This document provides instructions for analyzing statically indeterminate beams using the matrix force method. It discusses:
1. Formulating the problem by selecting redundant reactions and determining the deflections of the primary structure due to applied loads.
2. Assembling the flexibility matrix relating deflections to applied forces at various points.
3. Setting up compatibility equations equating total deflections to zero and solving the equations in matrix form to determine redundant reactions.
4. Using the determined redundant reactions and equilibrium equations to calculate all reactions and internal forces for a given statically indeterminate beam problem.
Two example problems are provided to demonstrate applying the procedure.
This document presents a new modified F-expansion method to obtain traveling wave solutions of the Benjamin-Bona-Mahony (BBM) equation and modified BBM equation. The method is applied to these nonlinear partial differential equations. Specifically:
1) The traveling wave solutions of the BBM equation are considered by substituting a transformation.
2) The solution is assumed to have the form of a polynomial in F(ξ) and its derivatives, where F(ξ) satisfies a Riccati equation.
3) Three explicit solutions for the BBM equation are obtained in terms of hyperbolic and trigonometric functions.
The document is an unsolved physics paper from 2008 for IITJEE (Indian Institute of Technology Joint Entrance Examination). It contains multiple choice and reasoning questions about physics concepts. The questions cover topics like mechanics, electromagnetism, thermodynamics, and waves. There are four sections with different question types including objective, reasoning, comprehension, and matrix-match questions. The document provides figures and diagrams to accompany some of the physics problems.
Mathematical models for a chemical reactorLuis Rodríguez
This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.
This document contains a summary of 8 mathematics questions on the topic of sets. Each question contains 1-4 parts asking students to shade regions on Venn diagrams, list set elements, or calculate set properties like union and intersection. The document also provides the answers to each question in point form for easy reference.
Module 13 Gradient And Area Under A Graphguestcc333c
1) The document provides examples and questions related to calculating gradient, area under graphs, speed, velocity, and distance from speed-time and distance-time graphs.
2) It includes 10 multi-part questions testing concepts like calculating rate of change of speed, uniform speed, total distance, meeting time, and average speed.
3) Detailed step-by-step answers are provided for each question at the end to demonstrate how to apply the concepts to calculate the requested values.
Electromagnetic theory mcq (1) removed (1) watermarkNikhilSharma1004
This document contains 10 multiple choice questions related to electromagnetic theory and vector properties. Specifically:
- Questions 1-5 cover topics like dot and cross products of vectors, Lorentz force, and Fleming's rules.
- Questions 6-10 cover vector functions in electromagnetics like gradient, divergence, curl, and the relationship between vector potential and field strength.
The questions test fundamental concepts in vector algebra and vector calculus as they relate to electromagnetic theory.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
- Root-locus plots show how the roots of a system change with variations in a system parameter like gain.
- The plot determines if the system will become unstable as parameters vary by checking if roots cross to the right half of the complex plane.
- A document section provides examples of how real and complex roots correspond to different system responses and stability.
This document presents a new modified F-expansion method to obtain traveling wave solutions of the Benjamin-Bona-Mahony (BBM) equation and modified BBM equation. The method is applied to these nonlinear partial differential equations. Specifically:
1) The traveling wave solutions of the BBM equation are considered by substituting a transformation.
2) The solution is assumed to have the form of a polynomial in F(ξ) and its derivatives, where F(ξ) satisfies a Riccati equation.
3) Three explicit solutions for the BBM equation are obtained in terms of hyperbolic and trigonometric functions.
The document is an unsolved physics paper from 2008 for IITJEE (Indian Institute of Technology Joint Entrance Examination). It contains multiple choice and reasoning questions about physics concepts. The questions cover topics like mechanics, electromagnetism, thermodynamics, and waves. There are four sections with different question types including objective, reasoning, comprehension, and matrix-match questions. The document provides figures and diagrams to accompany some of the physics problems.
Mathematical models for a chemical reactorLuis Rodríguez
This document presents a mathematical model for the concentration of a chemical in a reactor. It examines both steady state and time-dependent models. For steady state, the model is an ordinary differential equation that can be solved analytically. For time dependence, the model is a partial differential equation that requires numerical solution. Two numerical methods are presented: an implicit finite difference method and the finite element method.
This document contains a summary of 8 mathematics questions on the topic of sets. Each question contains 1-4 parts asking students to shade regions on Venn diagrams, list set elements, or calculate set properties like union and intersection. The document also provides the answers to each question in point form for easy reference.
Module 13 Gradient And Area Under A Graphguestcc333c
1) The document provides examples and questions related to calculating gradient, area under graphs, speed, velocity, and distance from speed-time and distance-time graphs.
2) It includes 10 multi-part questions testing concepts like calculating rate of change of speed, uniform speed, total distance, meeting time, and average speed.
3) Detailed step-by-step answers are provided for each question at the end to demonstrate how to apply the concepts to calculate the requested values.
Electromagnetic theory mcq (1) removed (1) watermarkNikhilSharma1004
This document contains 10 multiple choice questions related to electromagnetic theory and vector properties. Specifically:
- Questions 1-5 cover topics like dot and cross products of vectors, Lorentz force, and Fleming's rules.
