This presentation is part of the Solid Mechanics course taught in the second year of the Degree in Architecture at San Pablo CEU University Institute of Technology, in Madrid (Spain).
Three step by step examples of how to draw bending moment, shear and normal force diagrams in 2D, statically determinate structures, are depicted.
Feel free to comment!
Solid Mechanics. Reactions and internal forces. Step by step example . Part 2Maribel Castilla Heredia
This presentation is part of the Solid Mechanics course taught in the second year of the Degree in Architecture at San Pablo CEU University Institute of Technology, in Madrid (Spain).
This step by step example covers how to obtain reactions at supports and internal joints in 2D, statically determinate, multibody structural systems.
The second part uses the same example to learn how to draw axial (normal), shear and bending moment diagrams.
Feel free to comment!
Solid Mechanics. Reactions and internal force diagrams. Step by step example....Maribel Castilla Heredia
This presentation is part of the Solid Mechanics course taught in the second year of the Degree in Architecture at San Pablo CEU University Institute of Technology, in Madrid (Spain).
This step by step example covers how to obtain reactions at supports and internal joints in 2D, statically determinate, multibody structural systems.
The second part uses the same example to learn how to draw axial (normal), shear and bending moment diagrams.
Feel free to comment!
System shear connector digunakan sebagai aplikasi dalam konstruksi bangunan untuk menghasilkan kekuatan coran beton lebih kuat dan stabil sesuai dengan perhitungan engineering civil. Dalam hal ini ada 2 hal perhitungan kekuatan secara umum yaitu kekuatan kelengketan stud pada batang baja sesudah dilas. Dan yang kedua adalah kekuatan stud bolt yang digunakan.
Solid Mechanics. Reactions and internal forces. Step by step example . Part 2Maribel Castilla Heredia
This presentation is part of the Solid Mechanics course taught in the second year of the Degree in Architecture at San Pablo CEU University Institute of Technology, in Madrid (Spain).
This step by step example covers how to obtain reactions at supports and internal joints in 2D, statically determinate, multibody structural systems.
The second part uses the same example to learn how to draw axial (normal), shear and bending moment diagrams.
Feel free to comment!
Solid Mechanics. Reactions and internal force diagrams. Step by step example....Maribel Castilla Heredia
This presentation is part of the Solid Mechanics course taught in the second year of the Degree in Architecture at San Pablo CEU University Institute of Technology, in Madrid (Spain).
This step by step example covers how to obtain reactions at supports and internal joints in 2D, statically determinate, multibody structural systems.
The second part uses the same example to learn how to draw axial (normal), shear and bending moment diagrams.
Feel free to comment!
System shear connector digunakan sebagai aplikasi dalam konstruksi bangunan untuk menghasilkan kekuatan coran beton lebih kuat dan stabil sesuai dengan perhitungan engineering civil. Dalam hal ini ada 2 hal perhitungan kekuatan secara umum yaitu kekuatan kelengketan stud pada batang baja sesudah dilas. Dan yang kedua adalah kekuatan stud bolt yang digunakan.
Il presente volume affronta l’impostazione della progettazione e delle verifiche prestazionali e di
sicurezza per le costruzioni in acciaio secondo le Nuove Norme Tecniche e gli Eurocodici strutturali.
A una prima lettura, le normative europee e ora la normativa nazionale potrebbero sembrare piuttosto
complesse e a volte poco intuitive ma una volta fatti propri i concetti di base e chiarite le procedure di
calcolo, ci si rende conto che il loro utilizzo è meno difficile di quello che può sembrare.
Il presente testo propone un approccio elementare ma innovativo adatto a superare le difficoltà legate
a un primo utilizzo delle normative. Tale impostazione è stata concretizzata in una serie di diagrammi
di flusso che sintetizzano in forma ordinata le procedure di calcolo delle azioni sulle costruzioni e le
verifiche degli elementi strutturali in acciaio.
Per familiarizzare con le normative è inoltre importante svolgere dei calcoli a mano. A questo
proposito, nella parte applicativa del volume sono riportati nel dettaglio i calcoli relativi al
dimensionamento di un edificio multipiano in acciaio. Si fa comunque notare che in questa sede gli
argomenti sono presentati in forma elementare e richiedono studi e approfondimenti successivi.
I contenuti del presente testo sono destinati sia a studenti delle facoltà di Ingegneria e Architettura sia
ai tecnici professionisti che vogliano aggiornare le proprie competenze.
Mumbai University.
Mechanical Engineering
SEM III
Material Technology
Module 2.2
Fatigue Failure:
Definition of fatigue and significance of cyclic stress, Mechanism of fatigue and theories of fatigue failure, Fatigue testing, Test data presentation and statistical evolution, S-N Curve and its interpretation, Influence of important factors on fatigue, Notch effect, surface effect, Effect of pre-stressing, corrosion fatigue, Thermal fatigue.
The study of crystal geometry helps to understand the behaviour of solids and their
mechanical,
electrical,
magnetic
optical and
Metallurgical properties
Programma del Corso di Tecnica delle Costruzioni
A.A. 2021/22 - Prof. Ing. Franco Bontempi
Facoltà di Ingegneria Civile e Industriale
Università degli Studi di Roma la Sapienza
This document shows a valid approach to conduct the sizing and optimization of the members of a spaghetti truss. This competition is held in San Pablo CEU University's Institute of Technology in Madrid, Spain, during the second year in the Degree in Architecture.
Il presente volume affronta l’impostazione della progettazione e delle verifiche prestazionali e di
sicurezza per le costruzioni in acciaio secondo le Nuove Norme Tecniche e gli Eurocodici strutturali.
A una prima lettura, le normative europee e ora la normativa nazionale potrebbero sembrare piuttosto
complesse e a volte poco intuitive ma una volta fatti propri i concetti di base e chiarite le procedure di
calcolo, ci si rende conto che il loro utilizzo è meno difficile di quello che può sembrare.
Il presente testo propone un approccio elementare ma innovativo adatto a superare le difficoltà legate
a un primo utilizzo delle normative. Tale impostazione è stata concretizzata in una serie di diagrammi
di flusso che sintetizzano in forma ordinata le procedure di calcolo delle azioni sulle costruzioni e le
verifiche degli elementi strutturali in acciaio.
