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Large deflection and postbuckling behavior of Timoshenko beam–columns with semi-rigid connections including shear and axial effects
Engineering Structures 29 (2007) 991–1003 www.elsevier.com/locate/engstruct
Large deflection and postbuckling behavior of Timoshenko beam–columns with semi-rigid connections including shear and axial effects J. Dario Aristizabal-Ochoa ∗ School of Mines, National University, A.A. 75267, Medell´ın, Colombia Received 8 June 2006; received in revised form 24 July 2006; accepted 26 July 2006 Available online 20 September 2006
Abstract The nonlinear large deflection-small strain analysis and postbuckling behavior of Timoshenko beam–columns of symmetrical cross section with semi-rigid connections subjected to conservative and non-conservative end loads (forces and moments) including the combined effects of shear, axial and bending deformations, axial load eccentricities, lateral bracing and out-of-plumbness are developed in a simplified manner. A new set of stability functions based on the “modified shear equation” that includes the effects of shear deformations and the shear component of the applied axial forces is derived. Also, an expression for the axial displacement δb caused by the “bowing” of the beam–column subjected to end forces and moments with generalized end conditions is derived in a classic manner. The proposed method and corresponding nonlinear equations, although approximate, can be used in the tension and compression stability and nonlinear large deflection-small strain elastic analyses of Timoshenko beam–columns with rigid, semi-rigid, and simple connections. Analytical studies indicate that shear deformations increase the longitudinal and transverse deflections and reduce the buckling axial load capacities of beam–columns. The effects of shear deformations must be considered in the analysis of beam–columns with relatively low effective shear areas (like in short laced columns, columns with batten plates and with open webs) or low shear stiffness (like elastomeric bearings and short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E making the shear stiffness G As of the same order of magnitude as E I / h 2 ). The shear effects are also of great importance in the tension and compression stability and dynamic behavior of laminated elastomeric bearings used for seismic isolation of buildings. Four comprehensive examples are included that show the effectiveness of the proposed method and equations. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Beams; Beam–columns; Large deflections; P–∆ analysis; Nonlinear analysis; Semi-rigid connections; Stability; Structural analysis; Postbuckling behavior; Timoshenko beams
1. Introduction Advances in composite materials of high resilience capacities and low shear stiffness as well as the need for lighter and stronger structural members have created a great interest in their large-deflection analysis and postbuckling behavior. For instance, seismic isolators made of light polymer materials may undergo extremely large deflections under axial and transverse loads without exceeding their elastic limit. The nonlinear geometric elastic behavior of a slender beam–column with semi-rigid connections under conservative ∗ Corresponding address: National University of Colombia, Civil Engineering, Calle 9c # 15-165, Casa 2 Urbanizacion, Villaverde, Medell´ın, Antioquia, Colombia. Tel.: +57 42686218; fax: +57 44255152. E-mail address: [email protected].
c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.07.012
and nonconservative end loads has been investigated by Aristizabal-Ochoa [1,2] using the classical stability functions and the exact expression for the curvature in the differential equation of the deflection curve (i.e., the “Elastica” approach), respectively. Approximate methods based on the Finite Element Method (FEM) with large deflections and with or without large strains have been utilized by other researchers like Torkamani et al. [3] to solve the nonlinear geometric elastic behavior of slender beam–columns and frames with rigid connections. However, the exact solution for the large deflection behavior of Timohenko beam–columns under any type of end loads (conservative and nonconservative) including the combined effects of bending, shear and axial forces and their corresponding deformations is not known yet. The main objective of this publication is to present an approximate and practical method for the nonlinear large
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Nomenclature A cross-sectional area of the beam–column As effective shear area of the beam–column E and G elastic and shear moduli of the material, respectively Hz “bowing” bending function given by Eqs. (5) and (6) h initial length of the beam–column AB hr chord length of the beam–column AB after “bowing” caused by bending and shear actions I principal moment of inertia of the beam–column about its axis of bending M external bending moment applied at A Ma and Mb end moments of beam–column at A and B, respectively P applied vertical load at A (+ compression, − tension) Pe = π 2 E I / h 2 Euler load Pt axial load along the cord AB (+ compression, − tension) Q applied transverse load at A Ra and Rb stiffness indexes of the flexural connection at A and B, respectively ∆h and ∆v horizontal and vertical deflections of end A of beam–column AB, respectively κa and κb the flexural stiffness of the end connections at A and B, respectively ρa and ρb fixity factors at A and B of column AB, respectively θ angle of the cord AB with respect to the vertical axis (Fig. 1(a)) θo initial out-of-plumb angle with respect to the vertical axis (Fig. 1(a)) θa0 and θb0 end slopes of member AB measured with respect to its cord (Fig. 1(b)) u(x) lateral deflection of the column center line as shown by Fig. 2(c) ψ(x) rotation of the cross section as shown by Fig. 2(c) ∆a and ∆b lateral sway at A and B, respectively ∆ = ∆a − ∆b relative sidesway of column end A with respect to its bottom end B ψa and ψb rotations of cross sections at A and B due to bending, respectively deflection-small strain analysis and postbuckling behavior of Timoshenko beam–columns of symmetrical cross section with semi-rigid connections subjected to end loads and moments (conservative or nonconservative) including the combined effects of: (1) bending, axial and shear deformations; (2) the shear component of the applied axial forces (using the “modified shear equation” presented by Timoshenko and Gere [4]); (3) out-of-plumbness of the member’s longitudinal axis; and (4) axial load eccentricities. Also, an expression for the axial displacement δb caused by the “bowing” of the Timoshenko beam–column with generalized end conditions