- Questions 6-10 cover vector functions in electromagnetics like gradient, divergence, curl, and the relationship between vector potential and field strength.
The questions test fundamental concepts in vector algebra and vector calculus as they relate to electromagnetic theory.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
- Root-locus plots show how the roots of a system change with variations in a system parameter like gain.
- The plot determines if the system will become unstable as parameters vary by checking if roots cross to the right half of the complex plane.
- A document section provides examples of how real and complex roots correspond to different system responses and stability.
Let Me Tell You a Story: Finding the Narrative in Your Content MarketingJonathan Crossfield
Storytelling isn't just one of many creative options in the marketing toolkit. It is an essential part of communication. If your content doesn't guide the story, the reader's brain will create its own — and it might not be the story you want.
An exploration of how our brains really work, why story is the grammar of our minds and how you can get your message across more effectively by integrating storytelling techniques across everything you do.
For a talk given for a class on collections at Tufts University, I outline why social media is relevant to museums, best practices for posting, and a handful of excellent examples of cultural institutions engaging their audiences online.
Presentation for asbestos management system - ASKAMSLtd
Asbestos management system. Allows a dutyholder to store, edit, update and retrieve all of their information about the asbestos containing materials in a building.
- O HTML5 trará diversas melhorias na usabilidade e experiência do usuário através de novas tags e funcionalidades para mídia.
- Essas novas funcionalidades como microdados e marcação semântica podem ajudar os mecanismos de busca a entenderem melhor o conteúdo das páginas da web.
- À medida que o HTML5 se torne mais adotado, é possível que motores de busca passem a usar essas informações para melhorar os resultados de busca.
This document contains previews of various email marketing campaigns and event invitations created by Toolbox Design for clients such as Electrolux, Manitowoc Foodservice, and boutique hotels. It displays snippets of email content, images, and links to view full emails on topics like internal campaigns, events, promotions and recruitment. The document acts as a portfolio for Toolbox Design's email design work.
This document is a resume for Rakesh Singh Bhadoriya. It summarizes his professional experience, education, skills, and personal details. He has over 11 years of experience in power plant operation, commissioning, erection, and maintenance. Currently, he works as a Lead Engineer (Mechanical Maintenance) at Maithon Power Limited, where he oversees maintenance activities. He holds a B.E. in Mechanical Engineering and is proficient in Microsoft Office applications and SAP. His strengths include communication skills, project management, and problem solving abilities.
What can possibly get from the 11.11 - 2014ThoughtWorks Live ChinaThoughtworks
The well-known 11.11 has now become a popular atmosphere that drew everyone’s attention. Then what kind of indication that we can possibly get from the data? This topic mainly discusses the major opinion coming from the data reports on that day and introduces the ecosystem of Big Data. It also reveals how Big Data affects and applies in traditional enterprises in 3 main stages.
Additionally, we will use 2 industrial cases to illustrate how Big date has helped enterprises in business revolution such as customers attraction, stores operation and multi-industry cooperation.
3.3 Proyecto eTwinning Alfonso Mezquita 23042016Alfonso Mezquita
Este documento presenta un proyecto escolar sobre las tecnologías de la información y la comunicación (TIC) para estudiantes de 16 años. El proyecto tiene como objetivos explorar los temas de TIC de una manera transversal e interdisciplinaria, trabajando en equipos internacionales y utilizando las TIC. A lo largo de un curso académico, los estudiantes participarán en actividades como analizar los recursos TIC de sus escuelas, crear una página web y blogs para el proyecto, y examinar los conflictos gener
OFA 2015 - Online Resources to Improve Your Marketing & Your LifeSaffire
The document provides an overview of an event hosted by Aaron Pederson and Jeremy Emerson of Saffire Events to share online resources and tools to improve marketing and life. The agenda covers ten topics: social media, communication, productivity, organization, art & design, technology, life hacks, well-being, entertainment, and travel. For each topic, the document lists and briefly describes several apps and websites. It encourages attendees to share their business cards to receive presentation slides and provides contact information for Saffire Events.
Brand success in an era of Digital Darwinism
Companies adept at using digital tools along the consumer decision journey are gaining a sizable lead over competitors.
Building a better content experience is the future of content marketing. Through Uberflip's platform, entire organizations can work together to create exceptional experiences throughout the entire buyer journey. Learn more at http://uberflip.com.
Marc Sleen had geen hoge pet op van prins FilipThierry Debels
Marc Sleen kon het goed vinden met koning Albert II. Maar hoe dacht Sleen over Filip?
Begin 2003 gaat de kroonprins naar Frankrijk. Daar mag hij op het internationaal vermaarde stripfestival van Angoulême, een provinciestadje in de Charente (Frankrijk), de Hergéstraat en het standbeeld van Hergé, de geestelijke vader van Kuifje, onder massale belangstelling plechtig ingehuldigen.
Filip wil maar wat graag de plechtigheden met zijn aanwezigheid opluisteren. Ook Mathilde, Marc Sleen en Hervé Hasquin, de minister-president van de Franse Gemeenschap, zijn mee.
