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.
3 Key excerpts on "Deflection due to Bending"
eBook - ePubMechanical Engineering Design
Third Edition
- Ansel C. Ugural(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...4 Deflection and Impact 4.1 Introduction Strength and stiffness are considerations of basic importance to the engineer. The stress level is frequently used as a measure of strength. Stress in members under various loads was discussed in Chapter 3. We now turn to deflection, the analysis of which is as important as that of stress. Moreover, deflections must be considered in the design of statically indeterminate systems, although we are interested only in the forces or stresses. Stiffness relates to the ability of a part to resist deflection or deformation. Elastic deflection or stiffness, rather than stress, is frequently the controlling factor in the design of a member. The deflection, for example, may have to be kept within limits so that certain clearances between components are maintained. Structures such as machine frames must be extremely rigid to maintain manufacturing accuracy. Most components may require great stiffness to eliminate vibration problems. We begin by developing basic expressions relative to deflection and stiffness of variously loaded members using the equilibrium approaches. The integration, superposition, and moment-area methods are discussed. Then, the impact or shock loading and bending of plates are treated. The theorems based upon work–energy concepts, classic methods, and finite element analysis (FEA) for determining the displacement on members are considered in the chapters to follow. 4.1.1 Comparison of Various Deflection Methods When one approach is preferred over another, the advantages of each technique may be briefly summarized as follows. The governing differential equations for beams on integration give the solution for deflection in a problem. However, it is best to limit their application to prismatic beams, otherwise, considerable complexities arise...
eBook - ePub- Ken Wyatt, Richard Hough(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
...11 Deflections VALUES FOR SECOND MOMENT OF AREA The Second Moment of Area of the cross-sectional shape of a beam has a direct influence upon the amount that the beam deflects. The following table gives formulae for some common geometrical shapes. TABLE 11.1 I= bh 3 36 I= πd 4 64 I= (B 2 + 4 Bb+b 2) h 3 36 (B+b) I = 0.00686d 4 CONTENT OF CHAPTER 11 This chapter deals with the deflection of beams. After deducing the general form of the deflection equation, we consider its individual components, to understand the influence that each has on the overall deflection. The relative effects of different load configurations and support conditions are also considered. Mention is made of the parameters affecting dynamic, rather than just static, deflections. 11.1 GENERAL RULES It is quite a simple matter to determine experimentally the factors that influence the amount of deflection that a structure undergoes when a load is applied to it. If a simple beam made from a linear elastic material is subjected to a central point load, it will be found that: (a) the maximum deflection is proportional to the load; (b) the maximum deflection is proportional to the cube of the span; DEFLECTION OF A CANTILEVER The maximum deflection of a beam depends, as we see in paragraph 11.1, on load, span, cross-sectional shape and material of construction. Consider a cantilever beam supporting a point load at the free end. The beam will deflect to a smooth curve, with each small increment of length bending through a small angle. Consider one such increment of length, dx, bending through an angle θ. The angle between the perpendiculars to the beam will also be θ, and the radius of curvature is R...
eBook - ePub- Tianjian Ji(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...These four structural concepts can be summarised in a more concise and memorable manner and treated as rules of thumb as follows: 1. The more direct the internal force paths, the smaller the deflection. 2. The smaller the internal forces, the smaller the deflection. 3. The more uniform the distribution of internal forces, the smaller the deflection. 4. The more the bending moments are converted into axial forces, the smaller the deflection. In these statements, the form of a structure is not explicitly stated but is embedded. It has been shown in Section 1.2 that structural form, deflection and internal forces are closely related so that altering any one of the three will lead to a change of the other two. The four structural concepts provide a solid basis for creative applications. They will be examined and discussed further to gain a sound and thorough understanding. 2.7.2 Generality Equations 2.16 and 2.19 are derived from the principle of virtual work and are general and applicable to all types of truss and frame structures and include the structural concepts derived from equation 1.15 which are based on beam theory. The maximum bending moment of a uniform beam subjected to a uniformly distributed load is: M max = β q L 2 (2.29) For a simply supported beam, β = 1 / 8, and for a cantilever, β = 1 / 2. Substituting equation 2.29 into equation 1.15, the deflection can be alternatively expressed as: Δ max = α q L 4 E I = α M max 2 β 2 q E I (2.30) Equation 2.30 states that the maximum deflection is proportional to the maximum bending moment squared or in more general terms, the smaller the internal forces, the smaller the deflection, which is the second structural concept...
