MAE 343 - Intermediate Mechanics of Materials

Thursday, Sep. 2, 2004

Textbook Section 4.4

Bending of Symmetrical and Unsymmetrical Beams

Direct and Transverse Shear 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

zz

zx I

yM

Z

M

I

Mcmax

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.

AA

AAAaveA A

P

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)

Transverse Shear Stress Equations

ydyyXyyXI

zVc

yxxyzzy

1

1

)()(

)(

1

Stress distribution in section at “z” at distance y1 from neutral axis

AyyyXI

zVyy

xxyz )(

)()(

11

Area Moment method for calculating transverse shear stresses

ii

ixx

yz AyyyXI

zVyy

)(

)()(

11

Irregular Cross-Section

ydyyXyyXI

zVc

yxxyzzy

1

1

)()(

)(

1

Conclusions on Transverse Shearing Stress Calculations

• Maximum Value at Neutral Axis– Depends on Shape of Cross Section (Table 4.3)– Equal to Zero at Top and Bottom Boundaries

• Important for Short Beams– Wood Beams – if span/depth < 24– Metal Beams with Thin Webs- if span/depth<15– Metal Beams with Solid Section-if span/depth<8

MAE 343-Intermediate Mechanics of MaterialsHomework No. 2 - Thursday, Sep. 02, 2004

    

1) Textbook problems required on Thursday, Sep. 9, 2004: 

Problems 4.10 and 4.15  

2) Textbook problems recommended for practice before Sep. 9, 2004:

  Problems 4.7 – 4.18 (except 4.10 and 4.15)