1. Introduction

The study of elastic-plastic bending has a wide field of application. Engineering and mathematical theories of plastic bending are well studied in T. X. Yu at al. book[1]. The authors include many experimental verifications as well as theoretical results. Elastic-plastic analysis of beam-like structures of different cross-sections under bending is considered in[2-10]. The authors employ different models of non-linear stress-strain idealization. The most used models are linear and power hardening. M. Daunys performed theoretical analysis of rectangular cross-section linear hardening beam subjected to static and cyclic pure bending[11]. The author analyses the drift of the stress neutral axis from the centroidal axis of beam, if parameters of plastic tension and compression (proportional limits, modulus of hardening) are not equal. Circular cross-section linear hardening element under pure bending is studied in[12, 13]. Derived equations are fitted to analysis of cyclic pure bending.

Power hardening stress-strain curve model is more preferable in analytical researches of strength of structures than linear hardening one because of better accuracy. Therefore, using of such a kind of idealization in pure bending analysis has not only theoretical but practical importance and applications as well.

2. Pure Bending of Rectangular Cross-Section Element

It is assumed that in case of pure bending a transverse plane section perpendicular to the centroidal axis of the beam before deformation remains plane after deformation and that a cross-section is perpendicular to the centroidal axis after bending. It is also assumed that normal stresses and strains in the transverse and lateral directions are neglected and every fiber of the beam is either under uniaxial tension or under uniaxial compression. Therefore stress and strain distribution in the longitudinal cross-section of an element fit the schemes presented in Figs 1 and 2.

Stress and strain of longitudinal fibers are expressed in dimensionless units and , where and are the values corresponding to proportional limit. Monotonic stress-strain characteristics (tension or compression) with lower rate of proportional limit are marked by index 1. Therefore, and . Modulus of elasticity is the same for both cases. According to the law of equilibrium, we obtain

(1)

and

(2)

Figure 1. Stress distribution in the longitudinal cross-section of an element

Figure 2. Strain distribution in the longitudinal cross-section of an element

According to Fig. 2, . Hence, dimensional expression of distant is and . Substituting and in Eq. (1) we get

(3)

It is evident that and so Eq. (3) can be simplified to

(4)

In the plastically deformed fibers on the convex side of the beam, as in case of the monotonic tension, stress-strain curve with proportional limit is used, so

(5)

In the fibers on the concave side of the beam with proportional limit we get

(6)

In Eq. (5) and (6) and are strain hardening exponents. After dividing Eqs. (5) and (6) by and rearranging, we obtain dimensionless stress expressions

(7)

and

(8)

In Eq. (8), is a coefficient accounting for difference of proportional limits

(9)

If and , the position of the neutral axis coincides with the cross-section centroidal axis. If and (or) , the position of the neutral axis deviates from centroidal axis of an element by value . We mark the strain of fiber of centroidal axis as and transform Eqs. (7) and (8) to

(10)

and

(11)

It is evident, that if , the strains on the convex and concave sides of an element are not equal (). According to Fig. 1, and , where is the dimensionless angle of cross-sections deflection. In regard to this, final expressions of plastic stress are

(12)

and

(13)

In the fibers deformed elastically

(14)

Hence, integral (4) can be written as a sum of three integrals representing three different regions of strains

(15)

In case of pure bending, if (the fibers on the convex side of an element exceeded and the fibers on the concave side of an element are still under elastic conditions), Eq. (14) can by simplified

(16)

Dimensionless distant from extreme fiber with to the neutral axis is

(17)

By referring to integrals (15), (16) and Eq. (17), distant is found for the angle. Subsequently the strains and can be obtained.

The integral (2) can be transformed considering that , and

(18)

or

(19)

The pure bending moment corresponding to the proportional limit stress is

(20)

So, the integral (19) can be written in dimensionless form

(21)

Substituting (Eqs. (12)-(14)) in (21) we get:

(22)

In case, if

(23)

Integrals (21) and (22) allow to determine moment depending on maximum strain. Analytical expressions of integrals (15), (16), (22) and (23) exist, but are not presented.

Figs. 3 and 4 represent curves of functions and in the strain range for the fallowing values of parameters and : ; ; ; ; ; .

3. Pure Bending of Circular Cross-Section Element

Figure 3. The distance from extreme fiber with to the neutral axis of rectangular cross-section element vs. monotonic tension strain, for (a) and (b)

In case of elastic plastic pure bending of circular cross-section beam, the stress and strain distribution in the longitudinal cross-section also fit the schemes presented in Figs. 1 and 2. But in the transversal cross-section some differences are seen (see Fig. 5).

Figure 4. Dimensionless bending moment of rectangular cross-section element vs. monotonic tension strain, for (a) and (b)

Figure 5. The regions of elastic and plastic strains in the transversal cross-section of a circular element

The width of circular cross-section element depends on distance. Considering, that and we get

(24)

It is evident, that , so integral (1) can be expressed as

(25)

and simplified to

(26)

Using defined by Eqs. (12)-(14), allows to transform integral (26)

(27)

And in case, if

(28)

Next we transform the integral (2) to

(29)

and

(30)

In case of pure bending of circular cross-section element, the bending moment corresponding to the proportional limit is

(31)

Hence, dimensionless bending moment

(32)

The final expression of is

(33)

In case, if

(34)

Analytical expressions of integrals (27), (28), (33) and (34) do not exist. These integrals can be solved numerically.

Theoretical and curves for the same values of and are presented in Figs. 6 and 7.

The developed analytical solution of monotonic elastic plastic pure bending can be used to analyse the cyclic elastic plastic pure bending as it is presented by formulas[11, 12].

Figure 6. The distance from extreme fiber with to the neutral axis of circular cross-section element vs. monotonic tension strain, for (a) and (b)

Figure 7. Dimensionless bending moment of circular cross-section element vs. monotonic tension strain, for (a) and (b)

4. Conclusions

Elastic-power hardening idealization of uniaxial stress strain curve is used in theoretical research of rectangular and circular cross-section element under elastic plastic pure bending. Presented derivations allow to calculate deviation of stress neutral axis from centroidal axis of an element and pure bending moment versus monotonic strain if parameters of plastic tension and compression (proportional limits, exponents of hardening) are not equal. Derived expressions of bending moment can be used in theoretical research of cyclic elastic plastic pure bending.