This document provides an overview of the moment distribution method for analyzing continuous beams and rigid frames. It begins with definitions of key terms used in the method like stiffness factors, carry-over factors, and distribution factors. It then outlines the 5 step process for solving problems using moment distribution. As an example, it works through solving a continuous beam problem using the method in detail over multiple cycles of distribution. It also discusses adapting the method for structures with non-prismatic members.
This document provides an overview of the moment distribution method for analyzing statically indeterminate structures. It begins with introductions and definitions of key concepts like stiffness factors, distribution factors, and carry-over factors. It then outlines the step-by-step process of the method, which involves calculating fixed end moments, distributing moments at joints iteratively until convergence is reached, and determining shear forces and bending moments. Formulas are provided for prismatic beams. The document concludes by discussing how the method is adapted for non-prismatic members using design tables and graphs.
The document discusses the moment distribution method for analyzing statically indeterminate structures. It begins with an overview and introduction of the method. The basic principles are then stated, involving locking and releasing joints to determine fixed end moments and distributed moments through an iterative process. Key definitions are provided for stiffness factors, carry-over factors, and distribution factors. An example problem is then solved step-by-step using the moment distribution method. The document concludes with a discussion on extending the method to structures with non-prismatic members.
L15 analysis of indeterminate beams by moment distribution methodDr. OmPrakash
This document discusses the moment distribution method for analyzing indeterminate beams. It begins with an overview and introduction to the method, which was developed by Prof. Hardy Cross in 1932. It then describes the basic principles through a 5 step process: 1) joints are locked to determine fixed end moments, 2) joints are released allowing rotation, 3) unbalanced moments modify joint moments based on stiffness, 4) moments are distributed and modify other joints, 5) steps 3-4 repeat until moments converge. Key terms like stiffness and carry-over factors are also defined.
The document describes the moment distribution method, a technique for calculating bending moments in beams and frames that cannot be easily solved by other methods. It involves modeling joints between structural members as rigid and distributing applied moments between members based on their relative rotational stiffness. The method iterates between distributing moments at joints to balance them, until moments converge. Two example problems are worked through applying the method to determine bending moments at various points of indeterminate beams under loading.
This document describes Kani's method for analyzing indeterminate structures. Kani's method is an iterative approach that uses slope deflection to calculate member end moments. It involves calculating fixed end moments, relative member stiffnesses, rotation factors, and iterating to determine rotation contributions at joints until values converge. Two examples are provided to demonstrate applying Kani's method to analyze continuous beams, including calculating values, performing iterations, and determining final bending moments.
Aircraft Structures for Engineering Students 5th Edition Megson Solutions ManualRigeler
Full donwload : http://alibabadownload.com/product/aircraft-structures-for-engineering-students-5th-edition-megson-solutions-manual/ Aircraft Structures for Engineering Students 5th Edition Megson Solutions Manual
easy step on how to solve slope deflectionAlmasdan Alih
The document describes the displacement method of analysis using slope-deflection equations. It discusses general cases, derivation of stiffness coefficients, fixed-end moments, and analysis of beams and frames. Key steps include:
1) Developing slope-deflection equations relating displacements (slopes and translations) to applied loads and support reactions using beam stiffness properties.
2) Assembling equations into a matrix formulation relating displacement vectors to applied load vectors and fixed-end moment vectors using a stiffness matrix.
3) Solving the matrix equation to determine member displacements and internal forces for given loads.
This document provides an overview of the moment distribution method for analyzing statically indeterminate structures. It begins with introductions and definitions of key concepts like stiffness factors, distribution factors, and carry-over factors. It then outlines the step-by-step process of the method, which involves calculating fixed end moments, distributing moments at joints iteratively until convergence is reached, and determining shear forces and bending moments. Formulas are provided for prismatic beams. The document concludes by discussing how the method is adapted for non-prismatic members using design tables and graphs.
The document discusses the moment distribution method for analyzing statically indeterminate structures. It begins with an overview and introduction of the method. The basic principles are then stated, involving locking and releasing joints to determine fixed end moments and distributed moments through an iterative process. Key definitions are provided for stiffness factors, carry-over factors, and distribution factors. An example problem is then solved step-by-step using the moment distribution method. The document concludes with a discussion on extending the method to structures with non-prismatic members.
L15 analysis of indeterminate beams by moment distribution methodDr. OmPrakash
This document discusses the moment distribution method for analyzing indeterminate beams. It begins with an overview and introduction to the method, which was developed by Prof. Hardy Cross in 1932. It then describes the basic principles through a 5 step process: 1) joints are locked to determine fixed end moments, 2) joints are released allowing rotation, 3) unbalanced moments modify joint moments based on stiffness, 4) moments are distributed and modify other joints, 5) steps 3-4 repeat until moments converge. Key terms like stiffness and carry-over factors are also defined.
The document describes the moment distribution method, a technique for calculating bending moments in beams and frames that cannot be easily solved by other methods. It involves modeling joints between structural members as rigid and distributing applied moments between members based on their relative rotational stiffness. The method iterates between distributing moments at joints to balance them, until moments converge. Two example problems are worked through applying the method to determine bending moments at various points of indeterminate beams under loading.
This document describes Kani's method for analyzing indeterminate structures. Kani's method is an iterative approach that uses slope deflection to calculate member end moments. It involves calculating fixed end moments, relative member stiffnesses, rotation factors, and iterating to determine rotation contributions at joints until values converge. Two examples are provided to demonstrate applying Kani's method to analyze continuous beams, including calculating values, performing iterations, and determining final bending moments.
