Flexure Formula

4 Key excerpts on "Flexure Formula"

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  eBook - PDF

  Strength of Materials

  Fundamentals and Applications

  • T. D. Gunneswara Rao, Mudimby Andal(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  The radius of curvature is very large compared to the cross-sectional dimensions of the beam. 4.3 Flexure Formula In this section, we develop a formula to quantify the magnitude of stress developed due to bending. Consider a portion of the beam subjected to pure bending as shown in Figure 4.7, and consider an elemental portion of the beam. The magnified portion of the elemental portion is shown in Figure 4.8. Loaded beam in underformed condition dx M M M M dx dq R Loaded beam in derformed condition FIGURE 4.7 Beam. BENDING STRESS 135 O A B A¢ y N A R B¢ y dx dx dA FIGURE 4.8 From the deflected profile of the beam, it is clear that the top fibers of the beam are compressed, whereas the bottom side fibers are subjected to elongation. Thus within the beam, there exists one layer, which is subjected to neither extension nor compression due to bending. This layer is called neutral surface, within the cross-section of the beam it is referred as neutral axis. At neutral axis level, strain is zero. Consider an elemental area ‘dA’ located at a distance ‘y’ from the neutral axis within the cross-section. Let AB be the length of the fiber ‘dx’ before bending. After applying moment, let the fiber length AB be extended to A¢B¢. Strain at the level y distant from the neutral axis = . A B AB AB - ¢ ¢ Let ‘R’ be the radius of curvature of elastic profile of the beam A¢B¢ = (R + y)df AB = dx = Rdf (∵ at y = 0, i.e., at neutral axis length of the layer does not change) \ Strain at a distance y from neutral axis = ( ) ( )· R y R R df df df + - Œ= fi . y R Œ= (4.1) From Eq. (4.1), it is clear that, the strain variation is linear, if radius of curvature is constant. As per the seventh assumption, radius of curvature is constant. Strain variation is linear across the depth of the cross-section; this confirms the second assumption that plane section remains plane.

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  eBook - PDF

  Mechanics of Materials

  An Integrated Learning System

  • Timothy A. Philpot, Jeffery S. Thomas(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  This relationship is called the moment–curvature equation, and it shows that the beam curvature is directly related to the bending moment and inversely related to the quantity EI z . In general, the term EI is known as the flexural rigidity , and it is a measure of the bending resistance of a beam. Flexure Formula The relationship between the normal stress σ x and the curvature was developed in Equation (8.3), and the relationship between the curvature and the bending moment M is given by Equation (8.6). These two relationships can be combined, giving σ x = −Eκy = −E ( M _ EI z ) y to define the stress produced in a beam by a bending moment: σ x = − My _ I z Equation (8.7) is known as the elastic Flexure Formula or, simply, the Flexure Formula. As developed here, a bending moment M that acts about the z axis pro- duces normal stresses that act in the x direction (i.e., the longitudinal direction) of the beam. The stresses vary linearly in intensity over the depth of the cross section. The normal stresses produced in a beam by a bending moment are commonly re- ferred to as bending stresses or flexural stresses. Examination of the Flexure Formula reveals that a positive bending moment causes negative normal stresses (i.e., compression) for portions of the cross section above the neutral axis (i.e., positive y values) and positive normal stresses (i.e., tension) for portions below the neutral axis (i.e., negative y values). The opposite stresses occur for a negative bending moment. The distributions of bending stresses for both positive and negative bending moments are illustrated in Figure 8.6. In Chapter 7, a positive internal bending moment was defined as a moment that • acts counterclockwise on the right-hand face of a beam; or • acts clockwise on the left-hand face of a beam. (8.7) 8.3 Normal Stresses in Beams 213 This sign convention can now be enhanced by taking into account the bending stresses produced by the internal moment.

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  eBook - PDF

  Mechanics of Materials

  • Timothy A. Philpot(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  The integral term in Equation (8.5) can be replaced by the moment of inertia I z , where M EI z   to give an expression relating the beam curvature and its internal bending moment: 1 M EI z     (8.6) This relationship is called the moment–curvature equation, and it shows that the beam curvature is directly related to bending moment and inversely related to the quantity EI z . In general, the term EI is known as the flexural rigidity, and it is a measure of the bending resistance of a beam. Flexure Formula The relationship between normal stress  x and curvature was developed in Equation (8.3), and the relationship between curvature and bending moment M is given by Equation (8.6). These two relationships can be combined, giving x z E y E M EI y       to define the stress produced in a beam by a bending moment: x z My I    (8.7) Equation (8.7) is known as the elastic Flexure Formula or simply the Flexure Formula. As developed here, a bending moment M that acts about the z axis produces normal stresses that act in the x direction (i.e., the longitudinal direction) of the beam. The stresses vary linearly in intensity over the depth of the cross section. The normal stresses produced in a beam by a bending moment are commonly referred to as bending stresses or flexural stresses. Examination of the Flexure Formula reveals that a positive bending moment causes negative normal stresses (i.e., compression) for portions of the cross section above the neutral axis (i.e., positive y values) and positive normal stresses (i.e., tension) for portions below the neutral axis (i.e., negative y values). The opposite stresses occur for a negative bending moment. The distributions of bending stresses for both positive and negative bend- ing moments are illustrated in Figure 8.6. In the context of mechanics of materials, the area moment of inertia is usually referred to as simply the moment of inertia.

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  eBook - ePub

  Applied Strength of Materials

  • Robert L. Mott, Joseph A. Untener(Authors)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  N  = 2 to those values to determine the design stress.

  Composites have many advantages when applied to the design of beams because the placement of material can be optimized to provide efficient, lightweight beams. The resulting structure is typically not homogeneous, so the properties are highly anisotropic. Therefore, the Flexure Formula as stated in Equations (7–1) and (7–2) cannot be relied upon to give accurate values of stress. General approaches to using composites in beams will be discussed later in this chapter.

  7–8 Section Modulus and Design Procedures

  Stress analysis of beams has so far called for the use of the Flexure Formula:

  σ max

  =

  M c

  I

  A modified form is desirable for cases in which the determination of the dimensions of a section is to be done. Notice that both the moment of inertia I and the distance c are geometrical properties of the cross-sectional area of the beam. Therefore, the quotient I/c is also a geometrical property. For convenience, we can define a new term, “section modulus,” denoted by the letter S .

  ⇨ Section Modulus

  S =

  I c

  (7–4)

  The Flexure Formula then becomes

  σ max

  =

  M S

  (7–5)

  This is the most convenient form for use in design, as will be demonstrated in example problems that follow. Appendix A–1 gives formulas for S for common shapes. Appendixes A–4 through A–9 include the value of S for shapes often used for beams.

  Design Procedures

  Similar to the other forms of loading already studied, it is the responsibility of the designer to ensure that the applied stress is less than or equal to the design stress. Just as in the other types of loading, the applied stress is what the material will actually experience in use. The design stress is the amount of stress that the material handled in testing (its strength) divided by a factor the designer chooses to ensure a margin of safety. The design of beams begins by setting the applied stress equal to the design stress to determine the limit of the unknown variable, and then following through to determine the unknown value and make specifications. Example Problems 7–5 through 7–7

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