This document provides beam design formulas and shear and moment diagrams for various static loading conditions on wood beams. It includes formulas and diagrams for simple beams with uniform or partial uniform loading, cantilever beams, beams fixed at one or both ends, continuous beams over multiple spans, and beams with overhanging sections. A total of 32 figures are included with accompanying formulas to aid in wood beam design. The document is published by the American Wood Council and American Forest & Paper Association.
The lecture is in support of:
(1) The Design of Building Structures (Vol.1, Vol. 2), rev. ed., PDF eBook by Wolfgang Schueller, 2016
(2) Building Support Structures, Analysis and Design with SAP2000 Software, 2nd ed., eBook by Wolfgang Schueller,
The SAP2000V15 Examples and Problems SDB files are available on the Computers & Structures, Inc. (CSI) website: http://www.csiamerica.com/go/schueller
This document provides an overview of basic equations for the theory of plates and shells. It discusses the state of stress and strain at a point, including defining the six independent stress and strain components. It presents the relationships between strain and displacement, and discusses the equilibrium equations relating stress and body forces. Finally, it provides the equations for both Cartesian and cylindrical coordinate systems. The key concepts covered are the fundamental equations that form the basis of plate and shell theory.
Introduction-Plastic hinge concept-plastic section modulus-shape factor-redistribution of moments-collapse mechanism.
Theorems of plastic analysis - Static/lower bound theorem; Kinematic/upper bound theorem-Plastic analysis of beams and portal frames by equilibrium and mechanism methods.
This document discusses the design of reinforced concrete deep beams. It defines deep beams as having a span/depth ratio less than 2 or a continuous beam ratio less than 2.5. Deep beams behave differently than elementary beam theory due to non-linear stress distributions. Their behavior depends on loading type and cracking typically occurs between one-third to one-half of the ultimate load. Design considerations include checking for minimum thickness, flexural design, shear design, and anchorage of tension reinforcement.
This document provides an overview of analysis and design methods for concrete slabs, including:
1. Elastic analysis methods like grillage analysis and finite element analysis can be used to determine moments and shear forces in slabs.
2. Yield line theory is an alternative plastic/ultimate limit state approach for determining the ultimate load capacity of ductile concrete slabs. It involves assuming yield line patterns that divide the slab into rigid regions and equating external and internal work.
3. Examples are provided to illustrate yield line analysis for one-way spanning slabs and rectangular two-way slabs. Conventions, assumptions, and calculation procedures are explained.
A Picture Album of the Finite Element MethodRahul Leslie
FEM is still taught in some parts of the world as a dry subject: the faculty, after dealing with the Riley-Ritz, Galerkin’s and other numerical approaches, draws a small rectangle on the board, introduces it to the students as ‘this is an element’ and then quickly rushes into the derivations: polynomial representation, shape functions, strain energy equations, Gaussian quadrature, etc.
In those universities/colleges, the students might given a hands of experience in ANSYS or the like, for name sake, demonstrating the ‘plate with a hole’ sample (or the like, and may be a cantilever beam too), leaving the students unaware of the greater wider world of FEM (of course, unless the student is a web-miner, digging up and reading all those extra stuff, downloading some FE software himself, installing, and trying it out)
Here an effort is being made to present a “A Picture Album of the Finite Element Method”, so to say, which any FEM tutor can present to his class and introduce the range of applications that makes FEM such a wonderful tool, explaining each at his own capacity (which I’m confident of), before going into the dull ordeal of the underlying derivations. Such an introduction is sure to make the dull latter phase interesting to the students.
I’m also of the belief that hands of experience with an FEM package dealing with a range of problems, intending to give the student a deeper view into the versatility of FEM and the nearly unlimited things one can do with it makes it of interest enough, that I’m sure many of them will proclaim FEM to be ‘My hobby’ – at least some of the students.
Rahul Leslie
Jan ‘17
This document presents an example of analysis design of slab using ETABS. This example examines a simple single story building, which is regular in plan and elevation. It is examining and compares the calculated ultimate moment from CSI ETABS & SAFE with hand calculation. Moment coefficients were used to calculate the ultimate moment. However it is good practice that such hand analysis methods are used to verify the output of more sophisticated methods.
Also, this document contains simple procedure (step-by-step) of how to design solid slab according to Eurocode 2.The process of designing elements will not be revolutionised as a result of using Eurocode 2. Due to time constraints and knowledge, I may not be able to address the whole issues.
The lecture is in support of:
(1) The Design of Building Structures (Vol.1, Vol. 2), rev. ed., PDF eBook by Wolfgang Schueller, 2016
(2) Building Support Structures, Analysis and Design with SAP2000 Software, 2nd ed., eBook by Wolfgang Schueller,
The SAP2000V15 Examples and Problems SDB files are available on the Computers & Structures, Inc. (CSI) website: http://www.csiamerica.com/go/schueller
This document provides an overview of basic equations for the theory of plates and shells. It discusses the state of stress and strain at a point, including defining the six independent stress and strain components. It presents the relationships between strain and displacement, and discusses the equilibrium equations relating stress and body forces. Finally, it provides the equations for both Cartesian and cylindrical coordinate systems. The key concepts covered are the fundamental equations that form the basis of plate and shell theory.
Introduction-Plastic hinge concept-plastic section modulus-shape factor-redistribution of moments-collapse mechanism.
Theorems of plastic analysis - Static/lower bound theorem; Kinematic/upper bound theorem-Plastic analysis of beams and portal frames by equilibrium and mechanism methods.
This document discusses the design of reinforced concrete deep beams. It defines deep beams as having a span/depth ratio less than 2 or a continuous beam ratio less than 2.5. Deep beams behave differently than elementary beam theory due to non-linear stress distributions. Their behavior depends on loading type and cracking typically occurs between one-third to one-half of the ultimate load. Design considerations include checking for minimum thickness, flexural design, shear design, and anchorage of tension reinforcement.
This document provides an overview of analysis and design methods for concrete slabs, including:
1. Elastic analysis methods like grillage analysis and finite element analysis can be used to determine moments and shear forces in slabs.
2. Yield line theory is an alternative plastic/ultimate limit state approach for determining the ultimate load capacity of ductile concrete slabs. It involves assuming yield line patterns that divide the slab into rigid regions and equating external and internal work.
3. Examples are provided to illustrate yield line analysis for one-way spanning slabs and rectangular two-way slabs. Conventions, assumptions, and calculation procedures are explained.
