MOI UNIVERSIT
SCHOOL OF ENGINEERING
DEPARTMENT OF MECHANICAL & PRODUCTION ENGINEERING
MPE 222: SOLID MECHANICS I
LECTURE NOTES
By
Dr. Stephen M. Talai
(PhD. Mechanical Vibrations, University of Pretoria/Tshwane University of Technology – South Africa;
Msc. Energy Studies -Renewable Energy, Moi University - Kenya)
Email address:
Cell No.: +254 726 317 569
Technology Block: Room T64
© 2023
, MPE 222 – SOLID MECHANICS (3 units)
COURSE CONTENT
1. Stress and strain in tension and shear: definition of stress, uniaxial tension/compression. Members
with variables cross-section; compound members. Elastic constants.
2. Torsion analysis: Solid circular shafts, hollow circular shafts, thin-walled tubes, plastic torsion.
3. Bending moments and shearing forces: types of beams and loadings, Shear Force (S.F.) and
Bending Moment (B.M.) diagrams, relation to intensity of force. Stresses due to pure bending and
plastic bending.
REFERENCES
1. Strength of materials and Structures (4th Edition), Butterworth Heinemann, 1999, Case J.,
Chilver L. and Ross C.T.F
2. Strength of Materials (Mechanics of Solids), 2010, Er. R.K Rajput
ii
MPE 222: Solid Mechanics I, Lecture Notes; By Dr. SM Talai (Ph.D.)
, 1 CHAPTER 1
SIMPLE STRESSES AND STRAINS
1.1 Introduction
When a body is acted upon by some load or external force, it undergoes deformation (i.e., change
in shape or dimensions) which increases gradually. During deformation, the material of the body
resists the tendency of the load to deform the body, and when the load influence is taken over by
the internal resistance of the material of the body, it becomes stable. This internal resistance which
the body offers to meet with the load is called stress.
The various types of stresses may be classified as:
1. Simple or direct stress
i. Tension
ii. Compression
iii. Shear
2. Indirect stress
i. Bending
ii. Torsion
3. Combine stress any possible combination of 1 and 2.
1.2 Simple stress
Simple stress is often called direct stress because it develops under direct loading conditions. That
is, simple tension and simple compression occur when the applied force, called load, is in line with
the axis of the member (axial loading) (Fig. 1.1a and b) and simple shear occurs, when equal
parallel, and opposite forces tend to cause a surface to slide relative to the adjacent surface.
Fig. 1.1a: Tensile stress
Fig. 1.1b: Compressive stress
Simple stress 𝛿 (sigma) is calculated by
𝑃
𝜎=
𝐴
where,
𝜎 = 𝑆𝑡𝑟𝑒𝑠𝑠, 𝑘𝑁⁄𝑚2 𝑜𝑟 𝑁⁄𝑚𝑚2
1
MPE 222: Solid Mechanics I, Lecture Notes; By Dr. SM Talai (Ph.D.)
, 𝑃 = 𝑙𝑜𝑎𝑑, 𝑘𝑁 𝑜𝑟 𝑁
𝐴 = 𝐴𝑟𝑒𝑎 𝑜𝑣𝑒𝑟 𝑤ℎ𝑖𝑐ℎ 𝑠𝑡𝑟𝑒𝑠𝑠 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑠, 𝑚2 𝑜𝑟 𝑚𝑚2
1.3 Strain
The strain (𝑒) is the deformation produced by stress. The various types of strain are explained
below:
Tensile Strain
A piece of material, with uniform cross-section, subjected to a uniform axial tensile stress, will
increase its length from 𝑙 to (𝑙 + 𝛿𝑙) and the increment of length 𝛿𝑙 is the actual deformation of
the material. The fractional deformation or the tensile strain is given by
𝛿𝑙
𝑒𝑡 =
𝑙
Compressive Strain
Under compressive forces, a similar piece of material would be reduced in length from 𝑙 to (𝑙 −
𝛿𝑙). The fractional deformation gives the strain 𝑒𝑐 ,
𝛿𝑙
𝑒𝑐 = −
𝑙
Volumetric Strain
It is defined as the ratio between change in volume and original volume of the body, and is denoted
by 𝑒𝑣 .
𝐶ℎ𝑎𝑛𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒 𝛿𝑉
∴ 𝑒𝑣 = =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑉
The strains which disappear with the removal of load are termed as elastic strains and the body
which regains its original position on the removal of force is called an elastic body. The body is
said to be plastic if the strains exist even after the removal of external force. There is always a
limiting value of load up to which the strain totally disappears on the removal of load-the stress
corresponding to this load is called elastic limit.
Robert Hooke discovered experimentally that within elastic limit, stress varies directly as strain
𝑆𝑡𝑟𝑒𝑠𝑠
𝑖. 𝑒. 𝑆𝑡𝑟𝑒𝑠𝑠 ∝ 𝑆𝑡𝑟𝑎𝑖𝑛 𝑜𝑟 = 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑆𝑡𝑟𝑎𝑖𝑛
The constant is termed as Modulus of elasticity
2
MPE 222: Solid Mechanics I, Lecture Notes; By Dr. SM Talai (Ph.D.)
