1.0 PROPERTIES OF MATERIALS
1.1 Introduction
When an external force acts on a body, the body will tend to undergo some type of deformation. However,
due to the cohesive forces that act between the molecules of the body, the body will tend to resist the
deformation. This resistance is known as strength of material. Different materials have different levels of
strength. Within a certain limit i.e. the elastic stage, the resistance offered by the material is proportional
to the deformation brought out on the material by the external force and the resistance equals the
external force. Beyond the elastic stage, however, the resistance offered by the material is less than the
applied load and therefore the deformation will continue disproportionately until failure takes place.
1.2 Stress and Strain
As an introduction to your first topic in Structures I, we shall begin with the following definitions that shall
be important to your understanding of the properties of the common materials that you shall encounter
in Structural Engineering.
(i) Plasticity: this is the deformation of a material undergoing non-reversible changes of shape in
response to applied forces. Once a material undergoes plastic deformation, it can no longer be
returned to its original state. Plastic deformation is the permanent distortion that occurs when a
material is subjected to tensile, compressive, bending, or torsion stresses that exceed its yield
strength and cause it to elongate, compress, buckle, bend or twist. You can imagine a chewing
gum that can be stretched to many times its original dimensions.
(ii) Elasticity: this is the property of some deformed bodies to recover, at least partially, their initial
form after the withdrawal of the force that caused the deformation. So, elastic deformation is
the temporary distortion that occurs when a material is subjected to tensile, compressive,
bending, or torsional stresses that do not exceed its yield strength. A good example is a rubber
band that is stretched but quickly resumes its original dimensions upon release of the stretching
force.
(iii) Stress (σ): this is the restoring force per unit area. As discussed before, when some external force
acts on a body, different particles of the body will be displaced. These displaced particles will try
to come back to their original positions when the external force is withdrawn. Note that the
reaction set up in the body is equal and opposite to the applied force, so long as no permanent
change is produced in the body. Therefore, the restoring force is equal to the applied force. The
formula for stress is given below:
𝐹𝑜𝑟𝑐𝑒
𝑆𝑡𝑟𝑒𝑠𝑠 =
𝐴𝑟𝑒𝑎
𝑃
𝜎=
𝐴
𝑊ℎ𝑒𝑟𝑒: 𝜎 = 𝑠𝑡𝑟𝑒𝑠𝑠
𝑃 = 𝑓𝑜𝑟𝑐𝑒
𝐴 = 𝑎𝑟𝑒𝑎
, It would thus follow that the units of stress will be the units of Force (𝑘𝑁, 𝑜𝑟 𝑘𝑖𝑙𝑜𝑁𝑒𝑤𝑡𝑜𝑛𝑠)
divided by the units of Area (𝑚2 ), thus stress, σ, would be denoted in 𝑘𝑁/𝑚2 .
(iv) Strain (ε): this is the ratio of change in length or volume to the original length or volume. Since
strain is a ratio, it has no units. The formula for strain is given below:
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐿𝑒𝑛𝑔𝑡ℎ 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑉𝑜𝑙𝑢𝑚𝑒
𝑆𝑡𝑟𝑎𝑖𝑛 = =
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒
𝛿𝑙 𝛿𝑣
𝜀= =
𝐿 𝑉
𝑊ℎ𝑒𝑟𝑒: 𝜀 = 𝑠𝑡𝑟𝑎𝑖𝑛
𝛿𝑙 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐿𝑒𝑛𝑔𝑡ℎ
𝐿 = 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ
𝛿𝑣 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒
𝑉 = 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒
(v) Hooke’s Law: Hooke discovered a simple relationship between stress and strain. It states that
stress is proportional to strain within the elastic limits. Mathematically, it can be expressed as:
𝑆𝑡𝑟𝑒𝑠𝑠 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑆𝑡𝑟𝑎𝑖𝑛
𝑆𝑡𝑟𝑒𝑠𝑠 𝛼 𝑆𝑡𝑟𝑎𝑖𝑛
𝑆𝑟𝑒𝑠𝑠
∴ =𝐸
𝑆𝑡𝑟𝑎𝑖𝑛
𝜎
𝑜𝑟, =𝐸
𝜀
The constant of proportionality,𝐸, is called modulus of elasticity or coefficient of elasticity. Its
value depends on the nature of the material. Therefore, different materials have different values
of 𝐸.
(vi) Young’s Modulus (𝑬): this is the ratio of stress to longitudinal strain within the elastic limits. Let
us take a wire of length 𝐿 and let it change by 𝛿𝑙 under an applied load. The force 𝐹 acting on an
area of cross-section, 𝐴. Therefore:
𝛿𝑙
𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛 =
𝐿
𝑃
𝐴𝑛𝑑 𝑆𝑡𝑟𝑒𝑠𝑠 =
𝐴
𝑆𝑡𝑟𝑒𝑠𝑠
∴ 𝑌𝑜𝑢𝑛𝑔′ 𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝐸𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦, 𝐸 =
𝑆𝑡𝑟𝑎𝑖𝑛
𝑃
= 𝐴
𝛿𝑙
𝐿
1.1 Introduction
When an external force acts on a body, the body will tend to undergo some type of deformation. However,
due to the cohesive forces that act between the molecules of the body, the body will tend to resist the
deformation. This resistance is known as strength of material. Different materials have different levels of
strength. Within a certain limit i.e. the elastic stage, the resistance offered by the material is proportional
to the deformation brought out on the material by the external force and the resistance equals the
external force. Beyond the elastic stage, however, the resistance offered by the material is less than the
applied load and therefore the deformation will continue disproportionately until failure takes place.
