The document discusses influence lines, which graphically depict the variation of structural quantities (bending moment, shear force, etc.) due to a moving unit load. It provides examples of drawing influence line diagrams for bending moment and shear force at specific points on simply supported beams. The maximum value of a structural quantity occurs when the moving load is located at the point of maximum ordinate on the influence line diagram. Uniformly distributed loads produce the greatest effect when placed over regions where the ordinates have the same sign.
This document discusses lateral earth pressure and its importance in retaining wall design. It defines lateral earth pressure as the pressure soil exerts horizontally. Lateral earth pressure depends on soil shear strength, pore water pressure, and equilibrium state. It is important for designing structures like retaining walls, bridges, and tunnels. The document discusses coefficient of lateral earth pressure (K), and the three states: at-rest (Ko), active (Ka), and passive (Kp) pressure. It also presents Coulomb and Rankine theories for calculating earth pressure and describes investigation methods and lateral wall supports like gravity, cantilever, anchored, soil-nailed, and reinforced walls. Geofoam is discussed as a method to reduce lateral stresses in
The document discusses different methods of designing concrete structures, focusing on the limit state method. It describes the limit state method's goal of achieving an acceptable probability that a structure will not become unsuitable for its intended use during its lifetime. The document then discusses stress-strain curves for concrete and steel. It covers stress block parameters and equations for calculating the depth of the neutral axis and moment of resistance for singly reinforced concrete beams. The document concludes by providing examples of analyzing an existing beam section and designing a new beam section.
This document discusses dynamics of discrete systems, including:
- Single degree of freedom systems (SDOF), their equations of motion, and response to forced and free vibrations
- Damping mechanisms like viscous damping and how they affect system response
- Multiple degree of freedom systems and their equations of motion
It provides examples of undamped and damped SDOF system responses to different loading conditions.
This document summarizes the planning and design calculations for a pre-stressed concrete beam with the following parameters:
1. The required bending moment (Mt) is 350 ton-meters. The concrete compressive strength (f'c) is 47 MPa.
2. The initial dimensions of the beam are calculated as 200 cm height (h) and 4339.6 cm^2 cross-sectional area (Ab).
3. The final design meets the required bending moment of 350 ton-meters with a uniform prestress force (q) of 2285.71 kg/m distributed over the beam length. Stresses in the concrete are calculated to remain below the allowable limits.
This document contains 3 engineering problems:
1) A steel wire with a 1/16 in. gap must support a 50-lb block without contact; the location is needed.
2) Steel links in a structure will experience forces when a 600-lb load is applied; the forces and deflection must be found.
3) Steel reinforcing bars were heated to fit in a brass structure, then cooled; the required temperature change and resulting stress in the brass must be calculated.
Compressive strength and Flexural of Hardened Concrete | Jameel AcademyJameel Academy
This report details tests conducted to determine the compressive and flexural strength of hardened concrete. The compressive strength was tested on concrete cubes with an average result of 32.8 MPa, meeting the design strength of 24 MPa. The flexural strength was tested on concrete prisms and resulted in 6.4 MPa. While lower than compressive strength as expected, this shows the concrete can resist compression and tension loads required for construction projects. In conclusion, the concrete met design specifications and can be used safely in construction.
This document discusses vertical curves used in transportation design. Vertical curves provide a smooth transition between different road or rail grades. They are designed using parabolic equations to maintain a constant rate of change in slope. The key points are:
- Vertical curves connect two different grades using a parabolic shape.
- Their design ensures a constant rate of change in slope for driver comfort.
- The general parabolic equation and methods for computing curve elements like high/low points and elevations at different points are presented.
This document discusses lateral earth pressure and its importance in retaining wall design. It defines lateral earth pressure as the pressure soil exerts horizontally. Lateral earth pressure depends on soil shear strength, pore water pressure, and equilibrium state. It is important for designing structures like retaining walls, bridges, and tunnels. The document discusses coefficient of lateral earth pressure (K), and the three states: at-rest (Ko), active (Ka), and passive (Kp) pressure. It also presents Coulomb and Rankine theories for calculating earth pressure and describes investigation methods and lateral wall supports like gravity, cantilever, anchored, soil-nailed, and reinforced walls. Geofoam is discussed as a method to reduce lateral stresses in
The document discusses different methods of designing concrete structures, focusing on the limit state method. It describes the limit state method's goal of achieving an acceptable probability that a structure will not become unsuitable for its intended use during its lifetime. The document then discusses stress-strain curves for concrete and steel. It covers stress block parameters and equations for calculating the depth of the neutral axis and moment of resistance for singly reinforced concrete beams. The document concludes by providing examples of analyzing an existing beam section and designing a new beam section.
This document discusses dynamics of discrete systems, including:
- Single degree of freedom systems (SDOF), their equations of motion, and response to forced and free vibrations
- Damping mechanisms like viscous damping and how they affect system response
- Multiple degree of freedom systems and their equations of motion
It provides examples of undamped and damped SDOF system responses to different loading conditions.
This document summarizes the planning and design calculations for a pre-stressed concrete beam with the following parameters:
1. The required bending moment (Mt) is 350 ton-meters. The concrete compressive strength (f'c) is 47 MPa.
2. The initial dimensions of the beam are calculated as 200 cm height (h) and 4339.6 cm^2 cross-sectional area (Ab).
3. The final design meets the required bending moment of 350 ton-meters with a uniform prestress force (q) of 2285.71 kg/m distributed over the beam length. Stresses in the concrete are calculated to remain below the allowable limits.
This document contains 3 engineering problems:
1) A steel wire with a 1/16 in. gap must support a 50-lb block without contact; the location is needed.
2) Steel links in a structure will experience forces when a 600-lb load is applied; the forces and deflection must be found.
3) Steel reinforcing bars were heated to fit in a brass structure, then cooled; the required temperature change and resulting stress in the brass must be calculated.
Compressive strength and Flexural of Hardened Concrete | Jameel AcademyJameel Academy
This report details tests conducted to determine the compressive and flexural strength of hardened concrete. The compressive strength was tested on concrete cubes with an average result of 32.8 MPa, meeting the design strength of 24 MPa. The flexural strength was tested on concrete prisms and resulted in 6.4 MPa. While lower than compressive strength as expected, this shows the concrete can resist compression and tension loads required for construction projects. In conclusion, the concrete met design specifications and can be used safely in construction.
This document discusses vertical curves used in transportation design. Vertical curves provide a smooth transition between different road or rail grades. They are designed using parabolic equations to maintain a constant rate of change in slope. The key points are:
- Vertical curves connect two different grades using a parabolic shape.
- Their design ensures a constant rate of change in slope for driver comfort.
