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Lecture Strength of Materials I: Chapter 7 - PhD. Tran Minh Tu

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Lecture Strength of Materials I - Chapter 7: Bending. The following will be discussed in this chapter: Introduction, bending stress, shearing stress in bending, strength condition, sample problems, deflections of beam, statically indeterminate beams.

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Nội dung Text: Lecture Strength of Materials I: Chapter 7 - PhD. Tran Minh Tu

  1. STRENGTH OF MATERIALS 1/10/2013 TRAN MINH TU - University of Civil Engineering, 1 Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, Vietnam
  2. CHAPTER 7 BENDING 1/10/2013
  3. Contents 7.1. Introduction 7.2. Bending stress 7.3. Shearing stress in bending 7.4. Strength condition 7.5. Sample Problems 7.6. Deflections of beam 7.7. Statically indeterminate beams 1/10/2013 3
  4. 7.1. Introduction In previous charters, we considered the stresses in the bars caused by axial loading and torsion. Here we introduce the third fundamental loading: bending. When deriving the relationship between the bending moment and the stresses causes, we find it again necessary to make certain simplifying assumptions. We use the same steps in the analysis of bending that we used for torsion in chapter 6. 1/10/2013 4
  5. 7.1. Introduction Classification of Beam Supports 1/10/2013 5
  6. 7.1. Introduction  Limitation 1/10/2013 6
  7. 7.1. Introduction  Segment BC: Mx≠0, Qy=0 => Pure Bending  Segments AB,CD: Mx≠0, Qy≠0 => Nonuniform Bending 1/10/2013 7
  8. 7.1. Introduction Pure Bending: Prismatic members subjected to equal and opposite couples acting in the same longitudinal plane 1/10/2013 8
  9. 7.2. Bending stress  Simplifying assumptions 1/10/2013 9
  10. 7.2. Bending stress The positive bending moment causes the material within the bottom portion of the beam to stretch and the material within the top portion to compress. Consequently, between these two regions there must be a surface, called the neutral surface, in which longitudinal fibers of Neutral axis the material will not undergo a change in length. 1/10/2013 10
  11. 7.2. Bending stress  Compatibility 1 2 Consider a segment of the beam a b bounded by two cross-sections that c y d are separated by the infinitesimal distance dz. dz 2 1 Due to bending moment Mx caused d Neutral fiber by the applied loading, the cross-  1 2 section rotate relatively to each other a b by the amount of d. y c d The Normal strain of the longitudinal 1 2 fiber cd that lies distance y below the neutral surface. dz c ' d ' cd    y  d   d y y z     z  dz cd  d   1/10/2013  – radius of curvature of the neutral fiber. 11
  12. 7.2. Bending stress y 1 Following Hooke’s law, we have. z  E  ????    Equilibrium Mx x Because of the loads applied in the plane yOz, thus: Nz=My=0 and Mx≠0. x K z z y E dA N z    z dA   yd A  0 A  A y  yd A  S A x 0 x – neutral axis (the neutral axis passes through the centroid C of the E  M y  x z dA  A  A xyd A  0 cross-section).  xyd A  I A xy 0 y - axis – the axis of symmetry of the cross-section 1/10/2013 12
  13. 7.2. Bending stress E E M x   y z dA   y dA  I 2  – radius of neutral longitudinal fiber A  A  x 1 M Mx x  x  EI x x K z  Flexure formula – section modulus z y dA Mx – internal bending moment Mx y z  y EIx – stiffness of beam Ix y – coordinate of point Mx>0: stretch top portion Mx For convenient: z   y Belong to compressive zone Ix 1/10/2013 13
  14. 7.2. Bending stress • Stress distribution - Stresses vary linearly with the distance y from neutral axis • Maximum stresses at a cross-section Mx  max   t ymax Ix Mx  min   c ymax Ix ytmax – the distance from N.A to a point farthest away from N.A in the tensile portion ycmax – the distance from N.A to a point farthest away from N.A in the compressive portion
  15. 7.2. Bending stress min h t ymax  ymax c  h/2 2 Mx x Mx h Mx  max   I x 2 Wx h/2 z Mx h Mx max  min   y Ix 2 Wx  max   min Ix with Wx  called the section modulus of the beam h/2 1/10/2013 15
  16. 7.2. Bending stress 1/10/2013 16
  17. 7.2. Bending stress Properties of American Standard Shapes 1/10/2013 17
  18. 7.2. Bending stress 1/10/2013 18
  19. 7.2. Bending stress 1/10/2013 19
  20. 7.2. Bending stress 1/10/2013 20
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