Calculating Extension & Stress in Machined Steel Bar

AI Thread Summary
To calculate the total extension and tensile stress in a machined steel bar with varying diameters, the bar can be treated as three separate sections, each acting like a spring in series. The extension for each section can be determined using the formula k_i = A_iE/L_i, where A_i is the cross-sectional area, E is the modulus of elasticity, and L_i is the length of each section. The total extension is the sum of the individual extensions from each section under the axial load of 84 kN. The tensile stress in each portion can be calculated using the formula stress = force/area for the respective diameters. This approach simplifies the problem and allows for accurate calculations of both extension and stress.
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Homework Statement



1. A steel bar 375mm long is machined to give a diameter of 75mm for the first 175mm length, a diameter of 45mm for the next 100mm length and a diameter of 37mm for the final 100mm length. Determine:

(i) The total extension of the bar when it’s subjected to an axial tensile load of 84 kN.

(ii) The tensile stress in each portion of the bar.
Assume E = 200 GN/m2


Homework Equations





The Attempt at a Solution



can this be solved by working out each part as if it were 3 separate bars ?
 
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The simple answer is yes. A bar with varing diameter as noted is nothing more that 3 springs in "series" with 3 different spring constants "k". An axial force applied to springs in series would be equal to the sums of the k_i \cdot x_i.

k_i in this problem would be \frac{A_{i}E}{L_i}
 
thanks
 
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