Technology & Engineering
Deflection due to Bending
Deflection due to bending refers to the deformation of a structural element when subjected to bending moments. When a beam or any other structural member is loaded, it experiences bending stresses that cause it to deflect. The amount of deflection is influenced by the material properties, geometry, and the magnitude of the applied loads.
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eBook - ePub- B. Raghu Kumar(Author)
- 2022(Publication Date)
- CRC Press(Publisher)
CHAPTER 7 Deflection in Beams 7.1 INTRODUCTION In the preceding chapters we discussed about the stresses that are induced in the beams. This aspect may be used in design of beams from strength point of view. In this chapter we will be concerned about another aspect in the design of beams namely, deformation or the deflection of the beam. Since in some cases the design specification of a beam will generally includes the maximum possible deflection in association with or without maximum permissible stresses. For example, the building codes specify upper limit on deflections as well as stresses because the large deflections are associated with poor appearance and with too much flexibility. In addition knowledge of the deflection is required to analyze the statically indeterminate structures which will be dealt in the next chapter. A number of analytical methods are available to find the the deflection of beams. We consider three methods in this text. 7.2 DIFFERENTIAL EQUATION FOR DEFLECTION OF BEAMS When a beam is loaded the initially straight longitudinal axis beam deformed into curve called the differential curve of the beam. Double integration method uses a second order differential equation of deflection curve for a beam. To obtain the general equation of a beam, let us consider a cantilever beam loaded at free end as shown in the Fig. 7.1. The xy plane is considered as plane of bending. The deflection y at any point m l on beam at a distance of x from fixed end (origin) is the displacement of the point in the y – direction. Generally downward deflection is positive and upward deflection is negative. The y is expressed as a function of x, we have the equation of deflection curve. The angle of rotation θ of the beam at any point ml is the angle between the x – axis and the tangent to the deflection curve
eBook - PDFMechanics of Materials
An Integrated Learning System
- Timothy A. Philpot, Jeffery S. Thomas(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
331 CHAPTER 10 Beam Deflections 10.1 Introduction Important relations between applied load and both normal and shear stresses developed in a beam were presented in Chapters 8 and 9. However, a design is normally not complete until the deflection of the beam has been determined for its particular load. While they generally do not create a safety risk in themselves, excessive beam deflections may impair the successful functioning of a structure in other ways. In building construction, excessive deflections can cause cracks in walls and ceilings. Doors and windows may not close properly. Floors may sag or vibrate noticeably as people walk on them. In many machines, beams and flexural compo- nents must deflect just the right amount for gears or other parts to make proper contact. In sum, the satisfactory design of a flexural component usually includes a specified maximum deflection in addition to a minimum load-carrying capacity. The deflection of a beam depends on the stiffness of the material and the cross-sectional dimensions of the beam, as well as on the configuration of the ap- plied loads and supports. Three common methods for calculating beam deflections are presented here: (1) the integration method, (2) the use of discontinuity func- tions, and (3) the superposition method. In the discussion that follows, three coordinates will be used. As shown in Figure 10.1, the x axis (positive to the right) extends along the initially straight longitudinal axis of the beam. The x coordinate is used to locate a differential beam element, which has an undeformed width of dx. The v axis extends posi- tive upward from the x axis. The v coordinate measures the displacement of the beam’s neutral surface. The third coordinate is y, which is a localized coordinate v, y x P M w FIGURE 10.1 Coordinate system. 332 CHAPTER 10 Beam Deflections with its origin at the neutral surface of the beam cross section.
eBook - PDF- W. C. Bolton(Author)
- 2013(Publication Date)
- Wiley-Blackwell(Publisher)
Chapter 4 Deflections of beams 4.1 The deflection curve An effect of loading an initially straight beam is to deform it into a curved shape. The deformation can be expressed in terms of the deflection of the beam from its initially unloaded position. The convention is generally adopted of a downward deflection being positive and an upward deflection negative. Figure 4.1 shows part of a loaded beam. At A the deflection is y , at B it is y + δy . At A the angle between the tangent to the beam and the horizontal, i.e. the unloaded position, is θ while at B it is θ + δ θ . The angle subtended at the centre of the circle for which the curved beam between A and B is part of the circumference is thus δ θ . If δs is the arc length between A and B, then δs = Rδ θ , where R is the radius of curvature of the arc AB. Hence we can write 1 R = δ θ δs Because the deflections obtained with beams are generally very small (in figure 4.1 they have been considerably exaggerated) δx is a reasonable Fig. 4.1 Deflection curve of a beam. 97 98 Mechanical Science approximation for δs . Thus the equation becomes, in the limit when δ θ is made infinitesimally small, 1 R ≈ d θ d x [1] The slope of the straight line between A and B is δy/δx and, to a reasonable approximation, this is tan θ . Since θ is small then tan θ ap-proximates to θ . Thus, in the limit, we can write θ ≈ d y d x Hence, using equation [1], 1 R = d θ d x = d 2 y d x 2 [2] If the beam obeys Hooke’s law then the bending moment M = EI/R (equation [16], chapter 3), where E is the modulus of elasticity and I the second moment of area. For the beam shown the deflections y are downwards and defined as positive, thus the bending moment is negative since there is hogging. Thus equation [2] can be written as d 2 y d x 2 = − M EI [3] This is the basic differential equation of the deflection curve of a loaded beam. The product EI is often called the flexural rigidity of the beam.
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