Thin-Walled Tube Torsion: Bredt Formula

Understanding how structures twist under load is critical for designing everything from aircraft wings to bicycle frames. When dealing with hollow, thin-walled sections—ubiquitous in lightweight engineering—the classical formulas for solid shafts fall short. The Bredt formula, also known as the Bredt-Batho formula, provides an elegant and powerful method to analyze shear stress and angle of twist in these complex closed profiles, enabling safe and efficient design by focusing on the concept of shear flow.

The Limitation of Solid Shaft Formulas and the Shear Flow Concept

For a solid circular shaft, the torsional shear stress is maximum at the outer radius and zero at the center. This stress distribution is inefficient because the material near the center contributes little to resisting the torque. Thin-walled tubes solve this by placing material far from the center, where it is most effective. However, their analysis requires a different approach. The key is shear flow ($q$), defined as the shear force per unit length along the cross-section. In a thin-walled member under torsion, this shear flow is constant around the entire closed loop of the section, regardless of changes in wall thickness. This constant flow is analogous to the constant flow of water in a closed-loop pipe; the flow rate is the same at every point, but the velocity (like shear stress) changes if the pipe's diameter (like wall thickness) changes.

Deriving the Bredt-Batho Formula for Shear Stress

The primary result of this analysis is the Bredt-Batho formula for shear flow. It is derived by considering equilibrium and the behavior of shear stress in a thin wall. Consider a closed, thin-walled tube of arbitrary shape subjected to a pure torque $T$. By making a small imaginary cut in the tube wall, we can analyze the internal shear forces. Summing moments about a longitudinal axis leads to a remarkably simple relationship. The applied torque is equilibrated by the moment of the internal shear flow about the center.

The formula is: $$q=\frac{T}{2A_m}$$ Here, $q$ is the constant shear flow (in N/m or lb/in), $T$ is the applied torque, and $A_m$ is the area enclosed by the median line of the thin-walled section. The median line is the centerline of the wall thickness. This formula tells you that the shear flow is directly proportional to the torque and inversely proportional to twice the enclosed area. A larger enclosed area, for the same torque, results in a lower shear flow.

Shear flow is not the final answer for stress. To find the actual shear stress ($\tau$) at any point in the wall, you divide the constant shear flow by the local wall thickness ($t$): $$\tau =\frac{q}{t}=\frac{T}{2A_{m}t}$$ This reveals a crucial design insight: the shear stress becomes dangerously high where the wall is thinnest. In a tube with a constant wall thickness, the shear stress is uniform. In a complex section like an aircraft wing box, you must meticulously check the stress at locations of minimum thickness.

Calculating the Angle of Twist

Knowing the stress is only half the story; engineers must also limit deformation. The angle of twist per unit length ($\theta$) for a closed thin-walled section is given by another form of Bredt's formula, derived using energy methods or strain integration. The formula is: $$\theta =\frac{T}{4A_m^{2}G}\oint \frac{ds}{t}$$ Where $G$ is the material's shear modulus and the integral $\oint \frac{ds}{t}$ is taken around the entire closed perimeter of the median line. This integral sums up the contribution of each segment of the wall, weighted by the reciprocal of its thickness.

For a section with $n$ segments of constant thickness, the integral simplifies to a sum: $$\oint \frac{ds}{t}=\frac{s_1}{t_1}+\frac{s_2}{t_2}+...+\frac{s_n}{t_n}$$ Where $s_i$ is the length of a segment with constant thickness $t_i$. This calculation is straightforward but requires careful bookkeeping to ensure the entire closed perimeter is accounted for.

Application to Complex Sections and Multi-Cell Structures

The true power of the Bredt-Batho approach is its applicability to non-circular, complex cross-sections common in aircraft structures (like fuselage frames and wing spars) and automotive components (like space frames and subframes). As long as the section is closed and thin-walled, the formula holds. You simply determine the area enclosed by the median line, which for a complex shape might require breaking it into simpler geometric components (triangles, rectangles, etc.).

For advanced structures like multi-cell tubes (e.g., an aircraft wing with front and rear spars creating two or more closed compartments), the analysis extends logically. The constant shear flow concept still applies, but now each cell has its own constant shear flow. Equilibrium and compatibility conditions (requiring that the angle of twist be the same for all connected cells) are used to set up a system of equations that can be solved for the shear flow in each cell. This method efficiently handles geometries that would be intractable with elasticity theory.

Common Pitfalls

1. Using the Wrong Area ($A_m$): The most frequent error is using the outer area or inner area instead of the area enclosed by the median line. For a curved wall, the median line is the true center. For a section made of straight segments, you connect the midpoints of each wall. Always sketch the median line and calculate the area it encloses.
2. Applying it to Open Sections: The Bredt formulas are strictly for closed, thin-walled sections. An open section (like a C-channel or I-beam) has a much lower torsional stiffness and a completely different shear stress distribution. Applying the Bredt formula here will give wildly incorrect and non-conservative results.
3. Ignoring Variable Thickness in Twist Calculations: When calculating the angle of twist, it is essential to perform the contour integral $\oint \frac{ds}{t}$. Simply using an "average" thickness in the denominator will lead to an inaccurate result. You must account for each segment's specific thickness and length.
4. Forgetting Stress Concentrations: The formula gives the shear stress in the wall away from any discontinuities. At joints, corners, or attachments, stress concentrations will occur, and the nominal stress from $\tau =q/t$ must be multiplied by a stress concentration factor for a complete safety assessment.

Summary

• The Bredt-Batho formula ($q=T/(2A_m)$) provides the constant shear flow in any closed, thin-walled section subjected to torsion, where $A_m$ is the area enclosed by the wall's median line.
• The shear stress at any point is found by dividing the shear flow by the local wall thickness: $\tau =q/t$. This highlights that stress is highest where the wall is thinnest.
• The angle of twist is calculated using $\theta =(T/(4A_m^{2}G))\oint (ds/t)$, requiring integration around the entire perimeter of the section.
• This method is indispensable for analyzing efficient, lightweight structures like those in aerospace and automotive engineering, efficiently handling complex, non-circular cross-sections.
• Critical pitfalls to avoid include using the wrong enclosed area, misapplying the formula to open sections, and neglecting the proper integral for twist calculations in variable-thickness walls.