3.5 Supplementary Problems
87
Fig. 3.40 Cantilevered
Euler–Bernoulli beam with
prescribed tip displacement
3.22 Finite difference approximation of a cantilevered beam based on five
domain nodes with imposed tip displacement
Given is a cantilevered Euler–Bernoulli beam of length L with constant bending
stiffness E I Y as shown in Fig. 3.40. The beam is subjected to an enforced displacement −u 0 at the position X = L. Use five domain nodes of equidistant spacing,
i.e. =
L
4
, for the finite difference approximation. Use only centered difference
approximations of second order accuracy for the nodal evaluations and boundary
conditions. Determine the vertical displacements of the domain nodes.
References
1. Altenbach H, Öchsner A (eds) (2020) Encyclopedia of continuum mechanics. Springer, Berlin
2. Beer FP, Johnston ER Jr, DeWolf JT, Mazurek DF (2009) Mechanics of materials. McGrawHill, New York
3. Blaauwendraad J (2010) Plates and FEM: surprises and pitfalls. Springer, Dordrecht
4. Boresi AP, Schmidt RJ (2003) Advanced mechanics of materials. Wiley, New York
5. Budynas RG (1999) Advanced strength and applied stress analysis. McGraw-Hill Book, Singapore
6. Gere JM, Timoshenko SP (1991) Mechanics of materials. PWS-KENT Publishing Company,
Boston
7. Gould PL (1988) Analysis of shells and plates. Springer, New York
8. Hibbeler RC (2008) Mechanics of materials. Prentice Hall, Singapore
9. Melosh RJ (1961) A stiffness matrix for the analysis of thin plates in bending. J Aerosp Sci
1:34–64
10. Öchsner A (2014) Elasto-plasticity of frame structure elements: modeling and simulation of
rods and beams. Springer, Berlin
11. Öchsner A (2020) Computational statics and dynamics: an introduction based on the finite
element method. Springer, Singapore
12. Timoshenko S (1940) Strength of materials - part I elementary theory and problems. D. Van
Nostrand Company, New York
13. Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New York
14. Ventsel E, Krauthammer T (2001) Thin plates and shells: theory, analysis, and applications.
Marcel Dekker, New York
15. Winkler E (1867) Die Lehre von der Elasticität und Festigkeit mit besonderer Rücksicht auf
ihre Anwendung in der Technik. H. Dominicus, Prag
87
Fig. 3.40 Cantilevered
Euler–Bernoulli beam with
prescribed tip displacement
3.22 Finite difference approximation of a cantilevered beam based on five
domain nodes with imposed tip displacement
Given is a cantilevered Euler–Bernoulli beam of length L with constant bending
stiffness E I Y as shown in Fig. 3.40. The beam is subjected to an enforced displacement −u 0 at the position X = L. Use five domain nodes of equidistant spacing,
i.e. =
L
4
, for the finite difference approximation. Use only centered difference
approximations of second order accuracy for the nodal evaluations and boundary
conditions. Determine the vertical displacements of the domain nodes.
References
1. Altenbach H, Öchsner A (eds) (2020) Encyclopedia of continuum mechanics. Springer, Berlin
2. Beer FP, Johnston ER Jr, DeWolf JT, Mazurek DF (2009) Mechanics of materials. McGrawHill, New York
3. Blaauwendraad J (2010) Plates and FEM: surprises and pitfalls. Springer, Dordrecht
4. Boresi AP, Schmidt RJ (2003) Advanced mechanics of materials. Wiley, New York
5. Budynas RG (1999) Advanced strength and applied stress analysis. McGraw-Hill Book, Singapore
6. Gere JM, Timoshenko SP (1991) Mechanics of materials. PWS-KENT Publishing Company,
Boston
7. Gould PL (1988) Analysis of shells and plates. Springer, New York
8. Hibbeler RC (2008) Mechanics of materials. Prentice Hall, Singapore
9. Melosh RJ (1961) A stiffness matrix for the analysis of thin plates in bending. J Aerosp Sci
1:34–64
10. Öchsner A (2014) Elasto-plasticity of frame structure elements: modeling and simulation of
rods and beams. Springer, Berlin
11. Öchsner A (2020) Computational statics and dynamics: an introduction based on the finite
element method. Springer, Singapore
12. Timoshenko S (1940) Strength of materials - part I elementary theory and problems. D. Van
Nostrand Company, New York
13. Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New York
14. Ventsel E, Krauthammer T (2001) Thin plates and shells: theory, analysis, and applications.
Marcel Dekker, New York
15. Winkler E (1867) Die Lehre von der Elasticität und Festigkeit mit besonderer Rücksicht auf
ihre Anwendung in der Technik. H. Dominicus, Prag