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.
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.
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.
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.
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.
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 the derivation and application of three-moment equations for analyzing statically indeterminate continuous beams. The key points are:
1) Three-moment equations relate the bending moments at three successive supports to the applied loads on adjacent spans. They allow continuous beams to be analyzed by treating each span as simply supported with end moments.
2) The equations are derived by writing compatibility equations at each interior support in terms of the left, center, and right bending moments.
3) Examples show how to set up and solve the three-moment equations to determine support reactions and draw shear and moment diagrams for continuous beams with various loading conditions.
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.
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.
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.
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.
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 the derivation and application of three-moment equations for analyzing statically indeterminate continuous beams. The key points are:
1) Three-moment equations relate the bending moments at three successive supports to the applied loads on adjacent spans. They allow continuous beams to be analyzed by treating each span as simply supported with end moments.
2) The equations are derived by writing compatibility equations at each interior support in terms of the left, center, and right bending moments.
3) Examples show how to set up and solve the three-moment equations to determine support reactions and draw shear and moment diagrams for continuous beams with various loading conditions.
This document discusses statically determinate and indeterminate beams. It introduces the concept of continuous beams, which have at least one hinged support and roller supports. The key equations for analyzing continuous beams are presented, including the three-moment equation. This equation relates the bending moments at the ends of adjacent beam segments and is used to solve for unknown support reactions and draw shear and moment diagrams. An example problem demonstrates applying the three-moment equation to determine reactions for a continuous beam with a single load.
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.
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.
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 the moment distribution method for analyzing statically indeterminate structures. It begins with definitions of key terms used in the method like fixed end moments, distribution factors, carryover factors, and flexural stiffness. It then outlines the steps of the method, which involve calculating fixed end moments, distribution factors based on member stiffness, distributing moments at joints iteratively until equilibrium is reached, and calculating shear and bending moment diagrams. An example problem is then presented and solved using the moment distribution method.
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
Statically indeterminate beam moment distribution methodTHANINCHANMALAI
The method of moment distribution is this:
Imagine all joints in the structure held so that they cannot rotate and compute the moments at the ends of the member for this condition.
At each joint distribute the unbalanced fixed-end moment among the connecting members in proportion to the constant for each member defined as "stiffness“.
Multiply the moment distributed to each member at a joint by the carry-over factor at that end of the member and set this product at the other end of the member;
Distribute these moments just "carried over“.
Repeat the process until the moments to be carried over are small enough to be neglected.
Add all moments -- fixed-end moments, distributed moments, moments carried over -- at each end of each member to obtain the true moment at the end.
Statics and Strength of Materials Formula Sheetyasinabolfate
This document provides formulas and concepts related to statics and strength of materials. It includes:
1) Basic definitions and equations for statics including free body diagrams, force and moment balance, and methods for analyzing trusses and frames.
2) Cross-section geometry formulas for area, moment of inertia, and centroid location for common shapes.
3) Stress, strain, and Hooke's law relationships as well as stress and deformation equations for common cases like tension, torsion, bending and pressure vessels.
4) Miscellaneous formulas including buckling, Mohr's circle, and power in a shaft.
This document discusses the moment-area theorem for solving statically indeterminate structures. It begins by defining a statically indeterminate structure as one where the number of unknown reactions exceeds the number of equilibrium equations. It then lists the assumptions of the theorem as the beam being initially straight, elastically deformed in a way that causes small slope and deflection changes, and deformations caused by bending. The document goes on to explain the moment-curvature relationship and sign convention. It also describes how the moment-area theorem can be used to determine slope and displacement at different points by calculating the moment of the area under the moment diagram between those points. An example problem is shown to demonstrate applying the theorem.
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 the analysis of statically determinate structures, including defining idealized structures, explaining the principle of superposition and equations of equilibrium, classifying structures as determinate or indeterminate, and providing examples of determining reactions on beams and frames through applying the equations of equilibrium.
The document discusses methods for calculating deflections in structures, specifically the moment area method. It provides examples of using the moment area method to calculate slopes and deflections at various points along beams and frames by relating the bending moment diagram area to slope changes and vertical deflections using theorems. Sample problems are worked through step-by-step to demonstrate calculating slopes and deflections for beams under different loading conditions.
