# Validation of Bredt’s formulas for beams with hollow cross sections by the method of asymptotic splitting for pure torsion and their extension to shear force bending

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## Abstract

The equations governing pure torsion of prismatic beams with thin-walled closed cross sections, known as Bredt’s formulas, are verified using the method of asymptotic splitting. In particular, the strong formulation of the Saint-Venant problem of a straight beam is expanded asymptotically. We begin by validating well-known technical assumptions for the shear stress distribution. Furthermore, the influence of a transverse force acting on the beam is considered. This shear force causes a deformation of the cross section, and therefore an adaption of Bredt’s formulas is needed. Two distinct formulations of the shear center, called the kinematic and the energetic shear center, are obtained. The latter is verified in numerical experiments.

## Keywords

Bredt’s formulas Asymptotic splitting Single-cell cross section Shear center## 1 Introduction

*h*(

*s*) varies along the arc coordinate

*s*. The formulas are derived under the engineering assumption of a constant shear stress over the thickness. As a further assumption, shear stresses orthogonal to \(\varGamma \) are neglected. Equation(2) relates the shear stress \(\tau \) with the deformation. In [1], the same simplifications are used to obtain the position of the shear center by using Bredt’s formulas as an approximation even in the case of combined bending and torsion. We validate the engineering assumptions by an asymptotic analysis of an arbitrary single-cell cross section. Therefore, we consider the Saint-Venant problem of a linear elastic beam. The analytical description of linear elastic beams with arbitrary, also multiply connected cross sections, was done by Eliseev [2, 3]. In the limit of small thicknesses

*h*, represented by a formal small parameter \(\lambda \), the cross section can be characterized by its center line and a small dimension orthogonal to it. The strong form of the equations is developed asymptotically to obtain the principal terms of shear stresses and related quantities. The determination of the principal terms requires the solvability conditions of the minor terms, which is intrinsic to the procedure of asymptotic splitting. This method has also been applied to the actual problem but with constant wall thickness in [3] and to the problem of constrained warping of thin open cross sections as well as plate equations in [4, 5, 6]. Another possibility of an asymptotic analysis is shown in [7], in which also higher terms are derived, but again only constant thickness is considered. Further asymptotic solutions of the problem were found by the authors of [8, 9], who used the weak formulation of the problem.

## 2 Notations and formulation of the problem

*z*, and with the constant cross section placed orthogonal to it in the (

*x*,

*y*)-plane with a position vector \(\mathbf {x}=x\mathbf {e}_x+y\mathbf {e}_y\) (Fig. 1). In case of a thin-walled cross section, we describe this position vector by

*s*, the coordinate

*n*with \( -~\frac{h}{2}~\le ~n~\le ~~\frac{h}{2}\) orthogonal to the center line(with unit vector \(\mathbf {e}_n\)), and \(\lambda \) being a formal small parameter, representing the smallness of the thickness

*h*.

*a*being the mean axial stress, and \(\mathbf {b'}=\partial \mathbf {b}/\partial z\) determines the linear distribution of the axial stress in the cross section. We can write

## 3 The method of asymptotic splitting

*u*in the form of a power series in \(\lambda \), starting with \(\lambda ^{-1}\):

*u*finds the principal order (\(\lambda ^{-1}\)) term to

## 4 Bredt’s formulas of a single-cell cross section

First, the shear stress is determined to validate that it is constant over the thickness, and next the displacements are used to identify this constant.

