The document discusses the double integration method for calculating deflections in beams. It introduces the concept of using Macaulay's notation to write the bending moment expression in beams with point loads as a single equation using square brackets. This allows integrating the differential equation of the beam twice to obtain an expression for the deflection throughout the beam with just two integration constants, avoiding multiple equations that would otherwise be needed. Macaulay's notation makes the double integration method more efficient for problems involving point loads.
This document discusses the stress function approach for solving two-dimensional elasticity problems. It begins by presenting the general equations of elasticity, including stress-strain relationships, strain-displacement equations, and equilibrium equations. It then introduces the stress function method proposed by Airy, where a single function of space coordinates is assumed that satisfies all the elasticity equations. The key steps are: (1) choosing a stress function, (2) confirming it is biharmonic, (3) deriving stresses from its derivatives, (4) using boundary conditions to determine the function, (5) deriving strains, and (6) displacements. Examples of polynomial stress functions are also provided.
This document discusses beam design criteria and deflection behavior of beams. It covers two key criteria for beam design:
1) Strength criterion - the beam cross section must be strong enough to resist bending moments and shear forces.
2) Stiffness criterion - the maximum deflection of the beam cannot exceed a limit and the beam must be stiff enough to resist deflections from loading.
It then defines deflection, slope, elastic curve, and flexural rigidity. It presents the differential equation that relates bending moment, slope, and deflection. Methods for calculating slope and deflection including double integration, Macaulay's method, and others are also summarized.
This document provides an overview of the slope deflection method for analyzing statically indeterminate structures. It describes that the slope deflection method was developed in 1914 and can be used to analyze beams and frames. Key assumptions of the method are that joints are rigid and distortions from axial/shear stresses are neglected. The document outlines the application, sign convention, procedure, slope deflection equations, and provides examples for analyzing beams and frames using this method.
The document discusses column buckling and failure modes. It defines a strut and column, and describes failure due to direct stress, buckling stress, or a combination. It provides differential equations to model column buckling based on the column's end conditions, including fixed-fixed, fixed-free, and fixed-hinged. Solutions provide the critical buckling load. The document also discusses Rankine's formula which relates crushing load and buckling load for a column's total capacity.
This document provides information about shear stresses and shear force in structures. It includes:
- Definitions of shear force and shear stress. Shear force is an unbalanced force parallel to a cross-section, and shear stress develops to resist the shear force.
- Explanations of horizontal and vertical shear stresses that develop in beams due to bending moments. Shear stress is highest at the neutral axis and reduces towards the top and bottom of the beam cross-section.
- Derivations of formulas for calculating shear stress across different beam cross-sections. Shear stress is directly proportional to the shear force and beam geometry.
- Examples of calculating maximum and average shear stresses for various cross-sections
Mohr's circle is a graphical representation of the transformation of stresses on planes at a point in a material. It relates normal and shear stresses on inclined planes to the principal stresses. The circle is centered at the average stress and has a radius equal to the difference between the maximum and minimum principal stresses. Mohr's circle allows determination of stresses on any inclined plane from knowledge of the principal stresses and provides insight into failure conditions of materials.
This document discusses the flexibility method for structural analysis. The flexibility method involves determining flexibility coefficients by applying unit loads corresponding to redundant forces and calculating the resulting displacements. These flexibility coefficients are then used to calculate the redundant forces needed to satisfy compatibility conditions. The flexibility matrices for different structural elements are developed. Joint displacements, member end actions, and support reactions can be determined by incorporating the flexibility coefficients into the basic computations. Examples are provided to illustrate the flexibility method for a continuous beam with one redundant and for determining various outputs like redundants, joint displacements, and reactions.
This document discusses the stress function approach for solving two-dimensional elasticity problems. It begins by presenting the general equations of elasticity, including stress-strain relationships, strain-displacement equations, and equilibrium equations. It then introduces the stress function method proposed by Airy, where a single function of space coordinates is assumed that satisfies all the elasticity equations. The key steps are: (1) choosing a stress function, (2) confirming it is biharmonic, (3) deriving stresses from its derivatives, (4) using boundary conditions to determine the function, (5) deriving strains, and (6) displacements. Examples of polynomial stress functions are also provided.
This document discusses beam design criteria and deflection behavior of beams. It covers two key criteria for beam design:
1) Strength criterion - the beam cross section must be strong enough to resist bending moments and shear forces.
2) Stiffness criterion - the maximum deflection of the beam cannot exceed a limit and the beam must be stiff enough to resist deflections from loading.
It then defines deflection, slope, elastic curve, and flexural rigidity. It presents the differential equation that relates bending moment, slope, and deflection. Methods for calculating slope and deflection including double integration, Macaulay's method, and others are also summarized.
This document provides an overview of the slope deflection method for analyzing statically indeterminate structures. It describes that the slope deflection method was developed in 1914 and can be used to analyze beams and frames. Key assumptions of the method are that joints are rigid and distortions from axial/shear stresses are neglected. The document outlines the application, sign convention, procedure, slope deflection equations, and provides examples for analyzing beams and frames using this method.
The document discusses column buckling and failure modes. It defines a strut and column, and describes failure due to direct stress, buckling stress, or a combination. It provides differential equations to model column buckling based on the column's end conditions, including fixed-fixed, fixed-free, and fixed-hinged. Solutions provide the critical buckling load. The document also discusses Rankine's formula which relates crushing load and buckling load for a column's total capacity.
This document provides information about shear stresses and shear force in structures. It includes:
- Definitions of shear force and shear stress. Shear force is an unbalanced force parallel to a cross-section, and shear stress develops to resist the shear force.
- Explanations of horizontal and vertical shear stresses that develop in beams due to bending moments. Shear stress is highest at the neutral axis and reduces towards the top and bottom of the beam cross-section.
- Derivations of formulas for calculating shear stress across different beam cross-sections. Shear stress is directly proportional to the shear force and beam geometry.
- Examples of calculating maximum and average shear stresses for various cross-sections
Mohr's circle is a graphical representation of the transformation of stresses on planes at a point in a material. It relates normal and shear stresses on inclined planes to the principal stresses. The circle is centered at the average stress and has a radius equal to the difference between the maximum and minimum principal stresses. Mohr's circle allows determination of stresses on any inclined plane from knowledge of the principal stresses and provides insight into failure conditions of materials.
This document discusses the flexibility method for structural analysis. The flexibility method involves determining flexibility coefficients by applying unit loads corresponding to redundant forces and calculating the resulting displacements. These flexibility coefficients are then used to calculate the redundant forces needed to satisfy compatibility conditions. The flexibility matrices for different structural elements are developed. Joint displacements, member end actions, and support reactions can be determined by incorporating the flexibility coefficients into the basic computations. Examples are provided to illustrate the flexibility method for a continuous beam with one redundant and for determining various outputs like redundants, joint displacements, and reactions.
This document contains lecture notes on matrix methods of structural analysis. It discusses three methods for solving systems of linear equations that arise in structural analysis problems - Gauss elimination method, Gauss-Jordan method, and Gauss-Seidel iterative method. For each method, an example problem is provided and solved step-by-step to illustrate the application of the method.
