Write down the load deflection equation for each segment. Deflection analysis of beams with extension and shear piezoelectric patches using discontinuity functions to cite this article. Beam simply supported at ends concentrated load p at any point 22 1 pb l b. Lesson 42 discontinuity functions for beam deflections. The requirement is to obtain the shear load, moment, slope and deflection anywhere along the beam as shown. This approach is applied to static and vibration analyses of stepped beams. The statespace approach along with the jordan canonical form is used to obtain an analytical solution for the static deflection of smart beams with. Therefore the deflection of point c as determined for the left and right hand part of the beam must be equal. Using discontinuity functions, write the expression for the intensity qx of the equivalent distributed load acting on the beam. Obtain the equations for the equivalent distributed loads from the corresponding problems in section 9. The use of generalised functions in the discontinuous beam.
Discontinuous singularity functions for beam deflections the iiintegration methdhod becomes extremely tedious and time. This problem is solved by the use of generalized functions, among which is the well known dirac delta function. Beam deflection singularity function 12 beam deections. Even the very substantial end support block used in the 3d model still allows about a 3. F l x f m v lx a b figure 1 these equations permit us to find the deflection of the beam if m x is known as a function of x.
The method relies upon special functions which are quite unlike usual mathematical functions. Solutions of a simple beam deflection problem using a. Excessive deflection of a beam not only is visually disturbing but also may cause damage to other parts of the building. Also, the governing differential equations of a timoshenko beam with jump discontinuities in slope, deflection, flexural stiffness, and shear stiffness are obtained in the space of generalized functions. Deflections of beams and shafts chapter objectives use various methods to determine the deflection and slope at specific pts on beams and shafts. Deflection of flexural members macaulays method 3rd year structural engineering 20078.
The modulus of elasticity is 205 gpa and beam is a solid circular section. Em 324 session 20 beam deflection by discontinuity. The discontinuities considered in this development are associated with the jumps in the flexural rigidity and the applied loads. Integrate load deflection equation four times equations for vx, mx, v. Dec 09, 2014 this feature is not available right now. Deflection of beams introduction a deflection is the displacement of structural element under load.
Lesson 40 discontinuity functions for beam deflections. For this reason, building codes limit the maximum deflection of a beam to about 60 th of its spans. Beam deflection formulas beam type slope at ends deflection at any section in terms of x maximum and center deflection 6. The elastic curve ab of the segment has the same length dx as the undeformed segment. Pdf the use of generalised functions in the discontinuous. Assuming that the fuselage exerts only vertical reactions on the ends of the beam, determine the maximum deflection of the beam. In the case of the beams, we use this term for linear vertical displacement. Here we will solve a problem of differential equationin the space of generalized functions we solve the problem as a single beam using generalized functions therefore we. Cantilever example 22 beam deflection by integration. Consider the heaviside or step function hx given by. A more complex example of using singularity functions. Each of the functions mx will then lead to a different expression for the slope e x and for the deflection yxb since each of the expressions obtained for. The deflection curve for this beam is physically continuous at point c. If the deflection value is too large, the beam will bend and then fail.
Cantilever beam concentrated load p at any point 2 pa 2 e i lei 2 3for0 px yax xa 6 ei 2. Static, vibration analysis and sensitivity analysis of. The piezoelectric stress resultants are expressed in terms of heaviside discontinuity functions. The unit doublet is the distribution function representation for the applied. View notes beam deflection singularity function from mems meam 204 at university of pennsylvania. Cantilever beam concentrated load p at the free end 2 pl 2 e i nm 2 3 px ylx 6 ei 24 3 max pl 3 e i max 2. Solutions of a simple beam deflection problem using a variety of methods. Pdf deflection analysis of beams with extension and. Given a cantilevered beam with a fixed end support at the right end and a load p applied at the left end of the beam.
Momentarea method use the various methods to solve for the support reactions on a beam or shaft that is statically indeterminate. Examples of different beam loadings are presented for a complete use of d iscontinuity functions. Analytical solution for a finite eulerbernoulli beam with. What happens to the beam when there is a discontinuity in. Use of discontinuity functions for finding beam deflections 1 introduction from the theory of the bending of beams, we know that the bending moment m and the curvature 1. Different equations for bending moment were used at. The floor beam of the airplane is subjected to the loading shown.
Singularity functions are useful in the case of many beams where there is discontinuity in the loading of the beam 3. In conclusion we see that, in this example, the shear deflection adds about 2% to the deflection value. The beam is a steel wideflange section with e 28 106 psi and an allowable bending stress of 17,500 psi in both tension and compression. Deflection of beams free download as powerpoint presentation. Fbd of the entire beam do not need to enforce equilibrium 2. Related content smart beams with extension and thicknessshear piezoelectric actuators. Calculate the slope and deflection at the free end.
