define best practices and critical success factors essential for achieving sustainability goals within organizations.
CONTENTS
1. Introduction and Key Concepts of Sustainability
2. Principles and Practices of Sustainability
3. Measures and Reporting in Sustainability
4. Sustainability Implementation & Best Practices
To download the complete presentation, visit: https://www.oeconsulting.com.sg/training-presentations
Premium MEAN Stack Development Solutions for Modern BusinessesSynapseIndia
Stay ahead of the curve with our premium MEAN Stack Development Solutions. Our expert developers utilize MongoDB, Express.js, AngularJS, and Node.js to create modern and responsive web applications. Trust us for cutting-edge solutions that drive your business growth and success.
Know more: https://www.synapseindia.com/technology/mean-stack-development-company.html
What are the main advantages of using HR recruiter services.pdfHumanResourceDimensi1
HR recruiter services offer top talents to companies according to their specific needs. They handle all recruitment tasks from job posting to onboarding and help companies concentrate on their business growth. With their expertise and years of experience, they streamline the hiring process and save time and resources for the company.
Skye Residences | Extended Stay Residences Near Toronto Airportmarketingjdass
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Putting the SPARK into Virtual Training.pptxCynthia Clay
This 60-minute webinar, sponsored by Adobe, was delivered for the Training Mag Network. It explored the five elements of SPARK: Storytelling, Purpose, Action, Relationships, and Kudos. Knowing how to tell a well-structured story is key to building long-term memory. Stating a clear purpose that doesn't take away from the discovery learning process is critical. Ensuring that people move from theory to practical application is imperative. Creating strong social learning is the key to commitment and engagement. Validating and affirming participants' comments is the way to create a positive learning environment.
"𝑩𝑬𝑮𝑼𝑵 𝑾𝑰𝑻𝑯 𝑻𝑱 𝑰𝑺 𝑯𝑨𝑳𝑭 𝑫𝑶𝑵𝑬"
𝐓𝐉 𝐂𝐨𝐦𝐬 (𝐓𝐉 𝐂𝐨𝐦𝐦𝐮𝐧𝐢𝐜𝐚𝐭𝐢𝐨𝐧𝐬) is a professional event agency that includes experts in the event-organizing market in Vietnam, Korea, and ASEAN countries. We provide unlimited types of events from Music concerts, Fan meetings, and Culture festivals to Corporate events, Internal company events, Golf tournaments, MICE events, and Exhibitions.
𝐓𝐉 𝐂𝐨𝐦𝐬 provides unlimited package services including such as Event organizing, Event planning, Event production, Manpower, PR marketing, Design 2D/3D, VIP protocols, Interpreter agency, etc.
Sports events - Golf competitions/billiards competitions/company sports events: dynamic and challenging
⭐ 𝐅𝐞𝐚𝐭𝐮𝐫𝐞𝐝 𝐩𝐫𝐨𝐣𝐞𝐜𝐭𝐬:
➢ 2024 BAEKHYUN [Lonsdaleite] IN HO CHI MINH
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➢CHILDREN ART EXHIBITION 2024: BEYOND BARRIERS
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"𝐄𝐯𝐞𝐫𝐲 𝐞𝐯𝐞𝐧𝐭 𝐢𝐬 𝐚 𝐬𝐭𝐨𝐫𝐲, 𝐚 𝐬𝐩𝐞𝐜𝐢𝐚𝐥 𝐣𝐨𝐮𝐫𝐧𝐞𝐲. 𝐖𝐞 𝐚𝐥𝐰𝐚𝐲𝐬 𝐛𝐞𝐥𝐢𝐞𝐯𝐞 𝐭𝐡𝐚𝐭 𝐬𝐡𝐨𝐫𝐭𝐥𝐲 𝐲𝐨𝐮 𝐰𝐢𝐥𝐥 𝐛𝐞 𝐚 𝐩𝐚𝐫𝐭 𝐨𝐟 𝐨𝐮𝐫 𝐬𝐭𝐨𝐫𝐢𝐞𝐬."
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P
INTRODUCTION
We have so far discussed and worked with a coplanar all system of forces,
wherein the forces in a system would lie in a single plane i.e. it was a two
dimensional force system. In this chapter we deal with" a system of forces
lying in different planes forming a three dimensional force system. Such a
system is also referred as a space Force System and a vector approach is
required to deal with these problems. The chapter begins with the study of
basic operations with forces using a vector approach. Later on we will learn to
find the resultant of a space force system and finally we shall deal with
equilibrium problems.
