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Comments and discussion: “Elastic buckling of columns with end restraint effects”
Journal of Constructional Steel Research 91 (2013) 60–63
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Comment
Comments and discussion: “Elastic buckling of columns with end restraint effects” S.E. Djellab University of Science and Technology Houari Boumediene, Faculty of Civil Engineering, Algiers, Algeria
Adman and Saidani tackled a simple classical structural stability problem for beam-columns, elastically restrained at the ends. They established without demonstration a so-called general stability criterion to be used for an eventual elastic buckling analysis of a global structure. Firstly, the very unsettling results obtained by the authors, especially when compared to Eurocode 2, are either an innovative novelty or some other proposal. Secondly, these results contradict previous work by the writer as a co-supervisor [7]. This work used a completely different, more general, matrix approach different to the authors so that no confusion can be made between the two approaches. These are the main reasons that have stimulated the presented discussion. Unfortunately, the authors inconsistent formulation of the problem and the so-called general stability criterion was, in the writer's opinion, erroneous, misleadingly used and confusingly interpreted, as will be demonstrated below. The writers objectives are three fold: (a) A suggested correction to the authors approach, refocusing it in the appropriate direction. (b) To enlighten designers of the ‘robustness’ of the Eurocode formulae. (c) Questioning the proposed Eurocodes formulae is not an easy task: Experts have carefully considered their approach involving special task teams in: field design practices, experimental laboratory setups, theoretical background and numerical knowledge. Constrained to remain within the context of elastic stability, the primary topic of the paper by Adman and Saidani in the present discussion is organised around three main points to be efficient: 1. The main areas for discussion 1.1. The formulation of the problem Starting from a second order ordinary differential equation Eq. (3) with two unknown functions y(x) and M(x) is, in the writer's opinion, not the right way to approach the problem. The quoted ‘appropriate boundary conditions’ Eq. (8) presented by the authors associated with Eq. (3) are inappropriate. The problem at hand is by no means concerned about conditions placed on displacements and rotations at nodes, please see Eq. (8). Furthermore the displacement and rotation is the same for the beam-column and the translational/rotational stiffnesses at the beam ends, the sums in Eq. (8) have no physical meaning. Conditions on shear forces and moments at nodes must be imposed. Since the shear force is a third order derivative of the displacement 0143-974X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jcsr.2013.08.001
y(x), the authors neither can nor did in the second order ODE Eq. (3) associated with Eq. (8). A fourth order ODE has to be used with its appropriate conditions as indicated below. Lastly, since the element is uniform (not tapered), permuting the indices i, j in Eq. (12) should not change the criteria and this is not the case. 1.2. The use of the stability criteria From the first stability analysis principle [1–6], the roots of a characteristic equation or a stability criterion are the eigenvalues or buckling loads of the system. Numerical results such as ‘infinitely large displacements’ in a finite structure is a serious basic physics question, as the question raised about the length of the element and material behaviour. To the best of the writer's knowledge, no such field has yet been observed or computed even in astrophysics. So, no need to compute the ‘infinitely large displacements’ in one analysis and then using these in the erroneous ‘general stability criterion’ Eq. (12). It is sufficient to compute the smallest of the roots of the characteristic equation and the sought solution is obtained. If a displacement approach is adopted, then material nonlinearity has to be considered so and only then, that a stability-elastoplastic collapse of the system can be viewed and physically interpreted. The authors did not indicate under what loading they obtained the ‘infinitely large displacements’ though. However, it is well established that, in a second order elastic analysis, the critical load is independent on any lateral load or geometric imperfections that may be used [9]. 1.2.1. Analytical demonstration The first author is undoubtedly well aware of the existence of ref. [7]. In ref. [7], an exact beam-column on a two parameter elastic foundation of Pasternak type was formulated. The analyses involved either or both material (elastoplastic) and geometric (stability) nonlinearities for either reinforced concrete or steel beams. Furthermore, continuous elastic support underneath the beam and/or discretely lumped rigid or elastic (translational/rotational) supports at the nodes can be handled in a straightforward manner. It started from a fourth order differential equation and its exact homogeneous solution was used to set up the exact shape functions to deduce the exact second order stiffness matrix. It can be easily shown that the correct stability criterion for a general beam-column with elastic end restraints is shown in Fig. 1. Starting from potential energy principles in integral form, invoking Euler–Lagrange equations a fourth order differential equation with its very appropriate boundary conditions is obtained. With the homogenous
S.E. Djellab / Journal of Constructional Steel Research 91 (2013) 60–63
i P
j
Rj
Ri
61
P
Kj
Ki L
Fig. 1. A beam-column with elastic restraints at the ends.
