Chapter 4 Axial Load

4.1 Saint-Venant’s Principle Localized Deformation * Deformation is localized. * Far from the load, the stress is uniform,  = P/A. * Based on experiments, a general rule to determine the distance: D = largest dimension of the cross section

Saint-Venant’s Principle If load is replaced by a statically equivalent load, only localized region is affected. At a distance larger than D, the stress and strain distributions remain the same.

For two different loadings, the stresses at section c-c are the same.

A rubber membrane is stretched. The localized distortion is only at the grips.

4.2 Elastic Deformation of an Axially Loaded Member General Formulus Consider general loadings and changing cross-section At location x, cut a small segment dx. The internal force is P(x), and the extension is d.

Total extension: * Procedure: External forces  internal force  stress  strain  deformation * Compressive internal force P(x) < 0, , ,  < 0.

Constant Load and Cross-Section Area Piecewise Constant Load and Cross-Section Area

Sign Convention *  > 0: tensile stress;  < 0: compressive stress *  > 0: elongation;  < 0: contraction

Divide the bar into three segments:AB, BC, CD Cut at middle of each segment, draw FBD, and find internal force. Internal force diagram

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4.3 Principle of Superposition Several external loads: P1, P2, ……, Pn. Apply each load individually; P1  1, 1, d1 P2  2, 2, d2 …………… Pn  n, n, dn The total solution if all loads are applied: stress:  = 1 + 2 + …… + n strain:  =  1 + 2 + …… + n deformation: d = d1 + d2 + …… + dn

* Superposition principle is applicable only for cases in which the deformation is small and Hooke’s law can be applied. * For large deformation, superposition principle is not applicable.

4.4 Statically Indeterminate Axially Loaded Member Concept of Statically Indeterminacy * Equilibrium equation: FA +FB = P * 1 equation < 2 unknowns FA and FB cannot be solved. * Statically indeterminacy: # of equations < # of unknowns * Statically determinacy: # of equations = # of unknowns

Compatibility (Kinematic) Condition * Upper segment AC: internal force FA, deformation AC * Lower segment CB: internal force FB, deformation CB * Compatibility condition:

FA and FB are then solved as * Compatibility conditions: from load-deformation relationship * Solve indeterminate problem: equilibrium equations compatibility conditions

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4.5 The Force Method of Analysis for Axially Loaded Member (skip) 4.6 Thermal Stress * Materials expand when temperature increases. Materials contract when temperature decreases. * For a homogeneous and isotropic material, T = LT T - change in length, L – original length T – temperature change  - linear coefficient of thermal expansion unit: SI system 1/C°, FPS or IPS: 1/F°

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4.7 Stress Concentration Stress Concentration Stress concentration occurs at (i) location where cross section changes, or (ii) location where load is applied.

Reduction of Stress Concentration * Avoid abrupt and/or large-magnitude changes in cross section. * Avoid sharp corners using round and fillet.

Stress Concentration Factor * max is used for design. * K depends on geometry.

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