It is described in wikipedia, and in more detail, with numerous examples by colin. Analysis of beams to find the deflections is dealt in later chapter. Unit iv deflection of beams 9 double integration method macaulays method area moment method for computation of slopes and deflections in beams conjugate beam and strain energy maxwells reciprocal theorems. Macaulays method helps to simplify the calculations to find the deflection of beams subjected to point loads. Chapter 11 page 1 beam deflection and rotation using macaulays method 11. Macaulays method is a convenient method of applying beam slopedeflection equations to find the shear forces, bending moments, slopes and deflections of a structural beam subject to complex load conditions, including partial varying distributed loads and point loads and moments.
B max ymax a methods for finding slope and deflection of beams. This method is suitable for cantilevers and simply supported beams carrying symmetrical loadings and beams fixed at both ends i. Download ce6306 strength of materials som books lecture notes syllabus part a 2 marks with answers ce6306 strength of materials som important part b 16 marks questions, pdf books, question bank with answers key, ce6306. Mechanics of materials chapter 6 deflection of beams. Structural analysis by amit sir macaulays method problem 1 structural analysis. The deflection of beam elements is usually calculated on the basis of the eulerbernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory. K curvature which is the second dederivativ of the deflection functio. Here the basic equation for deflection is ei km where e is modulus of elasticity i the second moment of inertia. Three basic methods to find deflection for statically determinate beams. You will also learn and apply macaulays method to the solution for beams with a combination of loads. International journal of mechanical engineering education 2012 35. Macaulays method problem 1 slope and deflection of. Deflection of beam and cantilever a deflection of a. Can we find the slope and deflection of a cantilever beam.
Beam actions and deflections by macaulays method newton. Pdf macaulays method for a timoshenko beam researchgate. In the technical bending theory, we make two main assumptions that. Able to analyze determinate beam deflection and slope by macaulay method. Method of double integration the primary advantage of the double integration method is that it produces the equation for the deflection everywhere along the beams. Chapter 11 macaulays deflection and rotation method. A cantilever beam is 6 m long and has a point load of 20 kn at the free end. Beam, deflection, stress, strain equation and calculators strength mechanical of materials. Stiffness method, which forms the basis of most computer software currentlyused for stt ltructural analilysis. Deflection equation using macaulays physics forums. Slope and deflection of beams mechanical engineering mcq questions and answers home category mechanical engineering mcq questions and answers slope and deflection of beams 1 a simply supported beam carries uniformly distributed load of 20 knm over the length of 5 m.
The material of beam has modulus of elasticity as 200 gpa. Macaulays method the procedure of finding slope and deflection for simply supported beam with an eccentric load is very laborious. Use integration method to determine the values of slope and deflection at 2 m from the free end of the cantilever due to the imposed load as shown in figure a. The project also discusses various theoretical analysis techniques that can be implemented in developing a computer program. Pdf the macaulay bracket notation is familiar to many engineers for the deflection analysis of a eulerbernoulli beam subject to multiple or. Macaulays method in which we can write the different equation for bending moment for different sections. You will also learn and apply macaulays method to the solution for beams with a combination of. May 15, 2015 while this beam is a good exercise for showing how to calculate deflection using mccauley s method, it is a terrible choice because its deflections are much too large to satisfy the small slope approximation on which elastic beam theory is based. Use of macaulays technique is very convenient for cases of discontinuous andor discrete loading. A simply supported beam with a point load at the middle. Double integration method macaulays method area moment method for computation. Mechanics of materials civl 3322 mech 3322 deflection of beams the elastic curve. Fortunately, we have what we call discontinuity functions that allow us to have an elastic curve that fully defines the. While this beam is a good exercise for showing how to calculate deflection using mccauleys method, it is a terrible choice because its deflections are much too large to satisfy the small slope approximation on which elastic beam theory is based.