Per familiarizzare con le normative è inoltre importante svolgere dei calcoli a mano. A questo
proposito, nella parte applicativa del volume sono riportati nel dettaglio i calcoli relativi al
dimensionamento di un edificio multipiano in acciaio. Si fa comunque notare che in questa sede gli
argomenti sono presentati in forma elementare e richiedono studi e approfondimenti successivi.
I contenuti del presente testo sono destinati sia a studenti delle facoltà di Ingegneria e Architettura sia
ai tecnici professionisti che vogliano aggiornare le proprie competenze.
Mumbai University.
Mechanical Engineering
SEM III
Material Technology
Module 2.2
Fatigue Failure:
Definition of fatigue and significance of cyclic stress, Mechanism of fatigue and theories of fatigue failure, Fatigue testing, Test data presentation and statistical evolution, S-N Curve and its interpretation, Influence of important factors on fatigue, Notch effect, surface effect, Effect of pre-stressing, corrosion fatigue, Thermal fatigue.
The study of crystal geometry helps to understand the behaviour of solids and their
mechanical,
electrical,
magnetic
optical and
Metallurgical properties
Programma del Corso di Tecnica delle Costruzioni
A.A. 2021/22 - Prof. Ing. Franco Bontempi
Facoltà di Ingegneria Civile e Industriale
Università degli Studi di Roma la Sapienza
This document shows a valid approach to conduct the sizing and optimization of the members of a spaghetti truss. This competition is held in San Pablo CEU University's Institute of Technology in Madrid, Spain, during the second year in the Degree in Architecture.
Mecánica de Sólidos. Bloque A. Ejercicio paso a paso 1. Primera parte: Reacci...Maribel Castilla Heredia
Ejercicio resuelto paso a paso para la obtención de reacciones en apoyos externos de un sistema estructural reticulado, plano e isostático formado por varios subsistemas. Asignatura: Mecánica de Sólidos. Curso 2012/13. Grado en Arquitectura. Escuela Politécnica Superior. Universidad San Pablo CEU. Madrid.
http://www.eps.uspceu.es
Autor: Maribel Castilla Heredia. @maribelcastilla http://about.me/maribelcastilla
Presentación sobre un ejercicio realizado paso a paso de obtención de tensiones normales debidas a la acción de una carga aplicada excéntricamente en una sección rectangular. Empleada en el grupo 5 de Mecánica de Sólidos del Grado en Arquitectura de la Universidad San Pablo CEU de Madrid. Profesor: Maribel Castilla Heredia.
EKIT (B) Sdn Bhd is able to provide one stop solution for hydro-testing jobs for the Oil & Gas industry.
Unique Hydrostatic Pressure Testing.
www.ekitbrunei.com
Material de la asignatura Sistemas Estructurales del Grado en Arquitectura de la Universidad CEU San Pablo de Madrid.
---
Este documento muestra cómo dimensionar y optimizar las celosías de espagueti fabricadas por nuestros alumnos con motivo del concurso anual que se celebra en la asignatura. El concurso se retransmite cada año en directo por streaming.
Sistemas estructurales - Obtención de deformaciones en estructuras reticulada...Maribel Castilla Heredia
Ejercicio paso a paso para mostrar la aplicación del principio de los trabajos virtuales en el método de la carga unitaria para obtener deformaciones (desplazamientos y giros) en estructuras isostáticas planas.
Este método se explica en la asignatura de Sistemas Estructurales del Grado en Arquitectura de la Escuela Politécnica Superior de la Universidad CEU San Pablo de Madrid. www.eps.uspceu.es
Dimensionado de Estructuras de Edificación - Arriostramientos en naves indust...Maribel Castilla Heredia
Esta presentación constituye parte del material de la asignatura de Dimensionado de Estructuras de Edificación del Grado en Arquitectura de la Escuela Politécnica Superior de la Universidad CEU San Pablo de Madrid.
Colección de videotutoriales para aprender a manejar CypeCad y Nuevo Metal 3D a partir del enunciado de una práctica de la asignatura de Dimensionado de Estructuras de 4º curso del Grado en Arquitectura de la Escuela Politécnica Superior de la Universidad CEU San Pablo de Madrid
Ejercicio de obtención de tensiones paso a paso - Sistemas Estructurales - Gr...Maribel Castilla Heredia
Ejercicio paso a paso sobre la obtención de tensiones en un pórtico plano isostático empleado en el grupo 5 de Mecánica de Sólidos del Grado en Arquitectura de la Universidad San Pablo CEU de Madrid. Curso 12/13. Profesor: Maribel Castilla Heredia.
Mecánica de Sólidos. Bloque A. Ejercicio paso a paso 1. Segunda parte: Esfuer...Maribel Castilla Heredia
Ejercicio resuelto paso a paso para la obtención de diagramas de esfuerzos de un sistema estructural reticulado, plano e isostático formado por varios subsistemas. Asignatura: Mecánica de Sólidos. Curso 2012/13. Grado en Arquitectura. Escuela Politécnica Superior. Universidad San Pablo CEU. Madrid.
http://www.eps.uspceu.es
Autor: Maribel Castilla Heredia. @maribelcastilla http://about.me/maribelcastilla
Presentación empleada durante las sesiones de teoría de la clase de Mecánica de sólidos del grupo 3 del curso 13/14 en el Grado en Arquitectura de la Universidad CEU San Pablo de Madrid. La clase versa sobre estructuras trianguladas articuladas, isostáticas y planas.
Bloque A - Teoría de representación de diagramas de esfuerzos - Mecánica de s...Maribel Castilla Heredia
Teoría de obtención de esfuerzos en estructuras planas reticuladas isostáticas.
Asignatura: Mecánica de Sólidos. Curso 2012/13. Grado en Arquitectura. Escuela Politécnica Superior. Universidad San Pablo CEU. Madrid.
http://www.eps.uspceu.es
Autor: Maribel Castilla Heredia. @maribelcastilla http://about.me/maribelcastilla
In physics, a force is any interaction which tends to change the motion of an object.
