BEAM THEORY: TORSION

Engineering Fundamentals

Precision Point - Beam Theory Torsion

INTRODUCTION

When a straight beam is subjected to an axial moment, each cross section twists around its torsional center. Shear stresses occur within the cross sectional planes of the beam.

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Angular twist

For a torsionally loaded beam, the angular twist is described by:

\varphi =\frac{T{\cdot}l}{G{\cdot}J_T}

Precision Point - Beam Theory Torsion

G is the shear modulus. The relation between the shear modulus G and the elastic modulus E is defined by the following formula:

G=\frac E{2\left(1+v\right)}{\approx}0.38E    (For most metals)

Rotational stiffness

The rotational stiffness of a torsionally loaded beam is:

K_z=\frac T{\varphi }=\frac{G{\cdot}J_T} l

Maximum torque load

For a torsionally loaded beam, the maximum torque load can be calculated with:

T_{\mathit{max}}=\frac{J_T} r\tau _{\mathit{max}}

Precision Point - Beam Theory Torsion

J_T is the torsion constant. It is equal to the polar moment of inertia I_z if the cross section is circular.

For non-circular cross sections warping occurs which reduces the effective torsion constant. For these shapes, approximate solutions of the torsion constant are given in the table below.

Torsion constant

Cross section

Precision Point - Beam Theory Torsion

Torsion constant J_T

J_T=I_z=\frac{\pi } 2r^4

Cross section

Precision Point - Beam Theory Torsion

Torsion constant J_T

J_T=I_z=\frac{\pi } 2\left({r_o}^4-{r_i}^4\right)

Cross section

Precision Point - Beam Theory Torsion

Torsion constant J_T

J_T{\approx}\frac 9{64}w^4

Cross section

Precision Point - Beam Theory Torsion

Torsion constant J_T

J_T{\approx}t\left(w-t\right)^3

Cross section

Precision Point - Beam Theory Torsion

Torsion constant J_T

With h>w

J_T{\approx}hw^3\left(\frac 1 3-0.21\frac w h\left(1-\frac{w^4}{12h^4}\right)\right)

Cross section

Precision Point - Beam Theory Torsion

Torsion constant J_T

J_T{\approx}\frac{2t\left(h-t\right)^2\left(w-t\right)^2}{h+w-2t}

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