The document discusses bending stresses in beams. It describes how bending stresses are developed in beams to resist bending moments and shearing forces. The theory of pure bending is introduced, where only bending stresses are considered without the effect of shear. Equations for calculating bending stresses are derived based on the beam's moment of inertia, bending moment, and distance from the neutral axis. Several beam cross-section examples are provided to demonstrate how to calculate the maximum bending stress and section modulus.
Analysis of non sway frame portal frames by slopeand deflection methodnawalesantosh35
The slope deflection method is a displacement method used to analyze statically indeterminate beams and frames. It involves solving for the slope and deflection of members at their joints, which are the basic unknowns. Equations are developed relating the fixed end moments, slopes, and relative deflections of each member. These equations are set up and solved to determine the bending moments in each member. The method is demonstrated through examples solving for the bending moments in non-sway and sway frames.
This document discusses stresses in beams and beam deflection. It covers several methods for analyzing bending stresses and deflection in beams, including: [1] the engineering beam theory relating moment, curvature, and stress; [2] double integration and moment area methods for calculating slope and deflection; and [3] Macaulay's method, which simplifies calculations for beams with eccentric loads. Formulas are provided relating bending moment, shear force, curvature, slope, and deflection. Moment-area theorems are also described for relating bending moment to slope and deflection.
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.
Shear Force and Bending Moment DiagramsVishu Sharma
Short notes for shear force and bending moment diagrams in Strength Of materials for design calculation
Reference- Shigley’s Mechanical Engineering Design
Lecture 9 shear force and bending moment in beamsDeepak Agarwal
The document discusses stresses in beams. It covers topics like shear force and bending moment diagrams, bending stresses, shear stresses, deflection, and torsion. Beams are structural members subjected to transverse forces that induce bending. Stresses and strains are created within beams when loaded. Shear forces and bending moments allow determining these internal stresses and maintaining equilibrium. Formulas are provided for calculating shear forces and bending moments in different beam configurations like cantilevers, simply supported beams, and beams with various load types.
This document discusses strength of materials concepts related to shear force diagrams (SFD) and bending moment diagrams (BMD) for beams. It defines key terms like shear force, bending moment, and point of contraflexure. It also explains how to draw SFDs and BMDs for different beam types under various loading conditions and the relationships between loading, shear force, and bending moment. Application of the diagrams to reinforcement design is also mentioned.
The document discusses various numerical methods for analyzing mechanical components under applied loads, including the finite element method. It describes weighted residual methods like the Galerkin method and collocation method which approximate solutions by minimizing residuals. The variational or Rayleigh-Ritz method selects displacement fields to minimize total potential energy. The finite element method divides a structure into small elements and applies these methods to obtain approximate solutions for displacements and stresses at discrete points.
1. The document outlines key concepts in structural dynamics including idealization of structures as single-degree-of-freedom systems, formulation of the equation of motion, free and forced vibration of undamped and damped systems.
2. Key topics covered include natural frequency determination, Duhamel's integral, damping in structures, and methods for solving dynamic problems.
3. Examples of single-degree-of-freedom systems are presented including lumped mass systems, beams with distributed mass, and determination of effective stiffness.
This document discusses stresses in beams. It covers bending stresses, shear stresses, deflection in beams, and torsion in solid and hollow shafts. The key assumptions in beam bending theory are outlined. Bending stresses are explained, including the location of the neutral axis and how stresses vary through the beam cross-section based on the bending moment and geometry. Section modulus is defined as the ratio of the moment of inertia to the distance of the outermost fiber from the neutral axis. Composite beams made of different materials are also discussed.
The document discusses shear stresses in beams. It defines shear stress as being due to shear force and perpendicular to the cross-sectional area. Shear stress is derived as τ = F/A, where F is the shear force and A is the cross-sectional area. Shear stress varies across standard beam cross sections like rectangular, circular, and triangular. Shear stress is maximum at the neutral axis for rectangular and circular beams, and at half the depth for triangular beams. Sample problems are included to demonstrate calculating and graphing the variation of shear stress across specific beam cross sections.
Deflection of curved beam |Strength of Material LaboratorySaif al-din ali
SAIF ALDIN ALI MADIN
سيف الدين علي ماضي
S96aif@gmail.com
Experiment Name:- Deflection of curved beam
2. Introduction
The deflection of a beam or bars must be often be limited in order to provide
integrity and stability of structure or machine. Plus, code restrictions often require
these members not vibrate or deflect severely in order to safely support their
intended loading.
This experiment helps us to show some kind of deflection and how to calculate the
deflection value by using Castigliano’s Theorem and make a comparison between
result of the experiment and the theory.
This document discusses the slope-deflection method for analyzing beams and frames. It provides the theory and equations of the slope-deflection method. Examples are included to demonstrate how to use the method to determine support reactions, member end moments, and draw bending moment and shear force diagrams.
The document discusses various topics related to stress and strain including: principal stresses and strains, Mohr's stress circle theory of failure, 3D stress and strain, equilibrium equations, and impact loading. It provides examples of stresses acting in different planes including normal, shear, oblique, and principal planes. It also gives examples of calculating normal and tangential stresses on an oblique plane subjected to stresses in one, two, or multiple directions with and without shear stresses.
The document discusses buckling and its theories. It defines buckling as the failure of a slender structural member subjected to compressive loads. It provides examples of structures that can experience buckling. It explains Euler's theory of buckling which derived an equation for the critical buckling load of a long column based on its bending stress. The assumptions of Euler's theory are listed. Four cases of long column buckling based on end conditions are examined: both ends pinned, both ends fixed, one end fixed and one end pinned, one end fixed and one end free. Effective lengths are defined for each case and the corresponding critical buckling loads given. Limitations of Euler's theory are noted. Rankine's empirical formula for calculating ultimate
Columns are structural members that experience compression loads. They can buckle if loaded beyond their buckling (or critical) load. Short columns fail through crushing, while long columns fail through lateral buckling. The Euler formula calculates the buckling load of a long column based on its properties and end conditions. The Rankine-Gordon formula provides a more accurate calculation of buckling load that applies to all column types by accounting for both buckling and crushing. Proper design of columns involves ensuring they are loaded below their safe loads, which incorporate factors of safety applied to the theoretical buckling loads.
- This chapter discusses the deflection of beams under lateral loading. Deflection is the displacement of points on the beam's axis from its original, unloaded position.
- The basic differential equation relates the bending moment to the second derivative of the deflection curve. This equation can be directly integrated or used with methods like Macaulay's to determine deflections.
- Macaulay's method involves integrating the loading equation rather than the bending moment equation directly. It is useful when the bending moment is difficult to obtain.
- Special cases like loads not starting from the beginning or ending at the beam's end are also discussed.
Deflection of Beams _Chapter 5_2019_01_17!10_37_41_PM.pptSivarajuR
- This chapter discusses the deflection of beams under lateral loads. Deflection is the displacement of points on the beam's axis from its original, unloaded position.
- The basic differential equation relates the bending moment to the second derivative of the deflection curve. It can be directly integrated or used with methods like Macaulay's to determine deflections under different load cases.
- Macaulay's method allows integrating the loading function directly instead of first obtaining the bending moment. It is useful for concentrated or discontinuous loads.
- Special cases with loads not starting or ending at the beam's ends require modifying the integration limits.
