Large Deflection Analysis of Prismatic Cantilever Beam Comparatively by Using Combing Method and Iterative DQM
Yıl 2020,
, 111 - 120, 01.03.2020
Zekeriya Girgin
,
Faruk Emre Aysal
,
Hüseyin Bayrakçeken
Öz
There is no exactly analytical solution for the large
deflection problem of prismatic cantilever beams under general loading
conditions. In the case of considering a non-prismatic cantilever beam, the
difficulty of the larger deflection problem is increased. In this study, the
comparison of the Iterative Differential Quadrature Method (I-DQM) and the
Combining Method (CM) was performed. Numerical solution of the large deflection
problem was separately performed with both the I-DQM and the CM for prismatic
cantilever beams. The obtaining results show that both of these methods gave
more accurate solutions compared with a reliable semi-analytic method which was
introduced by Dado and Sadder (2005). Besides, it was demonstrated that the
I-DQM provided a more wide-range solution than the CM.
Kaynakça
- [1] AL-Sadder S. and AL-Rawi R. A. O., "Finite difference scheme for large-deflection analysis of non-prismatic cantilever beams subjected to different types of continuous and discontinuous loadings.", Arch Appl Mech, 75: 459-473, (2006).
- [2] Dado M., and Al-Sadder S., "A new technique for large deflection analysis of non-prismatic cantilever beams.", Mechanics Research Communications, 32: 692-703, (2005).
- [3] Barten H. J, "On the deflection of a cantilever beam". Quart. J. Appl. Math., 3: 275, (1945).
- [4] Bisshop K. E. ve Drucker D. C. "Large deflections of cantilever beams.", Quart. J. Appl. Math., 3: 272-275, (1945).
- [5] Freeman J. G. "Mathematical theory of deflection of beam", Philos. Mag, 37: 551, (1946).
- [6] Conway H. D., "Large deflection of simply supported beams", Philos. Mag, 38: 905, (1947).
- [7] Timoshenko S. P. and Gere J. M., "Theory of Elastic Stability." McGraw-Hill, New York, (1961).
- [8] Holden J. T., "On the finite deflections of thin beams.", Int. J. Solids Struct, 8:1051-1055, (1972).
- [9] Lau J. H., "Large deflections of beams with combined loads", ASCE J. Eng. Mech. Div,12:140, (1974).
- [10] Wang,C. Y. And Watson L. T, "On large deformations of C-shaped springs.", Int. J. Mech. Sci, 22: 395-400, (1980).
- [11] Chucheepsakul S., Buncharoen S. and Wang C. M, "Large deflection of beams under moment gradient.", ASCE J. Eng. Mech, 120: 1848, (1994).
- [12] Bona F. and Zelenika S, "A generalized elastica-type approach to the analysis of large displacements of spring-strips.", Proc. Inst. Mech. Engrs. Part C, 21: 509-517, (1997).
- [13] Wang X. W., and Gu H. Z., "Static Analysis of Frame Structures by The Differential Quadrature Element Method." International Journal for Numerical Method in Engineering, 40: 759-772, (1997).
- [14] Chucheepsakul S., Wang C. M. and He X. Q., "Double curvature bending of variable-arc-length elastica.", J. Appl. Mech, 66: 87-94, (1999).
- [15] Coffin D. W. and Bloom F., "Elastica solution for the hygrothermal buckling of a beam.", Int. J. Non-lin. Mech, 34: 935, (1999).
- [16] Kang T.J., Kim J.G., Kim J.H. , Hwang K.C., Lee B.W., Baek C.W., Kim C.W., Kwon D., Lee H.Y. and Kim Y.H., "Deformation characteristics of electroplated MEMS cantilever beams released by plasma ashing.", Sensors and Actuators A: Physical, 148: 407-415, (2008).
- [17] Tolou N. And Herder J. L., "A Semianalytical Approach to Large Deflections in Compliant Beams under Point Load.", Mathematical Problems in Engineering, 13 pages, (2009).