A Picture Album of the Finite Element MethodRahul Leslie
FEM is still taught in some parts of the world as a dry subject: the faculty, after dealing with the Riley-Ritz, Galerkin’s and other numerical approaches, draws a small rectangle on the board, introduces it to the students as ‘this is an element’ and then quickly rushes into the derivations: polynomial representation, shape functions, strain energy equations, Gaussian quadrature, etc.
In those universities/colleges, the students might given a hands of experience in ANSYS or the like, for name sake, demonstrating the ‘plate with a hole’ sample (or the like, and may be a cantilever beam too), leaving the students unaware of the greater wider world of FEM (of course, unless the student is a web-miner, digging up and reading all those extra stuff, downloading some FE software himself, installing, and trying it out)
Here an effort is being made to present a “A Picture Album of the Finite Element Method”, so to say, which any FEM tutor can present to his class and introduce the range of applications that makes FEM such a wonderful tool, explaining each at his own capacity (which I’m confident of), before going into the dull ordeal of the underlying derivations. Such an introduction is sure to make the dull latter phase interesting to the students.
I’m also of the belief that hands of experience with an FEM package dealing with a range of problems, intending to give the student a deeper view into the versatility of FEM and the nearly unlimited things one can do with it makes it of interest enough, that I’m sure many of them will proclaim FEM to be ‘My hobby’ – at least some of the students.
Rahul Leslie
Jan ‘17
This document presents an example of analysis design of slab using ETABS. This example examines a simple single story building, which is regular in plan and elevation. It is examining and compares the calculated ultimate moment from CSI ETABS & SAFE with hand calculation. Moment coefficients were used to calculate the ultimate moment. However it is good practice that such hand analysis methods are used to verify the output of more sophisticated methods.
Also, this document contains simple procedure (step-by-step) of how to design solid slab according to Eurocode 2.The process of designing elements will not be revolutionised as a result of using Eurocode 2. Due to time constraints and knowledge, I may not be able to address the whole issues.
Stiffness method of structural analysisKaran Patel
This method is a powerful tool for analyzing indeterminate structures. One of its advantages over the flexibility method is that it is conducive to computer programming.
Stiffness method the unknowns are the joint displacements in the structure, which are automatically specified.
1. The document provides formulas for calculating slope, deflection, and maximum deflection for various beam types under different loading conditions. It gives the equations for cantilever beams with concentrated loads, uniformly distributed loads, and varying loads. It also provides the equations for simply supported beams with these different load types and with couple moments applied. The equations relate the beam properties like length, load location, and intensity to the resulting slope and deflection values.
Seismic Design of RC Diaphragms, Chords, and CollectorsRuangRangka
This document summarizes a presentation on seismic design of reinforced concrete diaphragms. It outlines the presentation topics which include an introduction to diaphragms, an overview of a case study building, calculating diaphragm forces from different methods, reviewing diaphragm components, using section cuts and simplified beam methods for analysis, and designing chords, collectors, and shear reinforcement. It also provides biographies of the presenters and describes assigning forces in structural analysis software.
This publication provides a concise compilation of selected rules in the Eurocode 8, together with relevant Cyprus National Annex, that relate to the design of common forms of concrete building structure in the South Europe. It id offers a detail view of the design of steel framed buildings to the structural Eurocodes and includes a set of worked examples showing the design of structural elements with using software (CSI ETABS). It is intended to be of particular to the people who want to become acquainted with design to the Eurocodes. Rules from EN 1998-1-1 for global analysis, type of analysis and verification checks are presented. Detail design rules for steel composite beam, steel column, steel bracing and composite slab with steel sheeting from EN 1998-1-1, EN1993-1-1 and EN1994-1-1 are presented. This guide covers the design of orthodox members in steel frames. It does not cover design rules for regularities. Certain practical limitations are given to the scope.
This document provides an overview of shear and torsion behavior in reinforced concrete sections. It discusses several key topics:
1. There is no unified theory to describe shear and torsion behavior, which involves many interactions between forces. Current approaches include truss mechanisms, strut-and-tie models, and compression field theories.
2. Shear stresses are produced by shear forces, torsion, and combinations of these. The origin and distribution of shear stresses is explained.
3. Concrete alone cannot resist much shear or torsion due to its low tensile capacity. Reinforcement is needed to resist forces through truss action after cracking.
4. Design procedures from codes like ACI 318 are summarized
This document provides an overview of the history and purpose of the STAAD Pro software. It was one of the first structural analysis programs, originally developed for DOS systems. It has since evolved to have a graphical user interface for Windows systems. The document outlines the basic four step process for using STAAD Pro to analyze and design structural models: creating geometry, assigning properties/loads, performing analysis, and reviewing results. It serves as an introduction for new users to understand the capabilities and workflow of the STAAD Pro program.
Abstract (Dutch)
Samengestelde betonnen liggers vervaardigd van prefab voorgespannen- en/of gewapende elementen zijn zeer populair in de huidige praktijk van de civiele techniek. Twee betonnen, samengestelde delen van de ligger worden gestort op verschillende tijdstippen. Verschillende elasticiteitsmoduli, opeenvolgende belastingaanbrenging, en verschillend krimp en kruip veroorzaken een herverdeling van de normaalspanning en ongelijke rekken en spanningen in twee aansluitende vezels in het aansluitvlak.
Dit seminar richt zich op de berekening volgens de EN 1992-1-1 en EN 1992-2. De aannames met betrekking tot de berekening en de controle van de gewapende en/of voorgespannen samengestelde liggers en doorsnedes zal worden toegelicht.
Ook wordt er ingegaan op:
• De spanning/rek respons van de doorsnede belast door normaalkracht en buigende momenten,
• De principes van het gebruik van de “initiële toestand” in berekeningen van de uiterste grenstoestand en de bruikbaarheidsgrenstoestand,
• De controle van dwarskracht en wringing,
• De interactie tussen alle snedekrachten,
• De principes van de controles van de spanningbeperking,
• De achtergrond van de scheurwijdtecontrole
Speciale aandacht zal er worden gegeven aan de berekening van de schuifspanning in het aansluitvlak, en de beschouwing van de invloed van de verschillende leeftijd van de betonnen delen met betrekking tot de schuifspanningen. Een alternatieve berekeningsmethode ten opzichte van de Eurocode 2 zal worden voorgesteld en worden getest.
De praktische voorbeelden volgens de Eurocode 2 zullen worden uitgevoerd met behulp van de IDEA StatiCa software.
STAAD.Pro 2006 is a structural analysis and design software. The document provides information about installing and getting started with STAAD.Pro 2006, including hardware requirements, contents of the installation CD, installing the software, selecting a copy protection method, and running the STAAD.Pro program and related programs.