This document contains lecture notes on matrix methods of structural analysis. It discusses three methods for solving systems of linear equations that arise in structural analysis problems - Gauss elimination method, Gauss-Jordan method, and Gauss-Seidel iterative method. For each method, an example problem is provided and solved step-by-step to illustrate the application of the method.
The document discusses bending stresses in beams. It describes how bending stresses are developed in beams to resist bending moments and shearing forces. The theory of pure bending is introduced, where only bending stresses are considered without the effect of shear. Equations for calculating bending stresses are derived based on the beam's moment of inertia, bending moment, and distance from the neutral axis. Several beam cross-section examples are provided to demonstrate how to calculate the maximum bending stress and section modulus.
Analysis of non sway frame portal frames by slopeand deflection methodnawalesantosh35
The slope deflection method is a displacement method used to analyze statically indeterminate beams and frames. It involves solving for the slope and deflection of members at their joints, which are the basic unknowns. Equations are developed relating the fixed end moments, slopes, and relative deflections of each member. These equations are set up and solved to determine the bending moments in each member. The method is demonstrated through examples solving for the bending moments in non-sway and sway frames.
This document discusses stresses in beams and beam deflection. It covers several methods for analyzing bending stresses and deflection in beams, including: [1] the engineering beam theory relating moment, curvature, and stress; [2] double integration and moment area methods for calculating slope and deflection; and [3] Macaulay's method, which simplifies calculations for beams with eccentric loads. Formulas are provided relating bending moment, shear force, curvature, slope, and deflection. Moment-area theorems are also described for relating bending moment to slope and deflection.
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.
Shear Force and Bending Moment DiagramsVishu Sharma
Short notes for shear force and bending moment diagrams in Strength Of materials for design calculation
Reference- Shigley’s Mechanical Engineering Design
Lecture 9 shear force and bending moment in beamsDeepak Agarwal
The document discusses stresses in beams. It covers topics like shear force and bending moment diagrams, bending stresses, shear stresses, deflection, and torsion. Beams are structural members subjected to transverse forces that induce bending. Stresses and strains are created within beams when loaded. Shear forces and bending moments allow determining these internal stresses and maintaining equilibrium. Formulas are provided for calculating shear forces and bending moments in different beam configurations like cantilevers, simply supported beams, and beams with various load types.
This document discusses strength of materials concepts related to shear force diagrams (SFD) and bending moment diagrams (BMD) for beams. It defines key terms like shear force, bending moment, and point of contraflexure. It also explains how to draw SFDs and BMDs for different beam types under various loading conditions and the relationships between loading, shear force, and bending moment. Application of the diagrams to reinforcement design is also mentioned.
The document discusses various numerical methods for analyzing mechanical components under applied loads, including the finite element method. It describes weighted residual methods like the Galerkin method and collocation method which approximate solutions by minimizing residuals. The variational or Rayleigh-Ritz method selects displacement fields to minimize total potential energy. The finite element method divides a structure into small elements and applies these methods to obtain approximate solutions for displacements and stresses at discrete points.
1. The document outlines key concepts in structural dynamics including idealization of structures as single-degree-of-freedom systems, formulation of the equation of motion, free and forced vibration of undamped and damped systems.
2. Key topics covered include natural frequency determination, Duhamel's integral, damping in structures, and methods for solving dynamic problems.
3. Examples of single-degree-of-freedom systems are presented including lumped mass systems, beams with distributed mass, and determination of effective stiffness.
This document discusses stresses in beams. It covers bending stresses, shear stresses, deflection in beams, and torsion in solid and hollow shafts. The key assumptions in beam bending theory are outlined. Bending stresses are explained, including the location of the neutral axis and how stresses vary through the beam cross-section based on the bending moment and geometry. Section modulus is defined as the ratio of the moment of inertia to the distance of the outermost fiber from the neutral axis. Composite beams made of different materials are also discussed.
The document discusses shear stresses in beams. It defines shear stress as being due to shear force and perpendicular to the cross-sectional area. Shear stress is derived as τ = F/A, where F is the shear force and A is the cross-sectional area. Shear stress varies across standard beam cross sections like rectangular, circular, and triangular. Shear stress is maximum at the neutral axis for rectangular and circular beams, and at half the depth for triangular beams. Sample problems are included to demonstrate calculating and graphing the variation of shear stress across specific beam cross sections.
Deflection of curved beam |Strength of Material LaboratorySaif al-din ali
SAIF ALDIN ALI MADIN
سيف الدين علي ماضي
S96aif@gmail.com
Experiment Name:- Deflection of curved beam
2. Introduction
The deflection of a beam or bars must be often be limited in order to provide
integrity and stability of structure or machine. Plus, code restrictions often require
these members not vibrate or deflect severely in order to safely support their
intended loading.
This experiment helps us to show some kind of deflection and how to calculate the
deflection value by using Castigliano’s Theorem and make a comparison between
result of the experiment and the theory.
This document discusses the slope-deflection method for analyzing beams and frames. It provides the theory and equations of the slope-deflection method. Examples are included to demonstrate how to use the method to determine support reactions, member end moments, and draw bending moment and shear force diagrams.
The document discusses various topics related to stress and strain including: principal stresses and strains, Mohr's stress circle theory of failure, 3D stress and strain, equilibrium equations, and impact loading. It provides examples of stresses acting in different planes including normal, shear, oblique, and principal planes. It also gives examples of calculating normal and tangential stresses on an oblique plane subjected to stresses in one, two, or multiple directions with and without shear stresses.
The document discusses buckling and its theories. It defines buckling as the failure of a slender structural member subjected to compressive loads. It provides examples of structures that can experience buckling. It explains Euler's theory of buckling which derived an equation for the critical buckling load of a long column based on its bending stress. The assumptions of Euler's theory are listed. Four cases of long column buckling based on end conditions are examined: both ends pinned, both ends fixed, one end fixed and one end pinned, one end fixed and one end free. Effective lengths are defined for each case and the corresponding critical buckling loads given. Limitations of Euler's theory are noted. Rankine's empirical formula for calculating ultimate
Columns are structural members that experience compression loads. They can buckle if loaded beyond their buckling (or critical) load. Short columns fail through crushing, while long columns fail through lateral buckling. The Euler formula calculates the buckling load of a long column based on its properties and end conditions. The Rankine-Gordon formula provides a more accurate calculation of buckling load that applies to all column types by accounting for both buckling and crushing. Proper design of columns involves ensuring they are loaded below their safe loads, which incorporate factors of safety applied to the theoretical buckling loads.
- This chapter discusses the deflection of beams under lateral loading. Deflection is the displacement of points on the beam's axis from its original, unloaded position.
- The basic differential equation relates the bending moment to the second derivative of the deflection curve. This equation can be directly integrated or used with methods like Macaulay's to determine deflections.
- Macaulay's method involves integrating the loading equation rather than the bending moment equation directly. It is useful when the bending moment is difficult to obtain.
- Special cases like loads not starting from the beginning or ending at the beam's end are also discussed.
Deflection of Beams _Chapter 5_2019_01_17!10_37_41_PM.pptSivarajuR
- This chapter discusses the deflection of beams under lateral loads. Deflection is the displacement of points on the beam's axis from its original, unloaded position.