Calculation of slope and deflection by macaulays method. In engineering, deflection is the degree to which a structural element is displaced under a load due to its deformation. Otherwise methods such as virtual work, direct integration, castiglianos method, macaulays method or the direct stiffness method are used. Use macaulays method to determine the values of slope and deflection at 2m from the free end of the cantilever due to the imposed load as shown in figure 61 a. We all agree that the max deflection occurs in the middle of the 5mtr beam. Download link is provided and students can download the anna university ce6306 strength of materials som syllabus question bank lecture notes syllabus part a 2 marks with answers part b 16 marks question bank with answer, all the materials are listed below for the students to make use of it and score good maximum marks with our study materials. Use of macaulay s technique is very convenient for cases of discontinuous andor discrete loading. The method is, in essence, a rst exposure to and predates generalised functions e. Macaulays method slope and deflection of beams strength of. A cantilever beam, 15m long has a udl of 25nm acting along its entire length and. Deflection of beam theory at a glance for ies, gate, psu 5. The main theoretical methods used in this project are moment distribution and macaulays method. State the difference between strain energy method and unit load method in the determination of deflection of. Before macaulays paper of 1919, the equation for the deflection of beams could not be found in closed form.
The macaulay bracket notation is familiar to many engineers for the deflection analysis of a eulerbernoulli beam subject to multiple or discontinuous loads. Areamoment method calculate deflections in beams engineers. M y x 2 dx this is the starting point of the double integration method, which enables one to evaluate slope duzdx and deflection uz in a slender beam in bending note that in the. Before macaulays paper of 1919, the equation for the deflection of beams. I have calculated one of the 2 questions successfully with it being a simply supported beam, now i am stuck on the remaining question. A generalization of macaulays method with applications in structural mechanics.
In the case of the beams, we use this term for linear vertical displacement. Structural behaviour and detailing for construction jesmond agius. Macaulays method of developing deflection and slope equations at any section of the beam will be discussed in details and students will also be able to solve different problems as this method is most versatile. Using macaulays step functions, determine the deflection at l2 flexural rigidity ei ab, pb pa rr ll equilibrium. Beams deflection macaulays method beam structure stress. Dirac delta, step, and ramp, with meaning given over to the. Slope and deflection of beams mechanical engineering mcq questions and answers home category mechanical engineering mcq questions and answers slope and deflection of beams 1 a simply supported beam carries uniformly distributed load. View chapter 11 macaulay s deflection and rotation method. The double integration method, also known as macaulays method is a powerful tool in solving deflection and slope of a beam at any point because we will be able to get the equation of the elastic curve. The macaulay bracket notation is familiar to many engineers for the deflection analsyis of a eulerbernoulli beam subject to multiple or discontinuous loads. It is described in wikipedia, and in more detail, with numerous examples by colin caprani. Differential equation of slender beams in bending taking into account the correct sign convention for deflection and bending moment, we have.
This program was created using the relatively new actionscript language. Areamoment method to calculate deflections in beams. To prevent any attached brittle materials from cracking 2. Deflection can be calculated by standard, or by methods such as virtual work, direct integration, castiglianos method, macaulays method or the direct stiffness method, amongst others. Macaulays method the double integration method is a technique used in structural analysis to determine the deflection of eulerbernoulli beams. Apr 20, 2009 i have calculated one of the 2 questions successfully with it being a simply supported beam, now i am stuck on the remaining question. Different equations for bending moment were used at. Macaulays method problem 1 structural analysis 1 youtube. Previously i have used the double integration method.
Caprani deflection of flexural members macaulays method 3rd year structural engineering 200910 dr. Macaulays method problem 1 slope and deflection of beams. Macaulays method the macaulays method involves the general method of obtaining slopes and deflections i. Of beams chapter five deflection of beams when a beam with a straight longitudinal axis is loaded by lateral forces, the axis is deformed into a curve, called the deflection curve of the beam. Lecture 12 deflection in beams linkedin slideshare. Macaulays method for a timoshenko beam eprints soton. Slope and deflection of beams mechanical engineering mcq. The deflection of beam elements is usually calculated on the basis of the eulerbernoulli beam equation while that of a plate or shell element is calculated. Bernoullis hypothesis about the plane crosssections is valid. Slope and deflections are maximum at free end increases from point a towards b. What is the difference between double integration and. Calculate the slope and deflection at the free end. Macaulay s method is a familiar topic within many mechanics of solids modules forming part of mechanical, aerospace and civil engineering undergraduate degree programmes 1. Truss problem 1 video lecture from deflection of beams using energy method chapter of structural analysis 1 for.