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.
1) The document discusses using the moment area method to solve for the deflection of a statically indeterminate beam.
2) It provides the theorems of the moment area method and outlines the process of drawing the M/EI diagram, elastic curve, and using the theorems to calculate slope change and vertical deviation.
3) As an example, it then applies the method to find the maximum downward deflection of a small aluminum beam with an applied force of 100N, showing the steps of determining the redundant reaction, drawing the diagrams, and using the theorems.
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.
This document discusses beam deflection and the mechanics of solids. It defines key terms like deflection, angle of rotation, curvature, and slope as they relate to the bending of beams. Equations are presented that relate these variables to each other and describe the deflection curve of beams undergoing small rotations. The assumptions and sign conventions used in the equations are also outlined.
This document discusses different types of beams and how to calculate support reactions for various beam configurations. It defines beams as structural members subjected to lateral loads perpendicular to the axis. The main types of beams covered are simply supported, cantilever, overhanging, continuous, and propped cantilever beams. It provides examples of calculating the support reactions of simply supported, cantilever, and continuous beams using free body diagrams and the equations of static equilibrium. The document emphasizes that finding support reactions is the first step in beam analysis and allows determining the internal shear forces and bending moments.
This document contains an unsolved physics paper from 2007 for the IIT JEE entrance exam. It includes multiple choice, assertion-reason, and matrix-match style questions testing concepts in physics. The questions cover topics such as circuits, electrostatics, nuclear physics, optics, mechanics, and thermodynamics. The document provides context, diagrams, and multiple potential answers for 22 questions testing understanding of fundamental physics principles.
Problems on simply supported beams (udl , uvl and couple)sushma chinta
1) A simply supported beam is subjected to a uniformly distributed load (UDL) over part of its span and a couple moment at one end.
2) Shear force and bending moment diagrams are drawn by dividing the beam into sections and analyzing each section.
3) The maximum bending moment occurs where the shear force is zero and is calculated to be 3761.59 N-m located at 2.637 m from the left support.
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 discusses statically determinate and indeterminate beams. It introduces the concept of continuous beams, which have at least one hinged support and roller supports. The key equations for analyzing continuous beams are presented, including the three-moment equation. This equation relates the bending moments at the ends of adjacent beam segments and is used to solve for unknown support reactions and draw shear and moment diagrams. An example problem demonstrates applying the three-moment equation to determine reactions for a continuous beam with a single load.
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.
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.
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 the moment distribution method for analyzing statically indeterminate structures. It begins with definitions of key terms used in the method like fixed end moments, distribution factors, carryover factors, and flexural stiffness. It then outlines the steps of the method, which involve calculating fixed end moments, distribution factors based on member stiffness, distributing moments at joints iteratively until equilibrium is reached, and calculating shear and bending moment diagrams. An example problem is then presented and solved using the moment distribution method.
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
Statically indeterminate beam moment distribution methodTHANINCHANMALAI
The method of moment distribution is this:
Imagine all joints in the structure held so that they cannot rotate and compute the moments at the ends of the member for this condition.
At each joint distribute the unbalanced fixed-end moment among the connecting members in proportion to the constant for each member defined as "stiffness“.
Multiply the moment distributed to each member at a joint by the carry-over factor at that end of the member and set this product at the other end of the member;
Distribute these moments just "carried over“.
Repeat the process until the moments to be carried over are small enough to be neglected.
Add all moments -- fixed-end moments, distributed moments, moments carried over -- at each end of each member to obtain the true moment at the end.
Statics and Strength of Materials Formula Sheetyasinabolfate
This document provides formulas and concepts related to statics and strength of materials. It includes:
1) Basic definitions and equations for statics including free body diagrams, force and moment balance, and methods for analyzing trusses and frames.
2) Cross-section geometry formulas for area, moment of inertia, and centroid location for common shapes.
3) Stress, strain, and Hooke's law relationships as well as stress and deformation equations for common cases like tension, torsion, bending and pressure vessels.
4) Miscellaneous formulas including buckling, Mohr's circle, and power in a shaft.
