This document presents a mathematical analysis of the vibration response of the vertical support members (columns) of a portal frame bridge subjected to a moving concentrated load. The analysis involves:
1) Constructing free body diagrams of the portal frame and its members to determine the forces and moments acting on them due to the moving load.
2) Performing a quantitative force analysis to derive equations representing the time-varying forces acting on the two vertical support columns as the load moves across the bridge.
3) Discussing different approaches to model the vibration response of the columns considering distributed/lumped mass-spring-damper systems and different loading patterns including random, harmonic, and single load cases.
The
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 three moment equation theory of structure analysis. [1] It relates the internal moments in a continuous beam at three points of support to the applied loads between supports. [2] The theory is proved using the conjugate beam method by equating shear forces and summing moments. [3] The general three moment equation is developed and modified for common load cases like point and uniform loads. An example problem demonstrates solving for reactions at supports.
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 discusses shear forces and bending moments in beams. It provides examples of different types of beams and loads, and how to calculate reactions, shear forces, and bending moments. Equations are given relating loads, shear forces, and bending moments. Methods are described for constructing shear force and bending moment diagrams for beams with different load conditions like concentrated loads, uniform loads, and combinations of loads.
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.
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.
Lecture01 design of concrete deck slabs ( Highway Engineering )Hossam Shafiq I
This document discusses the design of deck slabs for bridges. It explains how truck loads are distributed across multiple stringers below the deck through load transfer mechanisms like the deck material, spacing of stringers and secondary members, and their relative stiffnesses. While load distribution can be calculated precisely, codes like AASHTO simplify this using distribution factors based on the deck type and stringer spacing. The document also mentions design considerations for reinforced concrete deck slabs, including drainage features and corrosion protection methods.
This document discusses transmission line models used to represent transmission lines of different lengths. It introduces three models: the short line approximation for lines less than 80 km, the medium line approximation for lines between 80-250 km, and the long line model for lines over 250 km. It then discusses the ABCD parameters used to relate voltages and currents at the two ends of transmission lines. The following models are described in detail: short line, medium line using nominal-π and nominal-T representations, and the distributed parameter model for long lines.
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 three moment equation theory of structure analysis. [1] It relates the internal moments in a continuous beam at three points of support to the applied loads between supports. [2] The theory is proved using the conjugate beam method by equating shear forces and summing moments. [3] The general three moment equation is developed and modified for common load cases like point and uniform loads. An example problem demonstrates solving for reactions at supports.
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 discusses shear forces and bending moments in beams. It provides examples of different types of beams and loads, and how to calculate reactions, shear forces, and bending moments. Equations are given relating loads, shear forces, and bending moments. Methods are described for constructing shear force and bending moment diagrams for beams with different load conditions like concentrated loads, uniform loads, and combinations of loads.
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.
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.
Lecture01 design of concrete deck slabs ( Highway Engineering )Hossam Shafiq I
This document discusses the design of deck slabs for bridges. It explains how truck loads are distributed across multiple stringers below the deck through load transfer mechanisms like the deck material, spacing of stringers and secondary members, and their relative stiffnesses. While load distribution can be calculated precisely, codes like AASHTO simplify this using distribution factors based on the deck type and stringer spacing. The document also mentions design considerations for reinforced concrete deck slabs, including drainage features and corrosion protection methods.
This document discusses transmission line models used to represent transmission lines of different lengths. It introduces three models: the short line approximation for lines less than 80 km, the medium line approximation for lines between 80-250 km, and the long line model for lines over 250 km. It then discusses the ABCD parameters used to relate voltages and currents at the two ends of transmission lines. The following models are described in detail: short line, medium line using nominal-π and nominal-T representations, and the distributed parameter model for long lines.
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.
Electrical current, voltage, resistance, capacitance, and inductance are a few of the basic elements of electronics and radio. Apart from current, voltage, resistance, capacitance, and inductance, there are many other interesting elements to electronic technology. ... Use Electronics Notes to learn electronics online.
This document provides instructions and questions for a structural design exam. It consists of 4 questions. Students must answer question 1 and any other two questions. Question 1 involves calculating bending moments, designing reinforcement, and determining shear capacity for concrete beams. Question 2 involves checking the adequacy of steel sections and designing a bolt connection. Question 3 uses force methods to determine reactions and draws shear and bending moment diagrams. Question 4 analyzes a frame under vertical and lateral loads to determine reactions and internal forces at specific points. The document also includes relevant design formulas and appendices on load combinations, bending moment coefficients, and steel design strengths.
