Analysis of Bars of Varying Sections

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The bars of varying sections, material properties, and dimensions, that are subjected to axial loads can be analyzed to determine their corresponding stress and strain values.

The analysis is performed based on the fundamental equation from Hooke's law, that relates the stress (f) and the strain (e) by

f = e.E ----------Eq.1

Where, 'E' is the modulus of elasticity. Here the analysis of strain or deformation of bars of varying sections is explained briefly.

Also Read: Different Elastic Constants and Their Relationships

Analysis of Bars of Varying Sections

Consider a bar of different length, area of cross-section and diameter, subjected to a axial load of 'P' as shown in figure-1.

The stress and deformation of length caused in each of the sections of the bar are different, even if the axial load acting on the bar is the same. The derivation is followed by the principle that,

Total Change in length, dL = Sum of Change in length of Section 1, Section 2 and Section 3.

dL = dL₁ + dL₂ +dL₃ ----------Eq.2

1. Analysis of Section-1

Stress in Section 1,

f1 = Load/Area of Section-1 = P/A1

Strain in Section 1,

e₁ = Change in length/Original Length = dL₁/L₁ = f₁/E = P/(A₁E) [From Eq.1]

Hence, Change in Length dL₁=PL₁/(A₁E) ----------Eq.3

2. Analysis of Section-2

Stress in Section 2,

f2 = Load/Area of Section-2 = P/A2

e₂ = Change in length/Original Length = dL₂/L₂ = f₂/E =P/(A₂E) [From Eq.1]

Hence, Change in Length dL₂= PL₂/(A₂E) ----------Eq.4

2. Analysis of Section-2

Stress in Section 3,

f3 = Load/Area of Section-3 = P/A3

e₃ = Change in length/Original Length = dL₃/L₃ = f₃/E=P/(A₃E) [From Eq.1]

Hence, Change in Length dL₃= PL₃/(A₃E) ----------Eq.5

From Eq.2,

dL = dL₁ + dL₂ +dL₃

Therefore, Eq.3 + Eq.4 + Eq.5,

dL = dL₁ + dL₂ +dL₃ = PL₁/(A₁E) +PL₂/(A₂E)+ PL₃/(A₃E)

The above equation is used when the sections of the bar have same modulus of elasticity. If the 'E' value is different, we get the relation,

Also Read: Stress -Strain Relationships of Concrete

Also Read: Stress -Strain Relationships of Steel Bars