# Vlasov torsion theory

## Assumptions

• Warping can be constrained
• Derivative of section rotation about the axis of the twist is not constant

## Differential Equation

The rotation of the beam cross-section follows the following differential equation (Hoogenboom 2006):

$\;\;\;EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\; GI_t\;\frac{d^2 \phi}{dx^2}\;=\; m_x$

Refer to torsion page for the notation used in the above equation.

## Stresses

According to Vlasov to the theory, the applied torque causes the following three types of stresses:

1. shear stresses due to unrestrained torsion
2. shear stresses due to restrained warping
3. normal stresses due to restrained warping

These stresses are combined with stresses due to axial, bending and shear loading. For example the total axial stress can be obtained as follows (Hoogenboom 2006):

$\;\;\;\sigma_{xx}\;=\; \frac{N}{A}\;+\; \frac{M_y}{I_y}\;z\;-\; \frac{M_z}{I_z}\;y\;+\; \frac{B}{I_{\omega}}\;\omega$

## References

• V.Z. Vlasov: "Thin-Walled Elastic Bars" (in Russian), 2nd ed., Fizmatgiz, Moscow, 1959.
• Zdenek P. Bazant: Nonuniform torsion of thin-walled bars of variable cross section, Publications, IABSE, 25, p. 245-167, 1965.