Table 1 Sample classical results. Exact γ
Eq. (1) α
Eq. (12) β [9]
π
3.1416
3.1416
π/2
1.5708
2.7984
π/2
1.5708
1.3138
solution and the boundary conditions written in matrix form, one obtains the correct general stability criterion as: 2 4 α 2ri rj ki kj þ ð−2ri rjki kj−ki kj ðri þ rjÞα þ ðki þ kjÞðri þ rjÞα Cos ½α 2 4 þ α ðri þ rj−ri rjÞki kj þ ðki kj þ ri rj ðki þ kjÞÞα −ðki þ kjÞα Sin ½α Þ
ð1Þ Where ki, kj, ri and rj are the adimensional elastic translational and rotational stiffnesses at the ends. Eq. (1) is a transcendental equation with an infinite number of roots. The roots are obtained using special numerical techniques so that no root is missed, or simply by a graphic plot of Eq. (1). From the smallest root α we deduce the first buckling load. The higher modes, if needed, are obtained by computing the remaining roots. In deriving the above correct stability criterion, no claim is made of originality or ‘novelty’ as the authors did, since it is a very simple stability problem in the writer's opinion. 2. Numerical demonstrations Below are some basic stability problems with or without elastic restraints at the ends. First: Sample classical results without elastic restraints are summarised in Table 1. Great care has been undertaken to confront Eq. (12) in ref. [9], Eq. (1) and the exact value of the first
buckling load γ2EI/L2. We already note the discrepancies between the simple classical cases and the proposed Eq. (12). Secondly, other samples of elastically restrained cases can be used as benchmarks. Classical problems as indicated in Tables 2 and 3 are examined. In Table 2, a sway cantilever with a partially fixed base and free at the other end is investigated. In Table 3, a cantilever fixed at the base and a translation elastically restrained at the other end. By varying the translational stiffness at the top from infinitesimal to large, representing the two extreme cases of fixed-free to fixed-pinned respectively, the numerical results are given. The numerical results of a beam-column partially fixed at the base and guide with full rotational restraint at the top are given in Table 4 and plotted in Fig. 2. In front of the harmony between the ‘smart’ Eurocode formulae and the numerical ones as shown, we note: The Eurocode superiority as less effort is needed to obtain such a practical accuracy. Finally, without a single validation example from the literature, the authors [9] challenged directly the formulation used in the Eurocode. The continuous Eurocode equations are represented discretely in Figs. 4, 5 and 6 while the discrete erroneous solution is plotted continuously. In view of the non-convincing ‘virtually identical results’ as quoted by the authors in Fig. 4 and the even worst discrepancies of Figs. 5 and 6, no further comments about the authors' results in accordance with the various tests above will be made since the authors' formulation failed to model the simplest standard cases. The general comments below are given: – It is demonstrated that the Eurocode formulae are superior to any complex second order elastic analytical solution even if it is correct, achieved by their simplicity and proven accuracy. – Civil engineering designers tackling stability problems must be comforted in regards to the confusing conclusions of the discussed paper. – The confusing conclusion means that, either the Eurocode underestimates the buckling load or the buckling load should be that obtained by the authors.