BASIC OPERATIONS USING VECTOR APPROACH
Since the forces have three dimensions, we employ a vector approach, which
simplifies the working. Here we will learn to represent a force vectorially, to
find vectorially moment of a force about a point and about a line, to find
magnitude and direction of a force given in vector form and other basic
operations useful in solution of a space force problem.
Force in vector form
Fig. shows a force of magnitude F in space passing through A (x1, y1, z1) and.
B (x2,y2, z2). The force in vector form is
AB
ˆF F.e
2 1 2 1 2 1
2 2 2
2 1 2 1 2 1
F F
x x i y y j z z k
x x y y z z
x y zF F i F j F k
Note: i, j and k printed in bold type denote unit
vectors along the x, y and z axis respectively.
Space Forces
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MAGNITUDE AND DIRECTION OF FORCE
Fig. shows a force x y zF F i+F j+F k making angles
x y and z, with the x, y
and z axis respectively.
Here Fx is the component of force in the x
direction. Similarly Fy and Fz are the force
components in the y and z direction.
The magnitude of the force is
2 2 2
x y zF F F F
Also,
x x
y y
z z
F F cos θ
F Fcosθ
F Fcos θ
Here x y and z are known as the force directions, the value of which lies
between 0 and 180. There is an important identity which relates them.
2 2 2
cos cos cos 1x y z
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MOMENT OF A FORCE ABOUT A POINT
This is a very important operation while dealing with forces. For coplanar
forces, moment about g point was the product of the force and the
distance. Here if the force is in space the moment calculation requires a
vector approach.
Fig. shows a force F in space passing through points A (x1, y1, z1) and B (x2,
y2, z2) on its line of action. Let C (x3, y3, z3) be the moment centre i.e. the point
about which we have to find the moment. The procedure of finding the
moment of the force about the point is as follows.
Step 1: Put the force in vector form i.e.
x y zF F i+F j+F k
Step 2: Find the position vector extending from the moment centre to any
point on the force i.e. x y zr r i+r j+r k .
Step 3: Perform the cross product of the position vector and the force vector
to get the moment vector i.e.
F
point
x y z
x y z
M
i j k
r r r
F F F
r F
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EXERCISE 1
1. A force of 50 N acts parallel to the y axis in the -ve direction. Put the force
in vector form.
2. A 130 KN force acts at B (12, O, O) and passes through C (0, 3, 4). Put the
force in vector form.
3. A force F = (3i -4j 12 + k) N acts at a point A (1,-2, 3) m. Find
a. moment of the force about origin.
b. moment of the force about point B (2, l,2)m.
4. A force F = 80 i + 50 j - 60 k passes through a point A (6, 2, 6). Compute its
moment about a point B (8, 1, 4) (M. U. Dec12)
5. A 700 N force passes through two points A (-5,-1, 4) towards B (1, 2, 6) m.
Find moment of the force about a point C (2,-2, 1)m.
6. A force of magnitude 1200 N passes from A (2, -4, 6) to B (-3, 2, 3).
Calculate the moment of this force about the origin. Also determine the
component of this force along the line AC where C has the position vector
defined by cr i 2j 3kN .
7. A rectangular plate is supported by
brackets to the wall at A and B by wire CD
as shown in figure. Knowing that tension in
wire is 2OO N determine the moment about
point A, of the force exerted by wire on
point C. All dimensions are in mm.
8. Find moment of force P = 500 N about point H.
9. Three forces act on a rectangular box as shown in figure. Determine the
moment of each force about the origin.
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RESULTANT OF CONCURRENT SPACE FORCE SYSTEM
Resultant of a concurrent space force system is a single force R , which acts
through the point of concurrence. Fig. (a) shows a concurrent system at point
P. The resultant of the system is shown 1n Fig. (b) and calculated as,
RESULTANT OF PARALLEL SPACE FORCE SYSTEM
The resultant of a parallel space force system is a single force R which acts
parallel to the force system. The location of the resultant can be found out
using Varignon's theorem.