SCHOOL OF ENGINEERING
DEPARTMENT OF MECHANICAL & PRODUCTION ENGINEERING
MPE 222: SOLID MECHANICS I
LECTURE NOTES
By
Dr. Stephen M. Talai
(PhD. Mechanical Vibrations, University of Pretoria/Tshwane University of Technology – South Africa;
Msc. Energy Studies -Renewable Energy, Moi University - Kenya)
Email address:
Cell No.: +254 726 317 569
Technology Block: Room T64
© 2023
, MPE 222 – SOLID MECHANICS (3 units)
COURSE CONTENT
1. Stress and strain in tension and shear: definition of stress, uniaxial tension/compression. Members
with variables cross-section; compound members. Elastic constants.
2. Torsion analysis: Solid circular shafts, hollow circular shafts, thin-walled tubes, plastic torsion.
3. Bending moments and shearing forces: types of beams and loadings, Shear Force (S.F.) and
Bending Moment (B.M.) diagrams, relation to intensity of force. Stresses due to pure bending and
plastic bending.
REFERENCES
1. Strength of materials and Structures (4th Edition), Butterworth Heinemann, 1999, Case J.,
Chilver L. and Ross C.T.F
2. Strength of Materials (Mechanics of Solids), 2010, Er. R.K Rajput
ii
MPE 222: Solid Mechanics I, Lecture Notes; By Dr. SM Talai (Ph.D.)
, 1 CHAPTER 1
SIMPLE STRESSES AND STRAINS
1.1 Introduction
When a body is acted upon by some load or external force, it undergoes deformation (i.e., change
in shape or dimensions) which increases gradually. During deformation, the material of the body
resists the tendency of the load to deform the body, and when the load influence is taken over by
the internal resistance of the material of the body, it becomes stable. This internal resistance which
the body offers to meet with the load is called stress.
The various types of stresses may be classified as:
1. Simple or direct stress
i. Tension
ii. Compression
iii. Shear
2. Indirect stress
i. Bending
ii. Torsion
3. Combine stress any possible combination of 1 and 2.
1.2 Simple stress
Simple stress is often called direct stress because it develops under direct loading conditions. That
is, simple tension and simple compression occur when the applied force, called load, is in line with
the axis of the member (axial loading) (Fig. 1.1a and b) and simple shear occurs, when equal
parallel, and opposite forces tend to cause a surface to slide relative to the adjacent surface.
Fig. 1.1a: Tensile stress
Fig. 1.1b: Compressive stress
Simple stress 𝛿 (sigma) is calculated by
𝑃
𝜎=
𝐴
where,
𝜎 = 𝑆𝑡𝑟𝑒𝑠𝑠, 𝑘𝑁⁄𝑚2 𝑜𝑟 𝑁⁄𝑚𝑚2
1
MPE 222: Solid Mechanics I, Lecture Notes; By Dr. SM Talai (Ph.D.)
, 𝑃 = 𝑙𝑜𝑎𝑑, 𝑘𝑁 𝑜𝑟 𝑁
𝐴 = 𝐴𝑟𝑒𝑎 𝑜𝑣𝑒𝑟 𝑤ℎ𝑖𝑐ℎ 𝑠𝑡𝑟𝑒𝑠𝑠 𝑑𝑒𝑣𝑒𝑙𝑜𝑝𝑠, 𝑚2 𝑜𝑟 𝑚𝑚2
1.3 Strain
The strain (𝑒) is the deformation produced by stress. The various types of strain are explained
below:
Tensile Strain
A piece of material, with uniform cross-section, subjected to a uniform axial tensile stress, will
increase its length from 𝑙 to (𝑙 + 𝛿𝑙) and the increment of length 𝛿𝑙 is the actual deformation of
the material. The fractional deformation or the tensile strain is given by
𝛿𝑙
𝑒𝑡 =
𝑙
Compressive Strain
Under compressive forces, a similar piece of material would be reduced in length from 𝑙 to (𝑙 −
𝛿𝑙). The fractional deformation gives the strain 𝑒𝑐 ,
𝛿𝑙
𝑒𝑐 = −
𝑙
Volumetric Strain
It is defined as the ratio between change in volume and original volume of the body, and is denoted
by 𝑒𝑣 .
𝐶ℎ𝑎𝑛𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒 𝛿𝑉
∴ 𝑒𝑣 = =
𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑉
The strains which disappear with the removal of load are termed as elastic strains and the body
which regains its original position on the removal of force is called an elastic body. The body is
said to be plastic if the strains exist even after the removal of external force. There is always a
limiting value of load up to which the strain totally disappears on the removal of load-the stress
corresponding to this load is called elastic limit.
Robert Hooke discovered experimentally that within elastic limit, stress varies directly as strain
𝑆𝑡𝑟𝑒𝑠𝑠
𝑖. 𝑒. 𝑆𝑡𝑟𝑒𝑠𝑠 ∝ 𝑆𝑡𝑟𝑎𝑖𝑛 𝑜𝑟 = 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑆𝑡𝑟𝑎𝑖𝑛
The constant is termed as Modulus of elasticity
2
MPE 222: Solid Mechanics I, Lecture Notes; By Dr. SM Talai (Ph.D.)
