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

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?

## 1 Answer

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.

• Dear, thank you for you reply. The question is not about the difference between Euler-Bernoulli and Timoshenko, rather thin-walled beams (as described in the book) and Timoshenko beam theories. Jun 29, 2021 at 3:12
• You can read Mechanics of Materials (Timoshenko & Gere) Ch 9.6 - Shear Stress in Beams of Thin-Walled Open Cross Section. The shear stress tau = VQ/It, and shear flow f = tau*t = VQ/I. Is this you wanted to compare?
– r13
Jun 29, 2021 at 3:30
• Shear flow can be divided by wall thickness to obtain shear stresses. The shear flows are obtained using theory of thin-walled beams (Please correct me if I am wrong). Another way to obtain shear stresses in a beam is to use Timoshenko beam theory. What are the differences between these two shear stresses? What are the similarities and differences between these two? Jun 29, 2021 at 3:33
• Note that the shear formula above was derived from beam theory, For thin-walled profiles, the shear distribution through thickness can be ignored, thus the "t" is dropped out from the shear formula. The textbook I mentioned is available in many libraries, if you are in the US, that you can borrow and read.
– r13
Jun 29, 2021 at 11:52
• @AliBaig the difference is how the two approximations handle the boundary conditions on the stress distribution in the complete structure. (Note, both approximations make assumptions that are not necessarily correct!) Jun 29, 2021 at 14:28