Chapter 1 the fundamental concepts

average normal stress in each rod and the average
shear stress in the pin A between the members


V. Strain
5.1. Deformation
F2

F1

F3

F4


V. Strain
F2

5.2. Strain

F3

F1

5.2.1. Normal strain

 AB

AB  AB

AB

F4


V. Strain
5.2.2. Shear Strain




2

 


V. Strain
5.2.3. Cartesian strain

The approximate lengths of the sides of the parallelepiped are

1   x x

1   y 1   z
y

z

The approximate angles between sides, again originally defined by the
sides Δx, Δy and Δz are


2

  xy


2

  yz


2

  xz


V. Strain
5.2.3. Cartesian strain
Notice that the normal strains cause a change in volume of
rectangular element, whereas the shear strain cause a change in shape


V. Strain
Example: Due to a loading, the plate is deformed into the dashed shape
shown in fig. a. Determine (a) the average normal strain along the side
AB, and (b) the average shear strain in the plate at A relative to the and y
axes.


V. Strain
Solutions

Part (a)

• Line AB, coincident with the y axis, becomes line after deformation,
thus the length of this line is
AB ' 

250  22  32

 248.018 mm

• The average normal strain for AB is therefore

 AB avg

AB ' AB 248.018  250


 7.93 10 3 mm/mm (Ans)
AB
240





• The negative sign indicates the strain causes a contraction of AB.


V. Strain
Solutions

Part (b)
• As noted, the once 900 angle BAC between the sides of the plate,
referenced from the x, y axes, changes to θ’ due to the displacement
of B to B’.
• Since  xy 


2

  ' then  xy is the angle shown in the figure.

• Thus,
 xy

 3 
 arctg 
  0.121 rad (Ans)
 250  2 


V. Strain
Problem: The rigid beam is supported by a pin at A and wires BD and
CE. If the load P on the beam causes the end C to be displaced 10 mm
downward, determine the normal strain developed in wires CE and BD.


V. Strain
Problem: The force applied at the handle of the rigid lever causes the
lever to rotate clockwise about the pin B through an angle of 20.
Determine the average normal strain developed in each wire. The wires

are unstretched when the lever is in the horizontal position.


V. Strain
Problem: The two wires are connected together at A. If the force P
causes point A to be displaced horizontally 2 mm, determine the normal
strain developed in each wire


V. Strain
Problem: The piece of plastic is originally rectangular. Determine the
shear strain γxy at corners A and B if the plastic distorts as shown by the
dashed lines

Problem: The piece of plastic is originally rectangular. Determine the
shear strain γxy at corners D and C if the plastic distorts as shown by the
dashed lines.