MAE 343-Intermediate Mechanics of MaterialsQUIZ No.1 - Thursday, Aug. 26, 2004

• List three possible failure modes of a machine element (5points)

• List the three-dimensional equations of equilibrium for a non-accelerating body (5 points)

MAE 343 - Intermediate Mechanics of Materials

Thursday, Aug. 26, 2004

Textbook Section 4.4

States of Stress

Bending of Symmetrical Beams

Stress and Deflection Analysis

• Identify probable modes of failure• Select “loading severity parameter”

– Stress, Strain, Strain Energy per unit Volume

• Critical strength property related to the probable failure mode– Yield strength, deflection limits, buckling load

• Prevent failure by assuring that:– Loading severity parameter<Critical strength property

States of Stress at a Point

• Triaxial state of stress at a point (Fig. 4.1)– Stress Vector on any cutting plane resolved in one Normal and two

Shear components– Infinitesimal cube centered at the point and aligned with a right-

handed cartesian coordinate system– Three normal and three shear stress components define completely

the state of stress at the point (stress tensor)– Stresses on any and all other cutting planes through that point can

then be found from “stress transformation” equations– Lead to “principal stresses”&“combined stress theories of failure”

• Biaxial and Uniaxial states of stress

Common Stress Patterns

• Direct Axial Stress (Tension or Compression)– Fig.4.2, Restrictions for uniform stress distributions

• Bending (Flexural) Stress– Shear and bending moment diagrams –graphical summary of internal

force distributions– Table 4.1 for typical cases, based on force and moment equilibrium

equations as a function of position variable, “x”– Note differential equations V = dM/dx, and q(x) = dV/dx

• Direct Shear and Transverse Shear Stress• Torsional Shear Stress• Surface Contact Stress

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)

MAE 343-Intermediate Mechanics of Materials Homework No.1 - Thursday, Aug. 26, 2004

• Textbook problems required on Thursday, Sep. 2, 2004:

– Problems 4.2 and 4.5• Textbook problems recommended for

practice before Sep. 2, 2004:

– Problems 4.3, 4.4 and 4.6