In other words, a force can cause an object with mass to change its velocity (which includes to begin moving from a state of rest), i.e., to accelerate.
Force can also be described by intuitive concepts such as a push or a pull.
A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.
The original form of Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time.
If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object.
As a formula, this is expressed as:
Related concepts to force include: thrust, which increases the velocity of an object; drag, which decreases the velocity of an object; and torque which produces changes in rotational speed of an object. In an extended body, each part usually applies forces on the adjacent parts; the distribution of such forces through the body is the so-called mechanical stress.
Pressure is a simple type of stress. Stress usually causes deformation of solid materials, or flow in fluids.
Aristotle famously described a force
The Indian Dental Academy is the Leader in continuing dental education , training dentists in all aspects of dentistry and
offering a wide range of dental certified courses in different formats.for more details please visit
www.indiandentalacademy.com
Dynamic force analysis – Inertia force and Inertia torque– D Alembert’s principle –Dynamic Analysis in reciprocating engines – Gas forces – Inertia effect of connecting rod– Bearing loads – Crank shaft torque
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
1. Solid Mechanics Academic Year 2014/2015
Solid Mechanics Block A. Internal forces diagrams.
Prof. Maribel Castilla Heredia @maribelcastilla Version 1.0 October 2014
Degree in Architecture. San Pablo CEU University – Institute of Technology.
2. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces
As seen during the classes, we’ll think of our structural members as if they consisted of thin slices along the axis of the bar. When external forces are applied to our structural system, the slices are subjected to internal forces.
These internal forces exerted on the slices appear in couples, as each of the slices must be in equilibrium itself.
For a plane structural member, there are three possible kinds of internal forces:
3. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Axial (or Normal) force It makes both sides of the slice separate one from another, keeping both sides parallel. Letter N represents axial force.
Internal forces
As seen during the classes, we’ll think of our structural members as if they consisted of thin slices along the axis of the bar. When external forces are applied to our structural system, the slices are subjected to internal forces.
These internal forces exerted on the slices appear in couples, as each of the slices must be in equilibrium itself.
For a plane structural member, there are three possible kinds of internal forces:
4. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces
As seen during the classes, we’ll think of our structural members as if they consisted of thin slices along the axis of the bar. When external forces are applied to our structural system, the slices are subjected to internal forces.
These internal forces exerted on the slices appear in couples, as each of the slices must be in equilibrium itself.
For a plane structural member, there are three possible kinds of internal forces:
Shear force It makes both sides of the slice slide one from another perpendicularly to the axis of the bar. Both sides keep parallel. Letter V represents shear.
Axial (or Normal) force It makes both sides of the slice separate one from another, keeping both sides parallel. Letter N represents axial force.
5. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces
As seen during the classes, we’ll think of our structural members as if they consisted of thin slices along the axis of the bar. When external forces are applied to our structural system, the slices are subjected to internal forces.
These internal forces exerted on the slices appear in couples, as each of the slices must be in equilibrium itself.
For a plane structural member, there are three possible kinds of internal forces:
Shear force It makes both sides of the slice slide one from another perpendicularly to the axis of the bar. Both sides keep parallel. Letter V represents shear.
Axial (or Normal) force It makes both sides of the slice separate one from another, keeping both sides parallel. Letter N represents axial force.
Bending moment It makes both sides of the slice rotate, so they stop being parallel. As for the upper and lower sides of the slice, one shortens and the other lengthens. Letter M represents bending moment.
6. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces
As seen during the classes, we’ll think of our structural members as if they consisted of thin slices along the axis of the bar. When external forces are applied to our structural system, the slices are subjected to internal forces.
These internal forces exerted on the slices appear in couples, as each of the slices must be in equilibrium itself.
For a plane structural member, there are three possible kinds of internal forces:
Shear force It makes both sides of the slice slide one from another perpendicularly to the axis of the bar. Both sides keep parallel. Letter V represents shear.
Axial (or Normal) force It makes both sides of the slice separate one from another, keeping both sides parallel. Letter N represents axial force.
Bending moment It makes both sides of the slice rotate, so they stop being parallel. As for the upper and lower sides of the slice, one shortens and the other lengthens. Letter M represents bending moment.
Check this out! If these slices are equilibrated, both acting forces have to be equal in magnitude but opposed in sense.
7. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces. Sign criteria.
Internal forces are always defined by couples of forces (or moments) that act on a slice that belongs to a structural member. Thus, the sign can’t be stablished just buy looking at the sense of one of these forces, but at the global effect that those two forces produce on the slice.
Sign criteria are arbitrary. Before analyzing any structural system we must be aware of the sign criteria we are expected to use. Likewise, if I want to read the results of the analysis of a structural system that someone (other than me) has calculated, I must know first which sign criteria this person has employed.
The following criteria is the one that we use throughout the course, but it doesn’t have to be the same as the one used in other courses, textbooks, etc.
This examples are valid for members with horizontal axis (horizontal bars).
8. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces. Sign criteria.
Internal forces are always defined by couples of forces (or moments) that act on a slice that belongs to a structural member. Thus, the sign can’t be stablished just buy looking at the sense of one of these forces, but at the global effect that those two forces produce on the slice.
Sign criteria are arbitrary. Before analyzing any structural system we must be aware of the sign criteria we are expected to use. Likewise, if I want to read the results of the analysis of a structural system that someone (other than me) has calculated, I must know first which sign criteria this person has employed.
The following criteria is the one that we use throughout the course, but it doesn’t have to be the same as the one used in other courses, textbooks, etc.
This examples are valid for members with horizontal axis (horizontal bars).
Axial force If both sides of the slice tend to separate one from another, the sign will be positive. If they tend to get closer, the sign will be negative. When axial force is positive we say that the member is subjected to tension. If negative, we say the member is compressed.
9. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces. Sign criteria.
Internal forces are always defined by couples of forces (or moments) that act on a slice that belongs to a structural member. Thus, the sign can’t be stablished just buy looking at the sense of one of these forces, but at the global effect that those two forces produce on the slice.
Sign criteria are arbitrary. Before analyzing any structural system we must be aware of the sign criteria we are expected to use. Likewise, if I want to read the results of the analysis of a structural system that someone (other than me) has calculated, I must know first which sign criteria this person has employed.