- The basic differential equation relates the bending moment to the second derivative of the deflection curve. It can be directly integrated or used with methods like Macaulay's to determine deflections under different load cases.
- Macaulay's method allows integrating the loading function directly instead of first obtaining the bending moment. It is useful for concentrated or discontinuous loads.
- Special cases with loads not starting or ending at the beam's ends require modifying the integration limits.
This document provides solutions to problems involving the natural frequency and motion of single degree of freedom oscillatory systems. It analyzes problems related to beams, frames and pendulums undergoing harmonic motion. The solutions involve calculating the system stiffness and natural frequency, and determining displacement, velocity and acceleration as a function of time using the equations of motion for an undamped harmonic oscillator. Key parameters like mass, stiffness, initial conditions and system geometry are provided to find the desired responses.
The document discusses the double integration method for determining beam deflections. It defines beam deflection as the displacement of the beam's neutral surface from its original unloaded position. The differential equation relating the bending moment, flexural rigidity, and slope of the elastic curve is derived. This equation is integrated twice to obtain expressions for the slope and deflection of the beam in terms of the bending moment and constants of integration. Several examples are provided to demonstrate solving for the slope and maximum deflection of beams under different loading conditions using this method.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
Macaulay's method provides a simplified way to analyze beam bending by writing one equation for the entire beam's bending moment rather than separate equations for each load section. This allows boundary conditions from any part of the beam to determine integration constants. Poisson's ratio is the ratio of lateral to axial strain in a loaded system, and standard equations are derived relating stress and strain in multiple dimensions.
ME101-Lecture11 civil engineers use ful and9866560321sv
The document discusses beams and the relationships between shear force, slope of shear force diagram, bending moment, and slope of bending moment diagram. It states that the slope of the shear diagram equals the negative of the applied loading, while the slope of the moment curve equals the shear force. The degree of bending moment in x is two higher than that of the applied loading w. Examples of determining support reactions and drawing shear force and bending moment diagrams for beams under various loading conditions are also presented. The document then discusses cables, defining key parameters like tension, span, sag and length. It presents the differential equation that defines the shape of a flexible cable in equilibrium. Specific cases of parabolic cables under uniform loading and catenary cables under their
Deflection of structures using double integration method, moment area method, elastic load method, conjugate beam method, virtual work, castiglianois second theorem and method of consistent deformations
2d beam element with combined loading bending axial and torsionrro7560
The document discusses beam theory and finite element modeling of beams and frames. It provides information on modeling beams using one-dimensional beam elements with cubic shape functions. The formulation describes defining the element stiffness matrix and calculating the element's contribution to the global structural stiffness matrix and force vector based on applied loads. Boundary conditions and sample problems are presented to demonstrate the element modeling approach.
This document discusses cables and arches used in structural engineering. It provides information on how cables carry loads primarily in tension, while arches carry loads in compression. The document outlines various assumptions made in analyzing cables, including that cables are flexible, inextensible, and resist only axial forces. It also discusses analyzing cables under concentrated and uniformly distributed loads. Additional considerations are provided for cable-supported structures, including wind forces that must be considered in design.
The document discusses methods for calculating beam deflection under loading. Deflection is the displacement of the beam axis from its original straight position. The basic differential equation relates the bending moment to the second derivative of deflection. Direct integration and Macaulay's method can be used to solve this equation for specific load cases like concentrated loads or uniform loads not starting from the beginning. Mohr's area-moment method provides an alternative approach using properties of the bending moment diagram.
it contains the basic information about the shear force diagram which is the part of the Mechanics of solid. there many numerical solved and whivh will give you detaild idea in S.f.d.
The document describes an automatic calibration algorithm for a three-axis magnetic compass module. The algorithm has two stages:
1) The first stage characterizes the magnetic environment and corrects for distortions using an upper triangular soft iron matrix and hard iron offset vector. This makes the magnetometer measurements orthogonal and gain matched.
2) The second stage refines the soft iron matrix estimation by determining a rotation matrix to align the magnetometer coordinate system with the accelerometer coordinate system, improving heading accuracy. Given successful calibration, heading accuracies of 2 degrees or better can be achieved.
Strain energy is a type of potential energy that is stored in a structural member as a result of elastic deformation. The external work done on such a member when it is deformed from its unstressed state is transformed into (and considered equal to the strain energy stored in it.
Chapter 4 shear forces and bending momentsDooanh79
This document discusses shear forces and bending moments in beams. It defines types of beams and loads, and describes how to calculate reactions, shear forces, and bending moments for beams subjected to various loads. Examples are provided to demonstrate how to construct shear force and bending moment diagrams for beams with concentrated loads, uniform loads, and combinations of loads. Key relationships discussed include the relationship between shear forces and rate of change of the bending moment, and the relationship between the area under the shear force diagram and change in bending moment.
This document discusses key properties of quadratic graphs and functions including:
1) The standard form of a quadratic function is ax^2 + bx + c. The a value determines the width and whether there is a maximum or minimum.
2) The vertex is the maximum or minimum point, which can be found using the formula x = -b/2a.
3) The axis of symmetry is the line that passes through the vertex and the parabola is symmetric about this line. The axis of symmetry is defined by the equation x = -b/2a.
Se aplica el método de doble integración usando funciones de singularidad y el método de superposición para realizar el análsiis de deformaciones en vigas. Se resuelven vigas estáticaticamente por medio de estos métodos
1. Mohr's circle is a graphical representation used to determine stress components acting on a rotated plane passing through a point on a part.
2. It relates normal and shear stresses on the original x-y plane to those on a rotated plane using equations that define a circle.
3. Key values like maximum and minimum principal stresses that correspond to points on the circle can be read off to understand how stress changes with rotation.
Mohr's circle is a graphical representation of the transformation between normal and shear stresses on planes at various angles to the original plane of reference in two-dimensional stress fields. It allows determination of principal stresses and maximum shear stress. The document discusses the theory behind Mohr's circle, how to construct it, and provides an example problem calculating principal stresses and maximum shear stress given normal and shear stresses on a reference plane.
- The document describes the conjugate beam method for analyzing beams with varying rigidities.
- The method involves drawing an imaginary conjugate beam that is loaded based on the bending moments of the real beam.
- The slope and deflection of points on the real beam can then be determined from the shear and bending moment of the corresponding points on the conjugate beam.
- An example problem is worked out in detail to demonstrate calculating the slope and deflection at the end of a cantilever beam with varying moment of inertia along its length using the conjugate beam method.
Similar to Structural Mechanics: Deflections of Beams in Bending (20)
This document provides an overview and recap of key concepts related to Fourier analysis that are relevant for structural dynamics and earthquake engineering. It defines Fourier series and the Fourier transform, explaining how they can be used to decompose signals into harmonic components in the frequency domain. It also reviews concepts such as the frequency response function and dynamic amplification factor for single-degree-of-freedom oscillators subjected to harmonic loading.
This document outlines a lecture on structural dynamics and earthquake engineering. It introduces the module, including its aims, assessment, and schedule. Examples are provided of structural failures caused by vibrations, like the collapse of the Tacoma Narrows Bridge and wobbling of the Millennium Bridge. Equations of motion are derived for a single-degree-of-freedom undamped oscillator. Students generally found the module and lecturer engaging the previous time it was taught, though some suggested improving the lecture rooms.