Figure (a) shows a parallel system of three forces F1, F2 and F3. The resultant
of the system is shown in figure (b) and is calculated as,
1 2 3R F F F
The resultant acts at P. The coordinates (x, y, z) of the point P can be
calculated by using Varignon's theorem, the moments for which can be taken
about any convenient point like point O. The equation of Varignon's theorem
for space forces is
F R
O OM M
0
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RESULTANT OF GENERAL SPACE FORCE SYSTEM
A general space force system is neither a concurrent nor a parallel system.
The resultant of such a system is a single force R and a moment M at any
desired point. Since the resultant contains one force and one moment, it is
also known as a Force Couple System. Fig. (a) shows a general system of four
forces F1, F2, F3 and F4. If it is desired to have the resultant at point O, then
as per figure (b).
The single force 41 2 3R F F F F
and the single moment 31 2 4FF F F
O O O OM M M M M
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EXERCISE 2
1. A force P1 = 10 N in magnitude acts along direction AB whose co-ordinates
of points A and B are (3, 2, -l) and (8, 5, 3) respectively. Another force P2 = 5
N in magnitude acts along BC where C has co-ordinates (-2, 11, -5).
Determine
a. The resultant of Pr and Pz.
b. The moment of the resultant about a point D (1, 1, 1).
2. Knowing that the tension in AC = 20 kN, determine the required values of
TAB and TAD so that the resultant of the three forces applied at A is
vertical. Also find the resultant.
3. A plate foundation is subjected to five vertical forces as shown. Replace
these five forces by means of a single vertical force and find the point of
replacement.
4. Determine the loads to be applied at A and F, if the resultant of all six loads
is to pass through the centre of the
foundation of hexagonal shape of side 3 m.
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5. Knowing that the resultant of the three parallel forces is 500 N acting in
positive y direction and passing through the centre of the rectangular plate,
determine the forces F1 , F2 and F3.
6. A square foundation mat supports four columns as shown in figure.
Determine magnitude and point of application of resultant of four loads.
7. A rectangular parallelepiped carries four
forces as shown in the figure. Reduce the
force system to a resultant force applied at
the origin and moment around the origin.
OA = 5 m, OB = 2m, OC = 4 m. (M. U
Dec 12)
8. Figure shows a rectangular
parallelepiped subjected to four
forces in the direction shown.
Reduce them to a resultant force at
the origin and a moment.
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9. A tetrahedron A B C D is loaded by forces
F1 = 100 N at A along DA, F2 = 2OO N at B
along CB and F3 = 300 N at C along DC as
shown in the figure. Replace the three
force system by a single resultant force R
at B and a single resultant moment vector
M at B. Take the co-ordinates in metre
units.
10. The following forces act on the block shown in
figure. F1 = 40 kN at point C along CD, F2 = 3O
kN at point D along DB and F3 = 1OO kN at point
D along DE. Find the resultant force and
resultant moment of these forces acting at O.
(SPCE Nov 12)
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EQUILIBRIUM OF SPACE FORCES
When the resultant of a system is zero, the system is said to be in
equilibrium. For the resultant to be zero, the resultant force F and the
resultant moment M should be zero.
This gives us six scalar equations of equilibrium viz.
0xF 0xM
0yF 0yM
0zF 0zM
SUPPORTS FOR SPACE STRUCTURE
a) Ball and Socket Support
This support allows rotation in all the three
direction, unlike a hinge which allows rotation
about one axis only. A ball and socket support
does not allow any linear movement in any
direction. This results in a reaction force having
components along x, y and z axis. Fig. shows a
ball and socket support offering three components of reaction force.
b) Fixed End Support
A fixed end will not allow rotation in any
direction, nor will allow any linear movement.
This results in a reaction moment and a
reaction force having components along x, y
and z axis.
Refer Fig.
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Equilibrium of Concurrent Space Force System
For a concurrent system, only the resultant force needs to be zero. This is the
necessary and sufficient condition of equilibrium. This gives us three scalar
equations of equilibrium, viz.
0xF
0yF
0zF
We can also use the other three scalar equations of equilibrium viz.
0xM
0yM
0zM
EQUILIBRIUM OF NON-CONCURRENT SPACE FORCE SYSTEM
For equilibrium of both parallel and general space force systems, all the six
scalar equations of equilibrium are applicable viz.