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 chapter discusses stress and strain in materials subjected to tension or compression. It defines stress as the load applied over the cross-sectional area. Strain is defined as the change in length over the original length. Hooke's law states that stress is proportional to strain for elastic materials. Young's modulus is the constant of proportionality between stress and strain. The chapter also discusses stress and strain calculations for materials with non-uniform cross-sections, as well as examples of stress and strain problems.
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
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 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 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.
A graphical method to determine the shear force and bending moment distribution along a simply supported beam is given. A suitable example is used to illustrate the major steps in the process.
A sample calculation for the determination of the maximum stress values is also given.
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.
The document discusses seismic base isolation as an earthquake-resistant technique for buildings in regions of low to moderate seismicity. It analyzes a 10-story residential building in Dhaka, Bangladesh using base isolators. Static and dynamic analyses were performed on the building both with and without isolators. The results showed that the use of isolators significantly reduced base shear and moment demands. While isolators increase initial costs, the reduction in reinforcement requirements overall makes base isolation more cost-effective. Thus, the document concludes that base isolation is a suitable technique even for buildings in areas of low seismicity.
This document summarizes solutions to three theoretical questions:
1) Describes how to use measurements of gravitational redshift to determine the mass and radius of a star.
2) Explains Snell's law and how it can be used to determine the path of light rays through a medium with a linear change in refractive index.
3) Analyzes the motion of a floating cylindrical buoy, determining equations for its vertical and rotational oscillations and relating the periods.
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.
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
1) This document discusses beam shear and bending moment diagrams. It provides examples of calculating reactions, shear, and bending moment at different points along simple beams with various load configurations.
2) The maximum shear value is identified as well as locations where the shear passes through zero.
3) Examples show constructing and analyzing shear and moment diagrams for beams with 1-3 loads, including point and distributed loads. Critical values like maximum bending moment are determined.
The document discusses cables and arches, which carry loads and develop stresses. Cables experience mainly tensile stresses, while arches experience compressive stresses. The document analyzes cables subjected to concentrated and uniformly distributed loads. It describes determining cable tensions, sags, and total length for a cable with concentrated loads. For uniformly distributed loads, the document derives the parabolic equation that describes the cable profile. It also discusses additional considerations for cable-supported structures like wind forces and ensuring stability.
1. Determine the reinforcement ratio ρ.
2. Calculate the modular ratio n based on concrete and steel properties.
3. Use an iterative process to locate the neutral axis depth kd by solving for the parameter k.
4. With k determined, calculate the moment arm j.
5. Compute the moment capacity as Mallow = R * b * d^2, where R is the resisting stress block parameter dependent on k and j.
call for papers, research paper publishing, where to publish research paper, journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJEI, call for papers 2012,journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, research and review articles, engineering journal, International Journal of Engineering Inventions, hard copy of journal, hard copy of certificates, journal of engineering, online Submission, where to publish research paper, journal publishing, international journal, publishing a paper, hard copy journal, engineering journal
This document summarizes a mathematical model of prey-predator populations in marine life. The model divides a bounded region into two circular patches with different conditions. Finite element methods are used to study how fish population densities change over time and position. Equations model the diffusion and growth of prey and predator populations between patches. The domain is discretized and a variational formulation is derived. Graphs of population density versus radial distance can be plotted for different times.
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.
Electrical current, voltage, resistance, capacitance, and inductance are a few of the basic elements of electronics and radio. Apart from current, voltage, resistance, capacitance, and inductance, there are many other interesting elements to electronic technology. ... Use Electronics Notes to learn electronics online.
This document provides instructions and questions for a structural design exam. It consists of 4 questions. Students must answer question 1 and any other two questions. Question 1 involves calculating bending moments, designing reinforcement, and determining shear capacity for concrete beams. Question 2 involves checking the adequacy of steel sections and designing a bolt connection. Question 3 uses force methods to determine reactions and draws shear and bending moment diagrams. Question 4 analyzes a frame under vertical and lateral loads to determine reactions and internal forces at specific points. The document also includes relevant design formulas and appendices on load combinations, bending moment coefficients, and steel design strengths.
