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Chapter 5 Analysis and Design of Beams for Bending 

Chapter 5 Analysis and Design of Beams for Bending . 5.1 Introduction. -- Dealing with beams of different materials: steel, aluminum, wood, etc. -- Loading: transverse loads  Concentrated loads  Distributed loads. -- Supports  Simply supported  Cantilever Beam

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Chapter 5 Analysis and Design of Beams for Bending 

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  1. Chapter 5 Analysis and Design of Beams for Bending 

  2. 5.1 Introduction -- Dealing with beams of different materials: steel, aluminum, wood, etc. -- Loading: transverse loads  Concentrated loads  Distributed loads

  3. -- Supports  Simply supported  Cantilever Beam  Overhanging  Continuous  Fixed Beam

  4. A. Statically Determinate Beams -- Problems can be solved using Equations of Equilibrium B. Statically Indeterminate Beams -- Problems cannot be solved using Eq. of Equilibrium -- Must rely on additional deformation equations to solve the problems. FBDs are sometimes necessary:

  5. FBDs are necessary tools to determine the internal (1) shear force V – create internal shear stress; and (2) Bending moment M – create normal stress From Ch 4: (5.1) (5.2) Where I = moment of inertia y = distance from the N. Surface c = max distance

  6. Recalling, elastic section modulus,S = I/c, (5.3) hence For a rectangular cross-section beam, (5.4) From Eq. (5.3),max occurs at Mmax  It is necessary to plot the V and M diagrams along the length of a beam.  to know where Vmaxor Mmax occurs!

  7. 5.2 Shear and Bending-Moment Diagrams • Determining of V and M at selected points of the beam

  8. Sign Conventions • The shear is positive (+) when external forces acting on the beam tend to shear off the beam at the point indicated in fig 5.7b 2. The bending moment is positive (+) when the external forces acting on the beam tend to bend the beam at the point indicated in fig 5.7c  Moment

  9. 5.3 Relations among Load, Shear and Bending Moment • Relations between Load and Shear Hence, (5.5)

  10. Integrating Eq. (5.5) between points C and D (5.6) VD – VC = area under load curve between C and D (5.6’) 1 (5.5’)

  11. Relations between Shear and Bending Moment or  (5.7)

  12. (5.7) MD – MC = area under shear curve between points C and D

  13. 5.4 Design of Prismatic Beams for Bending -- Design of a beam is controlled by |Mmax| (5.1’,5.3’) Hence, the min allowable value of section modulus is: (5.9)

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