The following criteria is the one that we use throughout the course, but it doesn’t have to be the same as the one used in other courses, textbooks, etc.
This examples are valid for members with horizontal axis (horizontal bars).
Axial force If both sides of the slice tend to separate one from another, the sign will be positive. If they tend to get closer, the sign will be negative. When axial force is positive we say that the member is subjected to tension. If negative, we say the member is compressed.
Shear force We will say shear force is positive if the rotation that both forces would produce on the slice is a counterclockwise one and negative if the rotation were a clockwise one.
10. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces. Sign criteria.
Internal forces are always defined by couples of forces (or moments) that act on a slice that belongs to a structural member. Thus, the sign can’t be stablished just buy looking at the sense of one of these forces, but at the global effect that those two forces produce on the slice.
Sign criteria are arbitrary. Before analyzing any structural system we must be aware of the sign criteria we are expected to use. Likewise, if I want to read the results of the analysis of a structural system that someone (other than me) has calculated, I must know first which sign criteria this person has employed.
The following criteria is the one that we use throughout the course, but it doesn’t have to be the same as the one used in other courses, textbooks, etc.
This examples are valid for members with horizontal axis (horizontal bars).
Axial force If both sides of the slice tend to separate one from another, the sign will be positive. If they tend to get closer, the sign will be negative. When axial force is positive we say that the member is subjected to tension. If negative, we say the member is compressed.
Shear force We will say shear force is positive if the rotation that both forces would produce on the slice is a counterclockwise one and negative if the rotation were a clockwise one.
Bending moment It will be considered positive if the lower side of the slice tends to lengthen. This implies that the upper side is compressed and the lower side is subjected to tension.. Values in bending moment diagrams will always be drawn beside the tensioned side of the member.
11. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Criterio de signos a emplear en esfuerzos (II)
Sign criteria is linked to the behaviour of slices. The orientation of the slices depend on the orientation of the member they belong to. That’s why sign criteria must be referred to a local coordinate system or to a representation of the slices in the different positions they can be found along the structural system.
The following criteria is the one we will be using throughout the course: the three types of internal forces are represented in their positive disposition on one single slice. Then, we represent this slice in each possible position we could find it in the problems that we will have to solve.
12. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Horizontal member
Criterio de signos a emplear en esfuerzos (II)
Sign criteria is linked to the behaviour of slices. The orientation of the slices depend on the orientation of the member they belong to. That’s why sign criteria must be referred to a local coordinate system or to a representation of the slices in the different positions they can be found along the structural system.
The following criteria is the one we will be using throughout the course: the three types of internal forces are represented in their positive disposition on one single slice. Then, we represent this slice in each possible position we could find it in the problems that we will have to solve.
13. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Horizontal member
Criterio de signos a emplear en esfuerzos (II)
Sign criteria is linked to the behaviour of slices. The orientation of the slices depend on the orientation of the member they belong to. That’s why sign criteria must be referred to a local coordinate system or to a representation of the slices in the different positions they can be found along the structural system.
The following criteria is the one we will be using throughout the course: the three types of internal forces are represented in their positive disposition on one single slice. Then, we represent this slice in each possible position we could find it in the problems that we will have to solve.
Vertical member
14. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Horizontal member
Criterio de signos a emplear en esfuerzos (II)
Sign criteria is linked to the behaviour of slices. The orientation of the slices depend on the orientation of the member they belong to. That’s why sign criteria must be referred to a local coordinate system or to a representation of the slices in the different positions they can be found along the structural system.
The following criteria is the one we will be using throughout the course: the three types of internal forces are represented in their positive disposition on one single slice. Then, we represent this slice in each possible position we could find it in the problems that we will have to solve.
Inclined member. Increasing slope.
Vertical member
15. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Horizontal member
Criterio de signos a emplear en esfuerzos (II)
Sign criteria is linked to the behaviour of slices. The orientation of the slices depend on the orientation of the member they belong to. That’s why sign criteria must be referred to a local coordinate system or to a representation of the slices in the different positions they can be found along the structural system.
The following criteria is the one we will be using throughout the course: the three types of internal forces are represented in their positive disposition on one single slice. Then, we represent this slice in each possible position we could find it in the problems that we will have to solve.
Inclined member. Increasing slope.
Vertical member
Inclined member. Decreasing slope.
16. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Diagramas de esfuerzos
To obtain the values of the internal forces in different points of the structure, we will “cut” the structure at the desired point. This “cut” will show three unknowns (the internal forces I can see in the visible side of the last slice before the cut). Using the equilibrium equations I will find the value of these unknowns, will give me the values for the axial and shear forces and the bending moment at that point.
We must learn how axial, shear and bending moment vary along each of the members of the structure, so that we become able to interpret the diagrams with just a glimpse.
17. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Diagramas de esfuerzos
To obtain the values of the internal forces in different points of the structure, we will “cut” the structure at the desired point. This “cut” will show three unknowns (the internal forces I can see in the visible side of the last slice before the cut). Using the equilibrium equations I will find the value of these unknowns, will give me the values for the axial and shear forces and the bending moment at that point.
We must learn how axial, shear and bending moment vary along each of the members of the structure, so that we become able to interpret the diagrams with just a glimpse. Overall procedure:
Obtain the reactions at the supports (or internal joints) and make sure that the structural system has been correctly equilibrated. Cut the structure at the needed points and obtain the values for the internal forces at each of those points. Depict these values on the correct side of the structural member. Represent the functions that will eventually form our diagrams.
18. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example.
Let’s limber up with the simply supported beam.
We apply a point load on it and obtain the reactions.
We will obtain the values of the internal forces in several sections of the bar and we will deduce which are the actual points at which we need to obtain these values.
Finally, we will draw the diagram for each internal force.
As we want to know how internal forces vary along the member, we must stablish an “x” axis and a local coordinate system.
19. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example.
Let’s limber up with the simply supported beam.
We apply a point load on it and obtain the reactions.
We will obtain the values of the internal forces in several sections of the bar and we will deduce which are the actual points at which we need to obtain these values.
Finally, we will draw the diagram for each internal force.