Table 2 A cantilever partially fixed at the base and free at one. fir
ri
Present α
1000 100 10 8 5 4 2 1 0.5 0.2 0.1 0.01 0.001
0.001 0.01 0.1 0.125 0.2 0.25 0.5 1 2 5 10 100 1000
[9]
2 Pcr αL2EI
0.03162 0.09983 0.31105 0.34645 0.43284 0.48009 0.65327 0.86033 1.07687 1.31384 1.42887 1.55525 1.56923
β 0.001 0.010 0.097 0.120 0.187 0.230 0.427 0.740 1.160 1.726 2.042 2.419 2.462
Exact Pcr ¼
2.798 2.798 2.798 2.798 2.798 2.798 2.798
β 2 EI L2
Pcr 7.8288 – – – – – – – – – – – 7.8288
Ri/L = 0.001EI/L2
π2 EI 4L2
62
S.E. Djellab / Journal of Constructional Steel Research 91 (2013) 60–63
Table 3 A cantilever fixed at one end and displacement partially restrained at the other. fjy
kj
Present study α
100 10 8 5 4 2 1 0.5 0.2 0.1 0.01 0.001
0.01 0.1 0.125 0.2 0.25 0.5 1 2 5 10 100 1000
Exact
Ref [9]
K
K ¼ βπ
2.0 – – – – – – – – – – 0.7
1.12243833 1.12059663 1.12008922 1.11857772 1.11757895 1.11269446 1.10342826 1.08662013 1.04624198 0.99897693 0.8013531 0.71336603
Eurocode
Eq. (12) 9]
K
K ¼ βπ
1.999 1.9901 1.90909 1.88889 1.83333 1.80000 1.66667 1.50000 1.33333 1.16667 1.09091 1.0099 1.001
2.39020714 2.48260894 2.71301731 2.75184837 2.84856117 2.90256632 3.11017984 3.39017741 3.75674005 4.39183345 4.99514676 7.86402793 12.4829347
K ¼ απ
1.57181 1.59485 1.60119 1.62004 1.63248 1.69318 1.80791 2.01581 2.52425 3.14492 4.42795 4.48437
1.99871 1.96984 1.96204 1.93921 1.92443 1.85544 1.73769 1.55848 1.24456 0.99894 0.70949 0.70056
β 2.7989 2.8035 2.80477 2.80856 2.81107 2.82341 2.84712 2.89116 3.00274 3.14481 3.92036 4.4039
Table 4 A beam-column partially fixed at the base and guided with full rotational restraint at the top. fir
ri
Present α
kj ri
0
Ri EI L
1000 100 10 8 5 4 2 1 0.5 0.2 0.1 0.01 0.001
0.001 0.01 0.1 0.125 0.2 0.25 0.5 1 2 5 10 100 1000
π α
K¼
1.57143 1.57714 1.63199 1.64656 1.68868 1.71551 1.8366 2.02876 2.28893 2.65366 2.86277 3.11049 3.13845
1.99919351 1.99195547 1.92500729 1.90797338 1.86038365 1.83128787 1.71054811 1.54852849 1.37251583 1.18387158 1.09739611 1.00999928 1.00100134
– Awareness and care must be taken when questioning well established principles. One has to be rigorous, concise with a full explanation and clear demonstrations proving the case. – From another writer's work [8] where it is still remembered that a simple RC beam failing in shear was quiet a defying challenge. One
1.8
1.6
K
Eurocode Equation (1)
1.4
1.2
1.0 0.01
0.1
1.31436 1.26544 1.15797 1.14163 1.10287 1.08235 1.0101 0.926675 0.836255 0.715326 0.628929 0.399489 0.251671
has to read the outstanding Vecchio and Collins international competition made for respectful international experts at that time. – The writer believes convincingly that: besides the present discussion, any elastic analysis even if it is correct should never be used beyond its limitations and context.
2.0
1E-3
β
1
10
ri Fig. 2. Effective length factor.
100
1000
S.E. Djellab / Journal of Constructional Steel Research 91 (2013) 60–63
References [1] Timoshenko SP. Theory of elastic stability. McGraw-Hill; 1961. [2] Bazant ZP, Cedolin L. Stability of structures, elastic, inelastic, fracture and damage theories. Dover Publications Inc.; 2003. [3] Galambos TV, Surovek AE. Structural stability of steel: concepts and applications for structural engineers. John Wiley & Sons Inc.; 2008. [4] Galambos TV. Guide to design stability criteria for metal structures. 5th ed. John Wiley & Sons Inc.; 1998.