- The general parabolic equation and methods for computing curve elements like high/low points and elevations at different points are presented.
The document provides information about a 21 meter long prestressed concrete pile driven into sand. The pile has an allowable working load of 502 kN, with an octagonal cross-section of 0.356 meters diameter and area of 0.1045 m^2. Skin resistance supports 350 kN of the load and point bearing the rest. The document requests calculating the elastic settlement of the pile given its properties, the load distribution, and soil parameters.
The document outlines a course plan for a foundation engineering course. It includes 9 units that will be covered: introduction and site investigation, earth pressure, shallow foundations, pile foundations, well foundations, slope stability, retaining walls, and soil stabilization. It provides details on the number of lectures for each unit and the topics that will be covered in each lecture. Some key topics include shallow foundation design methods, pile load testing, earth pressure theories, and slope stability analysis techniques. References for the course are also provided.
Chapter 4 earth work and quantities newBashaFayissa1
This document discusses earthwork quantities and calculations for highway construction projects. It covers topics such as classification of excavated materials, shrinkage and swell factors, methods for calculating cross-sectional areas and volumes, mass diagrams, distribution of excavated materials, limits of economical haul, and definitions of relevant terms. The key aspects are determining excavation, fill, and borrow quantities; accounting for shrinkage and swelling; and optimizing material distribution to minimize haul costs.
This document discusses reinforced concrete columns. Columns act as vertical supports that transmit loads to foundations. Columns may fail due to compression failure, buckling, or a combination. Short columns are more prone to compression failure, while slender columns are more likely to buckle. Column sections can be square, circular, or rectangular. The dimensions and bracing affect whether a column is classified as short or slender. Longitudinal reinforcement and links are designed to resist axial loads and moments based on the column's effective height and end conditions. Design charts are used to determine reinforcement for columns with axial and uniaxial bending loads. Examples show how to design column reinforcement.
This document discusses methods for determining areas, volumes, centroids, and moments of inertia of basic geometric shapes. It begins by introducing the method of integration for calculating areas and volumes. Standard formulas are provided for areas of rectangles, triangles, circles, sectors, and parabolic spandrels. Formulas are also provided for volumes of parallelepipeds, cones, spheres, and solids of revolution. The concepts of center of gravity, centroid, and center of mass are defined. Equations are given for calculating the centroids of uniform bodies, plates, wires, and line segments. Methods for finding centroids of straight lines, arcs, semicircles, and quarter circles are illustrated.
Lec06 Analysis and Design of T Beams (Reinforced Concrete Design I & Prof. Ab...Hossam Shafiq II
1) T-beams are commonly used structural elements that can take two forms: isolated precast T-beams or T-beams formed by the interaction of slabs and beams in buildings.
2) The analysis and design of T-beams considers the effective flange width provided by slab interaction or the dimensions of an isolated precast flange.
3) Two methods are used to analyze T-beams: assuming the stress block is in the flange and using rectangular beam theory, or using a decomposition method if the stress block extends into the web.
This document discusses the design of compression members subjected to axial load and biaxial bending. It introduces the concept of biaxial eccentricities and explains that columns should be designed considering possible eccentricities in two axes. The document outlines the method suggested by IS 456-2000, which is based on Breslar's load contour approach. It relates the parameter αn to the ratio of Pu/Puz. Finally, it provides a step-by-step process for designing the column section, which involves determining uniaxial moment capacities, computing permissible moment values from charts, and revising the section if needed. It also briefly mentions the simplified method according to BS8110.
- The document discusses stress analysis of composite beams made of two materials like concrete and steel.
- It explains the concept of transforming the cross-section of the composite beam into an equivalent cross-section of one material using the modular ratio.
- The maximum stresses in each material can then be calculated from the transformed section and adjusted using the modular ratio to get the true stresses.
Numerical problem bearing capacity terzaghi , group pile capacity (usefulsear...Make Mannan
A 1m wide strip footing is located 0.8m below ground in a c-φ soil. The soil properties are given. Using Terzaghi's analysis with a factor of safety of 3, the safe bearing capacity is calculated to be 112.1 kN/m^2.
A 2m x 3m rectangular footing at a depth of 1.5m in a different c-φ soil is considered. Using Terzaghi's analysis, the safe bearing capacities are calculated to be 471.7 kN/m^2 based on net ultimate capacity and 453.7 kN/m^2 based on ultimate capacity, both with a factor of safety of 3.
This document discusses stresses in beams, specifically shear stresses. It covers five lectures on related topics like bending moment and shear force diagrams, bending stresses, shear stresses, deflection, and torsion. For shear stresses in beams with rectangular cross-sections, it explains that both normal and shear stresses are developed when loads produce both bending moments and shear forces. The maximum shear stress occurs at the center of the beam and its distribution is parabolic. Equations are provided for calculating shear stress values.
This document discusses soil sampling and exploration. It describes different types of soil samples including disturbed, undisturbed, representative and non-representative samples. It discusses criteria for obtaining undisturbed samples and transporting and preserving samples. Different types of soil samplers are described. Factors related to planning a soil exploration program such as spacing and depth of borings are covered. Components of a soil exploration report are outlined.
Sample calculation for design mix of concreteSagar Vekariya
This document provides details on designing a concrete mix with a characteristic compressive strength of 35 MPa at 28 days. The mix uses M35 grade cement, medium sand, and a coarse aggregate of 20mm angular gravel mixed with 10mm gravel in a 70:30 ratio. The mix design calculations determine a water-cement ratio of 0.40, a cement content of 370 kg/m3, and aggregate contents of 1150 kg/m3 for 20mm gravel and 345 kg/m3 for 10mm gravel. The final concrete mix is specified with weight proportions of cement, water, fine aggregate, 20mm coarse aggregate, 10mm coarse aggregate, and admixture.
This document provides information and formulas for calculating bar bending schedules. It discusses hook length, bend length, overlap length, and how to prepare a bar bending schedule table. Formulas are given for calculating hook length, bend length for cranked and corner bars, and overlap length for tension and compression members. An example bar bending schedule is also shown for a sample RCC column, calculating the number of bars, cutting lengths, total bar lengths, and total steel weight required.
This document provides a comprehensive summary of strong-motion attenuation relationships published between 1969 and 2000 for predicting peak ground acceleration and spectral ordinates from earthquakes. It reviews 70 published attenuation models from researchers around the world, organized chronologically. Each model is briefly described in 1-2 sentences with the author(s) and year. The purpose is to aid engineers and seismologists in selecting appropriate ground motion prediction models for earthquake engineering and seismic hazard analyses.
