Main Steps of Beam Bending Analysis 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)