This document provides an introduction and list of 32 figures showing shear and moment diagrams for beams under various static loading conditions, such as uniformly distributed loads, concentrated loads, cantilever beams, continuous beams, and beams with overhangs. It is from the Western Woods Use Book and is presented by the American Wood Council as a reference for designing wood beams.
This document provides information about a software module for designing reinforced concrete beams and slabs. It describes the module's capabilities for analyzing continuous beams and slabs under pattern loading and moment redistribution. It also summarizes the module's design approach, code compliance, analysis methods, and output capabilities like bending schedules.
The document provides an overview of the ASCE 7 provisions for determining wind loads on structures. It discusses the three main design methods in ASCE 7: the simplified procedure, analytical procedure, and wind tunnel procedure. Key terms covered include basic wind speed, exposure categories, importance factor, velocity pressure coefficients, gust factor, and pressure coefficients. It also summarizes how to determine internal and external wind pressures on building components using equations and diagrams from ASCE 7.
Modeling and Design of Bridge Super Structure and Sub StructureAIT Solutions
This document discusses modeling and analysis techniques for bridge superstructures and substructures. It covers modeling bridge decks using various element types including beam, grid, plate-shell, and solid models. It also discusses modeling bridge piers and foundations using solid elements, beam elements, or springs to represent soil-structure interaction. The document emphasizes the importance of modeling both superstructure and substructure together to accurately capture their interaction, and discusses challenges like modeling bearings and soil.
CE 72.32 (January 2016 Semester) Lecture 6 - Overview of Finite Element AnalysisFawad Najam
The document discusses the fundamentals of finite element analysis (FEA) for structural analysis. It provides an overview of FEA, including the modeling process of discretizing the structure into finite elements, generating the stiffness matrix, and solving the algebraic equations to determine structural responses like displacements and stresses. The document also reviews some prerequisite concepts from solid mechanics like stress-strain relationships, material properties, and solution of systems of equations. It traces the history and development of FEA and discusses its wide range of engineering applications today.
This publication provides a concise compilation of selected rules in the Eurocode 8, together with relevant Cyprus National Annex, that relate to the design of common forms of concrete building structure in the South Europe. Rules from EN 1998-1-1 for global analysis, regularity criteria, type of analysis and verification checks are presented. Detail design rules for concrete beam, column and shear wall, from EN 1998-1-1 and EN1992-1-1 are presented. This guide covers the design of orthodox members in concrete frames. It does not cover design rules for steel frames. Certain practical limitations are given to the scope.
The document discusses mechanics of solid deflection in beams. It provides relationships between bending moment and curvature, as well as sign conventions for shear force, bending moment, slope and deflection. It then analyzes simply supported beams with central point loads and uniform distributed loads. Equations are derived for slope, deflection and bending moment at any section. Cantilevers with point loads and uniform distributed loads are also analyzed. Macaulay's method, a versatile technique for determining slope and deflection in beams under various loading conditions, is introduced. Examples applying the concepts to specific beam problems are included.
The document discusses the design of columns in concrete structures. It covers several topics related to column design including: member strength and capacity versus section capacity, moment magnification, issues regarding slenderness effects, P-Delta analysis, and effective design considerations. The key steps in column design are outlined, including determining loads, geometry, materials, checking slenderness, computing design moments and capacities, and iterating the design as needed. Factors that influence column capacity such as slenderness, bracing, and effective length and stiffness are also described.
CE 72.52 - Lecture 7 - Strut and Tie ModelsFawad Najam
The document discusses the strut-and-tie approach for analyzing concrete structures. It begins with background concepts such as Bernoulli's hypothesis, St. Venant's principle, and the lower bound theorem of plasticity. It then discusses how axial stresses, shear stresses, and the interaction of stresses affect concrete sections. The document outlines the ACI approach to shear-torsion design and provides equations from ACI 318 for calculating the concrete shear capacity. It introduces the concept of modeling concrete as a truss system and compares this to flexural behavior in beams. The strut-and-tie method is presented as a unified approach for considering all load effects. Guidelines are provided for developing an appropriate strut-and-tie model and
The document provides formulae for calculating slope and deflection of beams under different loading conditions:
1) It gives equations for calculating slope at free ends and deflection at any section for cantilever beams with various loads including concentrated load, uniform load, and uniformly varying load.
2) Equations are also provided for simply supported beams with the same loading types to determine slope at supports and maximum deflection in the center or at load location.
3) The document contains 10 sections that list the beam type, applied load, formulae for slope at free ends or supports, and expressions for deflection along the beam and point of maximum deflection.
The HYDRA analyzers are a family of nitrogen analyzers that measure ammonium or nitrate using a single integrated sensor. The HYDRA-NH4 and HYDRA-NO3 sensors measure these compounds in municipal and environmental water applications. Key features include automatic compensation for interfering ions and temperature, integrated spray cleaning, and configurable alarm outputs. The sensors have replaceable electrodes and require minimal maintenance.
Stiffness method of structural analysisKaran Patel
This method is a powerful tool for analyzing indeterminate structures. One of its advantages over the flexibility method is that it is conducive to computer programming.
Stiffness method the unknowns are the joint displacements in the structure, which are automatically specified.
1. The document provides formulas for calculating slope, deflection, and maximum deflection for various beam types under different loading conditions. It gives the equations for cantilever beams with concentrated loads, uniformly distributed loads, and varying loads. It also provides the equations for simply supported beams with these different load types and with couple moments applied. The equations relate the beam properties like length, load location, and intensity to the resulting slope and deflection values.
Seismic Design of RC Diaphragms, Chords, and CollectorsRuangRangka
This document summarizes a presentation on seismic design of reinforced concrete diaphragms. It outlines the presentation topics which include an introduction to diaphragms, an overview of a case study building, calculating diaphragm forces from different methods, reviewing diaphragm components, using section cuts and simplified beam methods for analysis, and designing chords, collectors, and shear reinforcement. It also provides biographies of the presenters and describes assigning forces in structural analysis software.
This publication provides a concise compilation of selected rules in the Eurocode 8, together with relevant Cyprus National Annex, that relate to the design of common forms of concrete building structure in the South Europe. It id offers a detail view of the design of steel framed buildings to the structural Eurocodes and includes a set of worked examples showing the design of structural elements with using software (CSI ETABS). It is intended to be of particular to the people who want to become acquainted with design to the Eurocodes. Rules from EN 1998-1-1 for global analysis, type of analysis and verification checks are presented. Detail design rules for steel composite beam, steel column, steel bracing and composite slab with steel sheeting from EN 1998-1-1, EN1993-1-1 and EN1994-1-1 are presented. This guide covers the design of orthodox members in steel frames. It does not cover design rules for regularities. Certain practical limitations are given to the scope.