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 chapter discusses stress and strain in materials subjected to tension or compression. It defines stress as the load applied over the cross-sectional area. Strain is defined as the change in length over the original length. Hooke's law states that stress is proportional to strain for elastic materials. Young's modulus is the constant of proportionality between stress and strain. The chapter also discusses stress and strain calculations for materials with non-uniform cross-sections, as well as examples of stress and strain problems.
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
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 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 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.
A graphical method to determine the shear force and bending moment distribution along a simply supported beam is given. A suitable example is used to illustrate the major steps in the process.
A sample calculation for the determination of the maximum stress values is also given.
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.
The document discusses seismic base isolation as an earthquake-resistant technique for buildings in regions of low to moderate seismicity. It analyzes a 10-story residential building in Dhaka, Bangladesh using base isolators. Static and dynamic analyses were performed on the building both with and without isolators. The results showed that the use of isolators significantly reduced base shear and moment demands. While isolators increase initial costs, the reduction in reinforcement requirements overall makes base isolation more cost-effective. Thus, the document concludes that base isolation is a suitable technique even for buildings in areas of low seismicity.
This document summarizes solutions to three theoretical questions:
1) Describes how to use measurements of gravitational redshift to determine the mass and radius of a star.
2) Explains Snell's law and how it can be used to determine the path of light rays through a medium with a linear change in refractive index.
3) Analyzes the motion of a floating cylindrical buoy, determining equations for its vertical and rotational oscillations and relating the periods.
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.
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
1) This document discusses beam shear and bending moment diagrams. It provides examples of calculating reactions, shear, and bending moment at different points along simple beams with various load configurations.
2) The maximum shear value is identified as well as locations where the shear passes through zero.
3) Examples show constructing and analyzing shear and moment diagrams for beams with 1-3 loads, including point and distributed loads. Critical values like maximum bending moment are determined.
The document discusses cables and arches, which carry loads and develop stresses. Cables experience mainly tensile stresses, while arches experience compressive stresses. The document analyzes cables subjected to concentrated and uniformly distributed loads. It describes determining cable tensions, sags, and total length for a cable with concentrated loads. For uniformly distributed loads, the document derives the parabolic equation that describes the cable profile. It also discusses additional considerations for cable-supported structures like wind forces and ensuring stability.
1. Determine the reinforcement ratio ρ.
2. Calculate the modular ratio n based on concrete and steel properties.
3. Use an iterative process to locate the neutral axis depth kd by solving for the parameter k.
4. With k determined, calculate the moment arm j.
5. Compute the moment capacity as Mallow = R * b * d^2, where R is the resisting stress block parameter dependent on k and j.
call for papers, research paper publishing, where to publish research paper, journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJEI, call for papers 2012,journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, research and review articles, engineering journal, International Journal of Engineering Inventions, hard copy of journal, hard copy of certificates, journal of engineering, online Submission, where to publish research paper, journal publishing, international journal, publishing a paper, hard copy journal, engineering journal
This document summarizes a mathematical model of prey-predator populations in marine life. The model divides a bounded region into two circular patches with different conditions. Finite element methods are used to study how fish population densities change over time and position. Equations model the diffusion and growth of prey and predator populations between patches. The domain is discretized and a variational formulation is derived. Graphs of population density versus radial distance can be plotted for different times.
This document discusses SQL injection attacks in banking transactions and methods to prevent them. It begins with an abstract discussing how SQL injections are a major security issue for banking applications and can be used to access secret information like usernames and passwords or bank databases. The document then provides examples of SQL injection attacks on banks, describes how hackers perform SQL injections, and discusses approaches like input validation, static query statements, and least privilege to prevent injections. It also introduces tools like Amnesia and the X-Log Authentication technique to detect and block injection attacks. The conclusion is that Amnesia and X-Log Authentication are effective techniques for preventing SQL injections in banking transactions.