‘Een grote eer,’ reageert een Sleen in Het Belang van Limburg. ‘Maar hoe weet de pers dat al eigenlijk? Ik moest daarover het grootste stilzwijgen bewaren. Ik ben inderdaad nog nooit in Angoulême geweest. Omdat dat altijd samenviel met mijn jaarlijkse safari. En de jongste jaren had ik de moed er niet meer voor. Ik zal dat daar eens bekijken, net zoals het prinsenpaar. Neen, ik verwacht geen hulde daar. Er is al genoeg hulde gebracht, toch?’
De 30ste jubileumuitgave van Angoulême wordt voorgezeten door de Belgische auteur François Schuiten, de laureaat van het jaar daarvoor.
Bij hun aankomst aan de rue Marengo, de belangrijkste winkelstraat van Angoulême die later officieel wordt omgedoopt tot rue Hergé, wordt het prinselijk gezelschap opgewacht door Fanny Rodwell, de weduwe van Hergé, en Freddy Thielemans, de burgemeester van Brussel.
Het prinsenpaar keert, samen met Marc Sleen, diezelfde dag al terug met een legervliegtuig. Marc Sleen aan een bron die actief is in de stripwereld: ‘Ik ben in Angoulême geweest. Met prins Laurent en Mathilde. Toen ze zwanger was. Met hun persoonlijk vliegtuig. Zonder iemand anders. En ik moest naast de koning gaan. Naast de koning. Prins! Hij was nog geen koning. En hij zal geen koning worden ook, denk ik. Enfin. Het was niet prins Laurent. Het was Filip!... Jongens toch, ik ben aan het vangen. Och, indertijd heeft Jan Spier ook een andere vrouw gehad. Dus laat Laurent en Filip ook maar eens met elkaar wisselen. Awel ja, ik moest naast prins Filip gaan en zo. In het vliegtuig zat ik recht over hem. En toen begon hij en vroeg hij mij... Maar dat wordt niet opgenomen, hè?’ (eigen cursivering)
We namen contact op met deze persoon om te weten wat Sleen vertelde over prins Filip. De man wenste niet te reageren. Hij had immers zijn woord gegeven aan Marc Sleen. Erg positief zal het niet geweest zijn aangezien Filip volgens Sleen niet de volgende koning zou worden.
This document provides an agenda for a conference on social networking platforms and applications. The conference will cover topics like the growth of social networks and user engagement, the deployment of social applications and platforms, and strategies for monetization, development, and marketing on social networks. Keynote speakers from Facebook, MySpace, and Forrester Research will provide insights. Sessions will also explore issues like privacy, feeds, and bringing your own platform to social networks.
1) The document discusses analyzing statically indeterminate structures when supports settle or temperatures change, which can induce stresses.
2) The force method is extended to include compatibility equations that account for support displacements and temperature effects.
3) Two examples demonstrate calculating reactions in beams due to support settlements and temperature differences using the modified force method equations.
This document discusses the analysis of statically indeterminate plane frames using the force method. It provides examples of how to: (1) analyze rigid frames by selecting redundant reactions, writing compatibility equations, and solving for member forces; (2) account for support settlements by including the predicted support displacement in the compatibility equation; and (3) calculate joint rotations and displacements to sketch the deformed shape of the frame under loading. The method involves releasing constraints to determine a primary structure, using the unit load method to calculate displacements, and applying the principle of superposition to determine final member forces and deformations.
This document provides an overview of analyzing transient responses in R-L-C circuits with DC excitation. It begins by establishing the objectives of understanding the differential equations that describe such circuits and the different types of responses: overdamped, critically damped, and underdamped. The document then examines the response of a series R-L-C circuit due to a DC voltage source in detail. It derives the second-order differential equation that describes the circuit and shows how to find the natural and complete responses. Based on the characteristics of the differential equation, it classifies the responses as overdamped, critically damped, or underdamped. Finally, it provides an example problem of calculating circuit variables at time t=0
Let Me Tell You a Story: Finding the Narrative in Your Content MarketingJonathan Crossfield
Storytelling isn't just one of many creative options in the marketing toolkit. It is an essential part of communication. If your content doesn't guide the story, the reader's brain will create its own — and it might not be the story you want.
An exploration of how our brains really work, why story is the grammar of our minds and how you can get your message across more effectively by integrating storytelling techniques across everything you do.
For a talk given for a class on collections at Tufts University, I outline why social media is relevant to museums, best practices for posting, and a handful of excellent examples of cultural institutions engaging their audiences online.
Presentation for asbestos management system - ASKAMSLtd
Asbestos management system. Allows a dutyholder to store, edit, update and retrieve all of their information about the asbestos containing materials in a building.
- O HTML5 trará diversas melhorias na usabilidade e experiência do usuário através de novas tags e funcionalidades para mídia.
- Essas novas funcionalidades como microdados e marcação semântica podem ajudar os mecanismos de busca a entenderem melhor o conteúdo das páginas da web.
- À medida que o HTML5 se torne mais adotado, é possível que motores de busca passem a usar essas informações para melhorar os resultados de busca.
This document contains previews of various email marketing campaigns and event invitations created by Toolbox Design for clients such as Electrolux, Manitowoc Foodservice, and boutique hotels. It displays snippets of email content, images, and links to view full emails on topics like internal campaigns, events, promotions and recruitment. The document acts as a portfolio for Toolbox Design's email design work.