Beam fixed at one end, supported at otherconcentrated load at center. Another method that can be used to determine the deflection is to use the discontinuity functions. Deflection of beam theory at a glance for ies, gate, psu 5. Em 324 session 20 beam deflection by discontinuity functions continuation 1 30 1. Deflection analysis of beams with extension and shear. Dividing the finite beam with single discontinuity into two uniform beam components, and implementing the continuity at the contact between two uniform components, the natural frequencies are obtained and the mode shape functions are normalized subsequently in this section. Any beam element of differential width dx at any position x may be perceived to have a left face and a right face. Since our aim is to find a single equation for the bending moments along the beam, we will use discontinuity functions to represent the loads. The beam models are based on the firstorder and higherorder shear deformation beam theories. Fortunately, we have what we call discontinuity functions that allow us to have an elastic curve that fully defines the.
The use of discontinuity functions will be expanded to axial l oadings, torsion of circular rods and particle dynamics. Em 324 session 19 beam deflection by discontinuity functions. Pdf deflection analysis of beams with extension and shear. Macaulays method is a means to find the equation that describes the deflected shape of a beam.
The labor involved in solving problems of this type can be diminished by. We design the cross section for the maximum possible shear. It does this by using the equations found in the table below and taking in consideration that the. Beam simply supported at ends concentrated load p at the center 2 1216 pl e i 2 2 2 3 px l l for 0yx x 12 4 2 ei 3 max pl 48 e i x 7. Based on the modal superposition method, the explicit formulations of the problem are obtained for beam deflection, velocity, acceleration, bending moment and shear force.
Whereas usual functions of variables are continuous, these functions have discontinuities. A number of analytical methods are available for determining the deflections of beams. Chapter 7 deflections of beams determine the deflection and slope at specific points on beams and shafts, using various analytical methods including. Representation of loads on beams by discontinuity functions. For example in case of a point load there is a sign change in shear but what do we do. A discontinuity in moment is similar to a discontinuity in shear. The equations above can be used to determine the shear load, moment, slope and deflection for the beam from x 0. Investigation of slopes and deflections of a stepped beam.
Equation form example 1, page 4 of 6 x 9 kip r a 10 kip a 6 kip r b 5 kip b pass a section through the beam at a point between the 6kip force and the right end of the beam. Momentarea methodarea method 2005 pearson education south asia pte ltd 1. Write the equation of the elastic curve for segment ab of the beam, determine the slope at support a, and determine the deflection at a point of the beam located 3 m from support a. The natural frequencies and corresponding modal shape functions of the beam are obtained by imposing continuity at the contact between different components of the beam. Mohrs theorems for slope and deflection state that if a and b are two points on the deflection curve of a beam and b is a point of zero slope, then m. The use of generalised functions in the discontinuous beam bending differential equations article pdf available in international journal of engineering education 183 january 2002 with 326 reads. In the technical bending theory, we make two main assumptions that. Beam deflection formulae beam type slope at free end deflection at any section in terms of x maximum deflection 1. Each of the functions mx will then lead to a different expression for the slope e x and for the deflection yxb since each of the expressions obtained for the deflection must contain two constants of integration, a large number of constants will have to be determined. Using singularity functions,810 we may write, for the beam ab in fig.
The governing differential equation is eulerbernoulli beams with jump discontinuities on displacements and rotations. This paper discusses the mathematical modeling for the mechanics of solid using the distribution theory of schwartz to the beam bending differential equations. Expanded use of discontinuity and singularity functions in. From this equation, any deflection of interest can be found. Analysis of statically indeterminate reactions and. Bernoullis hypothesis about the plane crosssections is valid. Similarly, the slopes found for each part of the beam must be equal at point c. These functions take into account the starting and ending positions of concentrated loads, concentrated moments, distributed loads, and distributed moments across the beam. On applications of generalized functions in the discontinuous.
The discontinuity functions are used to bypass the step of making a moment diagram to determine the moment. Deflection analysis of beams with extension and shear piezoelectric patches using discontinuity functions article pdf available in smart materials and structures 102. Ahmed a khdeir and osama j aldraihem 2001 smart mater. Before macaulays paper of 1919, the equation for the deflection of beams could not be found in closed form. A beam carries a distributed load that varies from zero at support a to 50 knm at its overhanging end, as shown in figure 7.
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