As we want to know how internal forces vary along the member, we must stablish an “x” axis and a local coordinate system.
The axis of the bar will be the “x” axis and the origin will be located at the left support..
The red point represents the point at which I’ll be cutting the structure.
20. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
Let’s obtain the values for the internal forces at different points of the member.
21. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
Let’s obtain the values for the internal forces at different points of the member.
X = 0
- I cut the structure in two infinitely near the left support and I keep the smallest part (this would be the last slice before the support). Once I cut, I can see the forces on the right side of the slice, which will be represented according to the positive sign criteria for a horizontal bar.
22. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
Let’s obtain the values for the internal forces at different points of the member.
X = 0
- I cut the structure in two infinitely near the left support and I keep the smallest part (this would be the last slice before the support). Once I cut, I can see the forces on the right side of the slice, which will be represented according to the positive sign criteria for a horizontal bar.
- I equilibrate the remaining forces and moments on the structure according to the direction of each of my unknowns:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
23. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
Let’s obtain the values for the internal forces at different points of the member.
X = 0
- I cut the structure in two infinitely near the left support and I keep the smallest part (this would be the last slice before the support). Once I cut, I can see the forces on the right side of the slice, which will be represented according to the positive sign criteria for a horizontal bar.
- I equilibrate the remaining forces and moments on the structure according to the direction of each of my unknowns:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction (perpendicular to the axis of the member) is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. We still have to decide on the sign. If we keep “V” in the sense it would have if it were positive, it’s easy to see that it wouldn’t be able to equilibrate the left reaction. This means that the sense of the shear force at that side of the slice must be opposite to the one we had supposed. Thus, the shear force is negative.
24. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
Let’s obtain the values for the internal forces at different points of the member.
X = 0
- I cut the structure in two infinitely near the left support and I keep the smallest part (this would be the last slice before the support). Once I cut, I can see the forces on the right side of the slice, which will be represented according to the positive sign criteria for a horizontal bar.
- I equilibrate the remaining forces and moments on the structure according to the direction of each of my unknowns:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction (perpendicular to the axis of the member) is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. We still have to decide on the sign. If we keep “V” in the sense it would have if it were positive, it’s easy to see that it wouldn’t be able to equilibrate the left reaction. This means that the sense of the shear force at that side of the slice must be opposite to the one we had supposed. Thus, the shear force is negative.
- Bending moment. We perform sum of moments at the point we have cut. All acting forces pass through the selected point, so the bending moment at that point has to be zero.
- Finally, I represent on each diagram the obtained values.
25. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
Let’s obtain the values for the internal forces at different points of the member.
X = 0
- I cut the structure in two infinitely near the left support and I keep the smallest part (this would be the last slice before the support). Once I cut, I can see the forces on the right side of the slice, which will be represented according to the positive sign criteria for a horizontal bar.
- I equilibrate the remaining forces and moments on the structure according to the direction of each of my unknowns:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction (perpendicular to the axis of the member) is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. We still have to decide on the sign. If we keep “V” in the sense it would have if it were positive, it’s easy to see that it wouldn’t be able to equilibrate the left reaction. This means that the sense of the shear force at that side of the slice must be opposite to the one we had supposed. Thus, the shear force is negative.
- Bending moment. We perform sum of moments at the point we have cut. All acting forces pass through the selected point, so the bending moment at that point has to be zero.
- Finally, I represent on each diagram the obtained values.
26. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
Let’s obtain the values for the internal forces at different points of the member.
X = 0
- I cut the structure in two infinitely near the left support and I keep the smallest part (this would be the last slice before the support). Once I cut, I can see the forces on the right side of the slice, which will be represented according to the positive sign criteria for a horizontal bar.
- I equilibrate the remaining forces on the structure according to the direction of each of my unknowns:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction (perpendicular to the axis of the member) is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. We still have to decide on the sign. If we keep “V” in the sense it would have if it were positive, it’s easy to see that it wouldn’t be able to equilibrate the left reaction. This means that the sense of the shear force at that side of the slice must be opposite to the one we had supposed. Thus, the shear force is negative.
- Bending moment. We perform sum of moments at the point we have cut. All acting forces pass through the selected point, so the bending moment at that point has to be zero.
- Finally, I represent on each diagram the obtained values.
Graphical representation
- We will use a Cartesian system in which “y” axis represents the values of each internal force.
- Positive values will be depicted under “x” axis, whereas negative values will be represented over it.
- Once we had obtained several values, we’ll link them with a curve that represents the variation of the internal forces along the structure.
27. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L/4
- I split the bar in two at x=L/4 and I keep the smallest part (the left one).
- I equilibrate the acting forces and moments on the left side of the cut with the unknown internal forces that appear at the right side of the slice, according to my positive sign criteria:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
28. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L/4
- I split the bar in two at x=L/4 and I keep the smallest part (the left one).
- I equilibrate the acting forces and moments on the left side of the cut with the unknown internal forces that appear at the right side of the slice, according to my positive sign criteria:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. If we keep “V” in the sense it would have if it were positive, it wouldn’t be able to equilibrate the forces on the left side. Thus, the shear force is negative .
29. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L/4
- I split the bar in two at x=L/4 and I keep the smallest part (the left one).
- I equilibrate the acting forces and moments on the left side of the cut with the unknown internal forces that appear at the right side of the slice, according to my positive sign criteria:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. If we keep “V” in the sense it would have if it were positive, it wouldn’t be able to equilibrate the forces on the left side. Thus, the shear force is negative
-Bending moment: The only acting force that can produce a moment is the vertical reaction. The value or this moment is PL/8, and its direction, clockwise. To compensate it a moment with the same sense as the one we supposed at first would be fine, therefore the bending moment is positive.
30. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L/4
- I split the bar in two at x=L/4 and I keep the smallest part (the left one).
- I equilibrate the acting forces and moments on the left side of the cut with the unknown internal forces that appear at the right side of the slice, according to my positive sign criteria:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. If we keep “V” in the sense it would have if it were positive, it wouldn’t be able to equilibrate the forces on the left side. Thus, the shear force is negative
-Bending moment: The only acting force that can produce a moment is the vertical reaction. The value or this moment is PL/8, and its direction, clockwise. To compensate it a moment with the same sense as the one we supposed at first would be fine, therefore the bending moment is positive.