This document is a resume for Rakesh Singh Bhadoriya. It summarizes his professional experience, education, skills, and personal details. He has over 11 years of experience in power plant operation, commissioning, erection, and maintenance. Currently, he works as a Lead Engineer (Mechanical Maintenance) at Maithon Power Limited, where he oversees maintenance activities. He holds a B.E. in Mechanical Engineering and is proficient in Microsoft Office applications and SAP. His strengths include communication skills, project management, and problem solving abilities.
What can possibly get from the 11.11 - 2014ThoughtWorks Live ChinaThoughtworks
The well-known 11.11 has now become a popular atmosphere that drew everyone’s attention. Then what kind of indication that we can possibly get from the data? This topic mainly discusses the major opinion coming from the data reports on that day and introduces the ecosystem of Big Data. It also reveals how Big Data affects and applies in traditional enterprises in 3 main stages.
Additionally, we will use 2 industrial cases to illustrate how Big date has helped enterprises in business revolution such as customers attraction, stores operation and multi-industry cooperation.
3.3 Proyecto eTwinning Alfonso Mezquita 23042016Alfonso Mezquita
Este documento presenta un proyecto escolar sobre las tecnologías de la información y la comunicación (TIC) para estudiantes de 16 años. El proyecto tiene como objetivos explorar los temas de TIC de una manera transversal e interdisciplinaria, trabajando en equipos internacionales y utilizando las TIC. A lo largo de un curso académico, los estudiantes participarán en actividades como analizar los recursos TIC de sus escuelas, crear una página web y blogs para el proyecto, y examinar los conflictos gener
OFA 2015 - Online Resources to Improve Your Marketing & Your LifeSaffire
The document provides an overview of an event hosted by Aaron Pederson and Jeremy Emerson of Saffire Events to share online resources and tools to improve marketing and life. The agenda covers ten topics: social media, communication, productivity, organization, art & design, technology, life hacks, well-being, entertainment, and travel. For each topic, the document lists and briefly describes several apps and websites. It encourages attendees to share their business cards to receive presentation slides and provides contact information for Saffire Events.
Brand success in an era of Digital Darwinism
Companies adept at using digital tools along the consumer decision journey are gaining a sizable lead over competitors.
Building a better content experience is the future of content marketing. Through Uberflip's platform, entire organizations can work together to create exceptional experiences throughout the entire buyer journey. Learn more at http://uberflip.com.
Marc Sleen had geen hoge pet op van prins FilipThierry Debels
Marc Sleen kon het goed vinden met koning Albert II. Maar hoe dacht Sleen over Filip?
Begin 2003 gaat de kroonprins naar Frankrijk. Daar mag hij op het internationaal vermaarde stripfestival van Angoulême, een provinciestadje in de Charente (Frankrijk), de Hergéstraat en het standbeeld van Hergé, de geestelijke vader van Kuifje, onder massale belangstelling plechtig ingehuldigen.
Filip wil maar wat graag de plechtigheden met zijn aanwezigheid opluisteren. Ook Mathilde, Marc Sleen en Hervé Hasquin, de minister-president van de Franse Gemeenschap, zijn mee.
‘Een grote eer,’ reageert een Sleen in Het Belang van Limburg. ‘Maar hoe weet de pers dat al eigenlijk? Ik moest daarover het grootste stilzwijgen bewaren. Ik ben inderdaad nog nooit in Angoulême geweest. Omdat dat altijd samenviel met mijn jaarlijkse safari. En de jongste jaren had ik de moed er niet meer voor. Ik zal dat daar eens bekijken, net zoals het prinsenpaar. Neen, ik verwacht geen hulde daar. Er is al genoeg hulde gebracht, toch?’
De 30ste jubileumuitgave van Angoulême wordt voorgezeten door de Belgische auteur François Schuiten, de laureaat van het jaar daarvoor.
Bij hun aankomst aan de rue Marengo, de belangrijkste winkelstraat van Angoulême die later officieel wordt omgedoopt tot rue Hergé, wordt het prinselijk gezelschap opgewacht door Fanny Rodwell, de weduwe van Hergé, en Freddy Thielemans, de burgemeester van Brussel.
Het prinsenpaar keert, samen met Marc Sleen, diezelfde dag al terug met een legervliegtuig. Marc Sleen aan een bron die actief is in de stripwereld: ‘Ik ben in Angoulême geweest. Met prins Laurent en Mathilde. Toen ze zwanger was. Met hun persoonlijk vliegtuig. Zonder iemand anders. En ik moest naast de koning gaan. Naast de koning. Prins! Hij was nog geen koning. En hij zal geen koning worden ook, denk ik. Enfin. Het was niet prins Laurent. Het was Filip!... Jongens toch, ik ben aan het vangen. Och, indertijd heeft Jan Spier ook een andere vrouw gehad. Dus laat Laurent en Filip ook maar eens met elkaar wisselen. Awel ja, ik moest naast prins Filip gaan en zo. In het vliegtuig zat ik recht over hem. En toen begon hij en vroeg hij mij... Maar dat wordt niet opgenomen, hè?’ (eigen cursivering)
We namen contact op met deze persoon om te weten wat Sleen vertelde over prins Filip. De man wenste niet te reageren. Hij had immers zijn woord gegeven aan Marc Sleen. Erg positief zal het niet geweest zijn aangezien Filip volgens Sleen niet de volgende koning zou worden.
