Chapter 5
Consideration of Euler–Bernoulli Beams
with Plastic Material Behavior
5.1 Basics of the Layered Approach
The derivations of the previous sections will be extended in the following to elastoplastic material behavior. For simplicity reasons, a simply supported beam under
constant moment loading as shown in Fig. 5.1a is considered to explain an approach
to consider plasticity for beams. The material is assumed to be linear-elastic/idealplastic as shown in Fig. 5.1b [4] and this assumption allows to easily compare the
numerical results with the analytical solutions provided in [3].
Let us use five domain nodes (i = 1, . . . , 5) of equidistant spacing ((X =
L
4
)
for the finite difference approach as shown in Fig. 5.2a.
As in the case of the elasto-plastic finite element approach (see [2, 3, 5]), the
external load is now applied in incremental steps of magnitude M, see Fig. 5.2b. The
indicated time t is not important since we consider only time-independent material
behavior and the expression ‘time’ can simply be replaced by ‘step’ or ‘load step’.
The following derivations do not consider the predictor-corrector scheme as in [2].
A more simplified approach is introduced which is subject to the assumption of
monotonic loading. Thus, the cases of unloading, load reversal and cyclic loading
will be not covered in the framework of the following derivations.
The major challenge in the case of beams is that the stress state is changing over
the height of the beam and a plastic layer is moving inwards with increasing load
in the elasto-plastic range. One possible approach for this problem is the so-called
layered approach
1 [1] where the cross section at the position of node i is subdivided
in a certain numbers of layers (see the grey cross section in Fig. 5.3).
Let us have now a closer look at such a cross section as sown in Fig. 5.4.
The number of the layers ranges between 1 ≤ k ≤ k max = 5 and it is for simplicity
assumed that each layer k has the same height h. Furthermore, it is assumed that k
is an odd number which implies that the center (neural axis) of the beam is located in
the middle of layer
k max +1
2
. The coordinates of the top and bottom face of each layer,
i.e. Z
k and Z
k−1 , can be expressed as
1 The layered approach can be applied in a similar manner in the framework of the finite element
method, see Ref. [5].
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2_5
103
Consideration of Euler–Bernoulli Beams
with Plastic Material Behavior
5.1 Basics of the Layered Approach
The derivations of the previous sections will be extended in the following to elastoplastic material behavior. For simplicity reasons, a simply supported beam under
constant moment loading as shown in Fig. 5.1a is considered to explain an approach
to consider plasticity for beams. The material is assumed to be linear-elastic/idealplastic as shown in Fig. 5.1b [4] and this assumption allows to easily compare the
numerical results with the analytical solutions provided in [3].
Let us use five domain nodes (i = 1, . . . , 5) of equidistant spacing ((X =
L
4
)
for the finite difference approach as shown in Fig. 5.2a.
As in the case of the elasto-plastic finite element approach (see [2, 3, 5]), the
external load is now applied in incremental steps of magnitude M, see Fig. 5.2b. The
indicated time t is not important since we consider only time-independent material
behavior and the expression ‘time’ can simply be replaced by ‘step’ or ‘load step’.
The following derivations do not consider the predictor-corrector scheme as in [2].
A more simplified approach is introduced which is subject to the assumption of
monotonic loading. Thus, the cases of unloading, load reversal and cyclic loading
will be not covered in the framework of the following derivations.
The major challenge in the case of beams is that the stress state is changing over
the height of the beam and a plastic layer is moving inwards with increasing load
in the elasto-plastic range. One possible approach for this problem is the so-called
layered approach
1 [1] where the cross section at the position of node i is subdivided
in a certain numbers of layers (see the grey cross section in Fig. 5.3).
Let us have now a closer look at such a cross section as sown in Fig. 5.4.
The number of the layers ranges between 1 ≤ k ≤ k max = 5 and it is for simplicity
assumed that each layer k has the same height h. Furthermore, it is assumed that k
is an odd number which implies that the center (neural axis) of the beam is located in
the middle of layer
k max +1
2
. The coordinates of the top and bottom face of each layer,
i.e. Z
k and Z
k−1 , can be expressed as
1 The layered approach can be applied in a similar manner in the framework of the finite element
method, see Ref. [5].
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. Öchsner, Structural Mechanics with a Pen,
https://doi.org/10.1007/978-3-030-65892-2_5
103