This document discusses the moment-area theorem for solving statically indeterminate structures. It begins by defining a statically indeterminate structure as one where the number of unknown reactions exceeds the number of equilibrium equations. It then lists the assumptions of the theorem as the beam being initially straight, elastically deformed in a way that causes small slope and deflection changes, and deformations caused by bending. The document goes on to explain the moment-curvature relationship and sign convention. It also describes how the moment-area theorem can be used to determine slope and displacement at different points by calculating the moment of the area under the moment diagram between those points. An example problem is shown to demonstrate applying the theorem.
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 the analysis of statically determinate structures, including defining idealized structures, explaining the principle of superposition and equations of equilibrium, classifying structures as determinate or indeterminate, and providing examples of determining reactions on beams and frames through applying the equations of equilibrium.
The document discusses methods for calculating deflections in structures, specifically the moment area method. It provides examples of using the moment area method to calculate slopes and deflections at various points along beams and frames by relating the bending moment diagram area to slope changes and vertical deflections using theorems. Sample problems are worked through step-by-step to demonstrate calculating slopes and deflections for beams under different loading conditions.
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.
1) The document discusses using the moment area method to solve for the deflection of a statically indeterminate beam.
2) It provides the theorems of the moment area method and outlines the process of drawing the M/EI diagram, elastic curve, and using the theorems to calculate slope change and vertical deviation.
3) As an example, it then applies the method to find the maximum downward deflection of a small aluminum beam with an applied force of 100N, showing the steps of determining the redundant reaction, drawing the diagrams, and using the theorems.
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.
This document discusses beam deflection and the mechanics of solids. It defines key terms like deflection, angle of rotation, curvature, and slope as they relate to the bending of beams. Equations are presented that relate these variables to each other and describe the deflection curve of beams undergoing small rotations. The assumptions and sign conventions used in the equations are also outlined.
This document discusses different types of beams and how to calculate support reactions for various beam configurations. It defines beams as structural members subjected to lateral loads perpendicular to the axis. The main types of beams covered are simply supported, cantilever, overhanging, continuous, and propped cantilever beams. It provides examples of calculating the support reactions of simply supported, cantilever, and continuous beams using free body diagrams and the equations of static equilibrium. The document emphasizes that finding support reactions is the first step in beam analysis and allows determining the internal shear forces and bending moments.
This document contains an unsolved physics paper from 2007 for the IIT JEE entrance exam. It includes multiple choice, assertion-reason, and matrix-match style questions testing concepts in physics. The questions cover topics such as circuits, electrostatics, nuclear physics, optics, mechanics, and thermodynamics. The document provides context, diagrams, and multiple potential answers for 22 questions testing understanding of fundamental physics principles.
Problems on simply supported beams (udl , uvl and couple)sushma chinta
1) A simply supported beam is subjected to a uniformly distributed load (UDL) over part of its span and a couple moment at one end.
2) Shear force and bending moment diagrams are drawn by dividing the beam into sections and analyzing each section.
3) The maximum bending moment occurs where the shear force is zero and is calculated to be 3761.59 N-m located at 2.637 m from the left support.
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.
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.
This document discusses approximate analysis methods for building frames subjected to both vertical and horizontal loads. For vertical loads, assumptions are made that points of zero moment occur at fixed distances from beam supports, reducing each beam to determinacy. The portal method is described for horizontal loads, assuming points of zero moment at midpoints and distributing shear between columns. Example problems demonstrate solving for member forces. The cantilever method also assumes midpoints of zero moment but distributes axial stress in columns by their distance from the storey's centroid.
This document provides instruction on analyzing three-hinged arches. It defines a three-hinged arch as a statically determinate structure with three hinges: two at the supports and one at the crown. The document describes how to determine the reactions of a three-hinged arch under a concentrated load using equations of static equilibrium. It presents an example problem showing how bending moment is reduced in a three-hinged arch compared to a simply supported beam carrying the same load.
- The document discusses shear force and bending moment in beams subjected to different types of loads. It defines shear force and bending moment, and explains how to calculate and draw shear force and bending moment diagrams.