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
call for papers, research paper publishing, where to publish research paper, journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJEI, call for papers 2012,journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, research and review articles, engineering journal, International Journal of Engineering Inventions, hard copy of journal, hard copy of certificates, journal of engineering, online Submission, where to publish research paper, journal publishing, international journal, publishing a paper, hard copy journal, engineering journal
This document discusses the impact of data mining on business intelligence. It begins by defining business intelligence as using new technologies to quickly respond to changes in the business environment. Data mining is an important part of the business intelligence lifecycle, which includes determining requirements, collecting and analyzing data, generating reports, and measuring performance. Data mining allows businesses to access real-time, accurate data from multiple sources to improve decision making. Using business intelligence and data mining techniques can help businesses become more efficient and make better decisions to increase profits and customer satisfaction. The expected results of applying business intelligence include improved decision making through accurate, timely information to support organizational goals and strategic plans.
This document discusses shear force and bending moment in beams. It defines different types of beams, loads, and supports. Equations for calculating shear force and bending moment are presented for various beam configurations under different loading conditions, including cantilever beams with point loads and uniform loads, and simply supported beams with point and uniform loads. Diagrams illustrating the variation of shear force and bending moment along beams are shown as examples.
The document discusses modifications to the roof bar (girder) support system used in underground coal mining. It describes issues with the existing design where bending stresses cause premature failure of the roof bar. The author proposes a modified design where wooden lagging is placed directly above props to transfer support resistance to the roof rock, rather than through the bending roof bar. This eliminates roof bar bending and increases support resistance to improve mine stability. Finite element analysis of the new design shows reduced stresses in the roof bar, suggesting it will provide safer, more economical roof support for thick seam underground mining.
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.
Lesson 04, shearing force and bending moment 01Msheer Bargaray
1) The document discusses shear forces and bending moments in beams subjected to different load types. It defines types of beams, supports, loads, and sign conventions for shear forces and bending moments.
2) Examples are provided to calculate shear forces and bending moments at different points along beams experiencing simple loading cases such as a uniformly distributed load on a cantilever beam.
3) Methods for determining the shear force and bending moment in an overhanging beam subjected to a uniform load and point load are demonstrated. Diagrams and free body diagrams are used to solve for the reactions and internal forces.
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.
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 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.
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 bending moment and shear force for beams. It contains 3 main sections:
1) An introduction to bending moment, shear force, and the relationship between loading, shear force and bending moment.
2) How to draw shear force diagrams and bending moment diagrams by calculating shear forces and bending moments at critical points along a beam.
3) How to calculate reactions for simply supported beams and cantilever beams by applying equations of equilibrium. Several examples of calculating reactions are provided.
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 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.
The document summarizes the analysis of reinforced concrete beam cross sections to determine their moment of resistance at the ultimate limit state. It outlines the key assumptions of the strength design method and describes the behavior of beams under small, moderate and ultimate loads. It also discusses balanced, under-reinforced and over-reinforced beam sections, and introduces the concept of the equivalent stress block to simplify calculations. Worked examples are provided to demonstrate how to determine the depth of the neutral axis and moment of resistance for various beam cross sections.
1) The document provides answers to examination-style questions about magnetic fields and forces. It defines common units like the tesla and newton and the equation for magnetic force F = BIl.
2) Sample questions are worked through, applying the magnetic force equation to problems involving wires in magnetic fields and the forces between current-carrying conductors. Diagrams are included.
3) Further examples calculate the radius of circular paths taken by moving charged particles in magnetic fields, and discuss how isotope separation can be achieved using velocity selectors that employ electric and magnetic fields.
4) The final examples calculate properties of different particles like antiprotons and pions moving in magnetic fields, and explain how the semic
This document provides an overview of Karnaugh maps, which are a systematic method for obtaining simplified Boolean expressions in sum-of-products form. It discusses Venn diagrams, 2-variable, 3-variable, 4-variable, and larger K-maps. The key aspects covered are how K-maps are organized based on variables, how to identify groups of adjacent 1's to obtain product terms, and how larger groups correspond to terms with fewer literals in the simplified expression. The goal of using K-maps is to obtain expressions with the fewest possible terms and literals.
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Lec 10-flexural analysis and design of beamnsCivil Zone
The document discusses the concepts of balanced steel ratio, tension controlled sections, transition sections, compression controlled sections, strength reduction factors, maximum steel ratio, and minimum reinforcement for flexural members in reinforced concrete beams. The balanced steel ratio corresponds to the amount of steel that yields at the same time as the concrete crushes. Tension controlled sections have a steel strain over 0.005 when concrete strain is 0.003. Transition sections have steel strain between yield and 0.005 when concrete is at 0.003. Compression controlled sections have steel strain under yield when concrete is at 0.003.