Good question lets start with the method of double integration. Macaulays method problem 1 video lecture from deflection of beams using geometrical method chapter of structural analysis 1 for civil. Frrp ab m ab pa r l mrz a 0 macaulay moment function. This chapter deals with the finding of shear force and bending beams. Ce8301 syllabus strength of materials 1 regulation 2017. A cantilever beam with a uniformly distributed load. Can we find the slope and deflection of a cantilever beam by.
Example of using macaulays method for concentrated loads example. To prevent any attached brittle materials from cracking 2 beam deflection by integration. Macaulay s method the double integration method is a technique used in structural analysis to determine the deflection of eulerbernoulli beams. Structural behaviour and detailing for construction beam deflection and. The beam crosssection is tshaped with flange width as 150 mm and flange thickness 10 mm, web height as 120 mm and web. Nov 12, 2011 beam actions and deflections by macaulays method posted on november 12, 2011 by dougaj4 macaulays method is a convenient method of applying beam slopedeflection equations to find the shear forces, bending moments, slopes and deflections of a structural beam subject to complex load conditions, including partial varying distributed loads. From this equation, any deflection of interest can be found. A simply supported beam with a uniformly distributed load.
The deflection distance of a member under a load can be calculated by integrating the function that mathematically describes the slope of the deflected shape of the member under that load. Unit iii deflection of beams ce8301 syllabus strength of materials 1 elastic curve governing differential equation double integration method macaulays method area moment method conjugate beam method for computation of slope and deflection of determinant beams. Beams deflection macaulays method free download as pdf file. Deflection method this method takes into account only the bending deformations. While macaulays method is basically used to determine deflection and slope at any section of a loaded beam when beam will be loaded with multiple loads. Macaulays method slope and deflection of beams strength. Using calculus to find expressions for the deflection of loaded beams see deflection of beams part. At first year level the number of methods considered is usually restricted to one or two and almost certainly the first method. Dec 25, 2016 best way to explain you is by an example, so here goes. Macaulay method beams materials engineering reference.
Momentarea method the moment area method is a semigraphical procedure. It is obvious therefore to study the methods by which we can predict the deflection of members under lateral loads or transverse loads, since it is this form of loading which will generally produce the greatest deflection of beams. Determine the deflection of statically determinate beam by using double integration method. Deflection is the displacement in the y direction of any point on the axis of the beam. Pdf deflection of flexural members macaulays method 3rd year. Slope and deflection of beams mechanical engineering. Deflection of beams introduction a deflection is the displacement of structural element under load. Feb, 2014 differential equation of slender beams in bending taking into account the correct sign convention for deflection and bending moment, we have. The deflection of a beam must often be limited in order to provide integrity and stability of a structure or machine, or. In calculus, the radius of curvature of a curve y fx is given by the radius of curvature of a beam is given as.
Double integration method and moment area method are basically used to determine deflection and slope at any section of a loaded beam when beam will be loaded with a single load. An expression for the internal bending moment, and hence curbature, is valid at all locations along the beam, and the deflection curve can be calculated by integrating twice with respect to the axial coordinate. A number of analytical methods are available for determining the deflections of beams. Deflection of beams civil engineering ce notes edurev. Best way to explain you is by an example, so here goes. Determine the deflection of statically determinate beam by using macaulays method. Analysis of beams involves the determination of shear force, bending sections. So there are other methods by which we find the deflection like 1. Macaulays method is a means to find the equation that describes the deflected shape of a beam. Macaulays method is a convenient method of applying beam slope deflection equations to find the shear forces, bending moments, slopes and deflections of a structural beam subject to complex load conditions, including partial varying distributed loads and point loads and moments. Calculation of deflections and redundant reactions for beams on two supports can be made by the use of the areamoment method. Structural analysis iii deflection of flexural members macaulays method 3rd year.