5.2 Supplementary Problems
113
5.3 Elasto-plastic finite difference solution: influence of layer number
Consider again the problem of the simply supported Euler–Bernoulli beam under
constant bending load as shown in Figs. 5.1 and 5.7. The beam should be discretized
by five equidistant nodes and the moment should be applied in the range M/M
pl
lim =
2/3 and 0.8 and in the range from 0.8 to 0.9 in ten equidistant steps. Calculate the
deformation for M/M
pl
lim = 0.9 with k max = 11, 101, 1001, 10001 and determine in
addition the relative error to the analytical solution.
5.4 Elasto-plastic finite difference solution: influence of load increment
Consider again the problem of the simply supported Euler–Bernoulli beam under
constant bending load as shown in Figs. 5.1 and 5.7. The beam should be discretized
by five equidistant nodes and the layers are k max = 101. The moment should be
applied in the range M/M
pl
lim = 2/3 and 0.8 and in the range from 0.8 to 0.9 in 5, 8
or 10 equidistant steps. Calculate the deformation for M/M
pl
lim = 0.9 and determine
in addition the relative error to the analytical solution in dependence of the size of
the load increment.
References
1. Al-Amery RIM, Roberts TM (1990) Nonlinear finite difference analysis of composite beams
with partial interaction. Comput Struct 35:81–87
2. Öchsner A, Merkel M (2018) One-dimensional finite elements: an introduction to the FE method.
Springer, Cham
3. Öchsner A (2014) Elasto-plasticity of frame structure elements: modeling and simulation of
rods and beams. Springer, Berlin
4. Öchsner A (2016) Continuum damage and fracture mechanics. Springer, Singapore
5. Owen DRJ, Hinton E (1980) Finite elements in plasticity: theory and practice. Pineridge Press
Limited, Swansea
6. Reddy JN (2004) An introduction to nonlinear finite element analysis. Oxford University Press,
Oxford
113
5.3 Elasto-plastic finite difference solution: influence of layer number
Consider again the problem of the simply supported Euler–Bernoulli beam under
constant bending load as shown in Figs. 5.1 and 5.7. The beam should be discretized
by five equidistant nodes and the moment should be applied in the range M/M
pl
lim =
2/3 and 0.8 and in the range from 0.8 to 0.9 in ten equidistant steps. Calculate the
deformation for M/M
pl
lim = 0.9 with k max = 11, 101, 1001, 10001 and determine in
addition the relative error to the analytical solution.
5.4 Elasto-plastic finite difference solution: influence of load increment
Consider again the problem of the simply supported Euler–Bernoulli beam under
constant bending load as shown in Figs. 5.1 and 5.7. The beam should be discretized
by five equidistant nodes and the layers are k max = 101. The moment should be
applied in the range M/M
pl
lim = 2/3 and 0.8 and in the range from 0.8 to 0.9 in 5, 8
or 10 equidistant steps. Calculate the deformation for M/M
pl
lim = 0.9 and determine
in addition the relative error to the analytical solution in dependence of the size of
the load increment.
References
1. Al-Amery RIM, Roberts TM (1990) Nonlinear finite difference analysis of composite beams
with partial interaction. Comput Struct 35:81–87
2. Öchsner A, Merkel M (2018) One-dimensional finite elements: an introduction to the FE method.
Springer, Cham
3. Öchsner A (2014) Elasto-plasticity of frame structure elements: modeling and simulation of
rods and beams. Springer, Berlin
4. Öchsner A (2016) Continuum damage and fracture mechanics. Springer, Singapore
5. Owen DRJ, Hinton E (1980) Finite elements in plasticity: theory and practice. Pineridge Press
Limited, Swansea
6. Reddy JN (2004) An introduction to nonlinear finite element analysis. Oxford University Press,
Oxford