Geotechnical Engineering-II [Lec #25: Coulomb EP Theory - Numericals]Muhammad Irfan
Class notes of Geotechnical Engineering course I used to teach at UET Lahore. Feel free to download the slide show.
Anyone looking to modify these files and use them for their own teaching purposes can contact me directly to get hold of editable version.
Chapter 5: Axial Force, Shear, and Bending MomentMonark Sutariya
1. A beam can experience three internal forces at a section - axial force, shear, and bending moment. Even for planar beams, all three forces may develop.
2. There are three types of supports - roller/link, pin, and fixed. Roller/link supports resist one force, pin supports resist two forces, and fixed supports resist two forces and a moment.
3. Beams can experience different load types - concentrated, uniform distributed, and varying distributed loads. Methods are presented to calculate the shear, axial, and bending effects of these loads on beams.
This presentation is about RCC. one can find most of the information about RCC with architecture in mind. Structure Design - 2 Semester 2 B. Arch Notes
The document provides information about stress distribution in soil due to self-weight and surface loads. It discusses Boussinesq's formula for calculating vertical stress in soil due to a concentrated surface load. The formula shows that vertical stress is directly proportional to the load, inversely proportional to depth squared, and depends on the ratio of radius to depth. A table of coefficient values used in the formula for different ratios of radius to depth is also provided.
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.
The document provides information about mechanics of solids-I, including:
1) It describes different types of supports like simple supports, roller supports, pin-joint supports, and fixed supports. It also describes different types of loads like concentrated loads, uniformly distributed loads, and uniformly varying loads.
2) It discusses shear force as the unbalanced vertical force on one side of a beam section, and bending moment as the sum of moments about a section.
3) It explains the relationship between loading (w), shear force (F), and bending moment (M) for an element of a beam. The rate of change of shear force is equal to the loading intensity, and the rate of change of bending
The document provides information about a 21 meter long prestressed concrete pile driven into sand. The pile has an allowable working load of 502 kN, with an octagonal cross-section of 0.356 meters diameter and area of 0.1045 m^2. Skin resistance supports 350 kN of the load and point bearing the rest. The document requests calculating the elastic settlement of the pile given its properties, the load distribution, and soil parameters.
The document outlines a course plan for a foundation engineering course. It includes 9 units that will be covered: introduction and site investigation, earth pressure, shallow foundations, pile foundations, well foundations, slope stability, retaining walls, and soil stabilization. It provides details on the number of lectures for each unit and the topics that will be covered in each lecture. Some key topics include shallow foundation design methods, pile load testing, earth pressure theories, and slope stability analysis techniques. References for the course are also provided.
Chapter 4 earth work and quantities newBashaFayissa1
This document discusses earthwork quantities and calculations for highway construction projects. It covers topics such as classification of excavated materials, shrinkage and swell factors, methods for calculating cross-sectional areas and volumes, mass diagrams, distribution of excavated materials, limits of economical haul, and definitions of relevant terms. The key aspects are determining excavation, fill, and borrow quantities; accounting for shrinkage and swelling; and optimizing material distribution to minimize haul costs.
This document discusses reinforced concrete columns. Columns act as vertical supports that transmit loads to foundations. Columns may fail due to compression failure, buckling, or a combination. Short columns are more prone to compression failure, while slender columns are more likely to buckle. Column sections can be square, circular, or rectangular. The dimensions and bracing affect whether a column is classified as short or slender. Longitudinal reinforcement and links are designed to resist axial loads and moments based on the column's effective height and end conditions. Design charts are used to determine reinforcement for columns with axial and uniaxial bending loads. Examples show how to design column reinforcement.
This document discusses methods for determining areas, volumes, centroids, and moments of inertia of basic geometric shapes. It begins by introducing the method of integration for calculating areas and volumes. Standard formulas are provided for areas of rectangles, triangles, circles, sectors, and parabolic spandrels. Formulas are also provided for volumes of parallelepipeds, cones, spheres, and solids of revolution. The concepts of center of gravity, centroid, and center of mass are defined. Equations are given for calculating the centroids of uniform bodies, plates, wires, and line segments. Methods for finding centroids of straight lines, arcs, semicircles, and quarter circles are illustrated.
Lec06 Analysis and Design of T Beams (Reinforced Concrete Design I & Prof. Ab...Hossam Shafiq II
1) T-beams are commonly used structural elements that can take two forms: isolated precast T-beams or T-beams formed by the interaction of slabs and beams in buildings.
2) The analysis and design of T-beams considers the effective flange width provided by slab interaction or the dimensions of an isolated precast flange.
3) Two methods are used to analyze T-beams: assuming the stress block is in the flange and using rectangular beam theory, or using a decomposition method if the stress block extends into the web.
This document discusses the design of compression members subjected to axial load and biaxial bending. It introduces the concept of biaxial eccentricities and explains that columns should be designed considering possible eccentricities in two axes. The document outlines the method suggested by IS 456-2000, which is based on Breslar's load contour approach. It relates the parameter αn to the ratio of Pu/Puz. Finally, it provides a step-by-step process for designing the column section, which involves determining uniaxial moment capacities, computing permissible moment values from charts, and revising the section if needed. It also briefly mentions the simplified method according to BS8110.
- The document discusses stress analysis of composite beams made of two materials like concrete and steel.
- It explains the concept of transforming the cross-section of the composite beam into an equivalent cross-section of one material using the modular ratio.
- The maximum stresses in each material can then be calculated from the transformed section and adjusted using the modular ratio to get the true stresses.
Numerical problem bearing capacity terzaghi , group pile capacity (usefulsear...Make Mannan
A 1m wide strip footing is located 0.8m below ground in a c-φ soil. The soil properties are given. Using Terzaghi's analysis with a factor of safety of 3, the safe bearing capacity is calculated to be 112.1 kN/m^2.
A 2m x 3m rectangular footing at a depth of 1.5m in a different c-φ soil is considered. Using Terzaghi's analysis, the safe bearing capacities are calculated to be 471.7 kN/m^2 based on net ultimate capacity and 453.7 kN/m^2 based on ultimate capacity, both with a factor of safety of 3.
This document discusses stresses in beams, specifically shear stresses. It covers five lectures on related topics like bending moment and shear force diagrams, bending stresses, shear stresses, deflection, and torsion. For shear stresses in beams with rectangular cross-sections, it explains that both normal and shear stresses are developed when loads produce both bending moments and shear forces. The maximum shear stress occurs at the center of the beam and its distribution is parabolic. Equations are provided for calculating shear stress values.