Aircraft Structures for Engineering Students 5th Edition Megson Solutions ManualRigeler
Full donwload : http://alibabadownload.com/product/aircraft-structures-for-engineering-students-5th-edition-megson-solutions-manual/ Aircraft Structures for Engineering Students 5th Edition Megson Solutions Manual
easy step on how to solve slope deflectionAlmasdan Alih
The document describes the displacement method of analysis using slope-deflection equations. It discusses general cases, derivation of stiffness coefficients, fixed-end moments, and analysis of beams and frames. Key steps include:
1) Developing slope-deflection equations relating displacements (slopes and translations) to applied loads and support reactions using beam stiffness properties.
2) Assembling equations into a matrix formulation relating displacement vectors to applied load vectors and fixed-end moment vectors using a stiffness matrix.
3) Solving the matrix equation to determine member displacements and internal forces for given loads.
The document discusses the moment distribution method for analyzing statically indeterminate structures. It begins by outlining the basic principles and definitions of the method, including stiffness factors, carry-over factors, and distribution factors. It then provides an example problem, showing the calculation of fixed end moments, establishment of the distribution table through successive approximations, and determination of shear forces and bending moments. Finally, it discusses extensions of the method to structures with non-prismatic members, including using tables to determine necessary values for analysis.
This document discusses structural stability, statical determinacy, and influence lines. It defines stability as a prerequisite for structures to carry loads, which depends on comparing equations and unknown forces through structural analysis. Statical determinacy determines if a structure remains in equilibrium through static concepts alone. The number of external reactions must exceed the number of equilibrium equations. Influence lines show the variation of reactions, shear, or bending moment due to moving loads and identify their critical positions producing greatest effects.
Slope deflection method for structure analysis in civil engineeringNagma Modi
This document provides the steps to determine moments at points A, B, C, and D using the slope-deflection method for multiple structures. For each structure, the equations relating slope and moment are written for each member. The slopes and moments are then solved for using these equations along with the boundary conditions of zero rotation and moment at supports. The final moments calculated at each point are provided at the end.
Deflection of structures using double integration method, moment area method, elastic load method, conjugate beam method, virtual work, castiglianois second theorem and method of consistent deformations
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal.[1] The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.
Structural engineering iii- Dr. Iftekhar Anam
Joint Displacements and Forces,Assembly of Stiffness Matrix and Load Vector of a Truss,Stiffness Matrix for 2-Dimensional Frame Members in the Local Axes System,Transformation of Stiffness Matrix from Local to Global Axes,Stiffness Method for 2-D Frame neglecting Axial Deformations,Problems on Stiffness Method for Beams/Frames,Assembly of Stiffness Matrix and Load Vector of a Three-Dimensional Truss,Calculation of Degree of Kinematic Indeterminacy (Doki)
Determine the doki (i.e., size of the stiffness matrix) for the structures shown below,Material Nonlinearity and Plastic Moment,
http://www.uap-bd.edu/ce/anam/
This document discusses the analysis of indeterminate beams using the moment distribution method. It provides the modified stiffness factor equation when the end of the beam is simply supported. It then gives an example problem and shows the step-by-step solution using the moment distribution method, including calculating the distribution factors, compiling the moment distribution table through multiple cycles, and determining the shear forces and bending moments. Finally, it briefly mentions analyzing nonprismatic members and using design tables with the moment distribution method.
Lecture 2( Moment distribution method with sway)Iqbal Hafeez
This document provides information about analyzing structural frames using the moment distribution method with sway. It discusses analyzing portal frames where the amount of sway is unknown by assuming initial fixed end moments. It provides equations to calculate the ratio of fixed end moments at column heads for different end conditions. As an example, it analyzes a portal frame with given properties and loadings using the moment distribution method accounting for sway. It calculates the fixed end moments, distribution factors, moment distribution table, reactions, and final bending moments at the joints.
The document discusses the slope deflection method of structural analysis. It begins by deriving the fundamental slope deflection equations that relate end moments, slopes, and deflections of a beam. It then presents an example problem demonstrating the full procedure of applying the slope deflection method, which involves writing slope deflection equations for each member, establishing joint equilibrium equations, solving for unknown displacements, and substituting these into the slope deflection equations to determine end moments. The method provides a general approach for the analysis of continuous beams and frames.
The document discusses the classification of structures based on stability and statical determinacy. It defines different types of supports and condition equations. A structure is stable and determinate if it has 3 reaction components that are neither parallel nor concurrent. It is stable but indeterminate if it has more than 3 non-parallel/concurrent reactions. Several examples of structures are classified. Structures with less than 3 reactions or with concurrent reactions are unstable. Closed panels require 3 internal condition equations to be stable internally.
This document provides examples and problems related to static equilibrium of structures. Example 1 shows applying the equations of equilibrium to a weight suspended by a rope over a pulley. Example 2 calculates the forces in ropes supporting a weighted crate. Problem 3.7 asks the minimum force P needed for equilibrium of a crate supported by three ropes meeting at a point.
This document discusses two approximate methods for analyzing building frames subjected to loads: the portal method and cantilever method. The portal method assumes inflection points at midpoints of beams and mid-heights of columns, and that interior columns carry twice the shear of exterior columns. The cantilever method assumes inflection points at beam midpoints and column mid-heights, and that column axial stresses are proportional to their distance from the storey's centroid. Examples demonstrate applying each method to determine member forces in frames.
