Limitations of Euler's theory

The general expression of bucking load for the long column as per Euler’s theory is given as,

P = Π ²E I / L²

σ = Π ²E / (Le / k) ²

We know that, Le / k = slenderness ratio

Limitation 1: The above formula is applied only for long columns

Limitation 2: As the slenderness ratio decreases the crippling stress increases. Consequently if the slenderness ratio reaches to zero, then the crippling stress reaches infinity, practically which is not possible.

Limitation 3 : if the slenderness ratio is less than certain limit, then crippling stress is greater than crushing stress ,which is not possible practically. Therefore, up to limiting extent Euler’s formula is applicable with crippling stress equal to crushing stress.

Euler's theory assumptions for long columns

Some of the assumptions of Euler's theory for long columns are

1. The geometric and material properties of the column is uniform throughout the section,i.e, flexural rigidity is constant all through.
2. The material is isotropic homogeneous and elastic.
3. The layers of the columns are assumed to be straight before the application of the load.
4. The load on the column is usually applied axially at its ends.
5. Since the slenderness ratio  for the long column is more ,it fails by buckling only.
6. Self weight of the column is neglected during the calculation of failure load on the column.
7. The linear dimension of the long column is much more compared to the lateral dimension.
8. Long column experiences bending stress at a higher extent than compared to the direct stress.

Basic Properties of Materials

Density: It is defined as mass per unit volume.  It is expressed as kg/m3.

Specific gravity: It is the ratio of density of a material to density of water.

Porosity: The term porosity is used to indicate the degree by which the volume of a material is occupied by pores.  It is expressed as a ratio of volume of pores to that of the specimen.

Strength: Strength of a material has been defined as its ability to resist the action of an external force without breaking.

Elasticity: It is the property of a material which enables it to regain its original shape and size after the removal of external load.

Plasticity: It is the property of the material which enables the formation of permanent deformation.

Hardness: It is the property of the material which enables it to resist abrasion, indentation, machining and scratching.

Ductility: It is the property of a material which enables it to be drawn out or elongated to an appreciable extent before rupture occurs.

Brittleness: It is the property of a material, which is opposite to ductility. Material, having very little property of deformation, either elastic or plastic is called Brittle.

Creep: It is the property of the material which enables it under constant load to deform slowly but progressively over a certain period.

Stiffness: It is the property of a material which enables it to resist deformation.

Fatigue: The term fatigue is generally referred to the effect of cyclically repeated stress. A material has a tendency to fail at lesser stress level when subjected to repeated loading.

Impact strength: The impact strength of a material is the quantity of work required to cause its failure per its unit volume.  It thus indicates the toughness of a material.

Toughness: It is the property of a material which enables it to be twisted, bent or stretched under a high stress before rupture.

Thermal Conductivity: It is the property of a material which allows conduction of heat through its body.  It is defined as the amount of heat in kilo calories that will flow through unit area of the material with unit thickness in unit time when difference of temperature on its faces is also unity.

Corrosion  Resistance: It is the property of a material to withstand the action of acids, alkalis gases etc., which tend to corrode (or oxidize).

Stress-Strain diagrams for ferrous and non-ferrous materials

Relationship between Stress and Strain are derived on the basis of the elastic behaviour of material bodies.

A standard mild steel specimen( ductile specimen) is subjected to a gradually increasing pull by Universal Testing Machine. 

The stress-strain curve obtained is as shown below.

Stress strain diagram for ductile material

A -Elastic Limit
B - Upper Yield Stress
C - Lower Yield Stress
D -Ultimate Stress
E -Breaking Stress

Point A : Elastic limit point or Proportionality point: Proportional limit is point on the curve up to which the value of stress and strain remains proportional. The stress up to this point can be also be known as proportional limit stress.Hooke’s law is obeyed between the point O to A.

Point B : Upper Yield point:  Yield strength or yield point is the material property defined as the stress at which a material begins to deform plastically.Upper yield point is the point wherein the   stress increases and correspondingly strain also increases.

Point C : Lower Yield point : It is the point where the load remains constant and strain is increased correspondingly.

Point D : Ultimate Stress Point/ Maximum Stress point: Ultimate stress point is the maximum strength that material have to bear stress before breaking. It can also be defined as the ultimate stress corresponding to the peak point on the stress strain graph. 

Point E : Breaking point /Failure point/ Fracture point : Breaking point or breaking stress is point where strength of material breaks.  The stress associates with this point  known as breaking strength or rupture strength.

Stress strain diagram for Brittle material

Point A : Elastic limit point or Proportionality point: Proportional limit is point on the curve up to which the value of stress and strain remains proportional. The stress up to this point can be also be known as proportional limit stress.Hooke’s law is obeyed between the point O to A.

Point B : Breaking point /Failure point/ Fracture point : Breaking point or breaking stress is point where strength of material breaks.  The stress associates with this point  known as breaking strength or rupture strength. 

Section Modulus

                                            Section modulus

The moment carrying capacity of an object is directly dependent on geometrical property (I) and material property (E) of an object,which is collectively termed as flexural rigidity(EI).Geometry of an object plays an important role in load bearing capacity of an object which is indicated by moment of inertia of a section. Therefore section modulus is the predominant factor which evidences the strength of an object and is defined as the ratio of the moment of inertia of the object about its centroidal axis to the distance of the extreme fibers of the object from the neutral axis.

Section modulus is generally denoted by Z.

Therefore ,  Z = I / Ymax 

where, I = Moment of inertia of a section.

          Ymax = Distance of the outer most fiber of the object from the neutral axis.

Section modulus can also be defined by using the simple bending theory as,

we know that, M / I = σ / Y

Therefore, Z = M / σ 

i.e, section modulus is also expressed as the ratio of bending moment to the bending stress of a given object within the elastic limit.

Significance of section modulus

1. Section modulus is the important factor for design of beam and flexural member
2. Higher the value of section modulus, higher will be the resistance of member to bending
3. It is required to calculate stresses in beams.
4. It is used to calculate strength of the steel structure
5. More the section modulus, it can withstand more load and it is also considered to be more tougher.