Strain and Stress-Strain Diagram

Strain

• Strain, also known as unit deformation, is the ratio of the change in length to the original length.
• Strain is caused by an applied force, and the resultant value is dimensionless.
• ε = δ/L, where ε is strain, δ is deformation, and L is the original length.

Stress-Strain Diagram

• The graph plots stress (σ) on the y-axis against strain (ε) on the x-axis.
• The stress-strain diagram differs depending on the material.
• A metal specimen is placed in a tension-compression testing machine.
• Axial load is gradually increased, and total elongation is measured incrementally.
• Stress (σ) and strain (ε) are obtained from original cross-sectional area and length measurements.

Ductile and Brittle Materials

• Metallic engineering materials are classified as ductile or brittle.
• Ductile materials exhibit large tensile strains up to the point of rupture (e.g., steel, aluminum).
• Brittle materials exhibit small strains up to the point of rupture (e.g., cast iron, concrete).
• 0.05 mm/mm is an arbitrary strain value that distinguishes ductile and brittle materials.

Proportional Limit (Hooke's Law)

• The stress-strain curve is a straight line between the origin (O) and the proportional limit.
• Within the proportional limit, stress is directly proportional to strain.
• Sir Robert Hooke first observed the linear relationship between elongation and axial force in 1678.
• σ = kε, or σ ∝ ε, where k is the constant of proportionality.
• The constant of proportionality k is also known as the Modulus of Elasticity (E) or Young’s Modulus.
• The Modulus of Elasticity (E) or Young’s Modulus is equal to the slope of the stress-strain diagram between O and P.
• The formula to obtain stress, simplified via Young's Modulus, is σ = Eε.

Elastic Limit

• The elastic limit represents the point beyond which a material no longer returns to its original shape upon unloading.
• No permanent or residual deformation happens below an elastic limit, once the loading is entirely removed.

Elastic and Plastic Ranges

• The elastic range is the region in the stress-strain diagram from the origin (O) to the proportional limit (P).
• The plastic range is the region in the stress-strain diagram from P to Rupture (R).

Yield Point

• The yield point is where the material begins to elongate or yield noticeably without any increase in load.

Ultimate Strength

• The maximum ordinate on the stress-strain diagram indicates the ultimate strength, or tensile strength.

Rupture Strength

• Rupture strength is the strength of a material at the point of rupture.
• Rupture Strength is otherwise known as breaking strength.

Modulus of Resilience

• Modulus of resilience is the work done on a unit volume of material as force is gradually increased from O to P (Nm/m³).
• Modulus of resilience can be calculated as the area under the stress-strain curve from the origin (O) up to the elastic limit (E).
• Resilience is the material's ability to absorb energy without permanent distortion.

Modulus of Toughness

• Modulus of toughness is the work done on a unit volume of material as force is gradually increased from O to R (Nm/m³).
• Modulus of toughness can be calculated as the area under the entire stress-strain curve (from O to R).
• A material's toughness is its ability to absorb energy without breaking.

Working Stress, Allowable Stress, and Factor of Safety

• Working stress is the actual stress experienced by a material under loading.
• Allowable stress is the maximum safe stress a material can carry.
• Allowable stress should not exceed the proportional limit.
• The allowable stress is determined as the yield point or ultimate strength divided by a factor of safety.
• Factor of safety expresses the ratio of ultimate or yield strength to allowable stress.

Axial Deformation

• The stress is proportional to strain in the linear part of the stress-strain diagram.
• σ = Eε, where σ is stress, E is the modulus of elasticity, and ε is strain.
• δ = (PL)/(AE) = (σL)/E, where δ is total elongation, P is applied load, L is original length, A is cross-sectional area and E is the modulus of elasticity.
• To use the formula for axial deformation: the load must be axial, the bar must have a uniform cross-sectional area, and stress must not exceed the proportional limit.
• Axial deformation of a non-uniform cross-sectional area is determined by the integral of dX/A.

Axial Deformation of Non-Uniform Material

• δ = ∫(P/E) (dx/A) from 0 to L, where A = ty if variable must be expressed in terms of x.
• δ = (ρgL^2) / 2E = (MgL) / 2AE, or a rod of unit mass p suspended vertically from one end, the total elongation due to its own weight is Where:
• ρ is in kg/m³
• L is the length of the rod in mm,
• M is the total mass of the rod in kg,
• A is the cross-sectional area of the rod in mm²,
• g is 9.81 m/s².

Stiffness

• Stiffness (k) is the ratio of steady force acting on an elastic body to displacement.
• Stiffness is expressed in N/mm
• k = P/δ, where P is the applied force and δ is the displacement.

Shearing Deformation

• Shearing forces cause shearing deformation where an element changes in shape. However the length does not change.
• Shear Strain Formula: γ = δs/L ,Where: γ is the shear strain, δs is the shear deformation or displacement and L is the original length perpendicular to the shearing force.
• Modulus of elasticity is shear or modulus of rigidity is the ratio of the shear stress τ and the shear strain γ, and is denoted as G, in MPa.
• G = τ/ γ, where G is the modulus of rigidity
• The relationship between the shearing deformation and the applied shearing force: δs = (VL)/(AsG) = (τL)/G, As is the are where is shearing takes place

Poisson's Ratio

• A bar under tensile loading increases in length but decreases in lateral dimensions.
• Poisson's ratio (ν) is the ratio of sidewise deformation to longitudinal deformation.
• Poisson’s ratio is 0.25 to 0.3 for steel and 0.20 for concrete.
• The formula is: v = -εy/εx = -εz/εx, where εx is strain in the x-direction and εy and εz are the strains in the perpendicular direction.
• The negative sign indicates a decrease in the transverse dimension when εx is positive.

Biaxial Deformation

• Axial Deformation formula: εx = σx/E - v(σy/E) or σx = -((εx + vεy)E) / (1-v^2)
• εx is the stain in the x direction
• σ is the stress for x or y direction
• The stress in the y direction formula: εy = σy/E - v(σx/E) or σy = -((εy + vεx)E) / (1-v^2)

Triaxial Deformation

• Triaxial Deformation formula:
• εx = (1/E) * [σx - v(σy + σz)]
• εy = (1/E) * [σy - v(σx + σz)]
• εz = (1/E) * [σz - v(σx + σy)]
• Tensile stresses and elongation are taken as positive.
• Compressive stresses and contraction are taken as negative.

Relationship Between E, G, and v

• The relationship between the modulus of elasticity E, shear modulus G, and Poisson's ratio v: G = E / 2(1+v)

Bulk Modulus of Elasticity

• Bulk modulus of elasticity (K) measures a material's resistance to volume change without shape change.
• K = E / 3(1-2v) = σ / (ΔV/V) Where:
• V is the volume
• ΔV is the change in volume.
• Volumetric strain = ΔV/ V = σ/K = 3(1-2v) / E

Thick Walled Pressure Vessels

• Equation Formula: σt = ρD / 2t
• Used particularly for pipes experiencing high pressures.

Statically Indeterminate Members

• A structure is statically indeterminate, when the reactive forces or the internal resisting forces over a cross section exceed the number of independent equations of equilibrium.
• Additional equations that depend on the elastic deformations in the members are required to solve these cases.