Analysis of indeterminate beam by slopeand deflection methodnawalesantosh35
Slope-deflection method ,Slope-deflection equations, equilibrium equation of
method, application to beams with and without joint translation and rotation, Sinking or yielding of support,
Ch06 07 pure bending & transverse shearDario Ch
This document contains chapter 6 from a textbook on mechanics of materials. It includes 13 multi-part example problems involving the calculation of shear and moment diagrams for beams and shafts subjected to different loading conditions. The problems cover statically determinate beams with various end supports and load configurations, including point loads, distributed loads, overhanging sections, and compound sections. The solutions show the application of the principles of equilibrium to draw shear and moment diagrams. Key steps include writing the shear and moment equations and evaluating the diagrams at specific locations.
This document contains solutions to mechanics of solids problems involving deflection of beams. The first problem involves calculating the slope and deflection of a steel girder beam with given properties under a central load. Subsequent problems calculate reactions, slopes, and deflections of beams with various support conditions and loadings using concepts such as bending moment diagrams, integration, and the conjugate beam method. The last problem determines the magnitude of a propping force required to keep a beam with a uniform distributed load level at the center.
The document discusses various methods for analyzing beam deflection and deformation under loading, including:
1) Deriving the differential equation for the elastic curve of a beam and applying boundary conditions to determine the curve and maximum deflection.
2) Using the method of superposition to analyze beams subjected to multiple loadings by combining the effects of individual loads.
3) Applying moment-area theorems which relate the bending moment diagram to slope and deflection, allowing deflection calculations for beams with various support conditions.
Structural analysis II by moment-distribution CE 313,turja deb mitun id 13010...Turja Deb
The document summarizes the solution to determining the reactions and drawing the shear and bending moment diagrams for a beam using the moment distribution method. Key steps include: 1) calculating the stiffness factors and distribution factors for each joint; 2) using these factors to calculate the fixed end moments in a moment distribution table; 3) iteratively solving the table to determine the internal moments at each joint; and 4) using the internal moments to calculate the reactions at each support and plot the shear and bending moment diagrams.
The document provides an outline for a presentation on the moment distribution method for structural analysis. It includes:
- An introduction to the moment distribution method and its use for analyzing statically indeterminate beams and frames.
- Definitions of important terms used in the method like stiffness, carry over factor, and distribution factor.
- Sign conventions for support moments, member rotations, and sinking of supports.
- Expressions for fixed end moments under different load cases including centric loading, eccentric loading, uniform loads, support rotations, and sinking of supports.
- Examples of applying the method to a simply supported beam and fixed supported beam with sinking support.
The document discusses the moment distribution method for analyzing statically indeterminate structures. It begins by outlining the basic principles and definitions of the method, including stiffness factors, carry-over factors, and distribution factors. It then provides an example problem, showing the calculation of fixed end moments, establishment of the distribution table through successive approximations, and determination of shear forces and bending moments. Finally, it discusses extensions of the method to structures with non-prismatic members, including using tables to determine necessary values for analysis.
This document discusses structural stability, statical determinacy, and influence lines. It defines stability as a prerequisite for structures to carry loads, which depends on comparing equations and unknown forces through structural analysis. Statical determinacy determines if a structure remains in equilibrium through static concepts alone. The number of external reactions must exceed the number of equilibrium equations. Influence lines show the variation of reactions, shear, or bending moment due to moving loads and identify their critical positions producing greatest effects.
Slope deflection method for structure analysis in civil engineeringNagma Modi
This document provides the steps to determine moments at points A, B, C, and D using the slope-deflection method for multiple structures. For each structure, the equations relating slope and moment are written for each member. The slopes and moments are then solved for using these equations along with the boundary conditions of zero rotation and moment at supports. The final moments calculated at each point are provided at the end.
Deflection of structures using double integration method, moment area method, elastic load method, conjugate beam method, virtual work, castiglianois second theorem and method of consistent deformations
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal.[1] The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.
Structural engineering iii- Dr. Iftekhar Anam
Joint Displacements and Forces,Assembly of Stiffness Matrix and Load Vector of a Truss,Stiffness Matrix for 2-Dimensional Frame Members in the Local Axes System,Transformation of Stiffness Matrix from Local to Global Axes,Stiffness Method for 2-D Frame neglecting Axial Deformations,Problems on Stiffness Method for Beams/Frames,Assembly of Stiffness Matrix and Load Vector of a Three-Dimensional Truss,Calculation of Degree of Kinematic Indeterminacy (Doki)
Determine the doki (i.e., size of the stiffness matrix) for the structures shown below,Material Nonlinearity and Plastic Moment,
http://www.uap-bd.edu/ce/anam/
This document discusses the analysis of indeterminate beams using the moment distribution method. It provides the modified stiffness factor equation when the end of the beam is simply supported. It then gives an example problem and shows the step-by-step solution using the moment distribution method, including calculating the distribution factors, compiling the moment distribution table through multiple cycles, and determining the shear forces and bending moments. Finally, it briefly mentions analyzing nonprismatic members and using design tables with the moment distribution method.
Lecture 2( Moment distribution method with sway)Iqbal Hafeez
This document provides information about analyzing structural frames using the moment distribution method with sway. It discusses analyzing portal frames where the amount of sway is unknown by assuming initial fixed end moments. It provides equations to calculate the ratio of fixed end moments at column heads for different end conditions. As an example, it analyzes a portal frame with given properties and loadings using the moment distribution method accounting for sway. It calculates the fixed end moments, distribution factors, moment distribution table, reactions, and final bending moments at the joints.