63
[5] Chen WF, LUI EM. Elastic stability, theory and implementation. Elsevier; 1987. [6] M. Hetenyi, Beams on elastic foundation, theory with application in the fields of civil and mechanical engineering, Ann Arbor: The University of Michigan press. [7] M. Mekhloufi, Nonlinear analysis of beams on inelastic foundation, MSc thesis (in French, can be made available in English). [8] Djellab SE. Nonlinear finite element analysis of reinforced concrete coupled shear walls. Glasgow University; 1991 [M.Sc thesis]. [9] Adman R, Saidani M. Elastic buckling of columns with end restraint effects. J Constr Steel Res 2013;87:1–5.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Comment
Comments and discussion: “Elastic buckling of columns with end restraint effects” S.E. Djellab University of Science and Technology Houari Boumediene, Faculty of Civil Engineering, Algiers, Algeria
Adman and Saidani tackled a simple classical structural stability problem for beam-columns, elastically restrained at the ends. They established without demonstration a so-called general stability criterion to be used for an eventual elastic buckling analysis of a global structure. Firstly, the very unsettling results obtained by the authors, especially when compared to Eurocode 2, are either an innovative novelty or some other proposal. Secondly, these results contradict previous work by the writer as a co-supervisor [7]. This work used a completely different, more general, matrix approach different to the authors so that no confusion can be made between the two approaches. These are the main reasons that have stimulated the presented discussion. Unfortunately, the authors inconsistent formulation of the problem and the so-called general stability criterion was, in the writer's opinion, erroneous, misleadingly used and confusingly interpreted, as will be demonstrated below. The writers objectives are three fold: (a) A suggested correction to the authors approach, refocusing it in the appropriate direction. (b) To enlighten designers of the ‘robustness’ of the Eurocode formulae. (c) Questioning the proposed Eurocodes formulae is not an easy task: Experts have carefully considered their approach involving special task teams in: field design practices, experimental laboratory setups, theoretical background and numerical knowledge. Constrained to remain within the context of elastic stability, the primary topic of the paper by Adman and Saidani in the present discussion is organised around three main points to be efficient: 1. The main areas for discussion 1.1. The formulation of the problem Starting from a second order ordinary differential equation Eq. (3) with two unknown functions y(x) and M(x) is, in the writer's opinion, not the right way to approach the problem. The quoted ‘appropriate boundary conditions’ Eq. (8) presented by the authors associated with Eq. (3) are inappropriate. The problem at hand is by no means concerned about conditions placed on displacements and rotations at nodes, please see Eq. (8). Furthermore the displacement and rotation is the same for the beam-column and the translational/rotational stiffnesses at the beam ends, the sums in Eq. (8) have no physical meaning. Conditions on shear forces and moments at nodes must be imposed. Since the shear force is a third order derivative of the displacement 0143-974X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jcsr.2013.08.001
y(x), the authors neither can nor did in the second order ODE Eq. (3) associated with Eq. (8). A fourth order ODE has to be used with its appropriate conditions as indicated below. Lastly, since the element is uniform (not tapered), permuting the indices i, j in Eq. (12) should not change the criteria and this is not the case. 1.2. The use of the stability criteria From the first stability analysis principle [1–6], the roots of a characteristic equation or a stability criterion are the eigenvalues or buckling loads of the system. Numerical results such as ‘infinitely large displacements’ in a finite structure is a serious basic physics question, as the question raised about the length of the element and material behaviour. To the best of the writer's knowledge, no such field has yet been observed or computed even in astrophysics. So, no need to compute the ‘infinitely large displacements’ in one analysis and then using these in the erroneous ‘general stability criterion’ Eq. (12). It is sufficient to compute the smallest of the roots of the characteristic equation and the sought solution is obtained. If a displacement approach is adopted, then material nonlinearity has to be considered so and only then, that a stability-elastoplastic collapse of the system can be viewed and physically interpreted. The authors did not indicate under what loading they obtained the ‘infinitely large displacements’ though. However, it is well established that, in a second order elastic analysis, the critical load is independent on any lateral load or geometric imperfections that may be used [9]. 1.2.1. Analytical demonstration The first author is undoubtedly well aware of the existence of ref. [7]. In ref. [7], an exact beam-column on a two parameter elastic foundation of Pasternak type was formulated. The analyses involved either or both material (elastoplastic) and geometric (stability) nonlinearities for either reinforced concrete or steel beams. Furthermore, continuous elastic support underneath the beam and/or discretely lumped rigid or elastic (translational/rotational) supports at the nodes can be handled in a straightforward manner. It started from a fourth order differential equation and its exact homogeneous solution was used to set up the exact shape functions to deduce the exact second order stiffness matrix. It can be easily shown that the correct stability criterion for a general beam-column with elastic end restraints is shown in Fig. 1. Starting from potential energy principles in integral form, invoking Euler–Lagrange equations a fourth order differential equation with its very appropriate boundary conditions is obtained. With the homogenous
S.E. Djellab / Journal of Constructional Steel Research 91 (2013) 60–63
i P
j
Rj
Ri
61
P
Kj
Ki L
Fig. 1. A beam-column with elastic restraints at the ends.
