110
5 Consideration of Euler–Bernoulli Beams with Plastic Material Behavior
Fig. 5.6 a Stress-strain diagram and b modulus as a function of displacement
which is known under the expression direct or Picard’s iteration [6]. In components,
we can write this scheme for our specific case as:
⎡
⎢
⎢
⎣
2
3
4
⎤
⎥
⎥
⎦
( j+1)
=
2
4
⎡
⎢
⎢
⎣
−3
(E I Y ) 2
−2
(E I Y ) 3
−1
(E I Y ) 4
−2
(E I Y ) 2
−4
(E I Y ) 3
−2
(E I Y ) 4
−1
(E I Y ) 2
−2
(E I Y ) 3
−3
(E I Y ) 4
⎤
⎥
⎥
⎦
( j) ⎡
⎢
⎢
⎣
2
3
4
⎤
⎥
⎥
⎦
( j)
(5.28)
(5.8)
=
M
pl
lim L
2
64E I Y
⎡
⎢
⎢
⎢
⎢
⎣
−3
α 2
−2
α 3
−1
α 4
−2
α 2
−4
α 3
−2
α 4
−1
α 2
−2
α 3
−3
α 4
⎤
⎥
⎥
⎥
⎥
⎦
( j) ⎡
⎢
⎢
⎢
⎢
⎣
2
M
pl
lim
3
M
pl
lim
4
M
pl
lim
⎤
⎥
⎥
⎥
⎥
⎦
( j)
,
(5.29)
or in normalized form as:
⎡
⎢
⎢
⎣
ˆ
u 2
ˆ
u 3
ˆ
u 4
⎤
⎥
⎥
⎦
( j+1)
=
1
M
pl
lim L 2
E I Y
⎡
⎢
⎢
⎣
2
3
4
⎤
⎥
⎥
⎦
( j+1)
=
1
64
⎡
⎢
⎢
⎢
⎢
⎣
−3
α 2
−2
α 3
−1
α 4
−2
α 2
−4
α 3
−2
α 4
−1
α 2
−2
α 3
−3
α 4
⎤
⎥
⎥
⎥
⎥
⎦
( j) ⎡
⎢
⎢
⎢
⎢
⎣
2
M
pl
lim
3
M
pl
lim
4
M
pl
lim
⎤
⎥
⎥
⎥
⎥
⎦
( j)
. (5.30)
Let us now evaluate this iteration scheme for different ratios of the normalized
moment in the pure elastic and the elasto-plastic range and compare these numerical
results with the exact analytical solution. As shown in Fig. 5.2a, the beam is sub-