- Key points covered include the relationships between loading, shear force and bending moment. Formulas and examples are provided for calculating reactions, shear forces and bending moments in cantilever beams and simply supported beams loaded with point loads and uniform loads.
- The concept of point of contraflexure is introduced for overhanging beams, where the bending moment changes sign from negative to positive.
Overhanged Beam and Cantilever beam problemssushma chinta
This document discusses shear force diagrams (SFD) and bending moment diagrams (BMD) for overhanging beams and cantilever beams under different loading conditions. It provides examples of overhanging beams with uniform distributed loading and analyses the reactions, shear forces, bending moments, and point of contraflexure. It also discusses cantilever beams and provides examples of cantilevers with point loads and uniform distributed loads, deriving the corresponding SFDs and BMDs.
Stucture Design-I(Bending moment and Shear force)Simran Vats
* Given: Maximum bending moment the beam can resist = 22 kNm
* Span of beam = 5 + 5 + 4 = 14 m
* Point loads = 1 + 2 + 1 = 4 kN
* To find: Maximum uniform load (w) the beam can carry
* Bending moment at center due to point loads
= (1 × 2.5) + (2 × 2.5) + (1 × 1.5) = 7.5 kNm
* Bending moment at center due to uniform load
= wl^2/8 = w × (14)^2/8 = w × 98 kNm
* Total bending moment should not exceed 22 kNm
7.5 + w
This document discusses the slope deflection method for analyzing continuous beams and rigid frames. It introduces the slope deflection method, describes the assumptions and sign conventions used, and provides the derivation of the slope deflection equations. It also outlines the steps to apply the method, including determining fixed end moments, writing slope deflection equations, solving for unknown rotations and deflections, determining end moments, and drawing shear and moment diagrams. An example problem is provided to illustrate the full method.
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.
The document discusses the double integration method for determining beam deflections. It defines beam deflection as the displacement of the beam's neutral surface from its original unloaded position. The differential equation relating the bending moment, flexural rigidity, and slope of the elastic curve is derived. This equation is integrated twice to obtain expressions for the slope and deflection of the beam in terms of the bending moment and constants of integration. Several examples are provided to demonstrate solving for the slope and maximum deflection of beams under different loading conditions using this method.
Spring and lever balancing mechanism, anglepoise lampskywalker_
This document discusses the analysis of spring-and-lever balancing mechanisms, specifically analyzing George Carwardine's Anglepoise lamp design. It presents two methods for analyzing a single-degree-of-freedom mechanism and shows that perfect balance can be achieved with a close-coiled spring having zero free length and appropriate stiffness. It then analyzes Carwardine's two-degree-of-freedom mechanism and shows how it achieves independent balancing of both arms using springs located at the base. The analysis is extended to account for the mass of the mechanism components.
This document discusses DC electric machines. It begins by introducing electric machines as devices that continuously convert electrical energy to mechanical energy or vice versa through electromechanical energy conversion. It then describes the basic principles of DC motors and generators, including how motion in a magnetic field induces voltage and how current in a magnetic field produces force. The document provides equations for induced voltage and electromagnetic force. It uses a simple loop generator model to illustrate how alternating current is produced and how a commutator converts it to direct current for a load. Finally, it discusses components and windings of practical DC generators.
The document discusses bending stresses in beams. It begins by outlining simplifying assumptions made in deriving the flexure formula to relate bending stresses to bending moments. These assumptions include plane sections remaining plane and perpendicular to the deformed beam axis. The neutral axis is defined as the axis where longitudinal fibers experience no deformation.
The derivation of the flexure formula is shown. Flexural stresses are proportional to the distance from the neutral axis and bending moment. Procedures for determining stresses at given points, as well as maximum stresses, are provided. Sample problems demonstrate applying the flexure formula and finding maximum stresses for different beam cross sections.
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.