This document provides an agenda for a conference on social networking platforms and applications. The conference will cover topics like the growth of social networks and user engagement, the deployment of social applications and platforms, and strategies for monetization, development, and marketing on social networks. Keynote speakers from Facebook, MySpace, and Forrester Research will provide insights. Sessions will also explore issues like privacy, feeds, and bringing your own platform to social networks.
1) The document discusses analyzing statically indeterminate structures when supports settle or temperatures change, which can induce stresses.
2) The force method is extended to include compatibility equations that account for support displacements and temperature effects.
3) Two examples demonstrate calculating reactions in beams due to support settlements and temperature differences using the modified force method equations.
This document discusses the analysis of statically indeterminate plane frames using the force method. It provides examples of how to: (1) analyze rigid frames by selecting redundant reactions, writing compatibility equations, and solving for member forces; (2) account for support settlements by including the predicted support displacement in the compatibility equation; and (3) calculate joint rotations and displacements to sketch the deformed shape of the frame under loading. The method involves releasing constraints to determine a primary structure, using the unit load method to calculate displacements, and applying the principle of superposition to determine final member forces and deformations.
This document provides an overview of analyzing transient responses in R-L-C circuits with DC excitation. It begins by establishing the objectives of understanding the differential equations that describe such circuits and the different types of responses: overdamped, critically damped, and underdamped. The document then examines the response of a series R-L-C circuit due to a DC voltage source in detail. It derives the second-order differential equation that describes the circuit and shows how to find the natural and complete responses. Based on the characteristics of the differential equation, it classifies the responses as overdamped, critically damped, or underdamped. Finally, it provides an example problem of calculating circuit variables at time t=0
- The document discusses transformations between wye (Y) and delta (Δ) connected resistive networks.
- It provides equations to convert between the two configurations by equating the resistances between corresponding terminals.
- Several examples demonstrate applying the transformations to simplify resistive circuits and calculate equivalent resistances and currents.
SCRUTINY TO THE NON-AXIALLY DEFORMATIONS OF AN ELASTIC FOUNDATION ON A CYLIND...P singh
This paper is devoted to homogenization of partial differential operators to use in special structure that is a plate allied to an elastic foundation when it is situated through the basic loads (especially with the harmonic forces) with a Non-axially deformation of the cantilever. Furthermore, it contains the Equations of motion that they can be derived from degenerate Non-linear elliptic ones. Through the mentioned processes, there exists many excess works related to computing the bounded conditions for this special application form of study (when the deformation phenomenon has occurred). At the end of the article whole results of the study on a circular plate are debated and new ways assigned to them are discussed. Afterwards all the processes are formulised with the collection of contracting sequences and expanding sequences integrable functions that are intrinsically joints with the characteristic functions to expanding the behaviour of an elastic foundation. Thenceforth all the resultant functions are sets and compared with the other ones (without the loads). Sample pictures and analysis of the study were employed with the ANSYS software to obtain the better observations and conclusions.
Question-no.docx
Chapter7
Question no’s: 2,3,4,5,6,8,10,13,14,15,17,18,19,20,21,27,28,29,31,32,33,36
Chapter 8
Question no’s: 1,2,3,4,6,7,9,13,14,15,19,20,21,22,24,26,28,29,30
ch7.pdf
Chapter 7 Laplace’s Equation: The Potential Produced by Surface Charge
7.13 Problems
7.1 Finding Charge From Potential
The potential in a spherical region r < R is '(x, y, z) = '0(z/R)
3. Find a volume charge density
Ω(r, µ) in the region r < R and a surface charge density æ(µ) on the surface r = R which together
produce this potential. Express your answers in terms of elementary trigonometric functions.
7.2 A Periodic Array of Charged Rings
Let the z-axis be the symmetry axis for an infinite number of identical rings, each with charge
Q and radius R. There is one ring in each of the planes z = 0, z = ±b, z = ±2b, etc. Exploit
the Fourier expansion in Example 1.6 to find the potential everywhere in space. Check that your
solution makes sense in the limit that the cylindrical variable Ω ¿ R, b. Hint: If IÆ(y) and KÆ are
modified Bessel functions,
I
0
Æ(y)KÆ(y) ° IÆ(y)K0Æ(y) = 1/y.
7.3 Two Electrostatic Theorems
Use the orthogonality properties of the spherical harmonics to prove the following for a function
'(r) which satisfies Laplace’s equation in and on an origin-centered spherical surface S of radius
R:
(a)
R
S
dS '(r) = 4ºR2'(0)
(b)
Z
S
dSz'(r) =
4º
3
R
4 @'
@z
ØØØØ
r=0
7.4 Make a Field Inside a Sphere
Find the volume charge density Ω and surface charge density æ which much be placed in and on a
sphere of radius R to produce a field inside the sphere of
E = °2V0
xy
R3
x̂ +
V0
R3
(y
2 ° x2)ŷ ° V0
R
ẑ.
