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Change in length for a bar with continuously varying load or dimensions

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sometimes the axial force and the cross-section area varies along the length of the bar. for this case the equation δ=(PL)/(AE) will not be applicable ,  figure 1 shows a bar with continuously varying load and cross-section area along the length L, for P(x) acting at point A and varying along the length of the bar with force of PB at point B. the force acting along the bar is function of distance (unit of pound per inch or newton per meter). A distributed axial load may be produced by such factors as centrifugal forces, friction forces, or the weight of a bar hanging in a vertical position.






Figure 1









To compute the change of length for this case, we should integrate the change in length of a  differential element of the bar and then integrate over the length of the bar. in general the force is the function of x(change in length) as shown in figure b, the cross-section area A(X) also function of X, we should derive the relation of A as a function of X then use the equation below to calculate the change in length of bar.

dδ=(N(X)*dx)/(A(X)*EA)



Example (click here)




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My name is Mohammad Hamdan, and I am a Civil/bridge Engineer with extensive professional background and experience in the field of construction. I have 10+ years of experience in the construction field. During my professional career, my emphasis has been placed on structural projects, mainly bridges, infrastructure and roads. Reading and practicing the design of structures is my passion.i was lucky to be a part of a big team that executed major highway interchanges. I believe in spreading the knowledge, thus, i compose these courses to share my experience and knowledge. My course focuses on the practical side of construction field. On other word, i am sharing some of what new engineers need to know about construction field. Beside working on construction field, i love writing some articles on my website ( curious civil engineer) Moreover, i wrote some engineering papers, you can check them  by searching my name on Google (Mohammad mamon Hamdan)