- [18] Lin H. P. and Chang S. C., "Forced responses of cracked cantilever beams subjected to a concentrated moving load", International Journal of Mechanical Sciences, 48: 1456-1463, (2006).
- [19] Batista M., "Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions", International Journal of Solids and Structures, 51: 2308-2326, (2014).
- [20] Wang K.F. and Wang. B.L., "A general model for nano-cantilever switches with consideration of surface effects and nonlinear curvature.", Physica E, 66:197-208, (2015).
- [21] Joseph R.P., Wang B.L. and Samali B., "Size effects on double cantilever beam fracture mechanics specimen based on strain gradient theory.", Engineering Fracture Mechanics, 169: 309-320, (2017).
- [22] Abu-Alshaikh I. M., "Closed-Form Solution of Large Deflected Cantilever Beam against Follower Loading Using Complex Analysis", Modern Applied Science, 11(12): 12-21, (2017).
- [23] Navaee, S. and Elling, R. E., "Equilibrium configurations of cantilever beam subjected to inclined end loads.", Trans. ASME, 59: 572-579, (1992).
- [24] Faulkner M. G., Lipsett A. W. and Tam V., "On the use of a segmental shooting technique for multiple solutions of planar elastica problems.", Comp Meth Appl Mech Eng, 110: 221-236, (1993).
- [25] Bellman R. and J. Casti J.. "Differential Quadrature and long term integration.", J. Math. Anal. Appl, 34 235-238, (1971).
- [26] Bellman R., Kashef B.G. and Casti J., "Differential Quadrature: a technique for the rapid solution of non-linear partial differential equations.", J. Comput. Phys., 10: 40-52, (1972).
- [27] Quan J. R., and Chang C. T., "New Insights in Solving Distributed System Equations by The Quadrature Methods I.", Computational Chemical Engineering, 13 779-788, (1989a).
- [28] Quan J. R., and Chang C. T., "New Insights in Solving Distributed System Equations by The Quadrature Methods. II.", Computational Chemical Engineering, 13: 1017-1024, (1989b).
- [29] Shu, C. and Richards B.E., "Application of generalized differential quadrature to solve two dimensional incompressible Navier Stokes equations.", Int. J. Numer. Methods Fluids, 15: 791-798, (1992).
- [30] Jiwari R., Pandit S., and Mittal R. C., "Numerical simulation of two-dimensional Sine-Gordon solitons by differential quadrature method.", Computer Physics Communications, 183: 600-616, (2012).
- [31] Ansari, R., Gholami R., Shojaei M. F., Mohammadi V. and Sahmani S., "Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory.", Composite Structures, 100: 385-397, (2013).
- [32] Alinaghizadeh, F. and Shariati M., "Geometrically non-linear bending analysis of thick two-directional functionally graded annular sector and rectangular plates with variable thickness resting on non-linear elastic foundation.", Composites Part B, 86: 61-83, (2016).
- [33] Yilmaz Y., Girgin Z., and Evran S. "Buckling Analyses of Axially Functionally Graded Nonuniform Columns with Elastic Restraint Using a Localized Differential Quadrature Method.", Mathematical Problems in Engineering, 12 pages, (2013).
- [34] Sahmani S., Aghdam M. M. and Bahrami M., "On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects.", Composite Structures, 121: 377-385, (2015).
- [35] Jang S. K., Bert C. W. and Striz A. G., "Application of Differential Quadrature to Static Analysis of Structural Components.", International Journal for Numerical Methods in Engineering, 28: 561-577, (1989).
- [36] Wang, X., and Bert, C. W. "A New Approach in Applying Differential Quadrature to Static and Free Vibrational Analyses of Beams and Plates.", Journal of Sound and Vibration, 162(3): 566-572, (1993).