- I represent the obtained values on the diagrams.
31. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L/4
- I split the bar in two at x=L/4 and I keep the smallest part (the left one).
- I equilibrate the acting forces and moments on the left side of the cut with the unknown internal forces that appear at the right side of the slice, according to my positive sign criteria:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. If we keep “V” in the sense it would have if it were positive, it wouldn’t be able to equilibrate the forces on the left side. Thus, the shear force is negative
-Bending moment: The only acting force that can produce a moment is the vertical reaction. The value or this moment is PL/8, and its direction, clockwise. To compensate it a moment with the same sense as the one we supposed at first would be fine, therefore the bending moment is positive.
- I represent the obtained values on the diagrams.
32. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L/2 (before the point load)
- I split the bar in two at x=L/2 and I keep the left part.
- I equilibrate the acting forces and moments on the left side of the cut with the unknown internal forces that appear at the right side of the slice, according to my positive sign criteria:
- Axial force: zero.
- Shear: Again, the only acting force in the shear direction is the vertical reaction. Therefore, the magnitude is P/2 and the sign, negative.
- Bending moment: Again, the only acting force that can produce a moment is the vertical reaction. But now it’s farther than before. The magnitude of this moment is PL/8, and its direction, clockwise Therefore the bending moment is positive.
- I represent the obtained values on the diagrams.
33. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L/2 (just after the point load)
- I cut the structure again at x=L/2 but this time just after the point load.
- I equilibrate forces and moments on one side and on the other side of the slice:
- Axial force: zero.
- Shear: The resultant of forces on the left side is P/2 but downwards. To equilibrate it, the value of the shear has to be P/2 as well. The sense of the force I had supposed within my positive sign criteria is suitable this time. Therefore the shear is positive in this case.
34. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L/2 (just after the point load)
- I cut the structure again at x=L/2 but this time just after the point load.
- I equilibrate forces and moments on one side and on the other side of the slice:
- Axial force: zero.
- Shear: The resultant of forces on the left side is P/2 but downwards. To equilibrate it, the value of the shear has to be P/2 as well. The sense of the force I had supposed within my positive sign criteria is suitable this time. Therefore the shear is positive in this case.
- Bending moment: As I’m performing sum of moments at the point I’ve cut, the only force producing moments is the left reaction. Therefore the value of the bending moment is the same as I had obtained before: + PL/4.
- Again, I represent the obtained values on my diagrams.
35. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L/2 (just after the point load)
- I cut the structure again at x=L/2 but this time just after the point load.
- I equilibrate the forces and moments one side and on the other side of the slice:
- Axial force: zero.
- Shear: The resultant of forces on the left side is P/2 but downwards. To equilibrate it, the value of the shear has to be P/2 as well. The sense of the force I had supposed within my positive sign criteria is suitable this time. Therefore the shear is positive in this case.
- Bending moment: As I’m performing sum of moments at the point I’ve cut, the only force producing moments is the left reaction. Therefore the value of the bending moment is the same as I had obtained before: + PL/4.
- Again, I represent the obtained values on my diagrams.
36. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = 3L/4
- I cut at x=3L/4 and keep the left side of the system (because I want to see the continuous variation).
- I equilibrate the forces on both sides of the cut:
- Axial force: zero.
- Shear: + P/2 .
- Bending moment: in this case both the reaction and the applied load produce moment about the point I have cut at. The value for the bending moment is + PL/2
37. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = 3L/4
- I cut at x=3L/4 and keep the left side of the system (because I want to see the continuous variation).
- I equilibrate the forces on both sides of the cut:
- Axial force: zero.
- Shear: + P/2 .
- Bending moment: in this case both the reaction and the applied load produce moment about the point I have cut at. The value for the bending moment is + PL/2
38. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L (just before the right support)
- I cut the structure at x=L just before the right support (so I’m not taking into account the right reaction)
- I equilibrate the forces on both sides of the cut:
- Axial force: zero. · Shear: + P/2.
- Bending moment: Both the left reaction and the point load keep on producing moment about the point I have cut at, but distances are longer now. The result for the sum of moments produced by acting forces is zero, so the value for the bending moment is zero as well.
39. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
X = L (just before the right support)
- I cut the structure at x=L just before the right support (so I’m not taking into account the right reaction)
- I equilibrate the forces on both sides of the cut:
- Axial force: zero. · Shear: + P/2.
- Bending moment: Both the left reaction and the point load keep on producing moment about the point I have cut at, but distances are longer now. The result for the sum of moments produced by acting forces is zero, so the value for the bending moment is zero as well.
40. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
How to plot the diagrams.
- I join with a segment each obtained point and extract some conclusions.
41. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
How to plot the diagrams.
- I join with a segment each obtained point and extract some conclusions.
- Shear values change when new forces perpendicular to the bar appear.
- Bending moment varies linearly: the farther I cut from the force that produces the moment, the bigger the produced moment is. Whenever new forces appear, the slope of the diagram changes.
- Conclusion: I only need to cut the members near the points where point forces are applied.
42. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. First example (cont).
How to plot the diagrams.
- I join with a segment each obtained point and extract some conclusions.
- Shear values change when new forces perpendicular to the bar appear.
- Bending moment varies linearly: the farther I cut from the force that produces the moment, the bigger the produced moment is. Whenever new forces appear, the slope of the diagram changes.
- Conclusion: I only need to cut the members near the points where point forces are applied.
Don’t miss this!
- The “skips” that appear in the shear diagram correspond to the value of the applied loads at that points.
- Whenever a point load is applied, a skip is produced in the shear diagram and a change of slope happens in the bending moment one. - Positive values in the bending moment diagram represent areas of the member in which the lower side of the slice is subjected to tension.
- Bending moment diagrams remember us of the deflection of the bar.
43. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example.
Simply supported beam with several loads applied on it.
Obtain reactions.
Obtain the values of the internal forces before and after each applied load.
Plot the axial, shear and bending moment diagrams.
44. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
Let’s obtain the values for the internal forces at different points of the member.