The document discusses the concepts of balanced steel ratio, tension controlled sections, transition sections, compression controlled sections, strength reduction factors, maximum steel ratio, and minimum reinforcement for flexural members in reinforced concrete beams. The balanced steel ratio corresponds to the amount of steel that yields at the same time as the concrete crushes. Tension controlled sections have a steel strain over 0.005 when concrete strain is 0.003. Transition sections have steel strain between yield and 0.005 when concrete is at 0.003. Compression controlled sections have steel strain under yield when concrete is at 0.003.
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Sheryar Bismil
Student of Mirpur University of Science & Technology(MUST).
Student of Final Year Civil Engineering Department Main campus Mirpur.
Here we Gonna to learn about the basic to depth wise study of Plan Reinforced Concrete-i.
From basis terminology to wide information about the analysis and design of Concrete member like column,Beam,Slab,etc.
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More from International Journal of Engineering Inventions www.ijeijournal.com (20)
International Journal of Engineering Inventions (IJEI)
1. International Journal of Engineering Inventions
ISSN: 2278-7461, www.ijeijournal.com
Volume 1, Issue 9 (November2012) PP: 13-19
Vibration Analysis of a Portal Frame Subjected To a Moving
Concentrated Load
Ms. S. J. MODAK1, H. V. HAZARE2,
1
Assistant Professor in Civil Engg., Ramdeobaba College of Engineering & Management, Nagpur – 440 009
2
Professor of Civil Engg., Priyadarshini College of Engineering, Nagpur – 440 019
Abstract:––The objective of the investigation is to mathematically simulate dynamics and vibrations of a portal frame
subjected to a concentrated load moving on it’s horizontal member with a certain constant velocity. This portal frame is a
basic structure of a low length single span bridge. The emphasis is on an approach to model forced vibrations of the vertical
members of the portal frame.
Keywords:––Bridges, Columns, Portal Frame, Influence Line, Vibrations.
I. CONSTRUCTION OF A SHORT LENGTH BRIDGE
Fig. 1 is a schematic presentation of a short length bridge. The length is so short that the basic structure of a bridge
is a simple one span portal frame 01 AB 02. The width of the bridge is also fairly small so that it could be considered as a
particular case of a girder bridge [1]. The material of the frame is Mild Steel (M.S.). The philosophy of the analysis is
explained through a representative small scale structure with dimensions length of AB = 1005 mm, width = 50 mm and
thickness is = 5mm. The vertical members 0 1A and 02B are geometry wise identical. The material of 0 1A & that of 02B is
also M.S. A vehicle with total weight W is moving on AB with a constant velocity.
W
X’
A B
01 02
L
Fig. 1 : schematics of a portal frame for a short bridge
The objective of the investigation is to estimate vibration response of 01A & 02B.
II. FREE BODY DIAGRAMS OF A PORTAL FRAME
The free body diagrams of the members of the portal frame described in Fig. 1 are detailed in Fig. 2.
W
X’
A B
x= 5mm
WA ----------990mm WB
Figure 2 (a)
Numbers in the square brackets denote references listed at the end of the paper.
ISSN: 2278-7461 www.ijeijournal.com P a g e | 13
2. Vibration Analysis of a Portal Frame Subjected…
WA WA’ WA’ WB WB’
A B
WA’X = M’x CW WB’
M’x CW
gives
MB’X ccw
01 02
Fig. 2(b) The net Column Fig.2(c) The net column
action = WA + WA’ action = WB + WB’
Fig. 2 : free body diagrams of a portal frame
Fig. 2(a) describes Free Body Diagram of Horizontal beam portion AB of the portal frame. W is a concentrated
load (in fact the weight of the vehicle) acting on AB at an arbitrary distance x’ from left end A. This x’ is all the while
changing as the vehicle moves from left to right with a constant velocity V. WA is a reaction of left support 01A on AB at A
whereas WB is a reaction of right support 02B on AB at B.
At a section of AB at a distance x = 5mm (in this specific case) anticlock wise couple is exerted by support 0 1A on
AB. x = 5mm because linear dimension of a cross section of 0 1A at the top end = 10.0 mm. This meant top section of
member 01A is exerting on section X in the C.C.W. direction a moment because member 01A is imposing a moment restraint
on AB to see that complete portion worth 10mm from end A of the member AB of-course on it’s bottom side has to remain
straight. Hence, the slope of section x of AB from A at 5mm on account of probable free displacement of neutral axis of AB
as a simply supported beam due to concentrated load W is prevented.