This document discusses soil sampling and exploration. It describes different types of soil samples including disturbed, undisturbed, representative and non-representative samples. It discusses criteria for obtaining undisturbed samples and transporting and preserving samples. Different types of soil samplers are described. Factors related to planning a soil exploration program such as spacing and depth of borings are covered. Components of a soil exploration report are outlined.
Sample calculation for design mix of concreteSagar Vekariya
This document provides details on designing a concrete mix with a characteristic compressive strength of 35 MPa at 28 days. The mix uses M35 grade cement, medium sand, and a coarse aggregate of 20mm angular gravel mixed with 10mm gravel in a 70:30 ratio. The mix design calculations determine a water-cement ratio of 0.40, a cement content of 370 kg/m3, and aggregate contents of 1150 kg/m3 for 20mm gravel and 345 kg/m3 for 10mm gravel. The final concrete mix is specified with weight proportions of cement, water, fine aggregate, 20mm coarse aggregate, 10mm coarse aggregate, and admixture.
This document provides information and formulas for calculating bar bending schedules. It discusses hook length, bend length, overlap length, and how to prepare a bar bending schedule table. Formulas are given for calculating hook length, bend length for cranked and corner bars, and overlap length for tension and compression members. An example bar bending schedule is also shown for a sample RCC column, calculating the number of bars, cutting lengths, total bar lengths, and total steel weight required.
This document provides a comprehensive summary of strong-motion attenuation relationships published between 1969 and 2000 for predicting peak ground acceleration and spectral ordinates from earthquakes. It reviews 70 published attenuation models from researchers around the world, organized chronologically. Each model is briefly described in 1-2 sentences with the author(s) and year. The purpose is to aid engineers and seismologists in selecting appropriate ground motion prediction models for earthquake engineering and seismic hazard analyses.
Geotechnical Engineering-II [Lec #25: Coulomb EP Theory - Numericals]Muhammad Irfan
Class notes of Geotechnical Engineering course I used to teach at UET Lahore. Feel free to download the slide show.
Anyone looking to modify these files and use them for their own teaching purposes can contact me directly to get hold of editable version.
Chapter 5: Axial Force, Shear, and Bending MomentMonark Sutariya
1. A beam can experience three internal forces at a section - axial force, shear, and bending moment. Even for planar beams, all three forces may develop.
2. There are three types of supports - roller/link, pin, and fixed. Roller/link supports resist one force, pin supports resist two forces, and fixed supports resist two forces and a moment.
3. Beams can experience different load types - concentrated, uniform distributed, and varying distributed loads. Methods are presented to calculate the shear, axial, and bending effects of these loads on beams.
This presentation is about RCC. one can find most of the information about RCC with architecture in mind. Structure Design - 2 Semester 2 B. Arch Notes
The document provides information about stress distribution in soil due to self-weight and surface loads. It discusses Boussinesq's formula for calculating vertical stress in soil due to a concentrated surface load. The formula shows that vertical stress is directly proportional to the load, inversely proportional to depth squared, and depends on the ratio of radius to depth. A table of coefficient values used in the formula for different ratios of radius to depth is also provided.
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.
The document provides information about mechanics of solids-I, including:
1) It describes different types of supports like simple supports, roller supports, pin-joint supports, and fixed supports. It also describes different types of loads like concentrated loads, uniformly distributed loads, and uniformly varying loads.
2) It discusses shear force as the unbalanced vertical force on one side of a beam section, and bending moment as the sum of moments about a section.
3) It explains the relationship between loading (w), shear force (F), and bending moment (M) for an element of a beam. The rate of change of shear force is equal to the loading intensity, and the rate of change of bending
This document discusses transverse shear stresses in beams. It begins by explaining how shear stresses develop within beams subjected to transverse loads and defines the internal shear force V. It then discusses how shear stresses cause shear strains that distort the beam's cross-section. The document proceeds to derive the shear formula that relates the shear stress to the internal shear force V and the beam's geometry. It provides examples of applying the shear formula to compute shear stresses in different beam cross-sections.
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 deformation spectra for single-degree-of-freedom (SDF) linear systems subjected to base excitation. It presents the equations of motion for an SDF system with a moving base and defines terms like relative displacement and pseudo-acceleration. Graphs of deformation spectra are shown for half-cycle acceleration and velocity pulses. Key aspects of the spectra under different inputs are described, including asymptotic behavior and sensitivity to displacement, velocity, and acceleration portions of the input.
1. The document discusses static equilibrium of coplanar force systems. It covers drawing free-body diagrams, identifying reaction forces, and applying the three equations of equilibrium.
2. Key steps for solving problems include drawing the free-body diagram, identifying known and reaction forces, and setting the sum of forces and moments equal to zero.
3. Examples show calculating unknown forces and reactions for beams, rods, and pulley systems in static equilibrium. Forces and moments are analyzed to determine the magnitude and direction of reaction forces.
Columns are structural elements that transmit loads in compression from beams and slabs above to other elements below. Columns can experience both axial compression and bending loads. Biaxial bending occurs when a column experiences simultaneous bending about both principal axes, such as in corner columns of buildings. The biaxial bending method permits analysis of rectangular columns under these conditions. The document provides details on analyzing a sample reinforced concrete column for adequacy using the reciprocal load method to check that factored loads do not exceed design capacity. Diagrams are presented showing interaction surfaces and stress distributions for concentrically and eccentrically loaded columns.
This document provides an overview of statics concepts including:
- Forces on particles in 2D and 3D space including addition and resolution of forces
- Equilibrium of particles and rigid bodies using free body diagrams
- Moments of forces about points and axes
- Force couples and equivalent force systems
- Example problems are provided to demonstrate applying concepts to determine tensions, components of forces, moments, and equivalent single forces.
This document provides an overview of friction and circular motion topics. It discusses frictional force, types of friction, angle of friction, minimum friction angle, and direction of friction. It also covers circular motion topics like centripetal acceleration, kinematics of circular motion including angular velocity and centripetal force, dynamics of circular motion. Non-uniform circular motion, banking of roads, vehicles taking turns, and vertical circular motion are also summarized. Sample problems related to these topics are included for practice. The document promotes Unacademy subscription plans for access to live classes, study material and tests for JEE preparation.
This document discusses bending moments and shear forces in beams. It defines different types of beams such as simply supported beams, cantilever beams, and beams with overhangs. It also defines types of loads like concentrated loads, distributed loads, and couples. It explains how to calculate the shear force and bending moment at any cross-section of a beam and discusses relationships between loads, shear forces and bending moments. It provides examples of drawing shear force and bending moment diagrams. Finally, it discusses bending stresses in beams and bending of beams made of two materials.