- [37] Shu C., and Du H., "Implementation of Clamped and Simply Supported Boundary Conditions in The GDQ Free Vibration Analysis of Beams and Plates.", International Journal of Solids and Structures, 34(7): 819-835, (1997).
- [38] Liu G. R., and Wu T. Y., "Numerical Solution for Differential Equations of Duffing-Type Non-Linearity Using The Generalized Differential Quadrature Rule.", Journal of Sound and Vibration, 237(5): 805-817, (2000).
- [39] Tomasiello S., "Simulating Non-Linear Coupled Oscillators by An Iterative Differential Quadrature Method.", Journal of Sound and Vibration, 265: 507-525, (2003).
- [40] Liu F. L., and Liew K. M., "Analysis of Vibrating Thick Rectangular Plates with Mixed Boundary Constrants Using Differential Quadrature Element Method.", Journal of Sound and Vibration, 222 (5) 915-934 (1999).
- [41] Chen C. N., "Differential Quadrature Element Analysis Using Extended Differential Quadrature.", Computer and Mathematic with Application, 39: 65-79, (2000).
- [42] Kol C., "Diferansiyel Kuadratür (Quadrature) Eleman Metodunun Plakalara Uygulanması.", Yüksek Lisans Tezi, Pamukkale Üniversitesi Fen Bilimleri Enstitüsü, (2003).
- [43] Girgin Z., "Combining Differential Quadrature Method with simulation technique to solve nonlinear differential equations.", Int. J. Numer. Methods Eng., 75, (6): 723-733, (2008).
- [44] Demir E., "Lineer Olmayan Titreşim Problemlerinin Çözümünde Birleşim (Diferansiyel Quadrature Ve Simülasyon) Metodu.", Doktora Tezi, Pamukkale Üniversitesi Fen Bilimleri Enstitüsü, (2009).
- [45] Girgin Z., Yilmaz Y. and Demir E. "A Combining Method for solution of nonlinear boundary value problems.", Applied Mathematics and Computation, 232: 1037-1045, (2014).
Large Deflection Analysis of Prismatic Cantilever Beam Comparatively by Using Combing Method and Iterative DQM
Yıl 2020,
, 111 - 120, 01.03.2020
Zekeriya Girgin
,
Faruk Emre Aysal
,
Hüseyin Bayrakçeken
Öz
There is no exactly analytical solution for the large
deflection problem of prismatic cantilever beams under general loading
conditions. In the case of considering a non-prismatic cantilever beam, the
difficulty of the larger deflection problem is increased. In this study, the
comparison of the Iterative Differential Quadrature Method (I-DQM) and the
Combining Method (CM) was performed. Numerical solution of the large deflection
problem was separately performed with both the I-DQM and the CM for prismatic
cantilever beams. The obtaining results show that both of these methods gave
more accurate solutions compared with a reliable semi-analytic method which was
introduced by Dado and Sadder (2005). Besides, it was demonstrated that the
I-DQM provided a more wide-range solution than the CM.
Kaynakça
- [1] AL-Sadder S. and AL-Rawi R. A. O., "Finite difference scheme for large-deflection analysis of non-prismatic cantilever beams subjected to different types of continuous and discontinuous loadings.", Arch Appl Mech, 75: 459-473, (2006).
- [2] Dado M., and Al-Sadder S., "A new technique for large deflection analysis of non-prismatic cantilever beams.", Mechanics Research Communications, 32: 692-703, (2005).
- [3] Barten H. J, "On the deflection of a cantilever beam". Quart. J. Appl. Math., 3: 275, (1945).
- [4] Bisshop K. E. ve Drucker D. C. "Large deflections of cantilever beams.", Quart. J. Appl. Math., 3: 272-275, (1945).
- [5] Freeman J. G. "Mathematical theory of deflection of beam", Philos. Mag, 37: 551, (1946).
- [6] Conway H. D., "Large deflection of simply supported beams", Philos. Mag, 38: 905, (1947).