1.2 Stress and Strain
As an introduction to your first topic in Structures I, we shall begin with the following definitions that shall
be important to your understanding of the properties of the common materials that you shall encounter
in Structural Engineering.
(i) Plasticity: this is the deformation of a material undergoing non-reversible changes of shape in
response to applied forces. Once a material undergoes plastic deformation, it can no longer be
returned to its original state. Plastic deformation is the permanent distortion that occurs when a
material is subjected to tensile, compressive, bending, or torsion stresses that exceed its yield
strength and cause it to elongate, compress, buckle, bend or twist. You can imagine a chewing
gum that can be stretched to many times its original dimensions.
(ii) Elasticity: this is the property of some deformed bodies to recover, at least partially, their initial
form after the withdrawal of the force that caused the deformation. So, elastic deformation is
the temporary distortion that occurs when a material is subjected to tensile, compressive,
bending, or torsional stresses that do not exceed its yield strength. A good example is a rubber
band that is stretched but quickly resumes its original dimensions upon release of the stretching
force.
(iii) Stress (σ): this is the restoring force per unit area. As discussed before, when some external force
acts on a body, different particles of the body will be displaced. These displaced particles will try
to come back to their original positions when the external force is withdrawn. Note that the
reaction set up in the body is equal and opposite to the applied force, so long as no permanent
change is produced in the body. Therefore, the restoring force is equal to the applied force. The
formula for stress is given below:
𝐹𝑜𝑟𝑐𝑒
𝑆𝑡𝑟𝑒𝑠𝑠 =
𝐴𝑟𝑒𝑎
𝑃
𝜎=
𝐴
𝑊ℎ𝑒𝑟𝑒: 𝜎 = 𝑠𝑡𝑟𝑒𝑠𝑠
𝑃 = 𝑓𝑜𝑟𝑐𝑒
𝐴 = 𝑎𝑟𝑒𝑎
, It would thus follow that the units of stress will be the units of Force (𝑘𝑁, 𝑜𝑟 𝑘𝑖𝑙𝑜𝑁𝑒𝑤𝑡𝑜𝑛𝑠)
divided by the units of Area (𝑚2 ), thus stress, σ, would be denoted in 𝑘𝑁/𝑚2 .
(iv) Strain (ε): this is the ratio of change in length or volume to the original length or volume. Since
strain is a ratio, it has no units. The formula for strain is given below:
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐿𝑒𝑛𝑔𝑡ℎ 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑉𝑜𝑙𝑢𝑚𝑒
𝑆𝑡𝑟𝑎𝑖𝑛 = =
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒
𝛿𝑙 𝛿𝑣
𝜀= =
𝐿 𝑉
𝑊ℎ𝑒𝑟𝑒: 𝜀 = 𝑠𝑡𝑟𝑎𝑖𝑛
𝛿𝑙 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐿𝑒𝑛𝑔𝑡ℎ
𝐿 = 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ
𝛿𝑣 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒
𝑉 = 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒
(v) Hooke’s Law: Hooke discovered a simple relationship between stress and strain. It states that
stress is proportional to strain within the elastic limits. Mathematically, it can be expressed as:
𝑆𝑡𝑟𝑒𝑠𝑠 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑆𝑡𝑟𝑎𝑖𝑛
𝑆𝑡𝑟𝑒𝑠𝑠 𝛼 𝑆𝑡𝑟𝑎𝑖𝑛
𝑆𝑟𝑒𝑠𝑠
∴ =𝐸
𝑆𝑡𝑟𝑎𝑖𝑛
𝜎
𝑜𝑟, =𝐸
𝜀
The constant of proportionality,𝐸, is called modulus of elasticity or coefficient of elasticity. Its
value depends on the nature of the material. Therefore, different materials have different values
of 𝐸.
(vi) Young’s Modulus (𝑬): this is the ratio of stress to longitudinal strain within the elastic limits. Let
us take a wire of length 𝐿 and let it change by 𝛿𝑙 under an applied load. The force 𝐹 acting on an
area of cross-section, 𝐴. Therefore:
𝛿𝑙
𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛 =
𝐿
𝑃
𝐴𝑛𝑑 𝑆𝑡𝑟𝑒𝑠𝑠 =
𝐴
𝑆𝑡𝑟𝑒𝑠𝑠
∴ 𝑌𝑜𝑢𝑛𝑔′ 𝑠 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝐸𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦, 𝐸 =
𝑆𝑡𝑟𝑎𝑖𝑛
𝑃
= 𝐴
𝛿𝑙
𝐿
