Module 2: Simple Strain and Deformation

Understanding how materials respond and change shape under applied loads.
This module covers the response of materials to applied stresses, specifically focusing on deformation and strain. Understanding how materials change shape under load is crucial for structural design, ensuring members don't just resist breaking, but also resist bending or stretching beyond functional limits (serviceability).

Key Concepts

Checklist

Deformation vs Strain

While deformation (δ\delta) is an absolute physical measure of how much a material stretches or compresses (e.g., 2mm2 mm), strain (ϵ\epsilon) is a relative measure representing the deformation per unit length (e.g., 0.001mm/mm0.001 mm/mm). This makes strain a dimensionless value that can be compared universally across structural members of vastly different sizes.

Axial Deformation and Concept of Strain

Measuring changes in length under axial loads
When a structural member is subjected to an axial load (tension or compression), it deforms (elongates or shortens). Normal Strain (ϵ\epsilon) is the ratio of this total deformation to the original, unloaded length.
The formula for simple average strain is:
ϵ=δL \epsilon = \frac{\delta}{L}
Where:
  • ϵ\epsilon = Strain (dimensionless, often expressed in m/mm/m, in/inin/in, or microstrain μϵ\mu\epsilon)
  • δ\delta = Change in length or total deformation
  • LL = Original length

Strain

A dimensionless measure of deformation representing the relative displacement between particles in a material body under stress.
Key Takeaways
  • Strain (ϵ=δ/L\epsilon = \delta/L) is the dimensionless ratio of deformation to original length.

The Stress-Strain Diagram

Visualizing material behavior
By plotting stress (σ\sigma) against strain (ϵ\epsilon) during a tensile test, we create a Stress-Strain Diagram. This graph reveals critical mechanical properties of a material. For ductile materials like structural steel, key points include:
  • Proportional Limit: The highest stress at which stress and strain are directly proportional (Hooke's Law applies).
  • Elastic Limit: The maximum stress a material can withstand without permanent deformation. Upon unloading, it returns to its original shape.
  • Yield Point: The stress at which the material begins to deform plastically (permanently) with little to no increase in stress.
  • Ultimate Strength: The absolute maximum stress the material can endure before necking and eventual failure.
  • Rupture Strength: The stress at the exact point of fracture.

Material Behavior States

Explore how different materials deform under stress

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ε = 0.0000
Current Stress0.0 MPa
Current StateElastic Region

Ductile Steel (e.g., Low Carbon Steel)

Exhibits a distinct elastic region, yield point, a plastic plateau, strain hardening, and necking before fracture. Highly ductile and tough.

What is happening now?

Elastic Region

Material deforms reversibly. Stress is directly proportional to strain (Hooke's Law applies). If the load is removed, the material returns exactly to its original shape. The bonds between atoms are stretched but not broken.

Microscopic View

Atoms are stretched uniformly like springs.

Key Takeaways
  • Understand the regions of the stress-strain curve: proportional limit, yield point, ultimate strength, and rupture.
  • Ductile materials undergo significant plastic deformation before failure, providing warning before collapse.

True Stress vs. Engineering Stress

Accounting for changes in cross-sectional area
The standard formula σ=P/A0\sigma = P/A_0 uses the original cross-sectional area (A0A_0). This is called Engineering Stress (or nominal stress).
However, as a ductile material stretches significantly (e.g., during necking), its actual cross-sectional area (AfA_f) decreases. The True Stress (σtrue\sigma_{true}) is calculated using the instantaneous cross-sectional area:
σtrue=PAf \sigma_{true} = \frac{P}{A_f}
For practical design in the elastic region, the difference is negligible, so engineering stress is typically used.
Key Takeaways
  • Engineering stress is based on the original area (A0A_0), while true stress accounts for the changing area during deformation.

Hooke's Law and Modulus of Elasticity

The elastic behavior of materials
For most structural materials, there is an initial linear region on the stress-strain curve where stress is directly proportional to strain. This fundamental relationship is known as Hooke's Law.
The constant of proportionality is the Modulus of Elasticity (EE), also known as Young's Modulus. It represents the intrinsic stiffness of the material. A higher EE means the material is stiffer and deforms less under the same stress.
σ=Eϵ \sigma = E \epsilon
Substituting the definitions of normal stress (σ=P/A\sigma = P/A) and strain (ϵ=δ/L\epsilon = \delta/L):
PA=E(δL) \frac{P}{A} = E \left(\frac{\delta}{L}\right)
Rearranging this gives us the highly useful formula for total axial deformation (δ\delta):
δ=PLAE \delta = \frac{PL}{AE}
Where:
  • δ\delta = Deformation
  • PP = Axial Load
  • LL = Original Length
  • AA = Cross-sectional Area
  • EE = Modulus of Elasticity

Hooke's Law

A principle of mechanics stating that for relatively small deformations of an object, the resulting strain is directly proportional to the applied stress.

Axial Deformation & Stress-Strain Curve

L = 2000mm + 0.00mm
0Yield (125 kN)Ultimate (200 kN)
Normal Stress (σ\sigma)
0.0 MPa
Deformation (δ)
0.000 mm

Stress-Strain Diagram

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Key Takeaways
  • Hooke's Law (σ=Eϵ\sigma = E\epsilon) relates stress to strain in the linear elastic region.
  • The Modulus of Elasticity (EE) is a measure of material stiffness.
  • Axial deformation can be calculated using δ=PL/AE\delta = PL/AE.