- [7] Timoshenko S. P. and Gere J. M., "Theory of Elastic Stability." McGraw-Hill, New York, (1961).
- [8] Holden J. T., "On the finite deflections of thin beams.", Int. J. Solids Struct, 8:1051-1055, (1972).
- [9] Lau J. H., "Large deflections of beams with combined loads", ASCE J. Eng. Mech. Div,12:140, (1974).
- [10] Wang,C. Y. And Watson L. T, "On large deformations of C-shaped springs.", Int. J. Mech. Sci, 22: 395-400, (1980).
- [11] Chucheepsakul S., Buncharoen S. and Wang C. M, "Large deflection of beams under moment gradient.", ASCE J. Eng. Mech, 120: 1848, (1994).
- [12] Bona F. and Zelenika S, "A generalized elastica-type approach to the analysis of large displacements of spring-strips.", Proc. Inst. Mech. Engrs. Part C, 21: 509-517, (1997).
- [13] Wang X. W., and Gu H. Z., "Static Analysis of Frame Structures by The Differential Quadrature Element Method." International Journal for Numerical Method in Engineering, 40: 759-772, (1997).
- [14] Chucheepsakul S., Wang C. M. and He X. Q., "Double curvature bending of variable-arc-length elastica.", J. Appl. Mech, 66: 87-94, (1999).
- [15] Coffin D. W. and Bloom F., "Elastica solution for the hygrothermal buckling of a beam.", Int. J. Non-lin. Mech, 34: 935, (1999).
- [16] Kang T.J., Kim J.G., Kim J.H. , Hwang K.C., Lee B.W., Baek C.W., Kim C.W., Kwon D., Lee H.Y. and Kim Y.H., "Deformation characteristics of electroplated MEMS cantilever beams released by plasma ashing.", Sensors and Actuators A: Physical, 148: 407-415, (2008).
- [17] Tolou N. And Herder J. L., "A Semianalytical Approach to Large Deflections in Compliant Beams under Point Load.", Mathematical Problems in Engineering, 13 pages, (2009).
- [18] Lin H. P. and Chang S. C., "Forced responses of cracked cantilever beams subjected to a concentrated moving load", International Journal of Mechanical Sciences, 48: 1456-1463, (2006).
- [19] Batista M., "Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions", International Journal of Solids and Structures, 51: 2308-2326, (2014).
- [20] Wang K.F. and Wang. B.L., "A general model for nano-cantilever switches with consideration of surface effects and nonlinear curvature.", Physica E, 66:197-208, (2015).
- [21] Joseph R.P., Wang B.L. and Samali B., "Size effects on double cantilever beam fracture mechanics specimen based on strain gradient theory.", Engineering Fracture Mechanics, 169: 309-320, (2017).
- [22] Abu-Alshaikh I. M., "Closed-Form Solution of Large Deflected Cantilever Beam against Follower Loading Using Complex Analysis", Modern Applied Science, 11(12): 12-21, (2017).
- [23] Navaee, S. and Elling, R. E., "Equilibrium configurations of cantilever beam subjected to inclined end loads.", Trans. ASME, 59: 572-579, (1992).
- [24] Faulkner M. G., Lipsett A. W. and Tam V., "On the use of a segmental shooting technique for multiple solutions of planar elastica problems.", Comp Meth Appl Mech Eng, 110: 221-236, (1993).
- [25] Bellman R. and J. Casti J.. "Differential Quadrature and long term integration.", J. Math. Anal. Appl, 34 235-238, (1971).
- [26] Bellman R., Kashef B.G. and Casti J., "Differential Quadrature: a technique for the rapid solution of non-linear partial differential equations.", J. Comput. Phys., 10: 40-52, (1972).
- [27] Quan J. R., and Chang C. T., "New Insights in Solving Distributed System Equations by The Quadrature Methods I.", Computational Chemical Engineering, 13 779-788, (1989a).
