Question

The piece of plastic is originally rectangular. Determine the shear strain gamma xy at corners D and C if the plastic distorts as shown by the dashed lines. The piece of plastic is originally rectangular. Determine the average normal strain that occurs along the diagonals AC and DB.

Question

7-98

7-99

Answer 1

Concepts and reason

Shear strain is the change in angle that occurs between two line segments that are initially perpendicular. Shear strain is denoted by and measured in radians.

Shear stress can be calculated using known values of shear modulus and shear strain.

Normal strain is a measure of longitudinal deformation that is elongation or contraction in the body. Normal strain is denoted by .

Normal stress can be calculated using known values of Young’s modulus and normal strain.

Both the normal stress and shear stress are used in various real time applications.

Fundamentals

The shear strain is calculated as follows:

ABCD represents initial position of the block and AECD represents final position of the block.

From the figure,

For small values of , .

The angle is called shear strain.

The relation between shear stress and shear strain is as follows:

$T=GY $

Here, G is shear modulus.

The normal strain is calculated as follows:

From the figure, initial length is and the final length is .

The normal strain is,

7-98P

Draw the deformed diagram of the rectangular plastic piece.

Draw the deformed diagram of the triangle .

From the geometry of the triangle,

Convert the units degree to radians as follows:

Draw the deformed diagram of the triangle .

From the geometry of the triangle ,

Convert the units from degrees to radians.

Write the relation to calculate the shear strain at corner C.

Substitute for and for .

Draw the deformed diagram of corner D and triangle

Calculate the shear strain at corner D.

Substitute for and for .

7-99P

Figure shows original and deformed rectangle piece:

Calculate the diagonal length from the figure of AC and BD.

Here, and are sides of the rectangle.

Substitute 400 mm for , and 300 mm for .

Consider the triangle to find the deformed length of the diagonal.

Here, is deformed length of the AD, is deformed length of DC.

Substitute for , and for .

Find the normal strain along the diagonal.

Substitute for and for .

Consider the triangle .

Consider the triangle, use Pythagoras theorem to calculate the length of .

Substitute for and for .

Calculate the normal strain along using the equation.

Substitute for and 500 mm for .

Ans:

The shear strain at corner C and D are and respectively.