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

• Strain is a dimensionless quantity (m/m, in/in, %)

• Elongation (휀 > 0); contraction (휀 < 0)

휀 =𝐶ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ

𝑈𝑛𝑖𝑡 𝑙𝑒𝑛𝑔𝑡ℎ

휀 =𝛿

𝐿

2. Strain

• Normal strains cause a change in volume

– The initial dimensions are

𝐿𝑥, 𝐿𝑦, 𝐿𝑧

– Each side deforms to

𝐿𝑥 + 𝛿𝑥 = 𝐿𝑥 1 + 휀𝑥

𝐿𝑦 + 𝛿𝑦 = 𝐿𝑦 1 + 휀𝑦

𝐿𝑧 + 𝛿𝑧 = 𝐿𝑧 1 + 휀𝑧

2. Strain

• Example of normal strain:• A bar of length 400 mm is stretched to 420 mm;

calculate the normal strain in the bar:

• 휀 =𝛿

𝐿=

420 −400

400=

20

400= 0.05 = 5%

400 mm

δ = 20 mm

2. Strain

• Shear strain: the change of angle

• Angle decreases (𝛾 > 0)

• Angle increases (𝛾 < 0)

2. Strain

• Shear strains cause a change in shape

– Initial angles: 𝜋

2, 𝜋

2, 𝜋

2

– Deformed angles𝜋

2− 𝛾𝑥𝑦,

𝜋

2− 𝛾𝑦𝑧,

𝜋

2− 𝛾𝑧𝑥

2. Strain

• Example of shear strain:

𝛾𝑥𝑦 = 𝑡𝑎𝑛−10.04

2.0= 0.020 𝑟𝑎𝑑

2. Strain

• 3 normal strains

• 3 shear strains

휀𝑥, 휀𝑦, 휀𝑧

𝛾𝑥𝑦, 𝛾𝑦𝑧, 𝛾𝑧𝑥

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