Learning Center
Plans & pricing Sign in
Sign Out

MAE Class West Virginia University


									MAE 343 - Intermediate Mechanics of
   Tuesday, Aug. 31, 2004

        Textbook Section 4.4
    Bending of Symmetrical and
       Unsymmetrical Beams
 Direct and Transverse Shear Stress
         Main Steps of Beam Bending
• Step 1 – Find Reactions at External Supports
   – Free Body Diagram (FBD) of Entire Beam
   – Equations of Force and Moment Equilibrium (3 in 2D)
• Step 2 – Shear and Bending Moment Diagrams
   – Cutting Plane and FBD of Part of the Beam
   – Use Equilibrium Eqs. to Express Internal Forces in Terms of
     Position Variable, “x”
• Step 3 – Stress Distributions at Critical Sections
   – Linear Distribution of Bending (Normal) Stresses
   – Transverse Shear Stress Distribution in Terms of “Area Moment”
           Pure Bending of Straight
             Symmetrical Beams
• Linear bending stress distribution, and no shear
  stress (Fig. 4.3)
   – Neutral axis passes through centroid of cross-section
   – Section modulus, Z=I/c, used for the case when the neutral
     axis is also a symmetry axis for the cross-section
• Table 4.2 for properties of plane sections
• Restrictions to straight, homogeneous beams loaded
  in elastic range and cutting planes sufficiently far
  from discontinuities
    Bending of Straight Symmetrical
    Beams Under Transverse Forces
• Any cut cross-section loaded by two types of
  stresses (if no torsion occurs):
   – Bending stress as in case of pure bending
   – Transverse shear stresses
• Direct and transverse shear stress
   – Direct average shear stress in pin and clevis joint (Fig.
     4.4) is smaller than maximum stress
   – Non-linear distributions are caused in reality by
     stiffnesses and fits between mating members, etc.
  Transverse Shear Stress Equations
• Bending of laminated beam explains existence of
  transverse shear (Fig. 4.5)
• Beam loaded in a vertical plane of symmetry
  – Elemental slab in equilibrium under differential bending and
    shear forces (Fig. 4.6)
  – Derived equation valid for any cross-sectional shape
  – Expressed in terms of “moment of area” about neutral axis,
    leading to the “area moment” method for calculating transverse
    shearing stresses
  – Irregular cross-sections can be divided into regular parts (4-25)

To top