- [28] Quan J. R., and Chang C. T., "New Insights in Solving Distributed System Equations by The Quadrature Methods. II.", Computational Chemical Engineering, 13: 1017-1024, (1989b).
- [29] Shu, C. and Richards B.E., "Application of generalized differential quadrature to solve two dimensional incompressible Navier Stokes equations.", Int. J. Numer. Methods Fluids, 15: 791-798, (1992).
- [30] Jiwari R., Pandit S., and Mittal R. C., "Numerical simulation of two-dimensional Sine-Gordon solitons by differential quadrature method.", Computer Physics Communications, 183: 600-616, (2012).
- [31] Ansari, R., Gholami R., Shojaei M. F., Mohammadi V. and Sahmani S., "Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory.", Composite Structures, 100: 385-397, (2013).
- [32] Alinaghizadeh, F. and Shariati M., "Geometrically non-linear bending analysis of thick two-directional functionally graded annular sector and rectangular plates with variable thickness resting on non-linear elastic foundation.", Composites Part B, 86: 61-83, (2016).
- [33] Yilmaz Y., Girgin Z., and Evran S. "Buckling Analyses of Axially Functionally Graded Nonuniform Columns with Elastic Restraint Using a Localized Differential Quadrature Method.", Mathematical Problems in Engineering, 12 pages, (2013).
- [34] Sahmani S., Aghdam M. M. and Bahrami M., "On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects.", Composite Structures, 121: 377-385, (2015).
- [35] Jang S. K., Bert C. W. and Striz A. G., "Application of Differential Quadrature to Static Analysis of Structural Components.", International Journal for Numerical Methods in Engineering, 28: 561-577, (1989).
- [36] Wang, X., and Bert, C. W. "A New Approach in Applying Differential Quadrature to Static and Free Vibrational Analyses of Beams and Plates.", Journal of Sound and Vibration, 162(3): 566-572, (1993).
- [37] Shu C., and Du H., "Implementation of Clamped and Simply Supported Boundary Conditions in The GDQ Free Vibration Analysis of Beams and Plates.", International Journal of Solids and Structures, 34(7): 819-835, (1997).
- [38] Liu G. R., and Wu T. Y., "Numerical Solution for Differential Equations of Duffing-Type Non-Linearity Using The Generalized Differential Quadrature Rule.", Journal of Sound and Vibration, 237(5): 805-817, (2000).
- [39] Tomasiello S., "Simulating Non-Linear Coupled Oscillators by An Iterative Differential Quadrature Method.", Journal of Sound and Vibration, 265: 507-525, (2003).
- [40] Liu F. L., and Liew K. M., "Analysis of Vibrating Thick Rectangular Plates with Mixed Boundary Constrants Using Differential Quadrature Element Method.", Journal of Sound and Vibration, 222 (5) 915-934 (1999).
- [41] Chen C. N., "Differential Quadrature Element Analysis Using Extended Differential Quadrature.", Computer and Mathematic with Application, 39: 65-79, (2000).
- [42] Kol C., "Diferansiyel Kuadratür (Quadrature) Eleman Metodunun Plakalara Uygulanması.", Yüksek Lisans Tezi, Pamukkale Üniversitesi Fen Bilimleri Enstitüsü, (2003).
- [43] Girgin Z., "Combining Differential Quadrature Method with simulation technique to solve nonlinear differential equations.", Int. J. Numer. Methods Eng., 75, (6): 723-733, (2008).
- [44] Demir E., "Lineer Olmayan Titreşim Problemlerinin Çözümünde Birleşim (Diferansiyel Quadrature Ve Simülasyon) Metodu.", Doktora Tezi, Pamukkale Üniversitesi Fen Bilimleri Enstitüsü, (2009).
- [45] Girgin Z., Yilmaz Y. and Demir E. "A Combining Method for solution of nonlinear boundary value problems.", Applied Mathematics and Computation, 232: 1037-1045, (2014).