This document provides an overview of shear and torsion behavior in reinforced concrete sections. It discusses several key topics:
1. There is no unified theory to describe shear and torsion behavior, which involves many interactions between forces. Current approaches include truss mechanisms, strut-and-tie models, and compression field theories.
2. Shear stresses are produced by shear forces, torsion, and combinations of these. The origin and distribution of shear stresses is explained.
3. Concrete alone cannot resist much shear or torsion due to its low tensile capacity. Reinforcement is needed to resist forces through truss action after cracking.
4. Design procedures from codes like ACI 318 are summarized
This document provides an overview of the history and purpose of the STAAD Pro software. It was one of the first structural analysis programs, originally developed for DOS systems. It has since evolved to have a graphical user interface for Windows systems. The document outlines the basic four step process for using STAAD Pro to analyze and design structural models: creating geometry, assigning properties/loads, performing analysis, and reviewing results. It serves as an introduction for new users to understand the capabilities and workflow of the STAAD Pro program.
Abstract (Dutch)
Samengestelde betonnen liggers vervaardigd van prefab voorgespannen- en/of gewapende elementen zijn zeer populair in de huidige praktijk van de civiele techniek. Twee betonnen, samengestelde delen van de ligger worden gestort op verschillende tijdstippen. Verschillende elasticiteitsmoduli, opeenvolgende belastingaanbrenging, en verschillend krimp en kruip veroorzaken een herverdeling van de normaalspanning en ongelijke rekken en spanningen in twee aansluitende vezels in het aansluitvlak.
Dit seminar richt zich op de berekening volgens de EN 1992-1-1 en EN 1992-2. De aannames met betrekking tot de berekening en de controle van de gewapende en/of voorgespannen samengestelde liggers en doorsnedes zal worden toegelicht.
Ook wordt er ingegaan op:
• De spanning/rek respons van de doorsnede belast door normaalkracht en buigende momenten,
• De principes van het gebruik van de “initiële toestand” in berekeningen van de uiterste grenstoestand en de bruikbaarheidsgrenstoestand,
• De controle van dwarskracht en wringing,
• De interactie tussen alle snedekrachten,
• De principes van de controles van de spanningbeperking,
• De achtergrond van de scheurwijdtecontrole
Speciale aandacht zal er worden gegeven aan de berekening van de schuifspanning in het aansluitvlak, en de beschouwing van de invloed van de verschillende leeftijd van de betonnen delen met betrekking tot de schuifspanningen. Een alternatieve berekeningsmethode ten opzichte van de Eurocode 2 zal worden voorgesteld en worden getest.
De praktische voorbeelden volgens de Eurocode 2 zullen worden uitgevoerd met behulp van de IDEA StatiCa software.
STAAD.Pro 2006 is a structural analysis and design software. The document provides information about installing and getting started with STAAD.Pro 2006, including hardware requirements, contents of the installation CD, installing the software, selecting a copy protection method, and running the STAAD.Pro program and related programs.
This document provides an introduction and list of 32 figures showing shear and moment diagrams for beams under various static loading conditions, such as uniformly distributed loads, concentrated loads, cantilever beams, continuous beams, and beams with overhangs. It is from the Western Woods Use Book and is presented by the American Wood Council as a reference for designing wood beams.
This document provides information about a software module for designing reinforced concrete beams and slabs. It describes the module's capabilities for analyzing continuous beams and slabs under pattern loading and moment redistribution. It also summarizes the module's design approach, code compliance, analysis methods, and output capabilities like bending schedules.
The document provides an overview of the ASCE 7 provisions for determining wind loads on structures. It discusses the three main design methods in ASCE 7: the simplified procedure, analytical procedure, and wind tunnel procedure. Key terms covered include basic wind speed, exposure categories, importance factor, velocity pressure coefficients, gust factor, and pressure coefficients. It also summarizes how to determine internal and external wind pressures on building components using equations and diagrams from ASCE 7.
Modeling and Design of Bridge Super Structure and Sub StructureAIT Solutions
This document discusses modeling and analysis techniques for bridge superstructures and substructures. It covers modeling bridge decks using various element types including beam, grid, plate-shell, and solid models. It also discusses modeling bridge piers and foundations using solid elements, beam elements, or springs to represent soil-structure interaction. The document emphasizes the importance of modeling both superstructure and substructure together to accurately capture their interaction, and discusses challenges like modeling bearings and soil.
CE 72.32 (January 2016 Semester) Lecture 6 - Overview of Finite Element AnalysisFawad Najam
The document discusses the fundamentals of finite element analysis (FEA) for structural analysis. It provides an overview of FEA, including the modeling process of discretizing the structure into finite elements, generating the stiffness matrix, and solving the algebraic equations to determine structural responses like displacements and stresses. The document also reviews some prerequisite concepts from solid mechanics like stress-strain relationships, material properties, and solution of systems of equations. It traces the history and development of FEA and discusses its wide range of engineering applications today.
This publication provides a concise compilation of selected rules in the Eurocode 8, together with relevant Cyprus National Annex, that relate to the design of common forms of concrete building structure in the South Europe. Rules from EN 1998-1-1 for global analysis, regularity criteria, type of analysis and verification checks are presented. Detail design rules for concrete beam, column and shear wall, from EN 1998-1-1 and EN1992-1-1 are presented. This guide covers the design of orthodox members in concrete frames. It does not cover design rules for steel frames. Certain practical limitations are given to the scope.
The document discusses mechanics of solid deflection in beams. It provides relationships between bending moment and curvature, as well as sign conventions for shear force, bending moment, slope and deflection. It then analyzes simply supported beams with central point loads and uniform distributed loads. Equations are derived for slope, deflection and bending moment at any section. Cantilevers with point loads and uniform distributed loads are also analyzed. Macaulay's method, a versatile technique for determining slope and deflection in beams under various loading conditions, is introduced. Examples applying the concepts to specific beam problems are included.
The document discusses the design of columns in concrete structures. It covers several topics related to column design including: member strength and capacity versus section capacity, moment magnification, issues regarding slenderness effects, P-Delta analysis, and effective design considerations. The key steps in column design are outlined, including determining loads, geometry, materials, checking slenderness, computing design moments and capacities, and iterating the design as needed. Factors that influence column capacity such as slenderness, bracing, and effective length and stiffness are also described.
