Simple stress and strains

Strength: The ability of a part or element of a structure to resist failure. 

Stiffness: The ability to resist deformation.

Assumptions made in strength calculations of structural analysis:

• The material of a body has a solid continuous structure.
• Within limits of the part of the body the material is homogeneous & isotropic.

Homogeneous: A body is said to be homogeneous if it has identical properties at all points in identical directions.

Isotropic: A material is said to be isotropic, if at any point it has identical elastic properties in all

directions.

Ex:

1. A fine grained material is mostly isotropic.
2. For materials like timber, concrete and stone assumptions is only approximate.

• There are no internal forces in a body prior to loading. 
• Principle of super position is valid (It is valid only when the deformations are small compared to dimensions of body and deformations are linear functions of acting force.)
• Saint Venant’s principle is valid.

Stress: The internal resistance offered by a body against the deformation is called stress.

Mathematically, 

Stress = Force / Area

Types:

• Normal or direct stress : The stress due to an axial force. It may be either direct tensile (or) compresssive. 
  
• Shear Stress : The tangential force per unit area.

                                   τ = F/A

F = Shear force applied

• Bending Stress: Can be Bending (flexural) tensile or compressive stresses. in the element shown below the layers above N.A. are subjected to flexural compressive stress whereas the layers below N.A. are subjected to flexural tensile stress.

Bending Stress  fb = M / Z

M = Bending Moment

Z = Section Modulus

• Stress due to Torsion: Torsion or twisting moment results in torsional shear stress

Strain:

• Longitudinal Strain (or)Direct Strain (E): The deformation per unit length caused by normal force in its direction.
• Shear Strain: In case of tangential force (shear force on a plane), change in angle measured in radius is called shear strain.
  

Shear Strain, Φ = BB’ / AB

• Volumetric Strain (Dialation: Ratio of change in volume to the original volume.
  

Hook’s Law : (Given by Sir Robert Hooke in 1678): Stress is directly proportional to strain, within elastic limit (strictly speaking, upto limit of proportionality)

The above is valid for uni-axial force only.

Stress Strain Diagrams:

• For ductile materials : A ductile material is one having a relatively large tensile strain up to the point of ruptures.

Mild steel: It has a specific yield point.

P : Proportional limit

E : Elastic limit

Yu : Upper yield point

YL : Lower yield point

YLS = Strain hardening zone

S = Ultimate stress

B = Rupture stress (or) Breaking stress 

• Upper yield pointing unreliable. It varies with the shape of specimen rate of loading. Hence generally not considered as yield stress
• Lower yield stress is a reliable one and hence considered as yield stress of mild steel.

Nominal stresses strain curve: The curve based on original area of cross section of specimen while calculating various stresses.

True stress strain curve : Curve based on actual area of a specimen at the same instant, a

force is applied to calculate stresses.

Gauge Distance: While a specimen is being tested to an increasing force a change in length between two points ‘A’ and ‘B’, on the specimen is observed. The initial distance between the two points is called

“Gauge length” = 5.65 (A)1/2

For Tor Steel:  Tor Steel and aluminum do not have specific yield point.

The stress strain diagram is as follows:

When a material such as Tor steel does not have an obvious yield point A and yet undergoes large strains after the proportional limit is exceeded, an arbitrary yield stress may be determined by the offset method. A line is drawn on the stress strain diagram parallel to the initial linear part of the curve (see fig) but is offset by some standard amount of strain, such as 0.002 (or 0.2%). The intersection of the offset line the stress strain curve (point A in the fig) defines the yield stress. Since this stress is determined by an arbitrary rule and is not an inherent physical property of material, it should be referred to as the offset yield stress.

• Brittle Materials: Materials that fail in tension at relatively low values of strain are classified as brittle materials.

Examples are concrete, stone, cast iron, glass, ceramic materials and many common metallic alloys. These materials fail with only little elongation after the proportional limit (point A in fig) is exceeded and the fracture stress (point B) is the same as the ultimate stress. High carbon steel behave in a brittle manner, they

may have a very high yield stress, but fracture occurs at an elongation of only few percent.

Ordinary glass is a nearly ideal brittle material, because it exhibits almost no ductility whatsoever The stress — strain curve for glass intension is essentially a straight line, with failure occurring before any yielding takes

place.

Mechanical Properties of Materials:

• Elastic Material: A material which regains its original size and shape on removal of stress is said to be elastic material.
• Plastic Material: A material which can undergo permanent deformation without rupture is aid to be plastic material. This property of the material is known as plasticity. Plasticity is important when a material is to be mechanically formed by causing the material to flow.
• Ductile Material: A material which an undergo considerable deformation without rupture is said to be ductile material. The major portion of deformation is plastic.
• Brittle Material: A material which ruptures with little or no plastic deformation is said to be brittle material.
• Set or Permanent set: The deformation or strain remaining in a body after removal of stress is known as permanent set. This is due to plastic property of material.
• Elastic Limit: The greatest stress that a material can take without permanent set on the removal of stress is known as elastic limit.
•  Proportionality Limit: The greatest stress that a material can take without deviation from straight line relation between stress and strain is known as proportionality limit.
•  Endurance limit or Fatigue limit: The greatest stress, applied infinite number of times, that a material can take without causing failure is known as endurance or fatigue limit.
• Ultimate strength: The maximum stress material can take is known as ultimate strength. This is equal to maximum load divided by original area of cross-section.
• Modulus of resilience: The energy stored per unit volume at the elastic limit is known as modulus of resilience.
• Modulus of toughness: The amount of  work required per unit volume to cause failure under static loading, is called modulus of toughness.
• Modulus of rupture: The ultimate strength in flexure or torsion is known as modulus of
  rupture.
• Strain hardening: The increase in strength after plastic zone due to rearrangement of molecules in the material.
• Proof Stress: The stress which is just sufficient to cause a permanent set (elongation) equal to a specified percentage of the original gauge length.
• Elastic Strain: It is the dimensional change that occur in a material due to the application of loads and disappears completely on the removal of the loads.
•  Plastic Strain: It is the dimensional change that occurs in a material due to the application of the loads and does not disappear after the removal of the loads.
• Ductility and Malleability: The plastic response of material to tensile force is known as ductility and plastic response to compressive force is  known as malleability. The elongation and reduction of area of a  test piece tested to failure in tension are generally taken as measures of ductility of material.
• Creep: The long term deflection due to sustained ( constant) load.
• Factor of safety: It is defined as follows
  For Ductile materials,
  F.O.S = yield stress / working stress
  For Brittle materials,
  F.O.S = ultimate stress / working tress
  Margin of safety: = Factor of safety = -1