### 4.1 Validation of the assumptions of the shear stress \(\varvec{\tau }\)

### 4.2 Consideration of the displacements

*z*and one part, which is independent of

*z*. Both must hold on their own. The part dependent of

*z*is

*z*, which reads

*W*from pure torsion; therefore, we can write

### 4.3 Shear center

*x*-coordinate

## 5 Numerical example

*x*-axis (Fig. 2). The first one is a thin hollow square profile with the right wall being \(\beta h\) thick and the other three walls being

*h*thick. The second one has the shape of a hollow C-profile with a constant wall thickness. For the first one, we study the convergence of the asymptotic solution; the second example was chosen to proof the statement that for thin single-cell cross sections with constant thickness the difference of the two shear centers vanishes is wrong [14]. The computation is done by the above analytical formulas as well as numerically. We approach the exact three-dimensional solution by using the method of finite differences. The numerical solution makes no assumption about the thickness of the cross section. Instead of solving the above Laplace equations for the stress functions (15) and (16), the basis of the finite difference scheme are the equations established by Lacarbonara [10] (only the

*x*-component is needed in the two symmetric examples):

*W*:

*W*is the well-known ”ordinary” warping function from the problem of pure torsion. It should be noted that the above equations for \(W_2\) are only valid if the coordinate axes are principal axes of inertia. The advantage of the displacement approach of Lacarbonara compared with equations (15) and (16) is that the boundary condition is a Neumann one, which is known at all boundaries, whereas the above formulation uses the Prandtl stress function, which is constant at the different boundaries, but a priori unknown and has to be calculated with compatibility conditions (see above for \(C^{(0)}\)).

### 5.1 Finite difference scheme

*i*,

*j*) inside the domain, the scheme requires five points and reads

*i*,

*j*) is a convex corner, two points have to be replaced. Concave corners are treated like inner nodes. Because there are only Neumann boundary conditions, the solution is determined up to an additive constant, and the solution for an arbitrary point is set to zero, e.g., the left-low corner \(W_2^{0,0}=0\). Note that in (75) the term

### 5.2 Numerical results

*h*. We choose \(\varDelta x=\varDelta y\) and at least six points per wall thickness of the thinner walls. Then, the convergence is tested by repeated reduction of \(\varDelta x\) and \(\varDelta y\) by the factor 2. As we can see in Fig. 3, for small wall thicknesses (large ratios

*a*/

*h*), the two centers, measured from the left outer edge, remain distinct, and the numerical solutions tend toward the asymptotic expressions. Note that \(x^*\) and \(x^{**}\) are computed from (70) and (72), added by \(\mathbf {x}_A\), measured from the left edge.

*h*exactly at the center of the cross section, a fact which would count for the kinematic shear center, too. With increasing \(\beta \), the shear center at first shifts to the right up to a maximum value and decreases again. The maximum value of the shear center of the curve \(h/a=0.05\) is at a point at which the thickness of the wall is about \(25\%\) of the square’s length.

*c*, height

*H*and the angle \(\gamma \) opposite to

*c*, with the

*x*-axis being the axis of symmetry. Then, the positions of \(\mathbf {x}_{\varGamma }\) and \(\mathbf {x}_A\) in the limit \(h\rightarrow 0\), counted from the basis, are

## 6 Conclusions

In this paper, we studied the asymptotic behavior of the solutions of the Saint-Venant problem for a linear elastic beam with a hollow cross section under the action of torsion and shear force bending. Assuming thin-walled cross sections, we introduced the small parameter \(\lambda \), representing the thinness of the walls. After expressing the strong form of the equations in terms of \(\lambda \) and merging terms of equal order, we first managed to validate the engineering assumptions. Further, we got well-known formulas, like for the shear flow and finally Bredt’s formulas for the torsion of rods. We noticed that if the Poisson effect is not neglected within the cross section, the second formula of Bredt has to be adapted by using the energetic twist rate \(\theta _z'\) instead of the kinematic one. The leading-order terms of the kinematic and energetic shear centers were derived assuming a variable thickness of the cross section. In the final numerical investigation of two hollow cross sections, the exact location of the shear centers was calculated for different thickness values. A good agreement with the obtained asymptotic formulas was observed when the thickness is small. Furthermore, it has been shown that the energetic and kinematic shear centers are in general distinct even in the case of a single-cell hollow cross section with a constant small thickness. The opposite statement, which is sometimes met in the literature [14], is thus proven to be incorrect.

## Notes

### Acknowledgements

Open access funding provided by TU Wien (TUW).

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