CE 72.52 - Lecture 7 - Strut and Tie ModelsFawad Najam
The document discusses the strut-and-tie approach for analyzing concrete structures. It begins with background concepts such as Bernoulli's hypothesis, St. Venant's principle, and the lower bound theorem of plasticity. It then discusses how axial stresses, shear stresses, and the interaction of stresses affect concrete sections. The document outlines the ACI approach to shear-torsion design and provides equations from ACI 318 for calculating the concrete shear capacity. It introduces the concept of modeling concrete as a truss system and compares this to flexural behavior in beams. The strut-and-tie method is presented as a unified approach for considering all load effects. Guidelines are provided for developing an appropriate strut-and-tie model and
The document provides formulae for calculating slope and deflection of beams under different loading conditions:
1) It gives equations for calculating slope at free ends and deflection at any section for cantilever beams with various loads including concentrated load, uniform load, and uniformly varying load.
2) Equations are also provided for simply supported beams with the same loading types to determine slope at supports and maximum deflection in the center or at load location.
3) The document contains 10 sections that list the beam type, applied load, formulae for slope at free ends or supports, and expressions for deflection along the beam and point of maximum deflection.
The HYDRA analyzers are a family of nitrogen analyzers that measure ammonium or nitrate using a single integrated sensor. The HYDRA-NH4 and HYDRA-NO3 sensors measure these compounds in municipal and environmental water applications. Key features include automatic compensation for interfering ions and temperature, integrated spray cleaning, and configurable alarm outputs. The sensors have replaceable electrodes and require minimal maintenance.
This document provides information about shear force diagrams. It defines shear force as unaligned forces pushing parts of a body in opposite directions. A shear force diagram indicates the shear force resisted by a beam along its length. To make a shear force diagram, reactions are first determined using equilibrium equations, then shear forces are calculated for different loading cases and plotted. Common loading cases are presented with example diagrams, such as a simply supported beam with uniform load or a cantilever beam with a concentrated load. The document outlines the general procedure of determining reactions, calculating shear forces, and drawing the diagram to analyze a beam.
The document discusses concepts related to shear force and bending moment in beams, including:
- Definitions of bending, beams, planar bending, and types of beams including simple, cantilever, and overhanging beams.
- Calculation sketches simplify beams, loads, and supports for analysis.
- Internal forces in bending include shear force and bending moment. Relations and diagrams relate these to external loads.
- Equations define shear force and bending moment at each beam section. Diagrams illustrate variations along the beam.
Fixed end moments are moments that occur at the ends of beams or other structural elements. These moments are caused by external forces or reactions that are applied at or very near the ends of the beam. Fixed end moments directly influence the maximum bending stress that will occur within the beam based on the applied loads and how they are transferred into or out of the ends of the beam.
This document provides an overview of the slope deflection method for analyzing statically indeterminate structures. It describes that the slope deflection method was developed in 1914 and can be used to analyze beams and frames. Key assumptions of the method are that joints are rigid and distortions from axial/shear stresses are neglected. The document outlines the application, sign convention, procedure, slope deflection equations, and provides examples for analyzing beams and frames using this method.
Lecture 9 shear force and bending moment in beamsDeepak Agarwal
The document discusses stresses in beams. It covers topics like shear force and bending moment diagrams, bending stresses, shear stresses, deflection, and torsion. Beams are structural members subjected to transverse forces that induce bending. Stresses and strains are created within beams when loaded. Shear forces and bending moments allow determining these internal stresses and maintaining equilibrium. Formulas are provided for calculating shear forces and bending moments in different beam configurations like cantilevers, simply supported beams, and beams with various load types.
This report analyzes and compares an OEM steering tie rod system versus a heim-joint steering system designed and built by students for a 2005 Jeep Wrangler. Finite element analysis was performed on both systems to determine stresses and likelihood of failure. The OEM tie rod was found to fail at 575 lbf of load, while the designed heim-joint system showed stresses well below the yield strength, indicating it would be suitable for the vehicle. The report includes the process, analysis, and conclusions.
This document provides information and formulas for calculating voltage drop in electrical systems. It begins with an introduction and tables of common voltage drop formulas. It then discusses factors like the National Electrical Code requirements. The main body of the document derives and explains the voltage drop calculation formulas for single-phase and three-phase systems. It includes phasor diagrams and discusses concepts like impedance and power factor. Examples are provided to demonstrate calculating voltage drop for different real-world scenarios.
Great plains parts manual 2 section folding drill PartCatalogs Net
This document is a parts manual for the Great Plains 2600-2S 26' 2-Section Folding Drill. It contains over 100 pages detailing the parts, assemblies, and components that make up the drill. The manual provides exploded diagrams and part numbers for all major systems including the mainframe, transport axle, toolbars, seed and fertilizer boxes, metering system, gauge wheels, drives, openers, and press wheels. Safety warnings are marked throughout and the manufacturer's trademarks and copyright information are included at the beginning.
The document provides step-by-step instructions for performing flow simulations in SolidWorks Flow Simulation. It describes how to open a SolidWorks model, create a flow simulation project, define boundary conditions and engineering goals, run the simulation, and view results through various plot types. The document is divided into multiple chapters that guide the user through simulations of different flow scenarios including ball valve design, conjugate heat transfer, porous media, hydraulic loss, and cylinder drag coefficient.
Great plains Parts manual 3 s 4000f section folding drillPartCatalogs Net
This document is a parts manual for a Great Plains 40' 3-section folding drill. It provides an overview of the manual and instructions on its use. The manual contains detailed diagrams and part listings for all components of the drill, including the tongue, transport axle, toolbars, fertilizer boxes, seed cups, gauge wheels, drives, openers, and press wheels. Safety warnings are prominently marked. The manufacturer provides the manual without warranty and reserves the right to revise it over time.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
This document discusses a study conducted by Auburn University students on using subcooled propane instead of RP-1 as the fuel for the Saturn V first stage. The students developed a trajectory code to model the Saturn V's performance and used it to compare propane and RP-1. They found that with optimized tank sizes and mixture ratios, a propane-fueled first stage could achieve higher performance than the original RP-1 design.
This document provides instructions for using COSMOSFloWorks software to simulate fluid flow and heat transfer. It outlines steps for opening a CAD model, setting up a project, defining boundary conditions and engineering goals, running simulations, and visualizing results through plots and trajectories. The document includes sections on topics like conjugate heat transfer, porous media, drag coefficient calculation, and parametric studies.
This document provides a tutorial on using Flow Simulation 2012 software. It covers topics such as opening SolidWorks models, creating flow simulation projects, specifying boundary conditions and engineering goals, running calculations, and viewing results through cut plots, surface plots, and other visualization tools. Step-by-step instructions are provided for analyzing examples involving a ball valve, conjugate heat transfer, porous media flow, hydraulic loss determination, drag coefficient calculation, and heat exchanger efficiency.
