# Bending of Beams

• Print
This chapter is from the book

## 5.15 Comparison of the Results of Various Theories

We now examine the solutions obtained in Sections 5.13 and 5.14 with results determined using the flexure formula for straight beams. To do so, we consider a curved beam of rectangular cross section and unit thickness experiencing pure bending. The tangential stress predicted by the elementary theory (based on a linear distribution of stress) is My/I. The Winkler approach, which leads to a hyperbolic distribution, is given by Eq. (5.70) or Eq. (5.71), while the exact theory results in Eqs. (5.67). In each case, the maximum and minimum values of stress are expressed by

Table 5.4 lists values of m as a function of b/a for the four cases cited [Ref. 5.1], in which b = ro and a = rj (see Figs. 5.24 and 5.25). As can be seen, there is good agreement between the exact and Winkler results. On this basis, as well as from more extensive comparisons, we may conclude that the Winkler approach is adequate for practical applications. Its advantage lies in the relative ease with which it may be applied to any symmetric section.

#### Table 5.4. The Values of m for Typical Ratios of Outer Radius b to Inner Radius a

Curved Beam Formula

Elasticity Theory

b/a

Flexure Formula

r = a

r = b

r = a

r = b

1.3

±66.67

−72.980

61.270

−73.050

61.350

1.5

±24.00

−26.971

20.647

−27.858

21.275

2.0

±6.00

−7.725

4.863

−7.755

4.917

3.0

±1.50

−2.285

1.095

−2.292

1.130

The agreement between the Winkler approach and the exact analyses is not as good in situations involved combined loading as it is for the case of pure bending. As might be expected, for beams of only slight curvature, the simple flexure formula provides good results while requiring only simple computation. The linear and hyperbolic stress distributions are approximately the same for b/a = 1.1. As the curvature of the beam increases, (b/a > 1.3) the stress on the concave side rapidly increases over the one given by the flexure formula.

### 5.15.1 Correction of σθ for Beams with Thin-Walled Cross Sections

Where I-beams, T-beams, or thin-walled tubular curved beams are involved, the approaches developed in this chapter will not accurately predict the stresses in the system. The error in such cases is attributable to the high stresses existing in certain sections such as the flanges, which cause significant beam distortion. A modified Winkler’s equation can be applied in such situations if more accurate results are required [Ref. 5.6]. The distortion, and thus the error in σθ, is reduced when the flange thickness is increased. Given that material yielding is highly localized, its effect is not of concern unless the curved beam is under fatigue loading.