Table 1 Sample classical results. Exact γ
Eq. (1) α
Eq. (12) β [9]
π
3.1416
3.1416
π/2
1.5708
2.7984
π/2
1.5708
1.3138
solution and the boundary conditions written in matrix form, one obtains the correct general stability criterion as: 2 4 α 2ri rj ki kj þ ð−2ri rjki kj−ki kj ðri þ rjÞα þ ðki þ kjÞðri þ rjÞα Cos ½α 2 4 þ α ðri þ rj−ri rjÞki kj þ ðki kj þ ri rj ðki þ kjÞÞα −ðki þ kjÞα Sin ½α Þ
ð1Þ Where ki, kj, ri and rj are the adimensional elastic translational and rotational stiffnesses at the ends. Eq. (1) is a transcendental equation with an infinite number of roots. The roots are obtained using special numerical techniques so that no root is missed, or simply by a graphic plot of Eq. (1). From the smallest root α we deduce the first buckling load. The higher modes, if needed, are obtained by computing the remaining roots. In deriving the above correct stability criterion, no claim is made of originality or ‘novelty’ as the authors did, since it is a very simple stability problem in the writer's opinion. 2. Numerical demonstrations Below are some basic stability problems with or without elastic restraints at the ends. First: Sample classical results without elastic restraints are summarised in Table 1. Great care has been undertaken to confront Eq. (12) in ref. [9], Eq. (1) and the exact value of the first
buckling load γ2EI/L2. We already note the discrepancies between the simple classical cases and the proposed Eq. (12). Secondly, other samples of elastically restrained cases can be used as benchmarks. Classical problems as indicated in Tables 2 and 3 are examined. In Table 2, a sway cantilever with a partially fixed base and free at the other end is investigated. In Table 3, a cantilever fixed at the base and a translation elastically restrained at the other end. By varying the translational stiffness at the top from infinitesimal to large, representing the two extreme cases of fixed-free to fixed-pinned respectively, the numerical results are given. The numerical results of a beam-column partially fixed at the base and guide with full rotational restraint at the top are given in Table 4 and plotted in Fig. 2. In front of the harmony between the ‘smart’ Eurocode formulae and the numerical ones as shown, we note: The Eurocode superiority as less effort is needed to obtain such a practical accuracy. Finally, without a single validation example from the literature, the authors [9] challenged directly the formulation used in the Eurocode. The continuous Eurocode equations are represented discretely in Figs. 4, 5 and 6 while the discrete erroneous solution is plotted continuously. In view of the non-convincing ‘virtually identical results’ as quoted by the authors in Fig. 4 and the even worst discrepancies of Figs. 5 and 6, no further comments about the authors' results in accordance with the various tests above will be made since the authors' formulation failed to model the simplest standard cases. The general comments below are given: – It is demonstrated that the Eurocode formulae are superior to any complex second order elastic analytical solution even if it is correct, achieved by their simplicity and proven accuracy. – Civil engineering designers tackling stability problems must be comforted in regards to the confusing conclusions of the discussed paper. – The confusing conclusion means that, either the Eurocode underestimates the buckling load or the buckling load should be that obtained by the authors.