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.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...DharmaBanothu
The Network on Chip (NoC) has emerged as an effective
solution for intercommunication infrastructure within System on
Chip (SoC) designs, overcoming the limitations of traditional
methods that face significant bottlenecks. However, the complexity
of NoC design presents numerous challenges related to
performance metrics such as scalability, latency, power
consumption, and signal integrity. This project addresses the
issues within the router's memory unit and proposes an enhanced
memory structure. To achieve efficient data transfer, FIFO buffers
are implemented in distributed RAM and virtual channels for
FPGA-based NoC. The project introduces advanced FIFO-based
memory units within the NoC router, assessing their performance
in a Bi-directional NoC (Bi-NoC) configuration. The primary
objective is to reduce the router's workload while enhancing the
FIFO internal structure. To further improve data transfer speed,
a Bi-NoC with a self-configurable intercommunication channel is
suggested. Simulation and synthesis results demonstrate
guaranteed throughput, predictable latency, and equitable
network access, showing significant improvement over previous
designs
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
Supermarket Management System Project Report.pdfKamal Acharya
Supermarket management is a stand-alone J2EE using Eclipse Juno program.
This project contains all the necessary required information about maintaining
the supermarket billing system.
The core idea of this project to minimize the paper work and centralize the
data. Here all the communication is taken in secure manner. That is, in this
application the information will be stored in client itself. For further security the
data base is stored in the back-end oracle and so no intruders can access it.
Determination of Equivalent Circuit parameters and performance characteristic...pvpriya2
Includes the testing of induction motor to draw the circle diagram of induction motor with step wise procedure and calculation for the same. Also explains the working and application of Induction generator
Build the Next Generation of Apps with the Einstein 1 Platform.
Rejoignez Philippe Ozil pour une session de workshops qui vous guidera à travers les détails de la plateforme Einstein 1, l'importance des données pour la création d'applications d'intelligence artificielle et les différents outils et technologies que Salesforce propose pour vous apporter tous les bénéfices de l'IA.
3. Instructional Objectives
After reading this chapter the student will be able to
1. Calculate stiffness factors and distribution factors for various members in
a continuous beam.
2. Define unbalanced moment at a rigid joint.
3. Compute distribution moment and carry-over moment.
4. Derive expressions for distribution moment, carry-over moments.
5. Analyse continuous beam by the moment-distribution method.
18.1 Introduction
In the previous lesson we discussed the slope-deflection method. In slope-
deflection analysis, the unknown displacements (rotations and translations) are
related to the applied loading on the structure. The slope-deflection method
results in a set of simultaneous equations of unknown displacements. The
number of simultaneous equations will be equal to the number of unknowns to be
evaluated. Thus one needs to solve these simultaneous equations to obtain
displacements and beam end moments. Today, simultaneous equations could be
solved very easily using a computer. Before the advent of electronic computing,
this really posed a problem as the number of equations in the case of multistory
building is quite large. The moment-distribution method proposed by Hardy Cross
in 1932, actually solves these equations by the method of successive
approximations. In this method, the results may be obtained to any desired
degree of accuracy. Until recently, the moment-distribution method was very
popular among engineers. It is very simple and is being used even today for
preliminary analysis of small structures. It is still being taught in the classroom for
the simplicity and physical insight it gives to the analyst even though stiffness
method is being used more and more. Had the computers not emerged on the
scene, the moment-distribution method could have turned out to be a very
popular method. In this lesson, first moment-distribution method is developed for
continuous beams with unyielding supports.
18.2 Basic Concepts
In moment-distribution method, counterclockwise beam end moments are taken
as positive. The counterclockwise beam end moments produce clockwise
moments on the joint Consider a continuous beam ABCD as shown in Fig.18.1a.
In this beam, ends A and D are fixed and hence, 0
=
= D
A θ
θ .Thus, the
deformation of this beam is completely defined by rotations B
θ and C
θ at joints B
and C respectively. The required equation to evaluate B
θ and C
θ is obtained by
considering equilibrium of joints B and C. Hence,
Version 2 CE IIT, Kharagpur
4. 0
=
∑ B
M ⇒ 0
=
+ BC
BA M
M (18.1a)
0
=
∑ C
M ⇒ 0
=
+ CD
CB M
M (18.1b)
According to slope-deflection equation, the beam end moments are written as
)
2
(
2
B
AB
AB
F
BA
BA
L
EI
M
M θ
+
=
AB
AB
L
EI
4
is known as stiffness factor for the beam AB and it is denoted
by . is the fixed end moment at joint B of beam AB when joint B is fixed.