In view of above detailed analysis for free body diagram of member AB of the portal frame the free body diagrams
of the other members 01A and 02B which are vertical supports of AB get deduced logically as described in Figures 2 (b) and
2(c) respectively for 01A & 02B.
As per the free body diagrams of Figures 2(b) and 2(c) the total axial load on left support 0 1A is summation of WA
& WA’ whereas the same for the support 02B is WB & WB’. The exact quantitative relationships for WA & WA’ and that of
WB & WB’ is derived in the following section.
III. QUANTITATIVE FORCE ANALYSIS OF THE PORTAL FRAME
3.1 Analysis of Member AB
Redrawing the Free Body Diagram of Member AB (Fig. 2(a) for the sake of ready reference) as shown.
W
X’
A B
x= 5mm
RA -------------- 990mm RB
1000mm
Fig. 2(a) : Pepete FDB of AB redrawn for read reference as above
Refer Fig. 2(a) Pepete. The reactions RA and RB would be as under. Taking moments about point A and applying the
condition of ∑ M=0 of static equilibrium.
-Wx’ + RB x L = 0
……………………… (3.1)
ISSN: 2278-7461 www.ijeijournal.com P a g e | 14
3. Vibration Analysis of a Portal Frame Subjected…
Similarly, it can be proved that
…………………….(3.2)
The bending moment at x = x’ = 5mm in this specific case in which numerically x = x’ = 5mm is denoted as Mx’
Mx’ = RAx
Substituting for RA from Equation (3.2)
………..……… (3.3)
Similarly, the bending moment near support B at x = x’ = 5mm, in this specific case in which numerically x = x’ = 5mm is
denoted as Mx’ (near support B)
Mx’ (near support B) = RB x
Substituting the value of RB from Equation – (3.1) in the above equation
………………….. (3.4)
These two bending moments are trying to induce slopes dy/dx at sections x = x’ from both the supports A & B
respectively in member AB treating AB as a simply supported beam. These moments and associated transverse deflection y
at these sections are such that they are trying to turn the cross sections of member AB respectively C.W. and C.C.W.
However, these elastic deformations are restrained by members 0 1A & 02B. Hence, top sides of 01A & 02B are trying to
resent these deformations. This is only possible if top side of 0 1A is exerting C.C. W. moment at x = x’ near A and if top
side of 02B is exerting C.W. moment at x = x’ near B respectively.
To this action of 01A & 02B on member AB, the member AB will exert equal & opposite reactions on 01A & 02B.
With the result, member AB will exert moments Mx’ given by Eq. (3.3) on top side of 0 1A C.W. and member AB will exert
moment Mx’ given by Eq. (3.4) C.C. W. on top side of member 0 2B.
These two bending moments are Mx’ near A and Mx’ (near support B) respectively will have senses C.W. and
C.C. W. for supports 01A and 02B as described in Figures 2(b) and 2(c) respectively.
Similarly, the axial forces as exerted by AB on supports 0 1A and 02B will be both downwards and with magnitudes
as given respectively by Equations (3.2) and (3.1) but denoted as WA’ and WB’ in figures 2(b) and 2(c) respectively.
Thus, total vertically downwards action of AB on 01A is if denoted by WTD01A
………………….. (3.5)
Similarly if WTD02B is total downwards action as exerted by AB on 02B then
………………………. (3.6)
In equations (3.5) and (3.6) while deciding force actions of moments exerted by AB on top side of member 01A
and 02B these moments are assumed to be equivalent to two forces of equal magnitudes and opposite in senses acting along
the axis of members 01A & 02B. These are shown in free body diagrams Fig. 2(b) and Fig.2(c) respectively by forces WA,
WA’, WB, WB’. The forces WA’ and WB are assumed to have moment arm slightly more than X say more only by 1mm.
Hence, they come out be equal to the second terms in both the equations (3.5) and (3.6) given above.
Further Equations (3.5) and (3.6) can be rearranged in the forms given below.