This document outlines key concepts in thermodynamics including:
- It defines 5 postulates that form the basis for thermodynamic analysis.
- It discusses variables, definitions, and relationships for constant composition systems, properties as functions of temperature and pressure/volume, and systems of variable composition including partial molar properties.
- It covers solution thermodynamics including ideal gas mixtures, fugacity, the ideal solution model, and property changes upon mixing.
- The document also addresses evaluation of properties using various formulations, criteria for equilibrium, phase equilibria, chemical reaction equilibria, and thermodynamic analysis of processes.
This document provides guidance on investigating and planning small hydropower projects in rural Myanmar. It outlines the need to estimate power demand through surveys of village populations, appliances, and existing infrastructure. Sample calculations are provided to estimate household, public, and industrial power demands based on unit numbers, consumption levels, and operating times. Guidance is given on classifying demand by season and time of day to determine load curves and size generation capacity accordingly. The document emphasizes accurately assessing power needs to properly identify suitable hydro potentials for powering isolated village grids through renewable energy.
Great plains parts manual 3 s 3000f section folding drillPartCatalogs Net
This document is a parts manual for the 30' 3-Section Folding Drill manufactured by Great Plains Manufacturing, Inc. It contains sections that describe the individual parts of the drill in detail for ordering replacement parts, including the tongue and mainframe, transport axle, tool bars, boxes and box frames, feeder cup channels, gauge wheels, drives, openers, and press wheels. The manual provides part numbers and diagrams to help identify specific components. It is intended to assist with maintenance and repair of the drill.
This manual provides guidelines for designing the Allan Block Fence system to withstand wind loads. It discusses how wind is measured and how this relates to design wind speeds and pressures. The manual also describes the key components of the Allan Block Fence system, including panels, posts, foundations, and how they work together structurally to transfer loads. Sample calculations are provided for determining required compressive strength, tensile strength, concrete shear capacity, and footing design for a given wind load and site configuration.
This document provides an overview of the Multisim 9 simulation software. It includes sections on installation, documentation conventions, the documentation set, and the license agreement. The documentation set includes the user guide, Multisim 9 for educators guide, component reference guide, and online help files. It is intended to help users understand how to install and make use of the various resources available with Multisim 9.
This document provides guidance for the optional classification notation "SafeHull-Dynamic Loading Approach" (SH-DLA) for vessels intended to carry oil, ore, bulk cargoes, containers, or liquefied gases. It describes the DLA analysis procedure, which uses first-principle direct calculations to determine design loads and structural strength criteria. The document outlines procedures for load case development, environmental conditions, response amplitude operator analysis, long-term response calculation, and equivalent design wave determination.
This document provides guidance for conducting a structural analysis using the SafeHull-Dynamic Loading Approach (SH-DLA) classification notation. SH-DLA allows for structural assessment based on direct calculations of design loads and strength criteria using first principles, as an alternative to the criteria specified in the ABS Rules for Building and Classing Steel Vessels. The analysis procedure involves determining load cases considering ship speed, loading conditions, dominant load parameters; calculating response amplitudes using linear and nonlinear seakeeping analysis; determining long-term responses; defining equivalent design waves; calculating external and internal pressures; modeling container and lightship loads; performing finite element analysis; and verifying acceptance criteria for yielding, buckling and ultimate strength.
This document provides instructions for updating Field Manual 5-125, Rigging Techniques, Procedures, and Applications. It specifies that certain pages within the manual should be removed and replaced with new pages. The changes are marked with a bar. Distribution of the updated manual is unlimited and approved for public release. It is signed by General Eric K. Shinseki, Chief of Staff of the Army, and countersigned by the Administrative Assistant to the Secretary of the Army.
This document is a parts manual for the Great Plains 3N-4010 & 3N-4020 No-Till Drill. It contains:
1) Introductory information such as how to use the manual, replacement parts ordering, and safety warnings.
2) Detailed diagrams and part lists for all areas of the drill including the frame, drive system, seed metering, openers, and hydraulics.
3) Copyright information and notes that optional equipment may be shown that is not included with standard units.
This document summarizes a new set of models called DriveSE for sizing and estimating the cost of wind turbine hub and drivetrain components. It includes models for the hub, low speed shaft, main bearings, gearbox, bedplate, and yaw system. The models range from semi-empirical to physics-based and are validated against industry data. Validation results show the models reasonably capture first-order sizing but with some deviation from actual component sizes. The models provide a simple yet useful first-order design tool for wind turbine drivetrain analysis.
This document discusses the design and analysis of flywheels. It begins by defining key parameters that describe flywheel performance such as coefficient of fluctuation of speed and energy. It then analyzes the stresses in a flywheel rim due to centrifugal force and restraint of the arms. Stresses in the flywheel arms are also examined. The document provides equations for designing components of the flywheel including the arms, shaft, hub and key. Examples are given to demonstrate flywheel performance calculations and stress analysis of the rim. The document serves as a reference for flywheel design, analysis of stresses, and selection of appropriate materials and dimensions.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
1. NATIONAL DESIGN SPECIFICATION®
FOR WOOD CONSTRUCTION
American Wood Council
American
Forest &
Paper
Association
N D S
2005 EDITION
®
ANSI/AF&PA NDS-2005
Approval Date: JANUARY 6, 2005
WITH COMMENTARY AND
SUPPLEMENT: DESIGN VALUES FOR WOOD CONSTRUCTION
ASD/LRFD
AmericanWAmericanWAmericanWAmericanWAmericanWoodCounciloodCounciloodCounciloodCounciloodCouncil
BEAM DESIGN FBEAM DESIGN FBEAM DESIGN FBEAM DESIGN FBEAM DESIGN FORMULASORMULASORMULASORMULASORMULAS
WITH SHEAR AND MOMENTWITH SHEAR AND MOMENTWITH SHEAR AND MOMENTWITH SHEAR AND MOMENTWITH SHEAR AND MOMENT
DIADIADIADIADIAGRAMSGRAMSGRAMSGRAMSGRAMS
AmericanAmericanAmericanAmericanAmerican
FFFFForest &orest &orest &orest &orest &
PPPPPaperaperaperaperaper
AssociationAssociationAssociationAssociationAssociation
w
R R
V
V
2 2
Shear
Mmax
Moment
x
ᐉ
ᐉ
ᐉ ᐉ
DESIGN AID NDESIGN AID NDESIGN AID NDESIGN AID NDESIGN AID Nooooo. 6. 6. 6. 6. 6
3. AMERICAN FOREST & PAPER ASSOCIATION
Figures 1 through 32 provide a series of shear
and moment diagrams with accompanying formulas
for design of beams under various static loading
conditions.
