Chapter 16
Solution of the Simplest Problems
for the Strain Theory of Plasticity
16.1 Pure Bending of a Straight Beam
Let a straight beam be subject to bending by two pairs of M as shown in Fig. 16.1a.
For simplicity, we will assume that the beam cross-section has two symmetry axes
(Fig. 16.1b), and there is an elongation diagram for the beam material (Fig. 16.1c).
Let us designate the main central axes of the beam cross-section as x and y and
align the axis z with the lateral axis of the beam so that zy is the bending plane. Let
us designate the cross-section height and width as 2h and b, respectively, and the
curvature of the deformed beam axis as χ .
The problem is solved in stresses. We have equilibrium equations, strain conformity equations, and boundary conditions. By assuming the hypotheses of plane
cross-sections [3], we have
ε z = ε 1 = χy.
(16.1)
The condition of material non-compressibility gives
ε x + ε y + ε z = 0.
(16.2)
For the entire cross-section, let us assume the Poisson ratio as 0.5. In the
following, we will frequently use this simplification since its effect on the results
is insignificant [2]. Taking into account assumptions for strains in the transverse
direction, we have
ε x = ε y = −
1
2
χy.
(16.3)
Since strains are linear functions y, conformity equations are identically satisfied.
We also note that formulas (16.1)–(16.3) represent a law of uniaxial strain. So only
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_16
219
Solution of the Simplest Problems
for the Strain Theory of Plasticity
16.1 Pure Bending of a Straight Beam
Let a straight beam be subject to bending by two pairs of M as shown in Fig. 16.1a.
For simplicity, we will assume that the beam cross-section has two symmetry axes
(Fig. 16.1b), and there is an elongation diagram for the beam material (Fig. 16.1c).
Let us designate the main central axes of the beam cross-section as x and y and
align the axis z with the lateral axis of the beam so that zy is the bending plane. Let
us designate the cross-section height and width as 2h and b, respectively, and the
curvature of the deformed beam axis as χ .
The problem is solved in stresses. We have equilibrium equations, strain conformity equations, and boundary conditions. By assuming the hypotheses of plane
cross-sections [3], we have
ε z = ε 1 = χy.
(16.1)
The condition of material non-compressibility gives
ε x + ε y + ε z = 0.
(16.2)
For the entire cross-section, let us assume the Poisson ratio as 0.5. In the
following, we will frequently use this simplification since its effect on the results
is insignificant [2]. Taking into account assumptions for strains in the transverse
direction, we have
ε x = ε y = −
1
2
χy.
(16.3)
Since strains are linear functions y, conformity equations are identically satisfied.
We also note that formulas (16.1)–(16.3) represent a law of uniaxial strain. So only
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_16
219