This document contains the teaching schedule and lecture topics for a course on complex strains taught by Dr. Alessandro Palmeri. The course covers various topics related to complex stress and strain analysis, including beam shear stresses, shear centres, virtual forces, compatibility methods, moment distribution methods, column stability, unsymmetric bending, and complex stress/strain analysis. Lectures are delivered by Dr. Palmeri and other staff members. Tutorial sessions are also included to provide examples and applications of the taught concepts. The schedule lists the topics to be covered in each week across the 12-week term, with exams occurring in the final two weeks.
This document contains information about a teaching schedule for a course on complex stresses. It will cover topics like beam shear stresses, shear centres, virtual forces, compatibility methods, moment distribution methods, column stability, unsymmetric bending, and complex stress/strain over 11 weeks. The lectures and tutorials will be led by various staff members. The document also provides motivations for studying complex stresses, which include the fact that failure often results from different stresses acting together, and discussing examples like welded connections, reinforced concrete, and concrete cylinder tests.
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
Using blurred images to assess damage in bridge structures?Alessandro Palmeri
Faster trains and augmented traffic have significantly increased the number and amplitude of loading cycles experienced on a daily basis by composite steel-concrete bridges. This higher demand accelerates the occurrence of damage in the shear connectors between the two materials, which in turn can severely affect performance and reliability of these structures. The aim of this talk is to present the preliminary results of theoretical and experimental investigations undertaken to assess the feasibility of using the envelope of deflections and rotations induced by moving loads as a practical and cost-effective alternative to traditional methods of health monitoring for composite bridges. Both analytical and numerical formulations for this dynamic problem are presented and the results of a parametric study are discussed. A novel photogrammetric approach is also introduced, which allows identifying vibration patterns in civil engineering structures by analysing blurred targets in long-exposure digital images. The initial experimental validation of this approach is presented and further challenges are highlighted.
- This document discusses a teaching schedule for a course on the failure of slender and stocky columns. It covers topics like column stability, unsymmetric bending, and complex stress/strain over 11 weeks.
- The key learning outcomes are to derive the Euler critical load for slender pinned-pinned columns under compression and to predict the failure mode of short and slender columns.
- The document motivates the importance of considering both the stiffness and strength of materials, and how the slenderness of a column affects its failure mode in compression.
This document provides information about structural analysis and mechanics modules taught at Loughborough University. It discusses the topics covered, teaching staff, assessment methods, and schedule. The key topics are shear stresses, shear center, column stability, unsymmetrical bending, and complex stresses and strains. The modules are assessed through a lab-based coursework and exam. Changes introduced this year include an additional tutorial, individual lab coursework submissions instead of group submissions, and expectations around group work.
1) The document discusses the state-space formulation and numerical solution of the equation of motion for a single-degree-of-freedom oscillator subjected to earthquake ground motion.
2) It presents the state-space representation using state variables of displacement and velocity, and defines the transition matrix used to solve the equation of motion incrementally over time.
3) The solution is obtained by dividing the time interval into steps of size Δt, and computing the state variables at each step using the transition matrix and integration matrices dependent on the transition matrix.
Structural Mechanics: Shear stress in Beams (1st-Year)Alessandro Palmeri
- The document discusses shear stress in beams, specifically focusing on Jourawski's formula for calculating shear stress.
- Jourawski's formula provides an approximate solution for the shear stress distribution over a beam cross-section using simple equilibrium considerations.
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- For T-sections and I-sections, the formula can be applied to the web, where it accurately models the shear stress as parabolic. The maximum shear stress occurs at the neutral axis.
This document summarizes key points from lectures on Fourier analysis and frequency response functions for single degree of freedom oscillators. It recaps concepts like natural frequency, damping ratio, and dynamic amplification factor. It introduces Fourier series as a way to decompose periodic signals into harmonic components. The Fourier transform is presented as a way to study the frequency content of both periodic and non-periodic signals. Examples are given to illustrate the effects of varying parameters in the time and frequency domains.
This document provides an introduction and overview of a module on Structural Dynamics and Earthquake Engineering. It discusses motivations for studying structural dynamics, including examples of structural failures like the Tacoma Narrows Bridge and Millennium Bridge that collapsed due to resonant vibrations. It outlines the aims and content of the module, which will cover equations of motion for single-degree-of-freedom and multi-degree-of-freedom oscillators, modal analysis, damped structures, and seismic analysis including response spectra. The document concludes with the assessment structure and schedule for the module.
This document provides an overview of a structural dynamics and earthquake engineering module being taught by Dr. Alessandro Palmeri. The module aims to develop knowledge of vibrational problems in structural engineering and provide tools to assess dynamic response, with emphasis on seismic analysis per Eurocode 8. It will cover dynamics of single-degree-of-freedom and multi-degree-of-freedom systems, as well as topics in earthquake engineering including response spectra, lateral force methods, and performance-based design. Students will be assessed through a 2-hour exam and a group coursework assignment. The module will be delivered over 12 weeks through lectures, tutorials, and assignments.
Toward an Improved Computational Strategy for Vibration-Proof Structures Equi...Alessandro Palmeri
This presentation has been delivered at the 15th World Conference on Earthquake Engineering in Lisbon (Portugal) on 28th September 2012, and shows some preliminary results on the dynamic analysis on non-linear viscoelastic structures.
Spectrum-Compliant Accelerograms through Harmonic Wavelet TransformAlessandro Palmeri
This presentation has been delivered at the 11th International Conference on Computational Structures Technology in Dubrovnik (Croatia) on 7th September 2012, and shows how the harmonic wavelet transform can be effectively used: first, to adjust a recorded accelerogram to match a given elastic design spectrum; second, to generate a number of fully non-stationary samples with the same probabilistic features.
This presentation is intended for year-2 BEng/MEng Civil and Structural Engineering Students. The main purpose is to present how characterise wind loading on simple building structures according to Eurocode 1
Toward overcoming the concept of effective stiffness and damping in the dynam...Alessandro Palmeri
In the current state-of-practice, the time-domain dynamic analysis of structures incorporating viscoelastic members is generally carried out through the Modal Strain Energy (MSE) method, or other procedures somehow based on the quite simplistic idea of substituting the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping.
This crude approximation in civil engineering applications is very often encouraged by manufacturers of the viscoelastic devices themselves, whose interest is to simplify as much as possible the design procedures for structures embedding their products. As an example, elastomeric seismic isolators are generally advertised and sold with a table listing the equivalent values of elastic stiffness and viscous damping ratio for different amplitudes of vibration. Unfortunately, many experimental and analytical studies confirm that the real dynamic behaviour of such devices is much more complicated, and cannot be bend to the interests of manufacturers and designers.
Despite the advances in the field made in the last two decades, two well-established beliefs continue to underpin use and abuse of the concepts of effective stiffness and damping for viscoelastically damped structures: first, MSE method and similar procedures are unconditionally assumed to provide good approximations, which are acceptable for design purposes; second, the implementation of more refined approaches is thought to be computationally too expensive, and hence suitable just for a few very important constructions.
