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Bending under conditions of plane stress.

Abstract: The general solution of the problem under conditions of plane stress in plastic region was given in the papers (Hasanbegovic, 1997, 1998). These solutions are limited to polar coordinates and non-hardening material. In the paper (Hasanbegovic, 2000) are found same individual solutions of some problems with state plane stresses in the plastic region on the basis the general solution.

In this paper, consider of plane stress in which plastic and elastic regions exist side by side. On the basis the resultants of three papers earlier mentioned, the principal normal stresses in the plastic and elastic region is calculated.

Key words: Bending, non-hardening material, plane stress, polar coordinates and principal normal stresses.


Consider the bending of a rectangular beam that the bending moments are loaded. It is suppose that material of a beam is non-hardening and the conditions of plane stress are satisfied. In the process of the bending, the sections of the piece of work with the longitudinal planes are defined the configuration which is identical for all parallel longitudinal planes. The sections of the longitudinal planes with the transversal planes determine the direction of the momentary center radii of curvatures. Consequently, the principal directions of normal stresses turn and translator move.

A spatial rectangular and a material polar coordinates have the origins in the point the momentary center radius of bending. The material coordinates are bound for the body and deform together with him. In this way is enabled the application of the original analytical solutions given in the papers (Hasanbegovic, 1997, 1998, 2000).


Under conditions of plane stress, component stresses [[sigma].sub.x], [[sigma].sub.y] i [tau]([tau] = [[tau].sub.xy] = [[tau].sub.yx]) can be expressed through the middle normal stress [sigma], the maximum tangential stress [[tau].sub.max] and the angle of inclination [phi] of the first principal normal stress [[sigma].sub.1] to respect axes x by

[[sigma].sub.x] = [sigma] + [[sigma].sub.1] - [[sigma].sub.2] / 2 cos 2[phi]

[[sigma].sub.y] = [sigma] - [[sigma].sub.1] - [[sigma].sub.2] / 2 cos 2[phi]}

[tau] = [[sigma].sub.1] - [[sigma].sub.2] / 2 sin 2[phi] (1)

According to Tresca's yield criterion, the condition of beginning plastic flow can be expressed as (Sokolovskij, 1969)

[[sigma].sub.1] - [[sigma].sub.2] = [+ or -] 4 [k.sub.s] [- or +] 2 [sigma] (2)

for plane stress with the principal normal stresses equal sign. Substituting (2) into (1) given

[[sigma].sub.x] = [sigma] + (2[k.sub.s] [- or +] [sigma]) cos 2[phi], [[sigma].sub.y] = [sigma] - (2[k.sub.s] [- or +] [sigma]) cos 2[phi],

[tau] = (2[k.sub.s] [- or +] [sigma]) sin 2[phi]. (3)

This is system of equations, which satisfied the condition of plastic flow.


The polar system coordinates (O; r, [phi]) is putted that coincide to the principal directions of normal stresses [[sigma].sub.1], [[sigma].sub.2]. The connection between fundamental x, y and polar coordinates r, [phi] is given by

x = r cos [phi], y = r sin [phi]. (4)

The differential equations of equilibrium of an element which is situated under conditions plane stress, in the coordinates x, y can be presented by

[partial derivative][[sigma].sub.x] / [partial derivative]x + [partial derivative][tau] / [partial derivative]y = 0, [partial derivative][tau] / [partial derivative]x + [partial derivative][[sigma].sub.y] / [partial derivative]y = 0. (5)

In the ideal conditions, without hardening, in polar system of the coordinates r, [phi] the equations (5) transform to equation (Hasanbegovic, 1997)

r [partial derivative][sigma] / [partial derivative]r - sin[phi] - cos[phi] / sin[phi] + cos[phi] [partial derivative][sigma] / [partial derivative]x = -2[k.sub.s] ([[sigma].sub.1] [[sigma].sub.2] [less than or equal to] 0). (6)

And equation (Hasanbegovic, 1998)

(1 [- or +] 1) r [partial derivative][omega] / [partial derivative]r + (1 [+ or -]) cos[phi] - sin[phi] / cos[phi] + sin[phi] [partial derivative][sigma] / [partial derivative][phi] = -2 ([+ or -][k.sub.s] [- or +] [sigma]) ([[sigma].sub.1] [[sigma].sub.2] [greater than or equal to] 0). (7)

The equations (6) and (7) are linear, non-homogenous and first order.


If [[sigma].sub.[phi]] > [[sigma].sub.r], the first independent integrals in (6) are given by (Hasanbegovic, 1997)

ln r+ ln(sin [phi]- cos [phi]) = [C.sub.1], - [sigma] - 2[k.sub.s] ln(sin [phi]-cos [phi]) = [C.sub.2]. (8)

Under condition [[sigma].sub.r] > [[sigma].sub.[phi]] in (7), the first independent integrals are (Hasanbegovic, 2000)

[phi] = [C.sub.3], [sigma] = [C.sub.4] / r - 2[k.sub.s] (9)


Consider the beam the rectangular cross section b, s the length L loaded to the bending moment (Figure 1). The deformation is such transverse planes continue to remain (Hill, 1950). The planes of the cross sections with the longitudinal planes cut on the direction, which the distances from the piece of work alter. The sections, the piece of work, with the longitudinal planes define configuration, in the process of bending, which is identical for all parallel longitudinal planes.