B Ending Moments And Shearing Forces In Beams2Amr Hamed
This document discusses bending moments and shear forces in beams. It defines different types of beams such as simply supported beams, cantilever beams, and beams with overhangs. It also defines types of loads like concentrated loads, distributed loads, and couples. It explains how to calculate the shear force and bending moment at any cross-section of a beam and discusses relationships between loads, shear forces and bending moments. It provides examples of drawing shear force and bending moment diagrams. Finally, it discusses bending stresses in beams and bending of beams made of two materials.
This document discusses bending moments and shear forces in beams. It defines different types of beams such as simply supported beams, cantilever beams, and beams with overhangs. It also defines types of loads like concentrated loads, distributed loads, and couples. It explains how to calculate the shear force and bending moment at any cross-section of a beam and discusses relationships between loads, shear forces and bending moments. It provides examples of drawing shear force and bending moment diagrams. Finally, it discusses bending stresses in beams and bending of beams made of two materials.
B Ending Moments And Shearing Forces In Beams2Amr Hamed
This document discusses bending moments and shear forces in beams. It defines different types of beams such as simply supported beams, cantilever beams, and beams with overhangs. It also defines types of loads like concentrated loads, distributed loads, and couples. It explains how to calculate the shear force and bending moment at any cross-section of a beam and discusses relationships between loads, shear forces and bending moments. It provides examples of drawing shear force and bending moment diagrams. Finally, it discusses bending stresses in beams and bending of beams made of two materials.
- 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.
(1) The document provides conceptual problems and their solutions related to oscillations and simple harmonic motion. (2) It examines the kinetic and potential energy of an object undergoing simple harmonic motion with a given amplitude. (3) It compares the maximum speeds of two simple harmonic oscillators with identical amplitudes but different masses attached to identical springs.
1) The document discusses torsion and torsional deformation of circular shafts. It derives the torsion formula which relates the shear stress in a shaft to the torque and geometry of the shaft's cross section.
2) Power transmission using shafts is discussed. The relationship between torque, angular velocity, and power is defined. Shaft design using the torsion formula and allowable shear stress is also covered.
3) Examples are presented to demonstrate calculating shear stresses and designing shafts given torque and power transmission information.
Forces acting on the beam with shear force & bending momentTaral Soliya
The document discusses different types of beams and how to analyze the shear forces and bending moments in beams. It defines beams as structural members subjected to lateral loads and describes various types of beams based on their support conditions, including simply supported beams, cantilever beams, and continuous beams. It also covers types of loads beams may experience, such as concentrated loads, distributed loads, and couples. The document then explains how to determine the shear forces and bending moments in beams by using cut sections and equilibrium equations. It provides examples of analyzing shear forces and bending moments in beams with different load conditions.
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 provides an overview of the topics and lectures covered in S K Mondal's Engineering Mechanics course for GATE and IAS exams. The course is divided into 8 modules covering topics such as laws of motion, vector algebra, equilibrium of bodies, trusses, friction, properties of surfaces, method of virtual work, motion in a plane, rotational dynamics, harmonic oscillators, and projectile motion. The document lists the specific lectures in each module, along with example problems and their solutions related to the engineering mechanics topics.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
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1. 5
Influence Lines
UNIT 1 INFLUENCE LINES
Structure
1.1 Introduction
Objectives
1.2 Definition
1.3 Properties of Influence Lines
1.4 Influence Line Diagram for Bending Moment
(Simply Supported Beam)
1.5 Influence Line Diagram for Shearing Force
(Simply Supported Beam)
1.6 Influence Line Diagram for a Cantilever
1.7 Summary
1.8 Answers to SAQs
1.1 INTRODUCTION
Study of Applied Mechanics and Strength of Materials have enabled you to
compute the reactions, shear force (SF), bending moments (BM), deflections etc.
in a beam and bar forces in a pin-jointed truss, subjected to a given static load
system which remains stationary. Quite frequently, a beam or a truss is subjected
to a load which is not exactly stationary and may be moving along a certain path
(which may be the length of the beam or top or bottom chord of a truss etc.). Such
moving loads are called live loads as opposed to stationary loads which may be
called dead loads. Instances of live loads are quite common, e.g. a railway train
moving across a rail bridge or a vehicle moving along a road bridge. Obviously,
the value of any of the desired quantities (e.g., shear force, bending moment or
bar force) depends upon the position of the load. For the design of the members, it
is important to find out the position of the loads for which the stresses caused in
the structure is maximum at any point or in any member. For this purpose, a
graphical representation (or a curve), depicting the values of the desired quantity
for various load positions, is drawn and is used for calculation. Such curves or
lines are called influence lines for the quantity.
Objectives
After studying this unit, you should be able to
• conceptualise and define influence line,
• calculate the variation of a particular quantity (BM, SF, axial force
etc.) due to a unit load moving across a structure,
• depict the variation of the quantity, graphically, through influence
lines,
• discuss the properties of the influence line and to interpret it for direct
use in structural analysis, and
• calculate the magnitude of the quantity under a given system of live
loads moving across the structure.
2. 6
Theory of Structures-I
1.2 DEFINITION
An influence line is a curve, the ordinate of which at any point is equal to the
value of some structural quantity, when a unit load is placed at that point.
The structural quantity could be external support reactions, (e.g. vertical or
horizontal reactive forces or bending moments), internal stress resultants (e.g.
axial force, SF or BM) or deformations (e.g. slope and deflections). The above
definition can be explained by the following simple examples.
A B
x
10 m
7 m
20 KN
0.3
R =
A
10 –
10
x
1
P
1.0
1.0
(a) Beam
(b) Influence Line Diagram of R
A
(c)
x
Figure 1.1
Figure 1.1(a) shows a beam which is simply supported at A and B and has a span
of 10 m. If we want to find the influence line for the reaction RA at the support A,
we have to place a unit load (e.g. 1 kN) at various points on the beam and to find
the corresponding values of RA. These values are plotted at the points where the
load is placed. For example we know that if the unit load is placed at A, the
reaction RA = 1, and if it is placed at B, RA = 0. Now if the moveable unit load is
placed at any point P on the beam, which is between A and B, at a distance of x
from A, then taking moments of all forces about B, we have
RA 10 – 1× (10 – x) = 0
ּ
⇒ giving
10
10 x
RA
−
= . . . (1.1)
Eq. (1.1), as can be seen, is the equation of a straight line shown in Figure 1.1(b).
Thus, Figure 1.1(b) is the influence line for the reaction RA of the beam.
Similarly, we may draw influence lines for bending moment or shearing forces at
a point on the beam. This will be shown in the next sections.
3. 7
Influence Lines
1.3 PROPERTIES OF INFLUENCE LINES
Here, we shall state certain important properties of an influence line, to show how
they can be utilized successfully in structural engineering.