Thus,
AB
k F
BA
M
B
AB
F
BA
BA K
M
M θ
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
=
2
C
B
BC
F
BC
BC K
M
M
θ
θ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
=
2
B
C
CB
F
CB
CB K
M
M
θ
θ
C
CD
F
CD
CD K
M
M θ
+
= (18.2)
In Fig.18.1b, the counterclockwise beam-end moments and produce
a clockwise moment on the joint as shown in Fig.18.1b. To start with, in
moment-distribution method, it is assumed that joints are locked i.e. joints are
prevented from rotating. In such a case (vide Fig.18.1b),
BA
M BC
M
B
M
0
=
= C
B θ
θ , and hence
F
BA
BA M
M =
F
BC
BC M
M =
F
CB
CB M
M =
F
CD
CD M
M = (18.3)
Since joints B and C are artificially held locked, the resultant moment at joints B
and C will not be equal to zero. This moment is denoted by and is known as
the unbalanced moment.
B
M
Version 2 CE IIT, Kharagpur
5. Thus,
F
BC
F
BA
B M
M
M +
=
In reality joints are not locked. Joints B and C do rotate under external loads.
When the joint B is unlocked, it will rotate under the action of unbalanced
moment . Let the joint B rotate by an angle
B
M 1
B
θ , under the action of . This
will deform the structure as shown in Fig.18.1d and introduces distributed
moment in the span BA and BC respectively as shown in the figure.
The unknown distributed moments are assumed to be positive and hence act in
counterclockwise direction. The unbalanced moment is the algebraic sum of the
fixed end moments and act on the joint in the clockwise direction. The
unbalanced moment restores the equilibrium of the joint B. Thus,
B
M
d
BC
d
BA M
M ,
,
0
=
∑ B
M (18.4)
0
=
+
+ B
d
BC
d
BA M
M
M
The distributed moments are related to the rotation 1
B
θ by the slope-deflection
equation.
Version 2 CE IIT, Kharagpur
6. 1
B
BA
d
BA K
M θ
=
1
B
BC
d
BC K
M θ
= (18.5)
Substituting equation (18.5) in (18.4), yields
( ) B
BC
BA
B M
K
K −
=
+
1
θ
BC
BA
B
B
K
K
M
+
−
=
1
θ
In general,
∑
−
=
K
M B
B1
θ (18.6)
where summation is taken over all the members meeting at that particular joint.
Substituting the value of 1
B
θ in equation (18.5), distributed moments are
calculated. Thus,
B
BA
d
BA M
K
K
M
∑
−
=
B
BC
d
BC M
K
K
M
∑
−
= (18.7)
The ratio
∑K
KBA
is known as the distribution factor and is represented by .
BA
DF
Thus,
B
BA
d
BA M
DF
M .
−
=
B
BC
d
BC M
DF
M .
−
= (18.8)
The distribution moments developed in a member meeting at B, when the joint B
is unlocked and allowed to rotate under the action of unbalanced moment is
equal to a distribution factor times the unbalanced moment with its sign reversed.
B
M
As the joint B rotates under the action of the unbalanced moment, beam end
moments are developed at ends of members meeting at that joint and are known
as distributed moments. As the joint B rotates, it bends the beam and beam end
moments at the far ends (i.e. at A and C) are developed. They are known as
carry over moments. Now consider the beam BC of continuous beam ABCD.
Version 2 CE IIT, Kharagpur
7. When the joint B is unlocked, joint C is locked .The joint B rotates by 1
B
θ under
the action of unbalanced moment (vide Fig. 18.1e). Now from slope-
deflection equations
B
M
B
BC
d
BC K
M θ
=
B
BC
BC K
M θ
2
1
=
1
2
d
CB BC
M M
= (18.9)
The carry over moment is one half of the distributed moment and has the same
sign. With the above discussion, we are in a position to apply moment-
distribution method to statically indeterminate beam. Few problems are solved
here to illustrate the procedure. Carefully go through the first problem, wherein
the moment-distribution method is explained in detail.
Example 18.1
A continuous prismatic beam ABC (see Fig.18.2a) of constant moment of inertia
is carrying a uniformly distributed load of 2 kN/m in addition to a concentrated
load of 10 kN. Draw bending moment diagram. Assume that supports are
unyielding.
