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Hamrock Fundamentals of Machine Elements

Chapter 4: Stresses and Strains

I am never content until I

have constructed a

mechanical model of thesubject I am studying. If I

succeed in making one, I

understand; otherwise I do

not.

William Thomson (Lord

Kelvin)

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Centroid of Area

Figure 4.1 Centroid of area.

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Figure 4.2 Rectangular hole within a

rectangular section used in Example

4.1.

Figure 4.3 Area with coordinates

used in describing area moment of

inertia.

Example 4.1

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Figure 4.4 Centroid of area.

Example 4.2

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Parallel-Axis Theorem

Figure 4.5 Coordinates and

distance used in describing

parallel-axis theorem.

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Figure 4.6 Triangular cross

section with circular hole in it,

used in Example 4.3.

Examples 4.3 and 4.4

Figure 4.7 Circular cross-

sectional area relative tox-y

coordinates, used in Example 4.4.

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Properties of Cross

Sections

Table 4.1 Centroid, area moment

of inertia, and area for seven cross

sections

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Figure 4.8 Mass element in

three-dimensional coordinates

and distance from the three axes.

Mass Element

Figure 4.9 Mass element in two-

dimensional coordinates and

distance from the two axes.

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Table 4.2 Mass and mass moment

of inertia of six solids.

Mass and Mass

Moment of Inertia

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Circular Bar with Tensile Load

Figure 4.10 Circular bar with tensile load applied.

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Twist Due to Torque

Figure 4.11 Twisting of member due

to applied torque.

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Figure 4.12 Bar made of elastomeric material to illustrate

effect of bending. (a) Undeformed bar; (b) deformed bar.

Deformation in Bending

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Figure 4.13 Bending occurring

in cantilevered bar, showing

neutral surface

Figure 4.14 Undeformed and

deformed elements in bending.

Neutral Surface and Deformation in Bending

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Figure 4.15 Profile view ofbending stress variation.

Stress in Bending

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Figure 4.16 U-shaped cross section experiencing

bending moment, used in Example 4.10

Example 4.10

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Figure 4.17 Curved member in

bending. (a) Circumferential view;

(b) cross-sectional view

Deformation of Member in Bending

Figure 4.18 Rectangular cross

section of curved member

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Figure 4.19 How transverse shear is developed. (a) Boards not

bonded together; (b) boards bonded together.

Transverse Shear

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Figure 4.20 Cantilevered bar made of highly deformable material

and marked with horizontal and vertical grid lines to show

deformation due to transverse shear. (a) Undeformed; (b) deformed.

Deformation Due to Transverse Shear

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Figure 4.21 Three-dimensional and profile views of moments and

stresses associated with shaded top segment of element that has been

sectioned atyabout neutral axis. Shear stresses have been omitted

for clarity. (a) three-dimensional view; (b) profile view.

Moments and Stresses on Element

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Table 4.3 Maximum shearstress for different beam cross

sections.

Maximum Shear

Stress for Different

Cross Sections

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Example 4.13

Figure 4.22 Shaft with loading

considered in Example 4.13.

Figure 4.23 (a) Shear force; (b) normal force and (c) bending

moment diagrams for the shaft in Fig. 4.22.

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Figure 4.24 Cross section of shaft atx=5 in., with

identification of stress elements considered in Example 4.13.

Stress Elements

in Example 4.13.

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Figure 4.25 Shear stress distributions. (a) Shear stress due to

a vertical shear force; (b) Shear stress due to torsion.

Shear Stress Distributions

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Figure 4.26 Design of

shaft for coil slitting

line. (a) Illustration of

coil slitting line; (b)

knife and shaft detail;(c) free-body diagram of

simplified shaft for case

study.

Coil Slitter

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Figure 4.27 Shear diagram (a) and

moment diagram (b) for idealized coilslitter shaft.

Figure 4.28 Mohr's circle for

location of maximum bendingstress.

Coil Slitter Results

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