There is no other charge anywhere. Express your answer in terms of trigonometric functions of µ
and ¡.
7.5 Green’s Formula
Let n̂ be the normal to an equipotential surface at a point P . If R1 and R2 are the principal
radii of curvature of the surface at P . A formula due to George Green relates normal derivatives
(@/@n ¥ n̂ · r) of the potential '(r) (which satisfies Laplace’s equation) at the equipotential surface
to the mean curvature of that equipotential surface ∑ = 1
2
(R°11 + R
°1
2 ):
@2'
@n2
+ 2∑
@'
@n
= 0.
Derive Green’s equation by direct manipulation of Laplace’s equation.
7.6 The Channeltron
c∞2009 Andrew Zangwill 278
Chapter 7 Laplace’s Equation: The Potential Produced by Surface Charge
The parallel plates of a channeltron are segmented into conducting strips of width b so the po-
tential can be fixed on the strips at staggered values. We model this using infinite-area plates, a
finite portion of which is shown below. Find the potential '(x, y) between the plates and sketch
representative field lines and equipotentials. Note the orientation of the x and y axes.
1 1
02 2
x
y
d
b
7.7 The Calculable Capacitor
The figure below shows a circle which has been divided into two pairs of segments with equal
arc length by a horizontal bisector and a vertical line. The positive x-axis bisects the segment
labe ...
This document discusses matrix algebra concepts such as determinants, inverses, eigenvalues, and rank. It provides the following key points:
- The determinant of a square matrix is a number that characterizes properties like singularity. It is defined as the sum of products of the matrix elements.
- Cramer's rule provides a formula for solving systems of linear equations using determinants, but it is only practical for small matrices up to 3x3 or 4x4 due to computational complexity.
- A matrix is singular if its determinant is zero, meaning its rows and columns are linearly dependent. The rank of a matrix is the size of the largest non-singular sub-matrix. A full rank matrix has
This document discusses matrix algebra concepts such as determinants, inverses, eigenvalues, and rank. It provides the following key points:
- The determinant of a square matrix is a number that characterizes properties like singularity. It is defined as the sum of products of the matrix elements.
- Cramer's rule provides a formula for solving systems of linear equations using determinants, but it is only practical for small matrices up to 3x3 or 4x4 due to computational complexity.
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M2l8
1. Module
2
Analysis of Statically
Indeterminate
Structures by the
Matrix Force Method
Version 2 CE IIT, Kharagpur
2. Lesson
8
The Force Method of
Analysis: Beams
Version 2 CE IIT, Kharagpur
3. Instructional Objectives
After reading this chapter the student will be able to
1. Solve statically indeterminate beams of degree more than one.
2. To solve the problem in matrix notation.
3. To compute reactions at all the supports.
4. To compute internal resisting bending moment at any section of the
continuous beam.
8.1 Introduction
In the last lesson, a general introduction to the force method of analysis is given.
Only, beams, which are statically indeterminate to first degree, were considered.
If the structure is statically indeterminate to a degree more than one, then the
approach presented in the previous example needs to be organized properly. In
the present lesson, a general procedure for analyzing statically indeterminate
beams is discussed.
8.2 Formalization of Procedure
Towards this end, consider a two-span continuous beam as shown in Fig. 8.1a.
The flexural rigidity of this continuous beam is assumed to be constant and is
taken as EI . Since, the beam is statically indeterminate to second degree, it is
required to identify two redundant reaction components, which need be released
to make the beam statically determinate.
Version 2 CE IIT, Kharagpur
5. The redundant reactions at A and B are denoted by R1 and R2 respectively.
The released structure (statically determinate structure) with applied loading is
shown in Fig. 8.1b. The deflection of primary structure at B and C due to
applied loading is denoted by (Δ L )1 and (Δ L )2 respectively. Throughout this
module (Δ L )i notation is used to denote deflection at i th redundant due to
applied loads on the determinate structure.
4
7 PL3
(Δ L )1 = − wL − (8.1a)
8 EI 12 EI
4
27 PL3
(Δ L )2 = − 7 wL − (8.1b)
24 EI 16 EI
In fact, the subscript 1 and 2 represent, locations of redundant reactions
released. In the present case R A (= R1 ) and RB (= R2 ) respectively. In the present
and subsequent lessons of this module, the deflections and the reactions are
taken to be positive in the upward direction. However, it should be kept in mind
that the positive sense of the redundant can be chosen arbitrarily. The deflection
of the point of application of the redundant should likewise be considered positive
when acting in the same sense.
For writing compatibility equations at B and C , it is required to know deflection of
the released structure at B and C due to external loading and due to redundants.
