Comparison of Shear Stress obtained from thin-walled beam theory and Timoshenko beam theory

Question

I am studying the book "Structural Analysis with Applications to Aerospace Structures" by Craig. In Chapter 8 titled "Thin-Walled Beams", the authors derived the governing differential equation

$$ \frac{\partial n}{\partial x_1} + \frac{\partial f}{\partial s} = 0 $$

where, $n$ is the axial flow, $f$ is shear flow, $x_1$ is axial coordinate, and $s$ is curvilinear coordinate along cross-section.

Substituting $n = \sigma_{xx} \times t(s)$, and $\sigma_{xx}$ from Euler-Bernoulli Beam theory, we have

$$ f = c + \frac{Q_3(s)H_{23}^c - Q_2(s)H_{33}^c}{H_{22}^c H_{33}^c - H_{23}^c H_{23}^c} V_3 - \frac{Q_3(s)H_{22}^c - Q_2(s)H_{23}^c}{H_{22}^c H_{33}^c - H_{23}^c H_{23}^c} V_2$$

with

$Q_2 = \int E x_3 t ds$ and $Q_3 = \int E x_2 t ds$.

where, $c$ is an arbitrary constant which is zero at open end of cross-sections, $H_{ij}^c$ are centriodal bending stiffnesses, $V_i$ are shear forces.

Another way to determine shear stress in a beam cross-section is to use Timoshenko beam theory.

Question: What are the difference between the shear stresses obtained from the two methods? Do both approach capture the same effect?

Answer 1

From Wikipedia (emphasis mine):

The Timoshenko–Ehrenfest beam theory was developed by Stephen Timoshenko and Paul Ehrenfest early in the 20th century. The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams, or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order but, unlike Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions.

The inclusion of shear deformation and rotational bending effects being the main difference between the Timoshenko beam and Euler–Bernoulli beam. But I don't think there is a difference between the two models in the derivation of the shear stresses.