In this presentation, as a further contribution to overcome these popular beliefs, a novel time-domain numerical scheme of dynamic analysis is proposed and numerically validated. After a brief review of the LPA (Laguerre’s Polynomial Approximation) technique for one-dimensional viscoelastic members of known relaxation function, the state-space equations of motion for linear structures with viscoelastic components are derived in the modal space. Aimed at making the proposed approach more general, the distribution of the viscoelastic components is allowed to be non-proportional to mass and elastic stiffness, in so removing the most severe limitation of previous formulations. Then, a cascade scheme is derived by decoupling in each time step traditional state variables (i.e. modal displacements and velocities) and additional internal variables. The joint use of modal analysis and improved cascade scheme permits to reduce the size of the problem and to keep low the computational burden. The illustrative application to the small-amplitude vibration of a cable beam made of different viscoelastic materials demonstrates the versatility of the proposed approach. The numerical results confirm a superior accuracy with respect to the classical MSE method, whose underestimate in the low-frequency range can be as large as 75%.
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 3)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
Lesson Outcomes:
- students will be able to identify and name various types of ornamental plants commonly used in landscaping and decoration, classifying them based on their characteristics such as foliage, flowering, and growth habits. They will understand the ecological, aesthetic, and economic benefits of ornamental plants, including their roles in improving air quality, providing habitats for wildlife, and enhancing the visual appeal of environments. Additionally, students will demonstrate knowledge of the basic requirements for growing ornamental plants, ensuring they can effectively cultivate and maintain these plants in various settings.
Get Success with the Latest UiPath UIPATH-ADPV1 Exam Dumps (V11.02) 2024yarusun
Are you worried about your preparation for the UiPath Power Platform Functional Consultant Certification Exam? You can come to DumpsBase to download the latest UiPath UIPATH-ADPV1 exam dumps (V11.02) to evaluate your preparation for the UIPATH-ADPV1 exam with the PDF format and testing engine software. The latest UiPath UIPATH-ADPV1 exam questions and answers go over every subject on the exam so you can easily understand them. You won't need to worry about passing the UIPATH-ADPV1 exam if you master all of these UiPath UIPATH-ADPV1 dumps (V11.02) of DumpsBase. #UIPATH-ADPV1 Dumps #UIPATH-ADPV1 #UIPATH-ADPV1 Exam Dumps
Hospital pharmacy and it's organization (1).pdfShwetaGawande8
The document discuss about the hospital pharmacy and it's organization ,Definition of Hospital pharmacy
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Information and Communication Technology in EducationMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 2)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐈𝐂𝐓 𝐢𝐧 𝐞𝐝𝐮𝐜𝐚𝐭𝐢𝐨𝐧:
Students will be able to explain the role and impact of Information and Communication Technology (ICT) in education. They will understand how ICT tools, such as computers, the internet, and educational software, enhance learning and teaching processes. By exploring various ICT applications, students will recognize how these technologies facilitate access to information, improve communication, support collaboration, and enable personalized learning experiences.
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐫𝐞𝐥𝐢𝐚𝐛𝐥𝐞 𝐬𝐨𝐮𝐫𝐜𝐞𝐬 𝐨𝐧 𝐭𝐡𝐞 𝐢𝐧𝐭𝐞𝐫𝐧𝐞𝐭:
-Students will be able to discuss what constitutes reliable sources on the internet. They will learn to identify key characteristics of trustworthy information, such as credibility, accuracy, and authority. By examining different types of online sources, students will develop skills to evaluate the reliability of websites and content, ensuring they can distinguish between reputable information and misinformation.
Decolonizing Universal Design for LearningFrederic Fovet
UDL has gained in popularity over the last decade both in the K-12 and the post-secondary sectors. The usefulness of UDL to create inclusive learning experiences for the full array of diverse learners has been well documented in the literature, and there is now increasing scholarship examining the process of integrating UDL strategically across organisations. One concern, however, remains under-reported and under-researched. Much of the scholarship on UDL ironically remains while and Eurocentric. Even if UDL, as a discourse, considers the decolonization of the curriculum, it is abundantly clear that the research and advocacy related to UDL originates almost exclusively from the Global North and from a Euro-Caucasian authorship. It is argued that it is high time for the way UDL has been monopolized by Global North scholars and practitioners to be challenged. Voices discussing and framing UDL, from the Global South and Indigenous communities, must be amplified and showcased in order to rectify this glaring imbalance and contradiction.
This session represents an opportunity for the author to reflect on a volume he has just finished editing entitled Decolonizing UDL and to highlight and share insights into the key innovations, promising practices, and calls for change, originating from the Global South and Indigenous Communities, that have woven the canvas of this book. The session seeks to create a space for critical dialogue, for the challenging of existing power dynamics within the UDL scholarship, and for the emergence of transformative voices from underrepresented communities. The workshop will use the UDL principles scrupulously to engage participants in diverse ways (challenging single story approaches to the narrative that surrounds UDL implementation) , as well as offer multiple means of action and expression for them to gain ownership over the key themes and concerns of the session (by encouraging a broad range of interventions, contributions, and stances).
