Question

The rod is made of A-36 steel and has a diameter of 0.22 in . If the rod is 4 ft long when the springs are compressed 0.7 in . and the temperature of the rod is T= 30 ∘F , determine the force in the rod when its temperature is T= 150 ∘F .

Answer 1

The force in the rod when the temperature is 150 °F is 718.72 pounds-force.

How to determine the resulting the resulting force due to mechanical and thermal deformation

Let suppose that rod experiments a quasi-static deformation and that both springs have a linear behavior, that is, force ($F$), in pounds-force, is directly proportional to deformation. Then, the elongation of the rod due to temperature increase creates a spring deformation additional to that associated with mechanical contact.

Given simmetry considerations, we derive an expression for the spring force ($F$), in pounds-force,  as a sum of mechanical and thermal effects by principle of superposition:

$F = k\cdot (\Delta x + 0.5\cdot \Delta l)$   (1)

Where:

• $k$ - Spring constant, in pounds-force per inch.
• $\Delta x$ - Spring deformation, in inches.
• $\Delta l$ - Rod elongation, in inches.

The rod elongation is described by the following thermal dilatation formula:

$\Delta l = \alpha \cdot L_{o}\cdot (T_{f}-T_{o})$   (2)

Where:

• $\alpha$ - Coefficient of linear expansion, in $\frac{1}{^{\circ}F}$.
• $L_{o}$ - Initial length of the rod, in inches.
• $T_{o}$ - Initial temperature, in degrees Fahrenheit.
• $T_{f}$ - Final temperature, in degrees Fahrenheit.

If we know $k = 1000\,\frac{lb}{in}$, $\Delta x = 0.7\,in$, $\alpha = 6.5\times 10^{-6}\,\frac{1}{^{\circ}F}$, $L_{o} = 48\,in$, $T_{o} = 30\,^{\circ}F$ and $T_{f} = 150\,^{\circ}F$, then the force in the rod at final temperature is:

$F = \left(1000\,\frac{lb}{in} \right)\cdot \left[0.7\,in + 0.5\cdot\left(6.5\times 10^{-6}\,\frac{1}{^{\circ}F} \right)\cdot (48\,in)\cdot (150\,^{\circ}F-30\,^{\circ}F)\right]$

$F = 718.72\,lbf$

The force in the rod when the temperature is 150 °F is 718.72 pounds-force. $\blacksquare$

To learn more on deformations, we kindly invite to check this verified question: brainly.com/question/13774755