106
5 Consideration of Euler–Bernoulli Beams with Plastic Material Behavior
(E I Y ) i =
k max
k = 1
E
k
i
E
E
1
12
bh
3
I Y
1
(k max )
3
+
12
k max
Z
k
c
h
2
i
(5.5)
=E I Y
k max
k = 1
E
k
i
E
1
(k max )
3
+
12
k max
Z
k
c
h
2
i
,
(5.6)
or in dimensionless form and under consideration of Eq. (5.3) as:
(E I Y ) i
E I Y
=
k max
k = 1
E
k
i
E
1
k 3
max
+
3
k max
2k − 1
k max
− 1
2
i
= α i ,
(5.7)
where it should be noted that E I Y is the bending stiffness of the entire cross section
or beam in the pure elastic range and E
k
i is the modulus of the kth layer, i.e. E
k
i = E
in the elastic range and E
k
i = 0 in the plastic range. Thus, the dimensionless factor
is in the pure elastic range α = 1 and in the elastic-plastic range 0 ≤ α < 1.
To decide if a layer is now considered as elastic (E
k
i = E) or plastic (E
k
i = 0),
the following assumption is done: If the center of a layer—which is geometrically
represented by the coordinate Z
k
c —is in the elastic range, the entire layer k is assumed
to be elastic. Vice versa, if the center of a layer is in the plastic range, the entire layer
k is assumed to be plastic. To develop now an appropriate strategy to decide if the
center of a layer is in the pure elastic or elasto-plastic range, one may use the classical
assumption that the strain is linearly distributed over the cross section even when the
material is in the elasto-plastic range, see Fig. 5.5. Thus, the strain in the center of
layer k at node i can be expressed according to the kinematics equation in Table 3.1
as:
(ε X ) k,i = −
Z
k
c ×
d
2 u Z
dX 2
i
,
(5.8)
where the second order derivative at node i can be replaced by a centered difference
scheme (truncation error of order X
2 ) as given in Table 1.1. Thus, the discretized
form of Eq. (5.8) at node i is given by:
(ε X ) k,i = −
Z
k
c ×
u i+1 − 2u i + u i−1
X 2
i
.
(5.9)
It should be noted here that the displacements u i are calculated for the neutral fiber,
i.e. for Z
k
c = 0.
The analytical solutions in [3] are presented in a normalized form as
ˆ
u Z (X ) =
u Z (X )
M
pl
lim L 2
E I Y
.
(5.10)
5 Consideration of Euler–Bernoulli Beams with Plastic Material Behavior
(E I Y ) i =
k max
k = 1
E
k
i
E
E
1
12
bh
3
I Y
1
(k max )
3
+
12
k max
Z
k
c
h
2
i
(5.5)
=E I Y
k max
k = 1
E
k
i
E
1
(k max )
3
+
12
k max
Z
k
c
h
2
i
,
(5.6)
or in dimensionless form and under consideration of Eq. (5.3) as:
(E I Y ) i
E I Y
=
k max
k = 1
E
k
i
E
1
k 3
max
+
3
k max
2k − 1
k max
− 1
2
i
= α i ,
(5.7)
where it should be noted that E I Y is the bending stiffness of the entire cross section
or beam in the pure elastic range and E
k
i is the modulus of the kth layer, i.e. E
k
i = E
in the elastic range and E
k
i = 0 in the plastic range. Thus, the dimensionless factor
is in the pure elastic range α = 1 and in the elastic-plastic range 0 ≤ α < 1.
To decide if a layer is now considered as elastic (E
k
i = E) or plastic (E
k
i = 0),
the following assumption is done: If the center of a layer—which is geometrically
represented by the coordinate Z
k
c —is in the elastic range, the entire layer k is assumed
to be elastic. Vice versa, if the center of a layer is in the plastic range, the entire layer
k is assumed to be plastic. To develop now an appropriate strategy to decide if the
center of a layer is in the pure elastic or elasto-plastic range, one may use the classical
assumption that the strain is linearly distributed over the cross section even when the
material is in the elasto-plastic range, see Fig. 5.5. Thus, the strain in the center of
layer k at node i can be expressed according to the kinematics equation in Table 3.1
as:
(ε X ) k,i = −
Z
k
c ×
d
2 u Z
dX 2
i
,
(5.8)
where the second order derivative at node i can be replaced by a centered difference
scheme (truncation error of order X
2 ) as given in Table 1.1. Thus, the discretized
form of Eq. (5.8) at node i is given by:
(ε X ) k,i = −
Z
k
c ×
u i+1 − 2u i + u i−1
X 2
i
.
(5.9)
It should be noted here that the displacements u i are calculated for the neutral fiber,
i.e. for Z
k
c = 0.
The analytical solutions in [3] are presented in a normalized form as
ˆ
u Z (X ) =
u Z (X )
M
pl
lim L 2
E I Y
.
(5.10)