……………… (3.5.a)
and
…………………. (3.6.a)
In fact if it is assumed that the velocity of the vehicle moving over the bridge is V and it is constant and t is the
time elapsed from the instant the vehicle has entered the bridge from left end till the time it has covered the distance x’
where of-course 0< x’ <L then one can say that
X’ = Vt …………………….. (3.7)
Substituting for x’ from Equation (3.7) in to equation (3.5.a) and (3.6.a) one gets final action on members 0 1A & 02B
simulated by the below stated equations
…………………….. (3.8)
……………………..(3.9)
ISSN: 2278-7461 www.ijeijournal.com P a g e | 15
4. Vibration Analysis of a Portal Frame Subjected…
IV. VIBRATION RESPONSE OF 01A AND 02B
The members 01A and 02B the supports of a bridge are subjected to time varying forces as described by Equations
(3.8) and (3.9) respectively. Since, the material of supports 0 1A & 02B is elastic and that it will experience material
hysteresis, the supports are subjected to longitudinal vibrations.
The longitudinal vibrations can be ascertained by various approaches as regards various distributed mass, elasticity and
damping of the material of column.
(1) Considering entire column represented by single mass, single stiffness, single damper system, popularly
abbreviated in the science of vibrations of structures as SDF system
OR
(2) Considering entire column represented by multi mass, multi stiffness, multi damper system say considering 5
Lumped mass system or 5 DOF system.
OR
(3) Considering distributed mass, distributed elasticity, distributed damping system.
In the three cases stated above the column excitation i.e. the external force acting on the two columns is as described in
Equations (3.8) and (3.9) respectively.
(4) In the force analysis discussed so far it is considered as if only one vehicle is entering the bridge and it moves over
the bridge with constant velocity.
It may so happen that the entry of vehicles on the bridge may be in a random manner, their weights may be
different, their velocities may be different, same may be moving with some acceleration and retardation over the bridge. The
accelerations or retardations may be constant or variable. For each one of the above cases the loading pattern on bridge
supports 01A and 02B may be different. Further, for all these loading patterns in view of all the three styles of vibrations
described earlier, it may become imperative to ascertain vibration response. This may be considered as situation of random
excitation & consequently the random vibrations of the columns of a bridge
(5) Harmonic Excitation : One more interesting pattern of vehicle entry could be such that vehicles of the same make
are getting admitted from left end with same velocities and with a definite spacing. In this case the loading pattern on 01A
and 02B could be as shown below in Fig. 3 and Fig. 4.
WTD01A
T’ T’ T’
Fig. 3 : Load pattern on 01A
T = time to complete travel on bridge
L = Length of the bridge
V1 = Vehicle Velocity
T’ = Constant time gap between two consecutive entries.
WTD01B
T’ T’ T’
Fig. 4 : Load pattern on 02B
T = time to complete travel on bridge
L = Length of the bridge
V1 = Vehicle Velocity
T’ = Constant time gap between two consecutive entries
WTD01A
WTD01B
WTD01A
WTD02B
Figure 5 : Influence lines for the bridge
ISSN: 2278-7461 www.ijeijournal.com P a g e | 16
5. Vibration Analysis of a Portal Frame Subjected…
As the load patterns described in Fig. 3 and Fig. 4 one can say that the load pattern is a periodically varying
function. Such a function can be represented by Harmonic Series [2]. In view of every harmonic component one can decide
what could probably be the resonant frequencies. It should then be seen that none of these resonant frequencies are close to
the natural frequencies of either 01A or 01B in order to avoid induction of higher value of stress under vibrations [3] due to
complete or partial resonance.
This aspect should be checked in view of all the 1 to 3 patterns of assumptions of distribution of mass, elasticity and
damping.
The paper essentially addresses the issue of vibrations of bridge column which is much less addressed so far [4 to 69].
V. CONCLUSION
The first section of the paper details the scope of bridge column vibrations of a fairly small length bridge which
may be considered as a portal frame. Sections 2 & 3 respectively detail qualitative and quantitative force analysis of a portal
frame with a concentrated load acting only such that the load changes continuously its position from one end to the other.
Section 4 details fairly detailed possibility of deciding vibration response of bridge columns. Detailing every possibility of
vibration excitation and OR mass-stiffness-damping, distribution of two columns will precipitate individual paper. This is
what is planned as a future extension of this work.
In addition the paper includes the influence line [70] of the bridge as depicted in Fig. 5.
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