Version 2 CE IIT, Kharagpur
8. Solution
Assuming that supports B and C are locked, calculate fixed end moments
developed in the beam due to externally applied load. Note that counterclockwise
moments are taken as positive.
2
2 9
1.5 kN.m
12 12
F AB
AB
wL
M
×
= = =
2
2 9
1.5 kN.m
12 12
F AB
BA
wL
M
×
= − = − = −
2
2
10 2 4
5 kN.m
16
F
BC
BC
Pab
M
L
× ×
= = =
2
2
10 2 4
5 kN.m
16
F
CB
BC
Pa b
M
L
× ×
= − = − = − (1)
Before we start analyzing the beam by moment-distribution method, it is required
to calculate stiffness and distribution factors.
3
4EI
K BA =
4
4EI
KBC =
At B: ∑ = EI
K 333
.
2
571
.
0
333
.
2
333
.
1
=
=
EI
EI
DFBA
429
.
0
333
.
2
=
=
EI
EI
DFBC
Version 2 CE IIT, Kharagpur
9. At C: ∑ = EI
K
0
.
1
=
CB
DF
Note that distribution factor is dimensionless. The sum of distribution factor at a
joint, except when it is fixed is always equal to one. The distribution moments are
developed only when the joints rotate under the action of unbalanced moment. In
the case of fixed joint, it does not rotate and hence no distribution moments are
developed and consequently distribution factor is equal to zero.
In Fig.18.2b the fixed end moments and distribution factors are shown on a
working diagram. In this diagram B and C are assumed to be locked.
Now unlock the joint C. Note that joint C starts rotating under the unbalanced
moment of 5 kN.m (counterclockwise) till a moment of -5 kN.m is developed
(clockwise) at the joint. This in turn develops a beam end moment of +5 kN.m
. This is the distributed moment and thus restores equilibrium. Now joint C
is relocked and a line is drawn below +5 kN.m to indicate equilibrium. When joint
C rotates, a carry over moment of +2.5 kN.m is developed at the B end of
member BC.These are shown in Fig.18.2c.
( CB
M )
When joint B is unlocked, it will rotate under an unbalanced moment equal to
algebraic sum of the fixed end moments(+5.0 and -1.5 kN.m) and a carry over
moment of +2.5 kN.m till distributed moments are developed to restore
equilibrium. The unbalanced moment is 6 kN.m. Now the distributed moments
and are obtained by multiplying the unbalanced moment with the
corresponding distribution factors and reversing the sign. Thus,
BC
M BA
M
Version 2 CE IIT, Kharagpur
10. 574
.
2
−
=
BC
M kN.m and 426
.
3
−
=
BA
M kN.m. These distributed moments restore
the equilibrium of joint B. Lock the joint B. This is shown in Fig.18.2d along with
the carry over moments.
Now, it is seen that joint B is balanced. However joint C is not balanced due to
the carry over moment -1.287 kN.m that is developed when the joint B is allowed
to rotate. The whole procedure of locking and unlocking the joints C and B
successively has to be continued till both joints B and C are balanced
simultaneously. The complete procedure is shown in Fig.18.2e.
The iteration procedure is terminated when the change in beam end moments is
less than say 1%. In the above problem the convergence may be improved if we
leave the hinged end C unlocked after the first cycle. This will be discussed in the
next section. In such a case the stiffness of beam BC gets modified. The above
calculations can also be done conveniently in a tabular form as shown in Table
18.1. However the above working method is preferred in this course.
Version 2 CE IIT, Kharagpur
11. Table 18.1 Moment-distribution for continuous beam ABC
Joint A B C
Member AB BA BC CB
Stiffness 1.333EI 1.333EI EI EI
Distribution
factor
0.571 0.429 1.0
FEM in
kN.m
+1.5 -1.5 +5.0 -5.0
Balance
joints C ,B
and C.O.
-1.713 -3.426
+2.5
-2.579
+5.0
0
-4.926 +4.926 -1.287
Balance C
and C.O.
+0.644 1.287
Balance B
and C.O.
