Mechanics of Deformable Bodies PDF

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Mechanics of Deformable Bodies 06-85-218 Chapter 2 Chapter 2: Strain 1. Deformation 2. Strain Introduction • Objectives: – Consider the deformation of structures – Define the concept of strain – Solve problems in which normal and shear strains need to be calculated 1. Deformation • External loads wi...

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Mechanics of Deformable Bodies Laurence Nan

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Mechanics of Deformable Bodies 06-85-218

Chapter 2

Chapter 2: Strain 1. Deformation 2. Strain

Introduction • Objectives: – Consider the deformation of structures – Define the concept of strain – Solve problems in which normal and shear strains need to be calculated

1. Deformation • External loads will cause a body to deform – Large deformations • Stretching of a rubber band • Deformation of metal in a wire drawing operation

– Small deformations • Deflections of the floor beams in a building • Elongation/contraction due to a temperature change

– Non-uniform deformations • e.g. bending of a beam

2. Strain • Normal strain: the change in length per unit length =

�ℎ� � �� � =

�

�ℎ

�ℎ

�

• Strain is a dimensionless quantity (m/m, in/in, %) • Elongation ( > ); contraction ( < )

2. Strain • Normal strains cause a change in volume – The initial dimensions are � ,� ,�

– Each side deforms to � +

� +

� +

=�

+

=�

+

=�

+

2. Strain • Example of normal strain: • A bar of length 400 mm is stretched to 420 mm; calculate the normal strain in the bar:

•

=

� �

=

4

4

−4

=

4

= .

= %

400 mm

δ = 20 mm

2. Strain • Shear strain: the change of angle

• Angle decreases ( > ) • Angle increases ( < )

2. Strain • Shear strains cause a change in shape � � �

– Initial angles: , , – Deformed angles �

−

,

�

−

,

�

−

2. Strain • Example of shear strain:

= ��

−

.

.

= .

��

2. Strain • 3 normal strains

,

,

• 3 shear strains

,

,

In-class problem (2-4) • Determine the normal strain in each wire after a 2°clockwise rotation around B.

In-class problem (2-12) • Determine the shear strain at A.

In-class problem (2-13) • Determine the normal strain along diagonal DB and side AD.

In-class problem (2-18) • Determine the shear strain at A and at B...