X = 0
- I cut the structure in two infinitely near the left support and I keep the smallest part (this would be the last slice before the support). Once I cut, I can see the forces on the right side of the slice, which will be represented according to the positive sign criteria for a horizontal bar.
- I equilibrate the remaining forces and moments on the structure according to the direction of each of my unknowns:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction (perpendicular to the axis of the member) is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. We still have to decide on the sign. If we keep “V” in the sense it would have if it were positive, it’s easy to see that it wouldn’t be able to equilibrate the left reaction. This means that the sense of the shear force at that side of the slice must be opposite to the one we had supposed. Thus, the shear force is negative.
- Bending moment. We perform sum of moments at the point we have cut. All acting forces pass through the selected point, so the bending moment at that point has to be zero.
- Finally, I represent on each diagram the obtained values.
45. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
Let’s obtain the values for the internal forces at different points of the member.
X = 0
- I cut the structure in two infinitely near the left support and I keep the smallest part (this would be the last slice before the support). Once I cut, I can see the forces on the right side of the slice, which will be represented according to the positive sign criteria for a horizontal bar.
- I equilibrate the remaining forces and moments on the structure according to the direction of each of my unknowns:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction (perpendicular to the axis of the member) is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. We still have to decide on the sign. If we keep “V” in the sense it would have if it were positive, it’s easy to see that it wouldn’t be able to equilibrate the left reaction. This means that the sense of the shear force at that side of the slice must be opposite to the one we had supposed. Thus, the shear force is negative.
- Bending moment. We perform sum of moments at the point we have cut. All acting forces pass through the selected point, so the bending moment at that point has to be zero.
- Finally, I represent on each diagram the obtained values.
46. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
X = L/4 (just before the first applied load)
- I split the bar in two at x=L/4 and I keep the smallest part (the left one).
- I equilibrate the acting forces and moments at the left side of the cut with the unknown internal forces that appear at the right side of the slice, according to my positive sign criteria:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is P/2. If we keep “V” in the sense it would have if it were positive, it wouldn’t be able to equilibrate the forces on the left side. Thus, the shear force is negative
-Bending moment: The only acting force that can produce a moment is the vertical reaction. The value or this moment is PL/8, and its direction, clockwise. To compensate it a moment with the same sense as the one we supposed at first would be fine, therefore the bending moment is positive.
- I represent the obtained values on the diagrams.
47. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
X = L/4 (just after the first point load)
- I split the bar in two at x=L/4 after the point load and keep the left side.
- I equilibrate the acting forces and moments at the left side of the cut with the unknown internal forces that appear at the right side of the slice, according to my positive sign criteria:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The resultant of forces on the left side is P/6 upwards. To equilibrate it, the value of the shear has to be P/6 as well. The sense of the force I had supposed within my positive sign criteria is not valid to compensate this force, so the shear is negative again.
-Bending moment: There are two forces now but only the vertical reaction can produce a moment about the point I’ve cut at. The value or this moment is, again, +PL/8.
- I represent the obtained values on the diagrams.
48. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
X = L/2 (just before the point load)
- I cut the structure at x=L/2 just before the applied load and keep the left side.
- I equilibrate acting forces and moments on both sides of the slice:
- Axial force: zero.
- Shear: the acting forces are the same as the ones in the previous cut, so the result will be the same as well: –P/6.
- Bending moment: the forces now are the same as in the previous cut, but they are farther this time. If I take moments about the point at which I’ve cut, the acting forces produce a clockwise, PL/6 kNm moment, so the bending moment is positive.
- Don’t forget to write down the obtained values on your diagrams.
49. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
X = L/2 (just after the point load)
- I split the structure in two at x=L/2 just after the point load and keep the left side of the cut.
- I equilibrate the forces and moments on both sides of the slice:
- Axial force: zero.
- Shear: The total amount of load perpendicular to the bar is P/6, downwards. On the right side of the slice I should have a force going upwards to compensate it, and that’s precisely what the positive sign criteria is providing me. Therefore the sign is positive.
- Bending moment: the new force is applied at the point about which I’m taking moments and the existing forces are at the same distance… so the value of the moment is the same one as in the previous cut: +PL/6.
50. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
X = 3L/4 (just before the point load)
- I cut the structure at x=3L/4, just before the third point load and keep the left side.
- I equilibrate forces and moments on both sides of the cut:
- Axial force: zero
- Shear: +P/6
- Bending moment: +PL/4
- I write these values down on my diagrams
51. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
X = 3L/4 (just after the third point load)
- I cut the structure at x=3L/4, just after the new point load and keep the left side.
- I equilibrate forces and moments on both sides of the slice:
- Axial force: zero.
- Shear: + P/2.
- Bending moment: +PL/8
52. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
X = L
- I cut the structure at x=L, just before the right support
- I equilibrate forces and moments on both sides of the slice:
- Axial force: zero.
- Shear: +P/2.
- Bending moment: zero.
53. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
- To draw the diagrams, I simply join the dots and extract some conclusions.
- Shear values change when new forces perpendicular to the bar appear.
- Bending moment varies linearly: the farther I cut from the force that produces the moment, the bigger the produced moment is. Whenever new forces appear, the slope of the diagram changes.
- Conclusion: I only need to cut the members near the points where point forces are applied.
54. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Second example (cont).
- To draw the diagrams, I simply join the dots and extract some conclusions.
- Shear values change when new forces perpendicular to the bar appear.
- Bending moment varies linearly: the farther I cut from the force that produces the moment, the bigger the produced moment is. Whenever new forces appear, the slope of the diagram changes.
- Conclusion: I only need to cut the members near the points where point forces are applied.
Don’t miss this!
- The “skips” that appear in the shear diagram correspond to the value of the applied loads at that points.
- Whenever a point load is applied, a skip is produced in the shear diagram and a change of slope happens in the bending moment one. - Positive values in the bending moment diagram represent areas of the member in which the lower side of the slice is subjected to tension.
- Bending moment diagrams remember us of the deflection of the bar.
55. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Third example
Simply supported beam subjected to uniform, distributed load.
Obtain reactions.
Obtain the values of the internal forces before and after each applied load.
Plot the axial, shear and bending moment diagrams. .
56. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Third example
Simply supported beam subjected to uniform, distributed load.
Obtain reactions.
Obtain the values of the internal forces before and after each applied load.
Plot the axial, shear and bending moment diagrams. .
We already now that we only have to obtain values at the points where new loads appear.
In this case, as the load is evenly distributed, we will obtain an intermediate value just to deduce how the variation in the diagrams is produced.
57. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Third example (cont).
X = 0
- I cut the structure in two infinitely near the left support and I keep the smallest part (this would be the last slice before the support). Once I cut, I can see the forces on the right side of the slice, which will be represented according to the positive sign criteria for a horizontal bar.
- I equilibrate the remaining forces and moments on the structure according to the direction of each of my unknowns:
- Axial force: There are no forces in the direction of the axial force for “N” to equilibrate, so the value will be zero.
- Shear: The only acting force in the shear direction (perpendicular to the axis of the member) is the vertical reaction. Therefore, the magnitude of the shear at the selected point of the member is q*L/2. We still have to decide on the sign. If we keep “V” in the sense it would have if it were positive, it’s easy to see that it wouldn’t be able to equilibrate the left reaction. This means that the sense of the shear force at that side of the slice must be opposite to the one we had supposed. Thus, the shear force is negative.
- Bending moment. We perform sum of moments at the point we have cut. All acting forces pass through the selected point, so the bending moment at that point has to be zero.
- Finally, I represent on each diagram the obtained values.
58. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Third example (cont).
X = L (just before the right support)
- I cut the member at x=L, just before the right support and keep the left side.
- I equilibrate forces and moments on both sides of the slice:
- Axial force: zero.
- Shear: + q*L/2
- Bending moment: zero.
59. Solid Mechanics – Academic Year 2014/2015 Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Bending moment: There are two forces producing moments about the point I have cut at: the reaction at the pinned support and the
distributed load until that point (a total load of q*L/2 times the distance: q*L/4). Therefore, the magnitude of the moment produced by
these forces is:
and it’s a clockwise moment
To compensate a clockwise moment on the left of the slice we need a counterclockwise one on the right. This coincidences with my
positive sign criteria. Therefore the bending moment is positive.
Internal forces diagrams. Third example (cont).
Intermediate point: X = L/2
- I cut the member at x=L/2 in order to understand how internal forces
vary between x=0 and x=L.
- I equilibrate forces and moments on both sides of the slice:
- Axial force: N=0
- Shear: the resultant of forces on the left side of the slice in the
direction of the shear is zero. If I think of the variation of the resultant of
the forces, I can easily see that this variation is linear (the more
distance, the more load).
2 1
2 2 2 4 8
q L L L L
q qL
60. Solid Mechanics – Academic Year 2014/2015 Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Bending moment: There are two forces producing moments about the point I have cut at: the reaction at the pinned support and the
distributed load until that point (a total load of q*L/2 times the distance: q*L/4). Therefore, the magnitude of the moment produced by
these forces is:
and it’s a clockwise moment
To compensate a clockwise moment on the left of the slice we need a counterclockwise one on the right. This coincidences with my
positive sign criteria. Therefore the bending moment is positive.
Internal forces diagrams. Third example (cont).
Intermediate point: X = L/2
- I cut the member at x=L/2 in order to understand how internal forces
vary between x=0 and x=L.
- I equilibrate forces and moments on both sides of the slice:
- Axial force: N=0
- Shear: the resultant of forces on the left side of the slice in the
direction of the shear is zero. If I think of the variation of the resultant of
the forces, I can easily see that this variation is linear (the more
distance, the more load).
2 1
2 2 2 4 8
q L L L L
q qL
Don’t miss this!
The variation of the shear is linearly dependent on the
distance, hence the function that represents these values
must be a line.
On the other hand, the variation of the bending moment due
to distributed, uniform load depends on the distance,
squared.
Therefore the function is not linear, but exponential.
61. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Third example (cont).
Plotting diagrams.
- I join the dots and extract some conclusions.
- Uniform, distributed loads make the shear function to be represented with a line. The slope tells me how much load is acting per unit of length..
- Uniform, distributed loads make the bending moment function to become a parabolic function. When the shear diagram reaches zero, I’ll find a change in the sign of the slope of the parabola (therefore, there’s a maximum or a minimum).
62. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Third example (cont).
Plotting diagrams.
- I join the dots and extract some conclusions.
- Uniform, distributed loads make the shear function to be represented with a line. The slope tells me how much load is acting per unit of length..
- Uniform, distributed loads make the bending moment function to become a parabolic function. When the shear diagram reaches zero, I’ll find a change in the sign of the slope of the parabola (therefore, there’s a maximum or a minimum).
Don’t miss this!
- When a member is subjected to uniform, distributed load, the shear diagram varies linearly and the bending moment does it exponentially (in second degree).
- The curvature of the bending moment diagram is similar to the one that a piece of fabric would have if I blew against it in the direction of the acting force). - If an structural system is symmetrical (both, geometry and loads), the bending moment diagram is symmetrical as well and the shear force diagram is antisymmetrical (one of the sides is symmetrical and reflected).
- Bending moment diagram reminds me of the deflection of the structure..
63. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Third example (cont).
When structural systems become more complex, each of the members will have its own internal forces diagram. You only must remember that each bar is the x axis and the axis perpendicular to “x” is the internal force axis.
Check carefully the sign criteria for vertical and inclined members.
64. Solid Mechanics – Academic Year 2014/2015
Prof: Maribel Castilla Heredia @maribelcastilla v.1.0 October 2014
Internal forces diagrams. Third example (cont).
Sign criteria along this course:
You can see that beside the slice there’s a tiny (+). That will remind you on which side of the element you should represent the positive values.
o1. Horizontal member: positive values for internal forces under the bar.
o2. Vertical member: positive values for internal forces on the right side of the bar.
o3. Inclined member (both positive or negative slope): positive values for internal forces under the bar. Example ->.
65. Solid Mechanics Academic Year 2014/2015
Solid Mechanics Block A. Internal forces diagrams.
Prof. Maribel Castilla Heredia @maribelcastilla Version 1.0 October 2014
Degree in Architecture. San Pablo CEU University – Institute of Technology.