Transcript Document

MAE 343 - Intermediate Mechanics of
Materials
Tuesday, Aug. 31, 2004
Textbook Section 4.4
Bending of Symmetrical and
Unsymmetrical Beams
Direct and Transverse Shear Stress
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)