4a. Axial Members
(SSES Ch. 4.1,4.2) 
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Discretely Varying Axial Member
To solve problems involving discretely (suddenly) varying loads, areas, or modulus, the stress and strain at any crosssection at position x are: To determine the overall change in length, break up the bar into segments of length L_{i} over which the values of force, area and modulus are all constant. Thus: 

Continuously Varying Axial Member
The length over which P(x), A(x) and E(x) are all constant is differential length dx. The stress and strain at any crosssection x are: The total elongation can be determined by integrating over the entire length of the member: 
Displacement Method

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