The document discusses the slope deflection method of structural analysis. It begins by deriving the fundamental slope deflection equations that relate end moments, slopes, and deflections of a beam. It then presents an example problem demonstrating the full procedure of applying the slope deflection method, which involves writing slope deflection equations for each member, establishing joint equilibrium equations, solving for unknown displacements, and substituting these into the slope deflection equations to determine end moments. The method provides a general approach for the analysis of continuous beams and frames.
The document discusses the classification of structures based on stability and statical determinacy. It defines different types of supports and condition equations. A structure is stable and determinate if it has 3 reaction components that are neither parallel nor concurrent. It is stable but indeterminate if it has more than 3 non-parallel/concurrent reactions. Several examples of structures are classified. Structures with less than 3 reactions or with concurrent reactions are unstable. Closed panels require 3 internal condition equations to be stable internally.
This document provides examples and problems related to static equilibrium of structures. Example 1 shows applying the equations of equilibrium to a weight suspended by a rope over a pulley. Example 2 calculates the forces in ropes supporting a weighted crate. Problem 3.7 asks the minimum force P needed for equilibrium of a crate supported by three ropes meeting at a point.
This document discusses two approximate methods for analyzing building frames subjected to loads: the portal method and cantilever method. The portal method assumes inflection points at midpoints of beams and mid-heights of columns, and that interior columns carry twice the shear of exterior columns. The cantilever method assumes inflection points at beam midpoints and column mid-heights, and that column axial stresses are proportional to their distance from the storey's centroid. Examples demonstrate applying each method to determine member forces in frames.
Analysis of indeterminate beam by slopeand deflection methodnawalesantosh35
Slope-deflection method ,Slope-deflection equations, equilibrium equation of
method, application to beams with and without joint translation and rotation, Sinking or yielding of support,
Ch06 07 pure bending & transverse shearDario Ch
This document contains chapter 6 from a textbook on mechanics of materials. It includes 13 multi-part example problems involving the calculation of shear and moment diagrams for beams and shafts subjected to different loading conditions. The problems cover statically determinate beams with various end supports and load configurations, including point loads, distributed loads, overhanging sections, and compound sections. The solutions show the application of the principles of equilibrium to draw shear and moment diagrams. Key steps include writing the shear and moment equations and evaluating the diagrams at specific locations.
This document contains solutions to mechanics of solids problems involving deflection of beams. The first problem involves calculating the slope and deflection of a steel girder beam with given properties under a central load. Subsequent problems calculate reactions, slopes, and deflections of beams with various support conditions and loadings using concepts such as bending moment diagrams, integration, and the conjugate beam method. The last problem determines the magnitude of a propping force required to keep a beam with a uniform distributed load level at the center.
The document discusses various methods for analyzing beam deflection and deformation under loading, including:
1) Deriving the differential equation for the elastic curve of a beam and applying boundary conditions to determine the curve and maximum deflection.
2) Using the method of superposition to analyze beams subjected to multiple loadings by combining the effects of individual loads.
3) Applying moment-area theorems which relate the bending moment diagram to slope and deflection, allowing deflection calculations for beams with various support conditions.
Structural analysis II by moment-distribution CE 313,turja deb mitun id 13010...Turja Deb
The document summarizes the solution to determining the reactions and drawing the shear and bending moment diagrams for a beam using the moment distribution method. Key steps include: 1) calculating the stiffness factors and distribution factors for each joint; 2) using these factors to calculate the fixed end moments in a moment distribution table; 3) iteratively solving the table to determine the internal moments at each joint; and 4) using the internal moments to calculate the reactions at each support and plot the shear and bending moment diagrams.
The document provides an outline for a presentation on the moment distribution method for structural analysis. It includes:
- An introduction to the moment distribution method and its use for analyzing statically indeterminate beams and frames.
- Definitions of important terms used in the method like stiffness, carry over factor, and distribution factor.
- Sign conventions for support moments, member rotations, and sinking of supports.
- Expressions for fixed end moments under different load cases including centric loading, eccentric loading, uniform loads, support rotations, and sinking of supports.
- Examples of applying the method to a simply supported beam and fixed supported beam with sinking support.
The document discusses the moment coefficient method for analyzing statically indeterminate structures. It provides definitions of statically indeterminate structures as those where there are more unknown reactions or internal forces than available equilibrium equations. The moment coefficient method uses coefficients provided in the ACI code that are based on elastic analysis but account for inelastic redistribution. The coefficients are multiplied by the total factored load and span length to determine bending moments. The method was first included in the 1963 ACI code and remains permissible for analyzing two-way slabs supported on all sides. Advantages include providing a more exact analysis and potential cost savings through more precise design.
The moment distribution method can be used to analyze statically indeterminate beams and frames. It involves solving the linear equations obtained in the slope-deflection method through successive approximations. The key aspects of the method are:
1. Stiffness factor is defined as the moment required to produce a unit rotation at a point, and is used to relate moments and rotations.
2. Carry-over factor is the ratio of the moment induced at the far end of a propped cantilever to the moment applied at the near end.