Statically Indeterminate Members

When statics alone is not enough
A member or structure is considered statically indeterminate when the unknown reactive forces or internal forces cannot be determined using only the basic equations of static equilibrium (ΣF=0\Sigma F = 0, ΣM=0\Sigma M = 0). There are simply more unknowns than available statics equations.
To solve for these unknowns, we must introduce compatibility equations. These equations are derived from the geometric constraints and deformations of the system. By understanding how the members must physically deform to fit together (and using δ=PL/AE\delta = PL/AE), we create the necessary additional equations.
Key Takeaways
  • Statically indeterminate members require compatibility equations (based on geometry and deformation) in addition to statics equations.

Poisson's Ratio

Lateral strain caused by axial stress
When a rubber band is stretched longitudinally, it visibly thins out laterally. This phenomenon occurs in almost all solid materials. When a material is stressed in one direction, it exhibits strain in the perpendicular (transverse) directions.
Poisson's Ratio (ν\nu) is the absolute value of the ratio of transverse strain to axial strain.
ν=−ϵtransverseϵaxial \nu = -\frac{\epsilon_{transverse}}{\epsilon_{axial}}

Poisson's Ratio

A material property that quantifies the Poisson effect: the ratio of relative contraction strain (transverse) to relative extension strain (axial) in the direction of the applied force.

Note

For most structural metals (like steel and aluminum), Poisson's ratio is typically around 0.300.30 to 0.330.33. Concrete is around 0.100.10 to 0.200.20. A perfectly incompressible material (like rubber) has a theoretical maximum ν=0.5\nu = 0.5.
Key Takeaways
  • Poisson's Ratio (ν\nu) quantifies the lateral contraction that occurs during axial stretching.

Thermal Stresses

Stresses induced by temperature changes
Changes in ambient temperature cause materials to expand (when heated) or contract (when cooled). If a member is free to expand or contract without any restriction, it experiences thermal strain but zero thermal stress.
However, if the member is constrained (e.g., a steel rail wedged tightly between two rigid walls), the restriction prevents the natural deformation, forcing internal thermal stresses to develop as the material pushes against its supports.
Unrestricted thermal deformation (δT\delta_T) is given by:
δT=αLΔT \delta_T = \alpha L \Delta T
Where:
  • α\alpha = Coefficient of thermal expansion (a material property, e.g., per ∘C^\circ C)
  • LL = Original length
  • ΔT\Delta T = Change in temperature (Tfinal−TinitialT_{final} - T_{initial})
If the member is fully constrained against this movement, the support must provide a force to push it back to its original length. The resulting thermal stress (σT\sigma_T) is:
σT=EαΔT \sigma_T = E \alpha \Delta T

Important

Thermal stresses can be massive. This is why bridges have expansion joints, and why long continuous piping systems include "expansion loops" to absorb thermal movement without overstressing the pipes or their anchor points.
Key Takeaways
  • Temperature changes cause unrestricted deformation δT=αLΔT\delta_T = \alpha L \Delta T.
  • Thermal stresses only develop if this deformation is constrained or prevented by supports (σT=EαΔT\sigma_T = E \alpha \Delta T).

Shear Strain

Angular deformation caused by shear forces
While normal strain describes changes in length, shear strain (γ\gamma) describes changes in shape or angle. When a shear force is applied to a material, it causes the planes of the material to slide past one another, distorting originally right angles.
If you imagine a rectangular element subjected to shear stresses, it deforms into a parallelogram. The angle of this distortion (measured in radians) is the shear strain.

Shear Strain

The angular distortion of a material element under shear stress, defined as the change in an originally right angle, measured in radians.
Just as normal stress and normal strain are related by Hooke's Law via the Modulus of Elasticity (EE), shear stress (τ\tau) and shear strain (γ\gamma) are proportional in the elastic region. The constant of proportionality is the Modulus of Rigidity (or Shear Modulus), denoted by GG.
G=τγ G = \frac{\tau}{\gamma}
Where:
  • GG = Modulus of Rigidity / Shear Modulus (Pa, psi, etc.)
  • τ\tau = Shear Stress
  • γ\gamma = Shear Strain (radians)

Relationship Between Elastic Constants

For isotropic materials, the Modulus of Elasticity (EE), the Modulus of Rigidity (GG), and Poisson's ratio (ν\nu) are mathematically related:
G=E2(1+ν) G = \frac{E}{2(1 + \nu)}
Key Takeaways
  • Shear strain (γ\gamma) is an angular distortion, not a change in length.
  • The Modulus of Rigidity (GG) relates shear stress and shear strain in the elastic region: G=τ/γG = \tau / \gamma.

Bulk Modulus and Volumetric Strain

Response of materials to uniform hydrostatic pressure

Volumetric Strain and Bulk Modulus

When a material is subjected to uniform hydrostatic pressure (equal normal stress in all three orthogonal directions), it undergoes a change in volume but no change in shape. The Volumetric Strain (ee or ϵv\epsilon_v) is the ratio of the change in volume (ΔV\Delta V) to the original volume (V0V_0): e=ΔV/V0e = \Delta V / V_0.
The Bulk Modulus (KK) relates the applied hydrostatic stress (σ\sigma) to the volumetric strain: K=σ/eK = \sigma / e. It is a measure of a material's resistance to uniform compression.
K=E3(1−2ν) K = \frac{E}{3(1 - 2\nu)}
Where:
  • KK = Bulk Modulus
  • EE = Modulus of Elasticity
  • ν\nu = Poisson's Ratio
Key Takeaways
  • Volumetric strain (e=ΔV/V0e = \Delta V/V_0) describes volume changes under hydrostatic pressure.
  • Bulk Modulus (KK) relates hydrostatic stress to volumetric strain.
  • KK is mathematically related to EE and Poisson's ratio (ν\nu).
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