Structural Mechanics: Deflections of Beams in Bending
1. Deflections in beams
Dr Alessandro Palmeri
Senior Lecturer in Structural Engineering
<A.Palmeri@lboro.ac.uk>
1/
39
2. Learning Outcomes
When we have completed this unit (3 lectures + 1
tutorial), you should be able to:
◦ Use the double integration technique to
determine transverse deflections in slender
beams under distributed and/or concentrated
loads
Schedule:
◦ Lecture #1: Double integration method
◦ Lecture #2: Macaulay’s notation
◦ Lecture #3: Numerical application
◦ Tutorial
2/
39
4. Introduction
Structural
members must have:
◦ Strength (ULS: Ultimate Limit State)
◦ Stiffness (SLS: Serviceability Limit State)
Need
to limit deflection because:
◦ Cracking
◦ Appearance
◦ Comfort
4/
39
Engineering Structures,Volume 56, 2013, 1346 - 1361
5. Introduction
Standards typically limit deflection of beams by fixing
the maximum allowable deflection in terms of span:
◦ e.g. span/360 for steel beams designed according to
Eurocode 3
Deflections in beams may occur under working loads,
where the structure is usually in the linear elastic
range
Theyare therefore checked using an elastic analysis
◦ no matter whether elastic or plastic theory has been used
in the design for strength
We’ll introduce some basic concepts of plastic analysis for ductile
beams in bending later this semester
5/
39
6. Introduction
Many
methods are available for calculating
deflection in beams, but broadly speaking
they are based on two different
approaches
a) Differential equation of beams in bending
This approach will be considered in this module
b) Energy methods
6/
39
e.g.Virtual Work Principle
7. Curvature
From
the simple theory of bending we
have:
1
M
=
R EI
where
◦ E is the Young’s modulus of the material
◦ I is the second moment of area
◦ 1/R is referred as beam’s curvature
7/
39
8. Curvature
For a plane curve uz(x) in the xz plane, the curvature 1/Ry
(about the orthogonal axis y) is given by:
x
d 2uz
dx 2
1
=
Ry ⎡ ⎛ du ⎞ 2 ⎤
⎢1+ ⎜ z ⎟ ⎥
dx ⎠ ⎥
⎢ ⎝
⎣
⎦
8/
39
y
Ry
z
If duz/dx is small, then (duz/dx)2 can be considered negligible
d 2uz
1
Thus:
≅
Ry
dx 2
And so:
My
E I yy
d 2uz
=
dx 2
9. Sign convention
Mostly vertical loads act vertically
◦ Downward deflection uz is +ve
Already chosen bending moment convention
◦ Sagging moment My is +ve
We must reconcile these two choices:
load
slope
x
z
9/
39
z
duz
>0
dx
curvature
x
x
z
d 2uz
2
>0
dx
But this is the
shape of hogging
bending moment,
i.e. My<0
10. Differential equation of slender
beams in bending
Taking into account the correct sign convention
for deflection and bending moment, we have:
d 2uz (x)
E I yy
= − M y (x)
2
dx
◦ This is the starting point of the double integration
method, which enables one to evaluate slope duz/dx
and deflection uz in a slender beam in bending
◦ Note that in the above equation:
10/
39
Iyy means second moment of area about the horizontal axis y
My means bending moment about the same axis (depends on x)
uz is the vertical deflection (also depends on x)
11. Double integration method
The differential equation of beams in bending
must be integrated twice with respect to the
abscissa x
◦ The minus sign in the right-hand side is crucial
11/
39
Since the bending moment My usually varies along
the beam, therefore we need to write the
mathematical expression of My=My(x)
As we are solving a 2nd-order differential
equation, 2 integration constants, C1 and C2, will
arise
12. Boundary conditions
The
integration constants C1 and C2 are
determined from the known boundary
conditions, i.e. conditions at the supports
Simple support
No deflection
uz=0
Fixed support
No deflection and no slope
12/
39
uz=0 and duz/dx=0
13. Worked example
Determine
deflection and slope at the free
end B of a cantilever beam of length L
subjected to a uniformly distributed load qz
◦ subscript z means that the load acts vertically
qz
MA
13/
39
A
z
RA
B
L
x
15. Worked example
qz
MA
A
z
RA
B
L
x
2nd, write down the
expression of the
bending moment My as
a function of the
abscissa x along the
beam’s axis:
qz x 2
M y = −M A + RA x −
2
qz L2
qz x 2
=−
+ qz L x −
2
2
(
15/
39
)
16. Worked example
The
differential equation for the beam’s
deflection reads:
d 2uz
qz L2
qz x 2
E I yy 2 = − M y =
− qz L x +
2
2
dx
3rd, we
can integrate twice:
2
2
3
duz qz L
qz L x
qz x
E I yy
=
x−
+
+ C1
dx
2
2
6
16/
39
qz L2 2 qz L x 3 qz x 4
E I yy uz =
x −
+
+ C1 x + C2
4
6
24
17. Worked example
4th, the known boundary conditions at the fixed support (i.e.
no deflection and no slope at left-hand side end A):
duz
= 0 @ x = 0 ⇒ C1 = 0
dx
u z = 0 @ x = 0 ⇒ C2 = 0
Substituting now the values of the integration constants C1
and C2, the expressions for slope and deflection throughout
the beam become:
2
duz
qz L x 2 qz x 3 ⎞
1 ⎛ qz L
=
⎜ 2 x− 2 + 6 ⎟
dx E I yy ⎝
⎠
17/
39
2
3
4
1 ⎛ qz L 2 qz L x qz x ⎞
uz =
⎜ 4 x − 6 + 24 ⎟
EI⎝
⎠
18. Worked example
5th, intuitively we
know that slope and
deflection in the
cantilever beam take
the maximum values at
the free end B
By substituting x=L in
the general expression
of the slope along the
beam, we get:
qz
MA
A
z
RA
18/
39
B
L
x
2
⎛ duz ⎞
qz L L2 qz L3 ⎞
qz L3
1 ⎛ qz L
(> 0, )
⎜ dx ⎟ = E I ⎜ 2 L − 2 + 6 ⎟ = 6 E I
⎝
⎠B
yy ⎝
⎠
yy
21. Beams under point loads
E.g. simply
supported beam with a single
concentrated load
2m
4m
Fz
A
B
C
z
21/
39
RA
6m
RB
x
22. Beams under point loads
2m
Σ M (A) = 0
⇒ − Fz 2 + RB 6 = 0
4m
Fz
A
B
C
z
RA
22/
39
Support reactions
6m
RB
x
2 Fz Fz
⇒ RB =
=
()
6
3
Σ M (B) = 0
⇒ − RA 6 + Fz 4 = 0
4 Fz 2
⇒ RA =
= Fz ()
6
3
23. Beams under point loads
0<x<2
A
RA
2
Fz
A
RA
23/
39
In principle, we need
two expression for
the bending moment
My:
◦ one for 0<x<2
M y = RA x
C
2<x<6
◦ one for 2<x<6
(
M y = RA x − Fz x − 2
)
24. Beams under point loads