(a) To obtain the maximum value of a structural quantity due to a single
concentrated moving load, the load should be placed at that point
where the ordinate to the influence line is maximum. (This property is
obvious from the definition.)
(b) The value of a structural quantity due to a single concentrated moving
load is equal to the product of the magnitude of the load and the value
of the corresponding ordinate of the influence line. Thus, in
Figure 1.1(c), the ordinate of the influence line for RA at a distance of
7 m from A is
10
7
10 −
= 0.3.
Now if a load of 20 kN is placed at this point the magnitude of
reaction RA will be 20 × 0.3 = 6 kN. This can be easily verified.
(c) To find the maximum value of a structural quantity due to a uniformly
distributed live load, the load should be placed over all those portions
of the structure where its influence line ordinates have the same sign.
A D
B C
q per unit length
(b) Position of for Maximum Positive Value
udl
A D
B C
q per unit length
(c) Position of for Maximum Negative Value
udl
A
B C
D
(a) Influence Line Diargam
Figure 1.2
For example, Figure 1.2(a) shows the influence line diagram of a
particular quantity for the beam ABCD. The ordinates are positive
between B and C; and are negative between A and B, and again
between C and D. Hence, for maximum positive value of the quantity
the uniformly distributed load should cover the entire portion from B
to C (Figure 1.2(b)). Similarly, for maximum negative value it should
cover the portion AB and CD (Figure 1.2(c)).
(d) The value of a structural quantity due to a uniformly distributed live
load is equal to the product of the loading intensity (q) and the net
area of the influence line diagram under that portion.
Here, in Figure 1.2, the value of the structural quantity due to the
uniformly distributed load covering portion AB only is given by
q × (area of IL diagram over AB as shown shaded).
4. 8
Theory of Structures-I
1.4 INFLUENCE LINE DIAGRAM FOR BENDING
MOMENT (SIMPLY SUPPORTED BEAM)
In the following examples, influence line diagram for bending moment and shear
force of some common structures are shown, and also how they are used in actual
practice.
Example 1.1
Draw the influence line diagram for bending moment at point P of the
simply supported beam AB, shown in Figure 1.3.
P
1
A B
(a) Unit Load between A and P
4 m
x
6 m
A B
P
(b) Unit Load between P and B
x
P″
A'
P'
B'
(c) Influence Line Diagram for M
P
2.4
Figure 1.3
Solution
Case (i) : Load between A and P (Figure 1.3 (a))
If the unit load lies between A and P taking moment of forces to left
of P, then BM at P is
MP = RA·4 – 1 × (4 – x) =
10
6
)
4
(
4
10
10 x
x
x
=
−
−
×
−
This is a straight line A΄P″ (Figure 1.3(c)) with IL ordinate = 0 at A
(where x = 0) and IL ordinate = 2.4 at point P (where x = 4 m).
Case (ii) : Load between P and B (Figure 1.3 (b))
When the load crosses the point P to the right, bending moment at P
is
MP = RA · 4 = 4
10
10
×
− x
5. 9
Influence Lines
This is again a straight line P″ B′ and the influence line ordinate is
10 4
4 2.4
10
−
× = at P; and at B the IL ordinate is 0
4
10
10
10
=
×
−
.
This is shown in Figure 1.3(c).
Example 1.2
Two connected wheels (wheel base = 3 m) cross the beam in Figure 1.4
from left to right. The front wheel is carrying a load of 20 kN and the rear
wheel 10 kN. Find the maximum bending moment at point P due to these
wheels.
A B
F
P
4 m 6 m
3 m
1 m
R
10 kN 20 kN
A' B'
P'
P'’
0.6
2.4
A B
4 m 6 m
3 m 3 m
R
10 kN 20 kN
A' B'
P'’
P'
2.4
F
P
1.2
(a) (b)
Figure 1.4
Solution
The influence line diagram is the most convenient method to solve such
problems. It is obvious that since the maximum ordinate of the IL diagram
is 2.4 (at point P), the maximum value of the bending moment will be
obtained when one of the wheels is placed on this point P. There can be two
possibilities, either the front wheel is at P (Figure 1.4 (a)) or the rear wheel
is at P (Figure 1.4 (b)). Both are discussed below :
Case (i) : Front Wheel at Point P (Figure 1.4(a))
Rear wheel R will be 3 m to left of it (i.e. 4 – 3 = 1 m to right of A).
The ordinate at R (from similar triangles) is equal to 6
.
0
1
4
4
.
2
=
× . Now by
Property (b) of the IL diagrams, the total BM at point P due to the two
wheels will be MP = 20 × 2.4 + 10 × 0.6 = 54 kN m.
Case (ii) : When Rear Wheel is at P (Figure 1.4(b))
The front wheel F will be at F, i.e. 3 m to right of it.
The ordinate at F will be
6
4
.
2
× 3 = 1.2
Hence, BM at P will be MP = 20 × 1.2 + 10 × 2.4 = 48 kN m.
Hence, Case (i) will give the bigger value of bending moment, i.e. as shown
in Figure 1.4(a) and it will be 54 kN m.
This will be the maximum bending moment at P due to the wheels crossing
across the span.
6. 10
Theory of Structures-I
Example 1.3
For the beam in Figure 1.5 if a uniformly distributed load (udl) of 3 kN/m
longer than the span crosses it from left to right, what will be the maximum
bending moment at P?
Solution
P
A'
P'
B'
6 m
4 m
2.4
q = 3 kN/m
(b) ILD for M
P
(a)
3 kN/m
A
B
Figure 1.5
In Figure 1.5, it can be seen that as the influence line diagram for bending
moment at P (MP) is positive over the whole span AB, the moving udl has to
cover the entire span for maximum value.
Hence, by Property (d) of the IL Diagram as stated above the maximum
MP = (Intensity of loading) × (Area of IL Diagram)
= 3
1
2.4 10 36 kNm
2
⎛ ⎞
× × × =
⎜ ⎟
⎝ ⎠
.
Example 1.4
If the uniformly distributed load crossing the span in Figure 1.5 is smaller
than the span, say 4 m long, find the maximum bending moment at P.
P
A'
P'
B'
6 m
4 m
(b) ILD for M
P
(a)
A B
N'
N
M'
M
x (4 – x)
2.4
P
Figure 1.6
7. 11
Influence Lines
Solution
Here, the moving load can cover only a portion of the whole span.
Obviously, the maximum value of MP will occur when the moving load is
passing over the region where the ordinates are largest, that is, near the
point P itself.
Let us assume that the load occupies a position MN covering either side of
P, such that MP = x, then PN = 4 – x (Figure 1.6(b)).