0xF 0xM
0yF 0yM
0zF 0zM
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EXERCISE 3
1. A boom AB supports a load of 1000 N as shown. Neglect weight of the
boom. Determine tension in each cable and the reaction at A.
2. A vertical load of 1100 N is supported by the three
rods shown in figure. Find the force in each rod.
Points C, O and D are in XZ plane while point B
lies 5 m above this plane.
3. A vertical load of 1OOO N is supported by three
bars as shown. Find the force in each bar. Point
C, O and D are in the x-z plane while B is 1.5 m
above this plane.
4. A vertical tower DC shown is subjected to a horizontal force P = 5O kN at its
top and is anchored by two similar guy wires BC and AC. Calculate
a. Tension in the guy wires.
b. Thrust in the tower pole.
Co-ordinates of the points are as below,
O (0, 0, 0), B (0, 0, -4), D (3, 0, 0), A (0, 0, 4), C (3, 20, 0)
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5. A uniform thin plate ABCD, square in shape is
suspended by strings GA (2a m length), GD (2a m
length), GE 3amlength and GH (a m length) so that it
remains in horizontal plane. If the size of the plate is
2a x 2a and E is the midpoint of BC, find the tension in
each of the string if the plate weighs 400 N.
6. A square steel Plate 2400 mm 2400 mm has a mass of 1800 kg with
mass centre at G. Calculate the tension in each of the three cables with
which the plate is lifted while remaining horizontal. Length DG = 2400 mm.
7. A crate is supported by three cables as shown. Determine the weight of the
crate, if the tension in the cable AB is 750 N.
8. Determine the tension in cable BC and BD and reactions at the ball socket
joint A for the mast as shown in figure.
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9. A vertical mast OD is having base ‘0’ with ball and socket. Three cables DA,
DB and DC keep the mast in equilibrium. If tension in the cable DA is 100
kN, find tensions in the cables DB and DC and force in the mast. Refer
figure.
10. A load of 500 N is held in equilibrium by means of two strings CA and CB
and by a force p as shown in figure. Determine tensions in strings and
magnitude of P.
11. A uniform equilateral triangular plate weighing 2.4
kN is held in horizontal plane by means of three
cables at A, B and C. An additional load of 1.2 kN
acts at midpoint of edge AC of the plate. Determine
the tensions in the three cables.
12. A plate ACED 1O mm thick weighs 76O0 kg/ms. It is held in horizontal
plane by three wires at A, B and C. find tensions in the wires.
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13. Figure shows steel Pipe ABCD of length 1.8 m having 900 bend at B. The
Pipe weighs 30 N/m and supported by three vertical wires attached to the
Pipe at A, C and D. Find tension in wires.
EXERCISE 4
THEORY QUESTIONS
Q.1 How is a space force represented in magnitude and direction.
Q.2 Write the equations of equilibrium to be satisfied for non co-planar
concurrent force system. (SPCE Nov 12)
Q.3 State conditions of equilibrium for non-concurrent forces in space.
UNIVERSITY QUESTIONS
1. A pole is held in place by three cables. If the force of each cable acting on
the pole is as shown in fig. determine the resultant. (6 Marks)
2. A force of 10 kN acts at a point P(2,3,5) m and has its line of action passing
through Q(10, -3, 4)m. Calculate moment of this force about a point
S(1,-10,3) m. (4 Marks)
3. Explain conditions for equilibrium for forces in spare. (6 Marks)
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4. Force F = (3i – 4j + 12k) N acts at point A (1,-2,3) Find -- (4 Marks)
(i) Moment of force about origin
(ii) Moment of force about point B (2,1,2)m.
5. A rectangular parallelepiped carries Three forces shown in fig. Reduce the
force system to a resultant force applied at the origin and a moment around
origin. (6 Marks)
6. The lines of action of three forces concurrent at origin ‘O’ pass respectively
through points A(-1,2,4), B (3,0,-3) and C(2,-2,4)m. The magnitude of
forces are 40N, 10N and 30N respectively. Determine the magnitude and
direction of their resultant. (6 Marks)
7. A force of 1200N acts along PQ, P(4,5,-2) and Q (-3,1,6)m. Calculate its
moment about a point A(3,2,0)m (4 Marks)
8. A force 80 50 60F i j k passes through a point A Compute its moment
about point B (8,1,4). (6 Marks)