Shear and moment diagrams and formulas are
excerpted from the Western Woods Use Book, 4th
edition, and are provided herein as a courtesy of
Western Wood Products Association.
IntrIntrIntrIntrIntroductionoductionoductionoductionoduction Notations Relative to “Shear and Moment
Diagrams”
E = modulus of elasticity, psi
I = moment of inertia, in.4
L = span length of the bending member, ft.
R = span length of the bending member, in.
M = maximum bending moment, in.-lbs.
P = total concentrated load, lbs.
R = reaction load at bearing point, lbs.
V = shear force, lbs.
W = total uniform load, lbs.
w = load per unit length, lbs./in.
Δ = deflection or deformation, in.
x = horizontal distance from reaction to point
on beam, in.
List of FigurList of FigurList of FigurList of FigurList of Figureseseseses
Figure 1 Simple Beam – Uniformly Distributed Load ................................................................................................ 4
Figure 2 Simple Beam – Uniform Load Partially Distributed..................................................................................... 4
Figure 3 Simple Beam – Uniform Load Partially Distributed at One End.................................................................. 5
Figure 4 Simple Beam – Uniform Load Partially Distributed at Each End ................................................................ 5
Figure 5 Simple Beam – Load Increasing Uniformly to One End .............................................................................. 6
Figure 6 Simple Beam – Load Increasing Uniformly to Center.................................................................................. 6
Figure 7 Simple Beam – Concentrated Load at Center ............................................................................................... 7
Figure 8 Simple Beam – Concentrated Load at Any Point.......................................................................................... 7
Figure 9 Simple Beam – Two Equal Concentrated Loads Symmetrically Placed....................................................... 8
Figure 10 Simple Beam – Two Equal Concentrated Loads Unsymmetrically Placed .................................................. 8
Figure 11 Simple Beam – Two Unequal Concentrated Loads Unsymmetrically Placed .............................................. 9
Figure 12 Cantilever Beam – Uniformly Distributed Load........................................................................................... 9
Figure 13 Cantilever Beam – Concentrated Load at Free End .................................................................................... 10
Figure 14 Cantilever Beam – Concentrated Load at Any Point .................................................................................. 10
Figure 15 Beam Fixed at One End, Supported at Other – Uniformly Distributed Load ............................................. 11
Figure 16 Beam Fixed at One End, Supported at Other – Concentrated Load at Center ........................................... 11
Figure 17 Beam Fixed at One End, Supported at Other – Concentrated Load at Any Point ..................................... 12
Figure 18 Beam Overhanging One Support – Uniformly Distributed Load ............................................................... 12
Figure 19 Beam Overhanging One Support – Uniformly Distributed Load on Overhang ......................................... 13
Figure 20 Beam Overhanging One Support – Concentrated Load at End of Overhang ............................................. 13
Figure 21 Beam Overhanging One Support – Concentrated Load at Any Point Between Supports........................... 14
Figure 22 Beam Overhanging Both Supports – Unequal Overhangs – Uniformly Distributed Load......................... 14
Figure 23 Beam Fixed at Both Ends – Uniformly Distributed Load........................................................................... 15
Figure 24 Beam Fixed at Both Ends – Concentrated Load at Center.......................................................................... 15
Figure 25 Beam Fixed at Both Ends – Concentrated Load at Any Point .................................................................... 16
Figure 26 Continuous Beam – Two Equal Spans – Uniform Load on One Span ....................................................... 16
Figure 27 Continuous Beam – Two Equal Spans – Concentrated Load at Center of One Span ................................. 17
Figure 28 Continuous Beam – Two Equal Spans – Concentrated Load at Any Point ................................................ 17
Figure 29 Continuous Beam – Two Equal Spans – Uniformly Distributed Load ....................................................... 18
Figure 30 Continuous Beam – Two Equal Spans – Two Equal Concentrated Loads Symmetrically Placed ............. 18
Figure 31 Continuous Beam – Two Unequal Spans – Uniformly Distributed Load ................................................... 19
Figure 32 Continuous Beam – Two Unequal Spans – Concentrated Load on Each Span Symmetrically Placed ..... 19
4. AMERICAN WOOD COUNCIL
w
R R
V
V
2 2
Shear
Mmax
Moment
x