The sections of the longitudinal planes with the transverse planes define the centers the radii of bending. The directions of principal normal stresses, in the region of deformation, are circles and the radii of theirs. Consequently, the principal directions of normal stresses turn and translator move.



Consider one's self as that the conditions plan stress are satisfied if the breadth of beam b is smaller of hers the dickens s. The fibers above the neutral plane are extended and these below compressed.

In the transversal planes, fibers are compressed in the both regions. The circumferential stress [[sigma].sub.[phi]] always is higher of the radial stress in the absolute value.


In this zone the principal normal stress are equal sign [[sigma].sub.[phi]] < [[sigma].sub.r] < 0. The condition of plastic flow can be expressed by

- [[sigma].sub.[phi]] = 2 [k.sub.s]. (10)

The worth of the middle normal stress [sigma] on the slip lines [phi] = const., on the base of general solution, is obtained by

[sigma] = c / r - 2[k.sub.s]. (11)

On the boundary for r=[r.sub.1] the radial stress [[sigma].sub.r] = 0, the middle normal stress [sigma] = - [k.sub.s] and the value C = [k.sub.s] [r.sub.1] from (9). The principal normal stresses are determined by

[[sigma].sub.[phi]] = -2[k.sub.s], [[sigma].sub.r] = -2[k.sub.s](1 - [r.sub.1] / r), (12)

On the base the precede analysis the principal normal stresses [[sigma].sub.r] and [[sigma].sub.[phi]] can be calculated and in the elastic zone. It is supposed that relation between the first and the second principal normal stress, from the limited plastic zone, is valid and in the elastic zone. In this way, the valid of the normal stresses is given by (Figure 2)

[[sigma].sub.[phi]] = -2[k.sub.s] [[rho].sub.n] - r / [[rho].sub.n] - [r'.sub.1], [[sigma].sub.r] = - 2[k.sub.s] [[rho].sub.n] - r / [[rho].sub.n] - [r'.sub.1] (1 - [r.sub.1] / [r'.sub.1]). (13)



In the zone of tension the principal radial stress [[sigma].sub.r] is compressivestress, and the circumferential stress [[sigma].sub.[phi]] is tensile stress, the middle normal [sigma] is positive. Toward (Hasanbegovic, 1997) the middle normal stress is

[sigma] = [k.sub.s](1 - 2ln [R.sub.1] / r). (14)

The values of principal normal stresses are obtained as

[[sigma].sub.[phi]] = 2[k.sub.s](1 - ln [R.sub.1] / r), [[sigma].sub.r] = - 2[k.sub.s]ln [R.sub.1] / r. (15)

Where is using the condition of plastic flow and the definition of the middle normal stress

[[sigma].sub.[phi]] - [[sigma].sub.r] = 2 [k.sub.s], [[sigma].sub.[phi]] + [[sigma].sub.r] = 2 [sigma], (16)

The value (15) of the principal normal stresses are valid in plastic zone under the ideal conditions the plastic flow.

The value of circumferential stress, for elastic zone [[rho].sub.n] [less than or equal to] r [less than or equal to] [R'.sub.1](Figure 2), analogues as in the preliminary case, is

[[sigma].sub.[phi]] = 2[k.sub.s][R'.sub.1] - [[rho].sub.n] / [R.sub.1] - [[rho].sub.n]. (17)

The value of the principal normal stress in the radial direction is



In the process of bending the rectangular beam with the moments, the directions of principal normal stresses are the circles and theirs radii. With the progress of process the principal normal stresses turn and translator move. The polar system of coordinates is bound for the body and deforms together with him, and the origin of him is in the centre of curvature. For the spatial system of coordinates is taken a rectangular Cartesian coordinate system.

Of such a kind of the coordinate systems enable the application the general solution of the plane problem which was given in the paper (Hasanbegovic, 1997, 1998). In this paper are given the particular solutions for the principal normal stresses in the plastic and elastic zone.


Sokolovskij, V.V. (1969). Teorija plastienosti, Vissaja skola, Moskva.

Hasanbegovic,S.(1997). Opce rjesenje ravninskog problema u polarnim koordinatama za plastieno podrueje, Masinstno 1(1), Zenica, (23-27).

Hasanbegovic, S. (1998). Opce rjesenje problema ravninskog naprezanja s istim predznacima glavnih normalnih naprezanja u polarnim koordinatama za plasticno podrucje, IV Medunarodni naucno-strucni skup "Tendencije u razvoju masinskih konstrukcija i tehnologija" 1(1), Zenica, (100-107).

Hasanbegovic, S. (2000). Solutions of some problems with State plane stress, 3 rd International Congress of Croatian Society of Mechanics, Cavtat,2000, (209-216).

Hill, R. (1950). The Mathematical Theory of Plasticity, At the Clarendon Press, Oxford.
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Author:Hasanbegovic, S.
Publication:Annals of DAAAM & Proceedings
Article Type:Technical report
Geographic Code:4EUAU
Date:Jan 1, 2005
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