1. Chapter 4
Axial Load

2. 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

3. 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.

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

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

6. 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.

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

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

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

10. 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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18. 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

19. * 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.

20. 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

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

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

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33. 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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40. 4.7 Stress Concentration Stress Concentration
Stress concentration occurs at
(i) location where cross section changes, or
(ii) location where load is applied.

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

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

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