By Property (d), the bending moment at P is given by,
MP = (Intensity of load) × (Hatched area of IL diagram below the load).
Now, the hatched area of the IL diagram is composed of two trapeziums
MM΄P΄P and NN΄P΄P.
Since the ordinate PP΄ = 2.4, by similar triangles
Ordinate MM΄ =
4
4
.
2
× (4 – x) = 0.6 (4 – x) and
Ordinate NN΄ =
6
4
.
2
[6 – (4 – x)] = 0.4 (2 + x)
∴ Area MM΄N΄N = )
4
(
2
)
2
(
4
.
0
4
.
2
2
)
4
(
6
.
0
4
.
2
x
x
x
x
−
+
+
+
−
+
= 6.4 + 1.6x – 0.5x2
∴ MP = (Load intensity) × (Area of IL diagram)
= 3 × (6.4 + 1.6 x – 0.5x2
) . . . (1.2)
Now MP will be maximum when 0
P
dM
dx
=
Differentiating Eq. (1.2), we get,
3 × (1.6 – 0.5 × 2x) = 0
giving x = 1.6 m.
Hence, when the maximum BM at P occurs 1.6 m of the moving load is
towards its left and (4 – 1.6) = 2.4 m is towards its right and the value of the
maximum blending moment will be given from Eq. (1.2) above.
MP = 3 × [6.4 + 1.6 × 1.6 – 0.5 (1.6)2
] = 45.312 kNm.
It is interesting to compare the value of the two end ordinates MM΄ and NN΄
of the influence line diagram when the load occupies the maximum BM
position.
MM΄ = 0.6 (4 – x) = 0.6 (4 – 1.6) = 1.44
NN΄ = 0.4 (2 + x) = 0.4 (2 +1.6) = 1.44
Thus, we see that under the above conditions the value of the influence line
ordinates at the two ends of the loads are equal.
This hints to determine the value of x easily without going through the
process of differentiation etc. For example,
MM΄ = 0.6 (4 – x) and NN΄ = 0.4 (2 + x)
For maximizing the area, we must have MM ′ = NN ′
From the above equation, x = 1.6 m.
And then we can proceed with finding out the areas etc.
8. 12
Theory of Structures-I
1.5 INFLUENCE LINE DIAGRAM FOR SHEARING
FORCE (SIMPLY SUPPORTED BEAM)
Next, we shall study how to draw the influence line diagram for shear force in a
simply supported beam. For illustration we take the same beam and the same
point P as in Example 1.1 (Figure 1.3), for which we now proceed to draw the
shear force influence line diagram.
Example 1.5
Draw the influence line diagram for shear force at point P in the simply
supported beam AB of span 10 m. P is 4 m from the support A.
P
6 m
4 m
(a)
A B
x 1
P
6 m
4 m
(b)
A B
x
A' B'
P'
P''
+ 0.6
– 0.4
P
⊕
(c) ILD for SF at P.
Figure 1.7
Solution
Let us assume that the moving unit load is at a distance x from A. the
reactions
10
10 x
RA
−
= , and
10
x
RB = as determined earlier.
Case (a) : If the unit load is between the points A and P, then considering
forces to left of P,
Shear force at P = RA – 1 =
10
1
10
10 x
x
−
=
−
−
. By our sign convention, a
shear force is negative when the left hand portion of the beam tends to
move downward. Hence, the shear force in this case will be negative and
will depend upon the value of x (i.e. its distance from A). The ordinate of
the diagram will be zero then the load is at A (x = 0) and it will be
9. 13
Influence Lines
4
.
0
10
4
−
=
− when the rolling load is at P (i.e. x = 4). The diagram will be a
straight line A΄P΄.
Case (b) : When the unit load crosses to the right of the point P and lies
between P and B, then considering forces to left of P the shear force at
P = RA =
10
10 x
−
, which is also a straight line P˝ B such that the ordinate at
P (when the unit load has just crossed to right) is equal to
10 4
0.6
10
−
= +
and it is zero when the unit load is at B, ordinate = 0
10
10
10
=
−
.
Thus, we see that the shear force influence line consists of two parts : the
part between A and P has negative ordinates and the part between P and B
has positive ordinates, showing that the SF changes sign as the unit rolling
load crosses the point P.
Example 1.6
Find the maximum positive and negative shear force at point P in beam of
Figure 1.8 which is crossed by two connected wheel loads 3 m aparts
moving from right to left. The front wheel carries a load of 20 kN and the
rear wheel 10 kN.
P
6 m
4 m
A B
3 m
10 kN 20 kN
–0.1
A' B'
0.6
P'
–0.4
(a)
Q
A B
(b)
(c)
A' B'
+ 0.6
P'
+ 0.3
– 0.1
A' B'
+ 0.6
P'
Q’
P
10 kN 20 kN
3 m
R
P
10 kN 20 kN
3 m
R
P
Figure 1.8
10. 14
Solution
Theory of Structures-I
Maximum Negative SF
For maximum negative shear force at P, the heavier wheel (20 kN)
should be placed just to the left of P, the other wheel (10 kN) will
then lie at Q which is 3 m to left of P (Figure 1.8(a)). The ordinate of
IL diagram at P is – 0.4 and that at Q is – 0.1 (by similar triangles).
Hence, maximum negative shear force at P,
VP = 20 × (– 0.4) + 10 (– 0.1) = – 9 kN.
Maximum Positive SF
Here, two cases need to be examined
(a) When the heavier wheel 20 kN is just crossed to right of
P, and the lighter wheel (10 kN) is at Q 3 m behind it
(Figure 1.8(b)). Hence the shear force
VP = 20 × (+ 0.6) + 10 × (– 0.1) = 11 kN.
(b) When the lighter wheel (10 kN) has just crossed to right
of P and the front wheel (20 kN) is 3 m to right of it at R
as in Figure 1.8 (c).
The ordinate of IL diagram at P is + 0.6 and at R it is + 0.3.
Hence, shear force VP = 20 × (+ 0.3) + 10 × (+ 0.6) = 12 kN.
The second position gives the higher value, hence, the maximum
positive SF will be 12 kN.
SAQ 1
(a) Draw the influence line diagram for the bending moment at point P of
the simply supported beam AB in Figure 1.9.
6 m
3 m
A B
P
Figure 1.9
Using this diagram find the maximum bending moment at P, due to
following moving loads
• A uniformly distributed load of 4 kN/m longer than the
span.
• A uniformly distributed load of 4 kN/m of 3 m length.
• Two connected wheel loads of 10 kN each, 3 m apart.
(b) Draw the influence line for shear force at the same point P of the
above beam (Figure 1.9).