-0.368 -0.276 -0.138
Balance C -0.184 -5.294 +5.294 0.138
C.O. +0.069 0
Balance B
and C.O.
-0.02 -0.039 -0.030 -0.015
Balance C +0.015
Balanced
moments in
kN.m
-0.417 -5.333 +5.333 0
Modified stiffness factor when the far end is hinged
As mentioned in the previous example, alternate unlocking and locking at the
hinged joint slows down the convergence of moment-distribution method. At the
hinged end the moment is zero and hence we could allow the hinged joint C in
the previous example to rotate freely after unlocking it first time. This
necessitates certain changes in the stiffness parameters. Now consider beam
ABC as shown in Fig.18.2a. Now if joint C is left unlocked then the stiffness of
member BC changes. When joint B is unlocked, it will rotate by 1
B
θ under the
action of unbalanced moment .The support C will also rotate by
B
M 1
C
θ as it is
free to rotate. However, moment 0
=
CB
M . Thus
B
BC
C
BC
CB
K
K
M θ
θ
2
+
= (18.7)
But, 0
=
CB
M
⇒
2
B
C
θ
θ −
= (18.8)
Now,
C
BC
B
BC
BC
K
K
M θ
θ
2
+
= (18.9)
Version 2 CE IIT, Kharagpur
12. Substituting the value of C
θ in eqn. (18.9),
B
BC
B
BC
B
BC
BC K
K
K
M θ
θ
θ
4
3
4
=
−
= (18.10)
B
R
BC
BC K
M θ
= (18.11)
The is known as the reduced stiffness factor and is equal to
R
BC
K BC
K
4
3
.Accordingly distribution factors also get modified. It must be noted that there is
no carry over to joint C as it was left unlocked.
Example 18.2
Solve the previous example by making the necessary modification for hinged end
C.
Fixed end moments are the same. Now calculate stiffness and distribution
factors.
EI
EI
K
EI
K BC
BA 75
.
0
4
3
,
333
.
1 =
=
=
Joint B: ,
∑ = ,
083
.
2
K 64
.
0
=
F
BA
D 36
.
0
=
F
BC
D
Joint C: ∑ = ,
75
.
0 EI
K 0
.
1
=
F
CB
D
All the calculations are shown in Fig.18.3a
Please note that the same results as obtained in the previous example are
obtained here in only one cycle. All joints are in equilibrium when they are
unlocked. Hence we could stop moment-distribution iteration, as there is no
unbalanced moment anywhere.
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13. Example 18.3
Draw the bending moment diagram for the continuous beam ABCD loaded as
shown in Fig.18.4a.The relative moment of inertia of each span of the beam is
also shown in the figure.
Solution
Note that joint C is hinged and hence stiffness factor BC gets modified. Assuming
that the supports are locked, calculate fixed end moments. They are
16 kN.m
F
AB
M =
16 kN.m
F
BA
M = −
7.5 kN.m
F
BC
M =
7.5 kN.m
F
CB
M = − , and
15 kN.m
F
CD
M =
In the next step calculate stiffness and distribution factors
8
4EI
KBA =
6
8
4
3 EI
K BC =
Version 2 CE IIT, Kharagpur
14. 6
8EI
KCB =
At joint B:
∑ =
+
= EI
EI
EI
K 5
.
1
0
.
1
5
.
0
0.5
0.333
1.5
F
BA
EI
D
EI
= =
1.0
0.667
1.5
F
BC
EI
D
EI
= =
At C:
0
.
1
, =
=
∑ F
CB
D
EI
K
Now all the calculations are shown in Fig.18.4b
This problem has also been solved by slope-deflection method (see example
14.2).The bending moment diagram is shown in Fig.18.4c.
Version 2 CE IIT, Kharagpur
15. Summary
An introduction to the moment-distribution method is given here. The moment-
distribution method actually solves these equations by the method of successive
approximations. Various terms such as stiffness factor, distribution factor,
unbalanced moment, distributing moment and carry-over-moment are defined in
this lesson. Few problems are solved to illustrate the moment-distribution method
as applied to continuous beams with unyielding supports.
Version 2 CE IIT, Kharagpur