7-36 A
ᐉ
ᐉ
ᐉ ᐉ
a b c
x
R1 R2
V1
V2
Shear
a + —
R1
w
Mmax
Moment
wb
7-36 B
ᐉ
Figure 1 Simple Beam – Uniformly Distributed Load
Figure 2 Simple Beam – Uniform Load Partially Distributed
5. AMERICAN FOREST & PAPER ASSOCIATION
R1 R2
V1
V2
Shear
x
R1
—w1
Mmax
Moment
a b c
w2c
w1a
7-37 B
ᐉ
x
R1 R2
V2
V1
Mmax
Moment
Shear
R1
—
w
a
wa
7-37 A
ᐉ
Figure 3 Simple Beam – Uniform Load Partially Distributed at One End
Figure 4 Simple Beam – Uniform Load Partially Distributed at Each End
6. AMERICAN WOOD COUNCIL
x
W
2
R R
V
V
Shear
Mmax
Moment
2
7-38 B
ᐉ
ᐉ ᐉ
W
R1 R2
V1
V2
Shear
Mmax
Moment
x
.57741
7-38 A
ᐉ
Figure 5 Simple Beam – Load Increasing Uniformly to One End
Figure 6 Simple Beam – Load Increasing Uniformly to Center
7. AMERICAN FOREST & PAPER ASSOCIATION
x P
a
R1 R2
V1
V2
Shear
Mmax
Moment
b
7-39-b
ᐉ
x P
2
R R
V
V
Shear
Mmax
Moment
2
7-39 A
ᐉᐉ
ᐉ
Figure 7 Simple Beam – Concentrated Load at Center
Figure 8 Simple Beam – Concentrated Load at Any Point
8. AMERICAN WOOD COUNCIL
x P
a
R1 R2
V1
V2
Shear
M1
Moment
P
b
M2
7-40 B
ᐉ
x P
a
R R
V
Shear
Mmax
Moment
P
a
7-40 A
V
ᐉ
Figure 9 Simple Beam – Two Equal Concentrated Loads Symmetrically Placed
Figure 10 Simple Beam – Two Equal Concentrated Loads Unsymmetrically
Placed
9. AMERICAN FOREST & PAPER ASSOCIATION
a
R1 R2
V1
V2
Shear
M1
Moment
b
M2
7-41-a
x
P1 P2
ᐉ
Figure 11 Simple Beam – Two Unequal Concentrated Loads Unsymmetrically
Placed
Figure 12 Cantilever Beam – Uniformly Distributed Load
x
R
V
Shear
Moment
w
Mmax
7-41- B
ᐉ
ᐉ
11. AMERICAN FOREST & PAPER ASSOCIATION
R1
R2
V1
V2
Shear
M1
2
M2
3
—
11
7-43 B
2
x P
ᐉ
ᐉ ᐉ
ᐉ
w
R1
7-43 A
R2
V1
V2
—
4
Shear
M1
Mmax
3
8
x
ᐉ
ᐉ
ᐉ
ᐉ
Figure 15 Beam Fixed at One End, Supported at Other – Uniformly Distributed
Load
Figure 16 Beam Fixed at One End, Supported at Other – Concentrated Load at
Center
12. AMERICAN WOOD COUNCIL
a b
R1
V2
Shear
x
Pa
—
R2
M1
Moment
R2
V1
P
M2
7-44 A
Figure 17 Beam Fixed at One End, Supported at Other – Concentrated Load at
Any Point
Figure 18 Beam Overhanging One Support – Uniformly Distributed Load
x1
Shear
M2
M1
R1
(1– a2
)
Moment
V3
V2
(1– a2
)2 2
2
x
a
w( +a)
R2
V1
7-44 B
ᐉ
ᐉ
ᐉ
ᐉ
ᐉ
ᐉ
13. AMERICAN FOREST & PAPER ASSOCIATION
V2
Mmax
x1x
a
Moment
Shear
R1 R2
V1
P
7-45 B
ᐉ
V2
Mmax
x1x
a
wa
Moment
Shear
R1 R2
V1
7-45 A
ᐉ
Figure 19 Beam Overhanging One Support – Uniformly Distributed Load on
Overhang
Figure 20 Beam Overhanging One Support – Concentrated Load at End of
Overhang
14. AMERICAN WOOD COUNCIL
R1 R2
w
a b c
V2
V1
V4
V3
Mx1
M2
M1
M3
X1
X
7-46 B
ᐉ
ᐉ
x
R1 R2
x1
a b
V1
V2
Mmax
Shear
Moment
7-46 A
ᐉ
Figure 21 Beam Overhanging One Support – Concentrated Load at Any Point
Between Supports
Figure 22 Beam Overhanging Both Supports – Unequal Overhangs – Uniformly
Distributed Load
15. AMERICAN FOREST & PAPER ASSOCIATION
x
V
V
2 2
Shear
Moment
R R
Mmax
PPP
Mmax
4
7-47 B
ᐉ
ᐉ ᐉ
ᐉ
x
V
V
2 2
Shear
M1
Moment
w
R R
Mmax
.2113
7-47 A
ᐉ
ᐉ
ᐉ ᐉ
ᐉ
Figure 23 Beam Fixed at Both Ends – Uniformly Distributed Load
Figure 24 Beam Fixed at Both Ends – Concentrated Load at Center
16. AMERICAN WOOD COUNCIL
Shear
V1
R3R2
16
7
M1
V2
Moment
Mmax
x
7-48 B
R1
V3
w
ᐉ ᐉ
ᐉ
ᐉ
x
Moment
PPP
V2
Ma
M2
R2
V1
M1
7-48 A
R1
Shear
a b
ᐉ
Figure 25 Beam Fixed at Both Ends – Concentrated Load at Any Point
Figure 26 Continuous Beam – Two Equal Spans – Uniform Load on One Span
17. AMERICAN FOREST & PAPER ASSOCIATION
Shear
V1
R3R2
M1
Moment
Mmax
PPP
V3
V2
R1
ba
7-49 B
ᐉ ᐉ
Shear
V1
R3R2
M1
Moment
PPP
V3
V2
R1
2 2
7-49 A
Mmax
ᐉ ᐉ
ᐉ ᐉ
Figure 27 Continuous Beam – Two Equal Spans – Concentrated Load at Center
of One Span
Figure 28 Continuous Beam – Two Equal Spans – Concentrated Load at Any
Point
18. AMERICAN WOOD COUNCIL
7-50 A
R1 R3R2
3l/8
0.4215 0.4215
M1
w w
maxΔ
V2
V3
l l
ll
ll
M2
3l/8
V1
V2
V1
R3R2
PPP
V2
R1
PPP
a a a a
Mx
x
V2
V3
M1
M2
l l
Figure 29 Continuous Beam – Two Equal Spans – Uniformly Distributed Load
Figure 30 Continuous Beam – Two Equal Spans – Two Equal Concentrated Loads
Symmetrically Placed
19. AMERICAN FOREST & PAPER ASSOCIATION
R1 R3R2
x1
Mx2
M1
V1
V4
V3
V2
1 2
1 2
w w
x2
Mx1
ll
ll
V1
R3R2
P
V3
R1
P
a a b b
Mm1
V4
M1
Mm2
21
V2
P2P1
ll
Figure 31 Continuous Beam – Two Unequal Spans – Uniformly Distributed Load
Figure 32 Continuous Beam – Two Unequal Spans – Concentrated Load on Each
Span Symmetrically Placed
20. A F & P A®
American FAmerican FAmerican FAmerican FAmerican Forest & Porest & Porest & Porest & Porest & Paper Associationaper Associationaper Associationaper Associationaper Association
American WAmerican WAmerican WAmerican WAmerican Wood Councilood Councilood Councilood Councilood Council
1111 19th S1111 19th S1111 19th S1111 19th S1111 19th Streetreetreetreetreet, NWt, NWt, NWt, NWt, NW
SuitSuitSuitSuitSuite 800e 800e 800e 800e 800
WWWWWashingtashingtashingtashingtashington, DC 20036on, DC 20036on, DC 20036on, DC 20036on, DC 20036
Phone: 202-463-4Phone: 202-463-4Phone: 202-463-4Phone: 202-463-4Phone: 202-463-4777771111133333
FFFFFax: 202-463-2ax: 202-463-2ax: 202-463-2ax: 202-463-2ax: 202-463-2797979797911111
aaaaawwwwwcinfcinfcinfcinfcinfo@afandpa.orgo@afandpa.orgo@afandpa.orgo@afandpa.orgo@afandpa.org
wwwwwwwwwwwwwww.a.a.a.a.awwwwwc.orgc.orgc.orgc.orgc.org 11-07