##### Document Text Contents

Page 19

9. Mechanical Engineering Design

r r Figure 3-20 ,

dy r d b - jI

,

3V

Shear sresses in a reclcnqular

I 'T =-o

- 2 A

b o o m .

= i = = T TM

,

( 0 ,N~o-j --,

I I,

C o l C b) C e l,

(')

If we now use this value of [for Eq. (3-32) and rearrange, we get

t: ~ 3 V ( 1 _ Y i )

2A c2

(3-331

We note that the maximum shear stress exists when Yl

tral axis. Thus

:= 0, which is at the bending neu-

(3-3AI

for a rectangular section. As we move away from the neutral axis, the shear stress

decreases parabolically until it is zero at the outer surfaces where YI =±c, as shown

in Fig. 3-20c. It is particularly interesting and significant here to observe that the

shear stress is maximum at the bending neutral axis, where the normal stress due to

bending is zero, and that the shear stress is zero at the outer surfaces, where the

bending stress is a maximum. Horizontal shear stress is always accompanied by

vertical shear stress of the same magnitude, and so the distribution can be dia-

grammed as shown in Fig. 3-20d. Figure 3-20c shows that the shear r on the verti-

cal surfaces varies with y. We are almost always interested in the horizontal shear, r

in Fig. 3-20d, which is nearly uniform with constant y. The maximum horizontal

shear Occurs where the vertical shear is largest. This is usually at the neutral axis but

may not be if the width b is smaller Somewhere else. Furthermore, if the section is

such that b can be minimized on a plane not horizontal, then the horizontal shear

stress OCcurs on an inclined plane. For example, with tubing, the horizontal shear

stress OCcurs on a radial plane and the corresponding "vertical shear" is not vertical,

bu t tangential.

Formulas for the maximum flexural shear stress for the most commonly used

shapes are listed in Table 3-2.

II

9. Mechanical Engineering Design

r r Figure 3-20 ,

dy r d b - jI

,

3V

Shear sresses in a reclcnqular

I 'T =-o

- 2 A

b o o m .

= i = = T TM

,

( 0 ,N~o-j --,

I I,

C o l C b) C e l,

(')

If we now use this value of [for Eq. (3-32) and rearrange, we get

t: ~ 3 V ( 1 _ Y i )

2A c2

(3-331

We note that the maximum shear stress exists when Yl

tral axis. Thus

:= 0, which is at the bending neu-

(3-3AI

for a rectangular section. As we move away from the neutral axis, the shear stress

decreases parabolically until it is zero at the outer surfaces where YI =±c, as shown

in Fig. 3-20c. It is particularly interesting and significant here to observe that the

shear stress is maximum at the bending neutral axis, where the normal stress due to

bending is zero, and that the shear stress is zero at the outer surfaces, where the

bending stress is a maximum. Horizontal shear stress is always accompanied by

vertical shear stress of the same magnitude, and so the distribution can be dia-

grammed as shown in Fig. 3-20d. Figure 3-20c shows that the shear r on the verti-

cal surfaces varies with y. We are almost always interested in the horizontal shear, r

in Fig. 3-20d, which is nearly uniform with constant y. The maximum horizontal

shear Occurs where the vertical shear is largest. This is usually at the neutral axis but

may not be if the width b is smaller Somewhere else. Furthermore, if the section is

such that b can be minimized on a plane not horizontal, then the horizontal shear

stress OCcurs on an inclined plane. For example, with tubing, the horizontal shear

stress OCcurs on a radial plane and the corresponding "vertical shear" is not vertical,

bu t tangential.

Formulas for the maximum flexural shear stress for the most commonly used

shapes are listed in Table 3-2.

II