Determine the maximum shear force at P for the same load
combinations as given in SAQ 1(a) above.
11. 15
Influence Lines
1.6 INFLUENCE LINE DIAGRAM FOR A
CANTILEVER
In this section, we shall explain how to draw the influence line diagrams for the
support reactions and BM and SF at any point in a cantilever beam. You
should verify all the steps yourself and try to solve the numerical examples given
in the SAQ below which is based on this diagram.
In Figure 1.10, the cantilever whose free end is A and fixed end B has a span of L.
(a) Influence Line Diagram for Support Reactions
If a unit load moves from A to B along the beam, the vertical reaction
RB at B remains constant and is equal to 1.0. However, the fixed end
moment M
B
B = – 1 × (L – x) = – (L – x) and hence the influence line
coordinates for MBB varies from – L at A to 0 at B. (You should
carefully observe that it is just opposite to the BM diagram due to an
unit load at A. Why?)
The IL Diagrams are shown at Figures 1.10(b) and (c).
L
a
P
A B
x 1
MB
RB
A'
A'
B'
B'
P’
P’
B'
B'
1.0
– L
⊕
– a
1.0
(e) ILD for Shear Force VP
(d) ILD for Bending Moment MP
(c) ILD for Support Moment MB
(b) ILD for Reaction RB
(a) Cantilever with Rolling Unit Load
Figure 1.10
(b) Influence Line Diagram for Bending Moment at P
Next, we shall draw the influence line diagram for bending moment
(MP) at a point P which is at a distance ‘a’ from end A :
When the load is between A and P the BM at P is
12. 16
MP = – 1 × (a – x) = – (a – x)
Theory of Structures-I
Hence, the ordinate of the IL diagram is – a when x = 0 (at point A)
and it is 0 when x = a (at point P).
As soon as the load crosses over P to the right hand side the BM at
P = 0 and remains as such till the end, as shown in Figure 1.10(d).
(c) Influence Line Diagram for Shear Force at P
Next, we draw the influence line for shear force (VP) at P :
When the load is between A and P, the shear force (considering loads
to left of P) is VP = – 1 (downwards ∴ negative)
As soon as the load crosses P to the right of P, there is no load to left
of it, hence VP = 0. This is shown in Figure 1.10(e).
SAQ 2
A cantilever 6 m span is free at end A and fixed at B. Using influence line
diagrams for a point P, 3 m from free end A, find the maximum SF and BM
at P due to the following moving loads :
(a) A uniformly distributed load of 3 kN/m longer than the span.
(b) A set of 3 connected wheel loads, shown in Figure 1.11, moving
from left to right.
10 kN 10 kN 30 kN
1 m
1 m
Figure 1.11
1.7 SUMMARY
In this unit, you have learnt how to find the values of a structural quantity (BM,
SF, axial force support reactions etc.) for a unit moving load by means of
influence line diagrams. You have, thus, learnt the properties of the influence line
diagram and its uses.
You have also learnt to draw the IL diagrams for support reactions, BM and SF at
any point for a simply supported beam; and a cantilever.
In the next unit, we will study theorem of three moments applied to fixed and
continuous beams.
13. 17
Influence Lines
1.8 ANSWERS TO SAQs
SAQ 1
(a)
P
A B
3 m
+ 2.0
+ 2.0
+ 1.0
+ 2.0
4 kN/m
M′
M
(i)
(ii)
(iii)
N
N′
4 kN/m
3 x
−
3 x
−
x 3 + x
3 m 3 m 3 m
10 kN 10 kN
4/3 4/3
ILD for MP
6 m
+ 2.0
(i) Position of moving load for maximum Mp
(max) (Area of ) (Intensity of loading)
P
M ILD
= ×
1
2.0 9 4 36 kNm
2
⎛ ⎞
= × × × =
⎜ ⎟
⎝ ⎠
(ii) Position of load for maximum MP
Ordinate MM′ = Ordinate NN′
2 2
(3 ) (3 )
3 6
x x
− = +
giving x = 1 m and ordinates = 4/3
∴ (max) (Shaded area) × (Load intensity)
P
M =
2
4
2
3 3 4 5 m 4 kN/m 20 kNm
2
⎛ ⎞
+
⎜ ⎟
= × × = × =
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
14. 18
(iii) Position of connected loads for maximum MP
(max) 10 2 10 1 30 kNm
P
M = × + × =
Theory of Structures-I
(b)
3 m
3 m
10 kN
10 kN
3 m
(i)
(ii)
(iii)
3 m
+ 2/3
+ 2/3
+ 1/3
+ 2/3
− 1/3
− 1/3
− 1/3
1/3
6 m
3 m 3 m
3 m 3 m
A
A
A
B
B
B
Q
P
P
(i) For udl
VP (+) is max when PB is covered
(max)
1 2
6 4 8 kN
2 3
p
V
⎛ ⎞
= × × × = +
⎜ ⎟
⎝ ⎠
VP (–) is max when AP is covered only
(max)
1 1
3 4 2 kN
2 3
p
V
⎛ ⎞
= − × × × = −
⎜ ⎟
⎝ ⎠
(ii) For udl of 3 m length PQ is covered
(max) Area × Intensity
p
V =
1 2
2 3 3 4 6 k
2
⎛ ⎞
×
⎜ ⎟
= × × =
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
N
(iii) For two connected conc. loads
(max)
2 1
10 10 10 kN
3 3
p
V = × + × =
SAQ 2
15. 19
Influence Lines
P
A
A
A
B
P
P
B
B
3 m
3 m 3 m
3 m
3 m
ILD for MP
ILD for VP
3 m
– 3.0
+ 1.0
Case (a)
(max)
1
3 3 3 13.5 kNm
2
P
M
⎛ ⎞
= × × × = −
⎜ ⎟
⎝ ⎠
(max) (1 3) 3 9 kN
P
V = × × =
Case (b)
Considering the following three cases for MP
P
A
A
A
B
P
P
B
B
1 m
10 kN
10 kN
10 kN
30 kN
30 kN
30 kN
1 m
1 m 1 m
2 m 3 m
3 m
3 m
3 m
– 3
– 3
– 3
– 1
– 2
– 2
In the first case, 10 3 10 2 30 1 80 kNm
P
M = − × − × − × = −
In the second case, 10 3 30 2 90 kNm
P
M = − × − × = −
In the third case, 30 3 90 kNm
P
M = − × = −
So the second and third cases produce maximum BM of – 90 kNm.
For SF, as the diagram is rectangle, all the loads have the same result
placed anywhere between A and P.
∴ (max) 10 1 10 1 30 1 50 kN
P
V = × + × + × =