3. Distribution factor is the ratio of a member's stiffness factor to the sum of stiffnesses of members meeting at a joint, and is used to distribute an
L18 analysis of indeterminate beams by moment distribution methodDr. OmPrakash
The document discusses the moment distribution method for analyzing indeterminate beams. It begins with an overview of the method and some basic definitions. It then describes the step-by-step process, which involves (1) computing fixed end moments by assuming locked joints, (2) releasing joints causing unbalanced moments, (3) distributing unbalanced moments according to member stiffnesses, (4) carrying moments over to other joints, and (5) repeating until moments converge. Key terms discussed include stiffness factors, carry-over factors, and distribution factors.
Strength of materials_by_r_s_khurmi-601-700kkkgn007
1. The document presents an example problem of calculating the moments and reactions for a continuous beam ABC that has span AB of 8m and span BC of 6m, with the beam fixed at A and simply supported at B and C.
2. The beam carries a uniformly distributed load of 1 kN/m along its entire length. Using Clapeyron's theorem of three moments, the fixing moments MA, MB, and MC are calculated.
3. The bending moment and shear force diagrams are drawn, showing the moments and reactions calculated at each support.
The document provides steps to solve two examples using the moment distribution method for beams. It first divides each beam into sections and calculates the fixed end moment for each section. It then calculates the stiffness and distribution factor for each section. Moment distribution tables are constructed and the distributing moment and carryover moment are calculated in cycles until equilibrium is reached. Finally, the bending moment diagram and shear force diagram are drawn.
- The document discusses beams, shear force, and bending moment diagrams. It provides examples of calculating reactions, shear forces, and bending moments at different points along a beam under various loading conditions.
- Key steps include drawing free body diagrams, applying equations of equilibrium, and deriving equations to determine shear force and bending moment as a function of distance along the beam.
- Shear force and bending moment diagrams can then be plotted by inputting values of distance into the equations to show variations in shear force and bending moment along the length of the beam.
- The document discusses beams, shear force, and bending moment diagrams. It provides examples of calculating reactions, shear forces, and bending moments at different points along a beam under various loading conditions.
- Key steps include drawing free body diagrams, applying equations of equilibrium, and deriving equations to determine shear force and bending moment as a function of distance along the beam.
- Shear force and bending moment diagrams can then be plotted by inputting values of distance into the equations to show variations in shear force and bending moment along the length of the beam.
Analysis of non sway frame portal frames by slopeand deflection methodnawalesantosh35
The slope deflection method is a displacement method used to analyze statically indeterminate beams and frames. It involves solving for the slope and deflection of members at their joints, which are the basic unknowns. Equations are developed relating the fixed end moments, slopes, and relative deflections of each member. These equations are set up and solved to determine the bending moments in each member. The method is demonstrated through examples solving for the bending moments in non-sway and sway frames.
The document discusses the stiffness matrix method for analyzing indeterminate beams and structures. It provides an introduction to the stiffness matrix method and outlines the steps to solve beams using this approach. These include determining the degrees of freedom, assigning coordinate numbers, imposing restraints, determining the stiffness matrix [S], and setting up and solving the equilibrium equation. The document then provides examples of applying the stiffness matrix method to analyze different beams, including continuous beams with various support conditions. The examples show determining the stiffness matrix, setting up the equilibrium equations, and calculating the reactions and moments at various joints.
This document describes the moment distribution method for analyzing statically indeterminate structures. It begins by stating the objectives of understanding the method. It then provides an introduction to the method, describing how it was developed as an alternative to slope-deflection when computers were not widely available. The basic concepts are explained, including defining distribution factors and moments. An example problem is worked through step-by-step and the concept of modified stiffness factors for hinged ends is described.
This document provides an overview of analyzing indeterminate structures using the force method. It discusses:
1) Satisfying equilibrium, compatibility, and force-displacement relationships when analyzing indeterminate structures.
2) The force method, which treats forces as unknowns, and two methods under it - method of consistent deformation and moment distribution.
3) How to apply the method of consistent deformation to analyze different types of indeterminate structures like beams, trusses, and structures with temperature/fabrication errors. Compatibility equations are set up and solved for redundant/unknown forces.
The document discusses beams, shear forces, bending moments, and provides examples of calculating shear force diagrams (SFD) and bending moment diagrams (BMD) for beams under different loading conditions. Key points:
- A beam is a structural element that is capable of withstanding load primarily by resisting bending.
- Shear force is the sum of all vertical forces acting on a beam section. Bending moment is the sum of moments of all forces acting on the beam section.
- SFD shows the variation of shear force along the beam length. BMD shows the variation of bending moment.
- Examples demonstrate how to calculate reactions, draw SFDs, and BMDs for beams with various
Using the slope-deflection and moment distribution methods, determine the reactions and bending moments in the members of the given frame structures.
For the first frame, the slope-deflection method is used to find the member end moments. Equilibrium equations are written at the joints to solve for the unknown reactions. The bending moment diagram is then drawn.
For the second frame, the moment distribution method is applied. Stiffness factors are calculated and distribution factors are used to solve for the unknown end moments. The bending moment diagram is drawn based on the calculated member end moments.
For the third frame, influence lines are constructed for the specified reactions and internal forces to indicate their variations with the moving load position.
This document contains chapter 6 from a textbook on mechanics of materials. It includes 13 multi-part example problems involving the calculation of shear and moment diagrams for beams and shafts subjected to different loading conditions. The problems cover statically determinate beams with various end supports and load configurations, including point loads, distributed loads, overhanging sections, and compound sections. The solutions show the application of the principles of equilibrium to draw shear and moment diagrams. Key steps include writing shear and moment equations and evaluating the diagrams at specific locations.