Table 2 A cantilever partially fixed at the base and free at one. fir
ri
Present α
1000 100 10 8 5 4 2 1 0.5 0.2 0.1 0.01 0.001
0.001 0.01 0.1 0.125 0.2 0.25 0.5 1 2 5 10 100 1000
[9]
2 Pcr αL2EI
0.03162 0.09983 0.31105 0.34645 0.43284 0.48009 0.65327 0.86033 1.07687 1.31384 1.42887 1.55525 1.56923
β 0.001 0.010 0.097 0.120 0.187 0.230 0.427 0.740 1.160 1.726 2.042 2.419 2.462
Exact Pcr ¼
2.798 2.798 2.798 2.798 2.798 2.798 2.798
β 2 EI L2
Pcr 7.8288 – – – – – – – – – – – 7.8288
Ri/L = 0.001EI/L2
π2 EI 4L2
62
S.E. Djellab / Journal of Constructional Steel Research 91 (2013) 60–63
Table 3 A cantilever fixed at one end and displacement partially restrained at the other. fjy
kj
Present study α
100 10 8 5 4 2 1 0.5 0.2 0.1 0.01 0.001
0.01 0.1 0.125 0.2 0.25 0.5 1 2 5 10 100 1000
Exact
Ref [9]
K
K ¼ βπ
2.0 – – – – – – – – – – 0.7
1.12243833 1.12059663 1.12008922 1.11857772 1.11757895 1.11269446 1.10342826 1.08662013 1.04624198 0.99897693 0.8013531 0.71336603
Eurocode
Eq. (12) 9]
K
K ¼ βπ
1.999 1.9901 1.90909 1.88889 1.83333 1.80000 1.66667 1.50000 1.33333 1.16667 1.09091 1.0099 1.001
2.39020714 2.48260894 2.71301731 2.75184837 2.84856117 2.90256632 3.11017984 3.39017741 3.75674005 4.39183345 4.99514676 7.86402793 12.4829347
K ¼ απ
1.57181 1.59485 1.60119 1.62004 1.63248 1.69318 1.80791 2.01581 2.52425 3.14492 4.42795 4.48437
1.99871 1.96984 1.96204 1.93921 1.92443 1.85544 1.73769 1.55848 1.24456 0.99894 0.70949 0.70056
β 2.7989 2.8035 2.80477 2.80856 2.81107 2.82341 2.84712 2.89116 3.00274 3.14481 3.92036 4.4039
Table 4 A beam-column partially fixed at the base and guided with full rotational restraint at the top. fir
ri
Present α
kj ri
0
Ri EI L
1000 100 10 8 5 4 2 1 0.5 0.2 0.1 0.01 0.001
0.001 0.01 0.1 0.125 0.2 0.25 0.5 1 2 5 10 100 1000
π α
K¼
1.57143 1.57714 1.63199 1.64656 1.68868 1.71551 1.8366 2.02876 2.28893 2.65366 2.86277 3.11049 3.13845
1.99919351 1.99195547 1.92500729 1.90797338 1.86038365 1.83128787 1.71054811 1.54852849 1.37251583 1.18387158 1.09739611 1.00999928 1.00100134
– Awareness and care must be taken when questioning well established principles. One has to be rigorous, concise with a full explanation and clear demonstrations proving the case. – From another writer's work [8] where it is still remembered that a simple RC beam failing in shear was quiet a defying challenge. One
1.8
1.6
K
Eurocode Equation (1)
1.4
1.2
1.0 0.01
0.1
1.31436 1.26544 1.15797 1.14163 1.10287 1.08235 1.0101 0.926675 0.836255 0.715326 0.628929 0.399489 0.251671
has to read the outstanding Vecchio and Collins international competition made for respectful international experts at that time. – The writer believes convincingly that: besides the present discussion, any elastic analysis even if it is correct should never be used beyond its limitations and context.
2.0
1E-3
β
1
10
ri Fig. 2. Effective length factor.
100
1000
S.E. Djellab / Journal of Constructional Steel Research 91 (2013) 60–63
References [1] Timoshenko SP. Theory of elastic stability. McGraw-Hill; 1961. [2] Bazant ZP, Cedolin L. Stability of structures, elastic, inelastic, fracture and damage theories. Dover Publications Inc.; 2003. [3] Galambos TV, Surovek AE. Structural stability of steel: concepts and applications for structural engineers. John Wiley & Sons Inc.; 2008. [4] Galambos TV. Guide to design stability criteria for metal structures. 5th ed. John Wiley & Sons Inc.; 1998.
63
[5] Chen WF, LUI EM. Elastic stability, theory and implementation. Elsevier; 1987. [6] M. Hetenyi, Beams on elastic foundation, theory with application in the fields of civil and mechanical engineering, Ann Arbor: The University of Michigan press. [7] M. Mekhloufi, Nonlinear analysis of beams on inelastic foundation, MSc thesis (in French, can be made available in English). [8] Djellab SE. Nonlinear finite element analysis of reinforced concrete coupled shear walls. Glasgow University; 1991 [M.Sc thesis]. [9] Adman R, Saidani M. Elastic buckling of columns with end restraint effects. J Constr Steel Res 2013;87:1–5.