The deflection at B and C due to external loading can be computed easily. Since
redundants R1 and R2 are not known, in the first step apply a unit load in the
direction of R1 and compute deflection, a11 at B , and deflection, a 21 at C , as
shown in Fig.8.1c. Now deflections at B and C of the given released structure
due to redundant R1 are,
(Δ R )11 = a11 R1 (8.2a)
(Δ R )21 = a 21 R1 (8.2b)
In the second step, apply unit load in the direction of redundant R2 and compute
deflection at B (point 1), a12 and deflection at C , a 22 as shown in Fig 8.1d. It may
be recalled that the flexibility coefficient a ij is the deflection at i due to unit value
of force applied at j . Now deflections of the primary structure (released
structure) at B and C due to redundant R2 is
(Δ R )12 = a12 R2 (8.3a)
(Δ R )22 = a 22 R2 (8.3b)
Version 2 CE IIT, Kharagpur
6. It is observed that, in the actual structure, the deflections at joints B and C is
zero. Now the total deflections at B and C of the primary structure due to applied
external loading and redundants R1 and R2 is,
Δ 1 = (Δ L )1 + a11 R1 + a12 R2 (8.4a)
Δ 2 = (Δ L )2 + a 21 R1 + a 22 R2 (8.4b)
The equation (8.4a) represents the total displacement at B and is obtained by
superposition of three terms:
1) Deflection at B due to actual load acting on the statically determinate
structure,
2) Displacement at B due to the redundant reaction R1 acting in the positive
direction at B (point 1) and
3) Displacement at B due to the redundant reaction R2 acting in the positive
direction at C .
The second equation (8.4b) similarly represents the total deflection at C . From
the physics of the problem, the compatibility condition can be written as,
Δ 1 = (Δ L )1 + a11 R1 + a12 R2 = 0 (8.5a)
Δ 2 = (Δ L )2 + a 21 R1 + a 22 R2 = 0 (8.5b)
The equation (8.5a) and (8.5b) may be written in matrix notation as follows,
⎧ (Δ L )1 ⎫ ⎡ a11 a12 ⎤ ⎧ R1 ⎫ ⎧0⎫
⎨ ⎬+ ⎨ ⎬=⎨ ⎬
(Δ L )2 ⎭ ⎢a 21 a 22 ⎥ ⎩ R2 ⎭ ⎩0⎭
(8.6a)
⎩ ⎣ ⎦
{(Δ L )1 } + [A]{R} = {0} (8.6b)
In which,
⎧( Δ ) ⎫ ⎡a a12 ⎤ ⎧ R1 ⎫
{( Δ ) } = ⎪( Δ ) ⎪ ; [ A] = ⎢a
L 1 ⎨
L 1
⎬
11
⎥ and {R} = ⎨ R ⎬
⎪
⎩ L 2 ⎪
⎭ ⎣ 21 a22 ⎦ ⎩ 2⎭
Solving the above set of algebraic equations, one could obtain the values of
redundants, R1 and R2 .
{R} = −[A]−1 {Δ L } (8.7)
Version 2 CE IIT, Kharagpur
7. In the above equation the vectors {Δ L } contains the displacement values of the
primary structure at point 1 and 2, [A] is the flexibility matrix and {R} is column
vector of redundants required to be evaluated. In equation (8.7) the inverse of the
flexibility matrix is denoted by [A] . In the above example, the structure is
−1
indeterminate to second degree and the size of flexibility matrix is 2 × 2 . In
general, if the structure is redundant to a degree n , then the flexibility matrix is of
the order n × n . To demonstrate the procedure to evaluate deflection, consider
the problem given in Fig. 8.1a, with loading as given below
w=w; P = wL (8.8a)
Now, the deflection (Δ L )1 and (Δ L )2 of the released structure can be evaluated
from the equations (8.1a) and (8.1b) respectively. Then,
4
7 wL4 17 wL4
(Δ L )1 = − wL − =− (8.8b)
8 EI 12 EI 24 EI
7 wL4 27 wL4 95wL4
(Δ L )2 =− − =− (8.8c)
24 EI 16 EI 48EI
The negative sign indicates that both deflections are downwards. Hence the
vector {Δ L } is given by
wL4 ⎧34⎫
{Δ L } = − ⎨ ⎬ (8.8d)
48 EI ⎩95⎭
The flexibility matrix is determined from referring to figures 8.1c and 8.1d. Thus,
when the unit load corresponding to R1 is acting at B , the deflections are,
L3 5 L3
a11 = , a 21 = (8.8e)
3EI 6 EI
Similarly when the unit load is acting at C ,
5L3 8L3
a12 = , a 22 = (8.8f)
6 EI 3EI
The flexibility matrix can be written as,
L3 ⎡2 5 ⎤
[A] = ⎢5 16⎥ (8.8g)
6 EI ⎣ ⎦
Version 2 CE IIT, Kharagpur
8. The inverse of the flexibility matrix can be evaluated by any of the standard
method. Thus,
16 − 5⎤
[A]−1 = 6 EI ⎡
3 ⎢ ⎥ (8.8h)
7 L ⎣− 5 2 ⎦
Now using equation (8.7) the redundants are evaluated. Thus,
⎧ R1 ⎫ 6 EI wL4 ⎡ 16 − 5⎤ ⎧34⎫
⎨ ⎬= 3 × ⎢− 5 2 ⎥ ⎨95⎬
⎩ R2 ⎭ 7 L 48 EI ⎣ ⎦⎩ ⎭
69 20
Hence, R1 = wL and R2 = wL (8.8i)
56 56
Once the redundants are evaluated, the other reaction components can be
evaluated by static equations of equilibrium.