SFD & BMD Shear Force & Bending Moment DiagramSanjay Kumawat
The document discusses shear force and bending moment in beams. It defines key terms like beam, transverse load, shear force, bending moment, and types of loads, supports and beams. It explains how to calculate and draw shear force and bending moment diagrams for different types of loads on beams including point loads, uniformly distributed loads, uniformly varying loads, and loads producing couples or overhangs. Sign conventions and the effect of reactions, loads and geometry on the shear force and bending moment diagrams are also covered.
This document discusses analysis of statically determinate structures. It covers idealized structure representation, principles of superposition and equilibrium equations. Examples are provided to classify structures as determinate or indeterminate, determine stability, and calculate reactions on beams, frames and compound structures by applying equilibrium equations. Unknown reactions are solved for as force components at supports.
This document provides guidance on calculating shear force and bending moment diagrams (SFD and BMD) for beams under different loading conditions. It begins by explaining the process for a sample problem, which involves a beam with uniform and point loads. The key steps are to determine support reactions, divide the beam into sections, then calculate the SFD and BMD for each section. Linear variation indicates a straight line SFD, while parabolic variation means a curved BMD. Interpretations are provided for different loading types and the shapes of the resulting diagrams. References for further reading are listed at the end.
1. The document provides examples of constructing influence lines for statically determinate beams and trusses. It defines influence lines and shows how to determine the influence line for reactions, shear, and bending moment at various points.
2. Example problems are worked out step-by-step to show how to construct influence lines for a simple beam and a beam with a hinge support. The influence lines provide the response of the structure due to a moving unit load.
3. Equilibrium equations are also used to determine influence lines by relating reactions, shears and moments. General expressions for shear and moment are developed for a beam with multiple spans.
1. The document provides examples of constructing influence lines for statically determinate beams and trusses. It defines influence lines and shows how to determine the influence line for reactions, shear, and bending moment at various points.
2. Example problems are worked out step-by-step to show how to construct influence lines for a simple beam and a beam with a hinge support. The influence lines provide the response of the structure due to a moving unit load.
3. Equilibrium equations are also used to determine influence lines by relating reactions, shears and moments. General expressions for shear and bending moment over a beam with multiple spans are presented.
6161103 7.2 shear and moment equations and diagramsetcenterrbru
1) Beams are structural members designed to support loads perpendicular to their axes. Simply supported beams are pinned at one end and roller supported at the other, while cantilevered beams are fixed at one end and free at the other.
2) Internal shear forces (V) and bending moments (M) must be determined for beam design. V and M diagrams graphically display these values and can be discontinuous where loads change.
3) The procedure involves determining support reactions, then calculating V and M values along the beam using the method of sections to draw the diagrams.
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CA in Dwarka is a team of professional Chartered Accountant which are providing best services like Company Registration, Income Tax Return, Sales Tax Consultants, Bank Audit and other services specially in Dwarka and Delhi NCR.
Css Founder is Website Designing Company working with the mission of Website For Everyone Website Start From 999/-* More Packages are available. we are best company in website designing company in Delhi, as we are also working in Website Designing company in Mumbai.
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3. 7.1 MOMENT DISTRIBUTION METHOD - AN
OVERVIEW
• 7.1 MOMENT DISTRIBUTION METHOD - AN OVERVIEW
• 7.2 INTRODUCTION
• 7.3 STATEMENT OF BASIC PRINCIPLES
• 7.4 SOME BASIC DEFINITIONS
• 7.5 SOLUTION OF PROBLEMS
• 7.6 MOMENT DISTRIBUTION METHOD FOR STRUCTURES
HAVING NONPRISMATIC MEMBERS
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4. 7.2 MOMENT DISTRIBUTION METHOD -
INTRODUCTION AND BASIC PRINCIPLES
7.1 Introduction
(Method developed by Prof. Hardy Cross in 1932)
The method solves for the joint moments in continuous beams and
rigid frames by successive approximation.
7.2 Statement of Basic Principles
Consider the continuous beam ABCD, subjected to the given loads,
as shown in Figure below. Assume that only rotation of joints
occur
at B, C and D, and that no support displacements occur at B, C and
D. Due to the applied loads in spans AB, BC and CD, rotations
occur at B, C and D.
150 kN
15 kN/m 10 kN/m
3 m
A B C D
I I I
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8 m 6 m 8 m
5. In order to solve the problem in a successively approximating manner,
it can be visualized to be made up of a continued two-stage problems
viz., that of locking and releasing the joints in a continuous sequence.
7.2.1 Step I
The joints B, C and D are locked in position before any load is
applied on the beam ABCD; then given loads are applied on the
beam. Since the joints of beam ABCD are locked in position, beams
AB, BC and CD acts as individual and separate fixed beams,
subjected to the applied loads; these loads develop fixed end
moments.
-80 kN.m 15 kN/m -80 kN.m
A B
8 m
-112.5kN.m 112.5 kN.m
B C 8 m
6 m
-53.33 kN.m
10 kN/m
C D
150 kN
53.33 kN.m
3 m
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6. In beam AB
Fixed end moment at A = -wl2/12 = - (15)(8)(8)/12 = - 80 kN.m
Fixed end moment at B = +wl2/12 = +(15)(8)(8)/12 = + 80 kN.m
In beam BC
Fixed end moment at B = - (Pab2)/l2 = - (150)(3)(3)2/62
= -112.5 kN.m
Fixed end moment at C = + (Pab2)/l2 = + (150)(3)(3)2/62
= + 112.5 kN.m
In beam AB
Fixed end moment at C = -wl2/12 = - (10)(8)(8)/12 = - 53.33 kN.m
Fixed end moment at D = +wl2/12 = +(10)(8)(8)/12 = + 53.33kN.m
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7. 7.2.2 Step II
Since the joints B, C and D were fixed artificially (to compute the the fixed-end
moments), now the joints B, C and D are released and allowed to rotate.