In
principle, we need to integrate two
differential equations:
⎧ RA x , 0 < x < 2
⎪
=⎨
2
dx
⎪ RA x − Fz x − 2 , 2 < x < 6
⎩
2
E I yy
d uz
(
)
This
is possible, but four integration
constants arise, i.e. two for each differential
equation
◦ For more than one points load, the procedure
becomes quite cumbersome
24/
39
25. Macaulay’s notation
It would be much more effective to have a single
mathematical expression for the bending moment
My along the beam
This is possible with the help of the so-called
Macaulay’s notation, i.e. square brackets [ ] with a
special meaning:
◦ If the term within square brackets is +ve, then it
is evaluated
◦ If the term within square brackets is –ve, then it
is ignored
25/
39
26. Macaulay’s notation
That
is:
⎧ x , if x > 0
[ x] = ⎨
⎩0 , if x ≤ 0
Let’s
26/
39
try the following examples:
⎡ 2.3⎤ = 2.3
⎣ ⎦
⎡0 ⎤ = 0
⎣ ⎦
⎡ −3 / 4 ⎤ = 0
⎣
⎦
28. Macaulay’s method
The differential equation of bending becomes:
E I yy
d 2uz
dx
2
= −M y = −RA x + Fz ⎡ x − 2 ⎤
⎣
⎦
This expression can be integrated twice,
importantly, without expanding the term into
square brackets:
2
2
du
2 Fz x ⎡ x − 2 ⎤ − 2 ⎤
dy z
x
⎣x ⎦ + C
⎣ F ⎡⎦
=−
E I yy = −RA + W + z
1
2
dxdx
23 2
2
2
2
28/
39
3
⎡ x − 2⎤
Fz x
⎦ +C x +C
E I yy uz = −
+ Fz ⎣
1
2
9
6
3
29. Macaulay’s method
Since we are integrating a single 2nd-order differential
equation, just 2 integration constants appear in the solution,
C1 and C2:
◦ These quantities can be determined by using the boundary
conditions, i.e. conditions at the supports
◦ Importantly, the square bracket term is only included if the
quantity inside is +ve
uz = 0 @ x = 0 ⇒
3
⎡ −2 ⎤
0 = −0 + Fz ⎣ ⎦ + 0 + C2 ⇒ C2 = 0
6
uz = 0 @ x = 6 ⇒
3
3
⎡4⎤
Fz 6
⎛
32 ⎞
0=−
+ Fz ⎣ ⎦ + C1 6 ⇒ 6 C1 = ⎜ 24 − ⎟ Fz
9
6
3⎠
⎝
29/
39
20 Fz
1 ⎛ 72 − 32 ⎞
⇒ C1 = ⎜
⎟ Fz = 9
6⎝ 3 ⎠
30. Macaulay’s method
30/
39
Starting from a single expression of the bending
moment My, we obtained a single expression
throughout the beam for the deflection uz, in which
we have the Macaulay’s brackets:
⎛ 3 ⎡ x − 2⎤3
⎞
Fz
x ⎣
20 x
⎦ +
⎜− +
⎟
uz =
EI⎜ 9
6
9 ⎟
⎝
⎠
We can now evaluate the deflection of the beam at
the position of the point load uz(C), i.e. uz @ x= 2 m
⎛ 3 ⎡0 ⎤ 3
⎞
Fz
2 ⎣ ⎦ 20 × 2
40 − 8 Fz
⎜− +
⎟=
uz (C) =
+
E I yy ⎜ 9
6
9 ⎟
9 E I yy
⎝
⎠
32 Fz
=
()
9 E I yy
31. Macaulay’s method
We have also a single expression throughout the beam for
the slope duz/dx:
⎛ 2 ⎡ x − 2⎤2
⎞
duz
Fz
x
⎦ + 20 ⎟
⎜− + ⎣
=
dx E I yy ⎜ 3
2
9⎟
⎝
⎠
The slopes at the supports A and B, i.e. duz/dx @ x= 0 and
x= 6 m take the values
2
⎛
⎞
⎡ −2 ⎤
⎛ duz ⎞
Fz
20 ⎟ 20 Fz
⎜ −0 + ⎣ ⎦ +
=
=
()
⎜ dx ⎟
2
9 ⎟ 9 E I yy
⎝
⎠ A E I yy ⎜
⎝
⎠
⎛ 2 ⎡4⎤2
⎞
⎛ duz ⎞
Fz
6 ⎣ ⎦ 20
−216 + 144 + 40 Fz
⎜− +
=
+ ⎟=
⎜ dx ⎟
2
9⎟
18
E I yy
⎝
⎠B E I ⎜ 3
⎝
⎠
31/
39
=−
16 Fz
()
9 E I yy
33. Numerical example
Find position and value of the maximum deflection in
the simply supported beam shown below
z
RA
33/
39
C
20 kN
A
2m
60 kN
1m 2m
B
D
x
5m
RB
The beam’s flexural rigidity is EIyy= 2.58×104 kN m2
34. Support reactions
z
RA
C
20 kN
A
2m
60 kN
1m 2m
B
D
5m
x
The first step is to
evaluate the support
reactions at points A
and B:
RB
Σ M ( A) = 0 Q
⇒ − 60 × 1 − 20 × 3 + RB × 5 = 0
34/
39
60 + 60
⇒ RB =
= 24 kN (#)
5
35. Support reactions
z
RA
C
20 kN
A
2m
60 kN
1m 2m
B
D
5m
x
The first step is to
evaluate the support
reactions at points A
and B:
RB
A
Σ M ( B) = 0 Q
⇒ − 60 × 1 − 20 × 34 +RB × 52= 00
RA × 5 + 60 × + 20 × =
34/
39
60 ++ 40
240 60
= = 56kN ##)
24 kN ( ()
⇒ RB =
A
55
36. Bending moment’s expression
z
RA
35/
39
C
20 kN
A
2m
60 kN
1m 2m
B
D
5m
RB
x
Once we have all the
external forces
applied to the beam
(external forces and
support reaction),
the second step is to
write down the
expression of the
bending moment My
along the beam
M y = 56 x − 60 ⎡ x −1⎤ − 20 ⎡ x − 3⎤
⎣
⎦
⎣
⎦
37. Double integration
d 2uz
EI yy 2 = − M y = −56 x + 60 ⎡ x − 1⎤ + 20 ⎡ x − 3⎤
⎣
⎦
⎣
⎦
dx
2
2
⎡ x − 1⎤
duz
x
⎣
⎦ + 20 ⎡ x − 3⎤ + C
⎣
⎦
EI yy
= − 56
+ 60
1
dx
2
2
2
28
30
2
3
10
3
⎡ x −1⎤
x
⎣
⎦ + 10 ⎡ x − 3⎤ + C x + C
⎣
⎦
EI yy uz = −28 + 30
1
2
3
3
3
3
36/
39
10
Boundary conditions (simply supports at points A and B) gives:
⎧u z = 0 @ x = 0 ⇒ C 2 = 0
⎪
⎨
⎪uz = 0 @ x = 5 ⇒ C1 = 100
⎩
38. Abscissa of maximum deflection
37/
39
Within a span, the maximum deflection will occur where the
slope of the beam is zero. So to find the position of the maximum
deflection, we can determine the value of the abscissa x that gives
duz/dx=0.
We have the mathematical expression of the slope, but it contains
two square brackets, and we must decide which of them should
be retained.
As the position of maximum deflection is never very far away
from the centre of the span, we can guess that it occurs between
x=1 and x=3 m. In this region the expression for the slope
becomes:
2
2
duz
2
= −28 x + 30 ⎡ x − 1⎤ + 10 ⎡ x − 3⎤ + 100
⎣
⎦
⎣
⎦
dx
39. Abscissa of maximum deflection
We can now solve the quadratic equation:
duz
2
2
= 0 ⇒ − 28 x + 30 ( x − 1) + 100 = 0
dx
−28 x 2 + 30x 2 − 60x + 30 + 100 = 0
2x 2 − 60x + 130 = 0
38/
39
60 ± 602 − 4 × 2 × 130
3,600 − 1,040
x=
= 15 ±
4
4
⎧
⎪
⎪ 27.65 → Root unacceptable
⎪
xmax = 15 ± 12.65 = ⎨
(outside the beam)
⎪
⎪2.35 → Root consistent with the
⎪
assumption 0 ≤ x ≤ 3
⎩
40. Maximum deflection
We can now evaluate the deflection at x=2.35 m:
uz ,max
3
3
⎧
⎫
⎡ xmax − 3⎤
xmax
3
1 ⎪
⎦ + 100 x ⎪
=
−28
+ 10 ⎡ xmax − 1⎤ + 10 ⎣
⎨
max ⎬
⎣
⎦
EI yy ⎪
3
3
⎪
⎩
⎭
3
⎧
⎫
3
⎡ 2.35 − 3⎤
3
1
2.35
⎪
⎦ + 235⎪
=
−28
+ 10 ⎡ 2.35 − 1⎤ + 10 ⎣
⎨
⎬
⎣
⎦
3
3
2.58 × 104 ⎪
⎪
⎩
⎭
−121.13+ 24.60 + 235
138.47
=
=
= 53.7 × 10−4 m = 5.4mm
2.58 × 104
2.58 × 104
39/
39
So maximum deflection is 5.4mm at 2.35m from the left
support
Now check that you can show that the deflections under the
60kN and 20kN loads are 3.5mm and 5.0mm, respectively.