Example 8.1
Calculate the support reactions in the continuous beam ABC due to loading as
shown in Fig. 8.2a. Assume EI to be constant throughout.
Version 2 CE IIT, Kharagpur
9. Select two reactions viz, at B(R1 ) and C (R2 ) as redundants, since the given beam
is statically indeterminate to second degree. In this case the primary structure is
a cantilever beam AC . The primary structure with a given loading is shown in Fig.
8.2b.
In the present case, the deflections (Δ L )1 , and (Δ L )2 of the released structure at
B and C can be readily calculated by moment-area method. Thus,
(Δ L )1 = − 819.16
EI
and (Δ L )2 =−
2311.875
(1)
EI
For the present problem the flexibility matrix is,
125 625
a11 = a 21 =
3EI 6 EI
625 1000
a12 = a 22 = (2)
6 EI 3EI
In the actual problem the displacements at B and C are zero. Thus the
compatibility conditions for the problem may be written as,
a11 R1 + a12 R2 + (Δ L )1 = 0
(3)
a 21 R1 + a 22 R2 + (Δ L )2 = 0
⎧ R1 ⎫ 3EI ⎡ 1000 − 312.5⎤ 1 ⎧ 819.16 ⎫
⎨ ⎬= ⎢ × ⎨ ⎬
125 ⎥ EI
(5)
⎩ R2 ⎭ 27343.75 ⎣− 312.5 ⎦ ⎩2311.875⎭
Substituting the value of E and I in the above equation,
R1 = 10.609kN and R2 = 3.620 kN
Using equations of static equilibrium,
R3 = 0.771 kN and R4 = −0.755 kN.m
Version 2 CE IIT, Kharagpur
10. Example 8.2
A clamped beam AB of constant flexural rigidity is shown in Fig. 8.3a. The beam
is subjected to a uniform distributed load of w kN/m and a central concentrated
moment M = wL2 kN.m . Draw shear force and bending moment diagrams by
force method.
Select vertical reaction (R1 ) and the support moment (R2 ) at B as the
redundants. The primary structure in this case is a cantilever beam which could
be obtained by releasing the redundants R1 and R2 . The R1 is assumed to be
positive in the upward direction and R2 is assumed to be positive in the
counterclockwise direction. Now, calculate deflection at B due to only applied
loading. Let (Δ L )1 be the transverse deflection at B and (Δ L )2 be the slope at B
due to external loading. The positive directions of the selected redundants are
shown in Fig. 8.3b.
Version 2 CE IIT, Kharagpur
12. The deflection (Δ L )1 and (Δ L )2 of the released structure can be evaluated from
unit load method. Thus,
4
3wL4 wL4
(Δ L )1 = − wL − =− (1)
8EI 8EI 2 EI
3
wL3 2 wL3
and (Δ L )2 = − wL − =− (2)
6 EI 2 EI 3EI
The negative sign indicates that (Δ L )1 is downwards and rotation (Δ L )2 is
clockwise. Hence the vector {Δ L } is given by
wL3 ⎧3L ⎫
{Δ L } = − ⎨ ⎬ (3)
6 EI ⎩4⎭
The flexibility matrix is evaluated by first applying unit load along redundant R1
and determining the deflections a11 and a 21 corresponding to redundants R1 and
R2 respectively (see Fig. 8.3d). Thus,
L3 L2
a11 = and a 21 = (4)
3EI 2 EI
Similarly, applying unit load in the direction of redundant R2 , one could evaluate
flexibility coefficients a12 and a 22 as shown in Fig. 8.3c.
L2 L
a12 = and a 22 = (5)
2 EI EI
Now the flexibility matrix is formulated as,
L ⎡2 L2 3L ⎤
[A] = ⎢ ⎥ (6)
6 EI ⎣ 3L 6⎦
The inverse of flexibility matrix is formulated as,
Version 2 CE IIT, Kharagpur
13. − 3L ⎤
[A]−1 = 6 EI ⎡
6
3 ⎢ ⎥
3L ⎣− 3L 2 L2 ⎦
The redundants are evaluated from equation (8.7). Hence,
⎧ R1 ⎫ 6 EI ⎡ 6 − 3L ⎤ ⎛ wL3 ⎞⎧3L ⎫
⎨ ⎬ =− 3 ⎢ ⎥ ×⎜− ⎟⎨ ⎬
⎩ R2 ⎭ 3L ⎣− 3L 2 L2 ⎦ ⎜ 6 EI ⎟⎩ 4 ⎭
⎝ ⎠
w ⎧ 6L ⎫
= ⎨ ⎬
3 ⎩− L2 ⎭
wL2
R1 = 2wL and R2 = − (7)
3
The other two reactions ( R3 and R4 ) can be evaluated by equations of statics.
Thus,
wL2
R4 = M A = − and R1 = R A = − wL (8)
6
The bending moment and shear force diagrams are shown in Fig. 8.3g and
Fig.8.3h respectively.
Summary
In this lesson, statically indeterminate beams of degree more than one is solved
systematically using flexibility matrix method. Towards this end matrix notation is
adopted. Few illustrative examples are solved to illustrate the procedure. After
analyzing the continuous beam, reactions are calculated and bending moment
diagrams are drawn.
Version 2 CE IIT, Kharagpur