Due to the joint release, the joints rotate maintaining the continuous nature of
the beam. Due to the joint release, the fixed end moments on either side of
joints B, C and D act in the opposite direction now, and cause a net
unbalanced moment to occur at the joint.
150 kN
15 kN/m 10 kN/m
3 m
A B C D
I I I
8 m 6 m 8 m
Released moments -80.0 +112.5 -112.5 +53.33 -53.33
Net unbalanced moment
+32.5 -59.17 -53.33 admission.edhole.com
8. 7.2.3 Step III
These unbalanced moments act at the joints and modify the joint moments at
B, C and D, according to their relative stiffnesses at the respective joints. The
joint moments are distributed to either side of the joint B, C or D, according
to their relative stiffnesses. These distributed moments also modify the
moments at the opposite side of the beam span, viz., at joint A in span AB, at
joints B and C in span BC and at joints C and D in span CD. This
modification is dependent on the carry-over factor (which is equal to 0.5 in
this case); when this carry over is made, the joints on opposite side are
assumed to be fixed.
7.2.4 Step IV
The carry-over moment becomes the unbalanced moment at the joints
to which they are carried over. Steps 3 and 4 are repeated till the carry-over
or distributed moment becomes small.
7.2.5 Step V
Sum up all the moments at each of the joint to obtain the joint
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9. 7.3 SOME BASIC DEFINITIONS
In order to understand the five steps mentioned in section 7.3, some words
need to be defined and relevant derivations made.
7.3.1 Stiffness and Carry-over Factors
Stiffness = Resistance offered by member to a unit displacement or rotation at a
point, for given support constraint conditions
MA
qA
MB
AA B
RA RB
L
E, I – Member properties
A clockwise moment MA is
applied at A to produce a +ve
bending in beam AB. Find qA
and MB.
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10. Using method of consistent deformations
L
DA
MA
A
B
L
fAA
A
B
1
M L A
EI
A 2
2
3
f =
L AA 3
D = + EI
Applying the principle of
consistent
deformation,
M
3
R f R A
D + = ® = - ¯
L
0
A A AA A 2
M L A A A
M L
EI
R L
A EI
2 EI
4
2
hence k M EI
M EI
= 4 ; = A
= 4
q = + = L
L
A
A A
q
q q
Stiffness factor = kq admission.edhole.com = 4EI/L
11. Considering moment MB,
MB + MA + RAL = 0
MB = MA/2= (1/2)MA
Carry - over Factor = 1/2
7.3.2 Distribution Factor
Distribution factor is the ratio according to which an externally applied
unbalanced moment M at a joint is apportioned to the various members
mating at the joint
+ ve moment M
A C
B
D
A
D
B C
MBA
MBC
MBD
At joint B
M - MBA-MBC-MBD = 0
I1
L1 I3
L3
I2
L2
M
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12. i.e., M = MBA + MBC + MBD
E I
E I
2 2
1 1
é
æ
+ ÷ ÷ø
æ
( K K K
)
= + +
æ
+ ÷ ÷ø
BA BC BD B
M
E I
( )
BA BC BD
ö
( . )
3 3
ö
ö
æ
M D F M
K
B
Similarly
æ
=
æ
=
å
M K
M D F M
K
M K
ù
ö
M
M D F M
K
M K K
K
K K K
L
L
L
BD
BD
BD
BC
BC
BC
BA
BA
BA BA B
B
( . )
( . )
4 4 4
3
2
1
= ÷ ÷
ø
ç ç
è
= ÷ ÷
ø
ç ç
è
= ÷ ÷
ø
ç ç
è
= =
=
+ +
=
÷ ÷ø
úû
êë
ç çè
ö
ç çè
ö
ç çè
=
å
å
å
q
q
q
q
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13. 7.3.3 Modified Stiffness Factor
The stiffness factor changes when the far end of the beam is simply-supported.
qA MA
A B
L
RA RB
As per earlier equations for deformation, given in Mechanics of Solids
text-books.
M L
K M EI
3 3
AB fixed
A
A
AB
A
A
K
EI
ö L
çè
L
EI
3
( )
4
4
ö 4
çè
3
=
÷ø
æ ÷ø
= = = æ
=
q
q
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14. 7.4 SOLUTION OF PROBLEMS -
7.4.1 Solve the previously given problem by the moment
distribution method
7.4.1.1: Fixed end moments
2 2
M = - M = - wl = - = -
kN m
AB BA
(15)(8)
M = - M = - wl = - = -
kN m
BC CB
(150)(6)
M M wl kN m
CD DC
53.333 .
(10)(8)
12
12
112.5 .
8
8
80 .
