The tool SM-13 computes the internal force and stress in bars due to fabrication errors and thermal effects.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.

**Example: Stress in bar by thermal effects**

The input in the tool is

After clicking "Run", the results will be shown as follows in another page.

Change in length, ΔL = α*ΔT*L = 5.0000E-8

Stress in the bar, σ = E*α*ΔT = -85.000E-4

Inernal force on a cross section of the bar, P = E*α*ΔT*A = -85.000E-6

(Positive (+): Tensile; Negative (-): Compressive)

### Stress in Bars by Thermal Effects and Fabrication Errors

Due to fabrication errors and thermal effects, a bar may be subject to changes in length, which in return can cause elastic deformation of the bar.

(a) Too long

(a) Too short

**Fig. 1** Fabrication error

### Fabrication Errors

If the bar is manufactured too long by an amount ΔL, as shown in Fig. 1a, it will cause a compression internal force. On the other hand,
if the bar is too short, a tension internal force will be caused. Both the tension and compression stress is given by

where

E: Young's modulus

A: Area of the cross-section

L: Length of the bar

And the internal force P is in the form of

### Thermal Effects

If a bar of length L subject to a temperature increase ΔT, it will undergo an increase in length

where

α: Coefficient of thermal expansion

If the boundary condition of the bar is fixed-fixed, the internal stress is determined by

For the boundary condition fixed-spring or spring-fixed, the internal stress is

If both boudanries are springs, the internal stress is given by

### References

1. Yang, B., 2005, Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB Toolboxes, Elsevier Science.

2. Young, W.C., 1989, Roark's Formulas for Stress and Strain, 6th edition, McGraw-Hill, New York.