4.3 Supplementary Problems
101
(a) In a first approach, a centered difference approximation should be written
for the inner nodes. Thus, a system of equations with the dimension dim. =
2 × (domain nodes − 2) is obtained. The values of u and φ at the right-hand boundary X = L should be replaced by appropriate boundary conditions for M Y and Q Z
based on backward difference schemes of O((X
2
).
(b) In a second approach, write a centered difference approximation which includes
in addition the node at X = L. Thus, a system of equations with the dimension
dim. = 2 × (domain nodes − 1) is obtained. Replace the values for u and φ at the
fictitious node based on appropriate boundary conditions for M Y and Q Z based on
centered difference schemes of O((X
2
).
References
1. Altenbach H, Öchsner A (eds) (2020) Encyclopedia of continuum mechanics. Springer, Berlin
2. Blaauwendraad J (2010) Plates and FEM: surprises and pitfalls. Springer, Dordrecht
3. Chen WF, Saleeb AF (1982) Constitutive equations for engineering materials. Volume 1: Elasticity and modelling. Wiley, New York
4. Cowper GR (1966) The shear coefficient in Timoshenko’s beam theory. J Appl Mech 33:335–
340
5. Gould PL (1988) Analysis of shells and plates. Springer, New York
6. Gruttmann F, Wagner W (2001) Shear correction factors in Timoshenko’s beam theory for
arbitrary shaped cross-sections. Comput Mech 27:199–207
7. Levinson M (1981) A new rectangular beam theory. J Sound Vib 74:81–87
8. Mindlin RD (1951) Influence of rotary inertia and shear on flexural motions isotropic, elastic
plates. J Appl Mech-T ASME 18:1031–1036
9. Öchsner A (2014) Elasto-plasticity of frame structure elements: modeling and simulation of
rods and beams. Springer, Berlin
10. Reddy JN (1997) Mechanics of laminated composite plates: theory and analysis. CRC Press,
Boca Raton
11. Reddy JN (1997) On locking-free shear deformable beam finite elements. Comput Method
Appl M 149:113–132
12. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates.
J Appl Mech-T ASME 12:A68–A77
13. Timoshenko SP (1921) On the correction for shear of the differential equation for transverse
vibrations of prismatic bars. Philos Mag 41:744–746
14. Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill Book
Company, New York
15. Wang CM (1995) Timoshenko beam-bending solutions in terms of Euler-Bernoulli solutions.
J Eng Mech-ASCE 121:763–765
16. Wang CM, Reddy JN, Lee KH (2000) Shear deformable beams and plates: relationships with
classical solution. Elsevier, Oxford
17. Weaver W Jr, Gere JM (1980) Matrix analysis of framed structures. Van Nostrand Reinhold
Company, New York
18. Winkler E (1867) Die Lehre von der Elasticität und Festigkeit mit besonderer Rücksicht auf
ihre Anwendung in der Technik. H. Dominicus, Prag
101
(a) In a first approach, a centered difference approximation should be written
for the inner nodes. Thus, a system of equations with the dimension dim. =
2 × (domain nodes − 2) is obtained. The values of u and φ at the right-hand boundary X = L should be replaced by appropriate boundary conditions for M Y and Q Z
based on backward difference schemes of O((X
2
).
(b) In a second approach, write a centered difference approximation which includes
in addition the node at X = L. Thus, a system of equations with the dimension
dim. = 2 × (domain nodes − 1) is obtained. Replace the values for u and φ at the
fictitious node based on appropriate boundary conditions for M Y and Q Z based on
centered difference schemes of O((X
2
).
References
1. Altenbach H, Öchsner A (eds) (2020) Encyclopedia of continuum mechanics. Springer, Berlin
2. Blaauwendraad J (2010) Plates and FEM: surprises and pitfalls. Springer, Dordrecht
3. Chen WF, Saleeb AF (1982) Constitutive equations for engineering materials. Volume 1: Elasticity and modelling. Wiley, New York
4. Cowper GR (1966) The shear coefficient in Timoshenko’s beam theory. J Appl Mech 33:335–
340
5. Gould PL (1988) Analysis of shells and plates. Springer, New York
6. Gruttmann F, Wagner W (2001) Shear correction factors in Timoshenko’s beam theory for
arbitrary shaped cross-sections. Comput Mech 27:199–207
7. Levinson M (1981) A new rectangular beam theory. J Sound Vib 74:81–87
8. Mindlin RD (1951) Influence of rotary inertia and shear on flexural motions isotropic, elastic
plates. J Appl Mech-T ASME 18:1031–1036
9. Öchsner A (2014) Elasto-plasticity of frame structure elements: modeling and simulation of
rods and beams. Springer, Berlin
10. Reddy JN (1997) Mechanics of laminated composite plates: theory and analysis. CRC Press,
Boca Raton
11. Reddy JN (1997) On locking-free shear deformable beam finite elements. Comput Method
Appl M 149:113–132
12. Reissner E (1945) The effect of transverse shear deformation on the bending of elastic plates.
J Appl Mech-T ASME 12:A68–A77
13. Timoshenko SP (1921) On the correction for shear of the differential equation for transverse
vibrations of prismatic bars. Philos Mag 41:744–746
14. Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill Book
Company, New York
15. Wang CM (1995) Timoshenko beam-bending solutions in terms of Euler-Bernoulli solutions.
J Eng Mech-ASCE 121:763–765
16. Wang CM, Reddy JN, Lee KH (2000) Shear deformable beams and plates: relationships with
classical solution. Elsevier, Oxford
17. Weaver W Jr, Gere JM (1980) Matrix analysis of framed structures. Van Nostrand Reinhold
Company, New York
18. Winkler E (1867) Die Lehre von der Elasticität und Festigkeit mit besonderer Rücksicht auf
ihre Anwendung in der Technik. H. Dominicus, Prag