12
12
2 2
= - = - = - = -
7.4.1.2 Stiffness Factors (Unmodified Stiffness)
K K EI
= = = =
AB BA
4 (4)( )
K K EI
= = = =
BC CB
4 (4)( )
K EI EI EI
CD
4
ù
= = úû
= é
êë
K EI EI
EI EI
L
EI EI
L
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DC
0.5
8
4
0.5
8
8
4
0.667
6
0.5
8
= =
15. 7.4.1.3 Distribution Factors
BA
K
BA
BC
CB
CD
1.00
0.4284
EI
0.5
EI
0.5
0.667
EI
0.667
EI
EI
0.500
0.667 0.500
0.5716
0.667 0.500
0.5716
0.5 0.667
0.4284
0.5 0.667
0.0
0.5 ( )
K
K
K
K
K
= =
=
+
=
+
=
=
+
=
+
=
=
+
=
+
=
=
+
=
+
=
=
+ ¥
=
+
=
DC
DC
DC
CB CD
CD
CB CD
CB
BA BC
BC
BA BC
BA
BA wall
AB
K
DF
EI EI
K K
DF
EI EI
K K
DF
EI EI
K K
DF
EI EI
K K
DF
wall stiffness
K K
DF
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16. 7.4.1.4 Moment Distribution Table
Joint A B C D
Member AB BA BC CB CD DC
Distribution Factors 0 0.4284 0.5716 0.5716 0.4284 1
Computed end moments -80 80 -112.5 112.5 -53.33 53.33
Cycle 1
Distribution 13.923 18.577 -33.82 -25.35 -53.33
Carry-over moments 6.962 -16.91 9.289 -26.67 -12.35
Cycle 2
Distribution 7.244 9.662 9.935 7.446 12.35
Carry-over moments 3.622 4.968 4.831 6.175 3.723
Cycle 3
Distribution -2.128 -2.84 -6.129 -4.715 -3.723
Carry-over moments -1.064 -3.146 -1.42 -1.862 -2.358
Cycle 4
Distribution 1.348 1.798 1.876 1.406 2.358
Carry-over moments 0.674 0.938 0.9 1.179 0.703
Cycle 5
Distribution -0.402 -0.536 -1.187 -0.891 -0.703
Summed up -69.81 99.985 -99.99 96.613 -96.61 0
moments
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17. 7.4.1.5 Computation of Shear Forces
15 kN/m 10 kN/m
150 kN
B C
I I I
8 m 3 m 3 m 8 m
A
Simply-supported 60 60 75 75 40 40
reaction
End reaction
due to left hand FEM 8.726 -8.726 16.665 -16.67 12.079 -12.08
End reaction
due to right hand FEM -12.5 12.498 -16.1 16.102 0 0
Summed-up 56.228 63.772 75.563 74.437 53.077 27.923
moments
D
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18. 7.4.1.5 Shear Force and Bending Moment Diagrams
56.23
3.74 m
75.563
63.77
52.077
74.437
27.923
2.792 m
-69.806
98.297
35.08
126.704
-96.613
31.693
Mmax=+38.985 kN.m
Max=+ 35.59 kN.m
3.74 m
84.92
-99.985
48.307
2.792 m
S. F. D.
B. M. D
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19. Simply-supported bending moments at center of span
Mcenter in AB = (15)(8)2/8 = +120 kN.m
Mcenter in BC = (150)(6)/4 = +225 kN.m
Mcenter in AB = (10)(8)2/8 = +80 kN.m
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20. 7.5 MOMENT DISTRIBUTION METHOD FOR
NONPRISMATIC MEMBER (CHAPTER 12)
The section will discuss moment distribution method to analyze
beams and frames composed of nonprismatic members. First
the procedure to obtain the necessary carry-over factors,
stiffness factors and fixed-end moments will be outlined. Then
the use of values given in design tables will be illustrated.
Finally the analysis of statically indeterminate structures using
the moment distribution method will be outlined
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21. 7.5.1 Stiffness and Carry-over Factors
Use moment-area method to find the stiffness and carry-over
factors of the non-prismatic beam.
A
D
PA MB
B
MA
qA
P = (K ) D ( )
A A AB A M K
= q q
A AB A
M =
C M
B AB A
CAB= Carry-over factor of moment MA from A to B
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22. A
MB
B
M¢A=CBAMB
M =CBAKB M¢B(ºKB) B=CABMA
=CABKA
MA(ºKA)
qA (= 1.0)
MA qB (= 1.0)
A
B
(a) (b)
Use of Betti-Maxwell’s reciprocal theorem requires that the work
done by loads in case (a) acting through displacements in case (b) is
equal to work done by loads in case (b) acting through displacements in
case (a)
( M ) + ( M ) = ( M ¢ ) + ( M
¢
)
(0) (1) (1.0) (0.0)
A B A B
K =
K
AB BA C C
A B
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23. 7.5.2 Tabulated Design Tables
Graphs and tables have been made available to determine fixed-end
moments, stiffness factors and carry-over factors for common
structural shapes used in design. One such source is the Handbook of
Frame constants published by the Portland Cement Association,
Chicago, Illinois, U. S. A. A portion of these tables, is listed here as
Table 1 and 2
Nomenclature of the Tables
aA ab = ratio of length of haunch (at end A and B to the length
of span
b = ratio of the distance (from the concentrated load to end A)
to the length of span
hA, hB= depth of member at ends A and B, respectively
hC = depth of member admission.edhole.com at minimum section
24. Ic = moment of inertia of section at minimum section = (1/12)B(hc)3,
with B as width of beam
kAB, kBC = stiffness factor for rotation at end A and B, respectively
L = Length of member
MAB, MBA = Fixed-end moments at end A and B, respectively; specified in
tables for uniform load w or concentrated force P
r = h - h
A C
A h
C
r = h - h
B C
B h
C
Also
K k EI K k EI
BA C
L
A = , =
L
B
AB C
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