Beam Tension, Compression and Torsion

The total extension or contraction of a uniform bar in pure tension or compression is given by:

$$\delta =\frac{Fl}{AE}$$

If we have $\delta =y$ and use the equation $k=\frac{F}{y}$, the spring constant of an axially loaded bar is:

$$k=\frac{AE}{l}$$

where:

• $k$ is the spring constant
• $A$ is cross-sectional area
• $\delta$ and $y$ are deformation
• $E$ is modulus of elasticity
• $l$ is the initial length of the bar
• $F$ is axial load of tension or compression

The angular deflection of a uniform solid or hollow round bar subjected to a twisting moment $T$ is

$$\theta =\frac{Tl}{GJ}=\frac{T}{k}$$

This can be re-arranged to get:

$$k=\frac{T}{\theta }=\frac{GJ}{l}$$

We can also write $J=\pi d^4/32$ for a round bar and convert to radians to get:

$$\theta =\frac{583.6Tl}{G{d}^4}$$