subjected to end forces and moments is derived in a classic manner. The proposed method and corresponding nonlinear equations, although approximate, can be used in the tension and compression stability and postbuckling behavior as well as in the nonlinear large deflection-small strain elastic analyses of Timoshenko beam–columns with rigid, semi-rigid, and simple connections. The proposed method and derived equations are extensions of those previously proposed by AristizabalOchoa [1] for slender beam–columns. The proposed method and equations can also be utilized in the large deflectionsmall strain inelastic analysis and postbuckling behavior of Timoshenko beam–columns whose connections suffer from flexural degradation or, on the contrary, flexural stiffening. The advantages of the proposed method are: (1) the effects of semi-rigid connections are condensed into a single nonlinear equation with a single unknown (for tension or compression axial loads) without introducing additional degrees of freedom and equations; (2) the proposed method, which is based on the “modified shear equation” for Timoshenko beam–columns with semi-rigid connections, is more accurate than any other approximate method available and capable of capturing the phenomena of tension buckling in members with low shear stiffness; (4) the method can be incorporated into the large deflection elastic analysis and postbuckling behavior of Timoshenko beam–columns without major difficulties; and (5) extension of the method to general elastic–plastic analysis requires the development and solution of incremental equations and the handling of the spread of plasticity throughout the volume of the members, including the interaction between normal and shear stresses. This is a complex extension and it is beyond the scope of this paper. Four comprehensive examples are included that show the effectiveness of the proposed method and equations. 2. Structural model 2.1. Assumptions Consider a 2-D prismatic beam–column that connects points A and B as shown in Fig. 1(a). The element is made up of the beam–column itself AB and the flexural connections κa and κb at the top and bottom ends, respectively. It is assumed that the member AB of span h bends about the principal axis of its cross section with a moment of inertia I , gross sectional area A, and effective shear area As and: (1) is made of a homogeneous linear elastic material with Young and shear moduli E and G, respectively; (2) its centroidal axis is a straight line with an initial out-of-plumbness θo with respect to the vertical axis; and (3) is loaded at extreme A with P (vertical load), Q (transverse load), and M (M = Pe + M 0 , where M 0 is an additional moment applied at A). Each one of these three loads can vary independently and are applied in the plane of bending. The lateral bracing at A is provided by a linear displacement spring of magnitude S∆ with one extreme connected to end A of the beam–column and the other end on vertical rollers that slide on a frictionless rigid vertical wall as shown in Fig. 1(a). The dimensions of S∆ are in force/distance. The flexural
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Fig. 1. Model of column with sidesway partially inhibited and with rotational and lateral end restraints. (a) Structural model with an initial out-of-plumb angle θo ; (b) end moments, forces, rotations and deflection moments in the plane of bending; and (c) internal forces and external loads on deformed element.
connectors have stiffnesses κa and κb (whose dimensions are in force × distance/radian) in the plane of bending of the beam–column. The ratios Ra = κa /(E I / h) and Rb = κb /(E I / h) are denoted as the stiffness indices of the flexural connections. These indices vary from zero (i.e., Ra = Rb = 0) for simple connections (i.e., pinned) to infinity (i.e., Ra = Rb = ∞) for fully restrained connections (i.e., rigid). Notice that the proposed algorithm can be utilized in the inelastic analysis of beam–columns when the inelastic behavior is concentrated at the connections. This can be carried out by updating the flexural stiffness of the connections κa and κb for each load increment in a linear-incremental fashion or including the corresponding moment–rotation relationship of the connection as shown by Aristizabal-Ochoa [1,2]. For convenience the following two parameters are introduced: ρa =
1 1+
3 Ra
and
ρb =
1 1+
3 Rb
(1a–b)
where ρa and ρb are called the fixity factors. For perfectly hinged connections, both the fixity factor ρ and the rigidity index R are zero; but for rigid connections (i.e., perfectly clamped), the fixity factor is 1 and the rigidity index is infinity. Since the fixity factor can only vary from 0 to 1 for elastic connections (while the rigidity index R may vary from 0 to ∞), it is more convenient to use in the elastic analysis of structures with semi-rigid connections. 3. Proposed method and equations The large deflection-small strain equations are developed in the plane of bending about one of the principal axes. The proposed method consists in solving simultaneously the
nonlinear equations (2) and (3). Ma + M b + P sin θ + Q − S∆ h h hr hr sin θ − sin θo cos θ =0 × h h Pt Hz hr / h = 1 − − E A 4Pt2 h 2
(2) (3)
where: hr Pt = P cos θ + S∆ h sin θ − sin θo − Q sin θ. h By substituting Eq. (3) into Eq. (2), only the following nonlinear equation has to be solved: Ma + M b + P sin θ + Q − S∆ h h Pt Hz × 1− − sin θ − sin θo cos θ E A 4Pt2 h 2 Pt Hz − = 0. (4) × 1− E A 4Pt2 h 2 Eq. (2) represents the condition of moment equilibrium of the beam–column shown in Fig. 1(b) about B, and Eq. (3) the geometric condition of the actual distance between the two extremes of the beam–column (expression that includes the effects of the axial strain and the “bowing” caused by its bending and shear deformations). Notice that the axial load on the beam–column Pt is taken as the sum of the actual components of the vertical and horizontal forces at A along its chord. The magnitudes of the end moments Ma and Mb and the “bowing” factor Hz depend on the actual sign of Pt , as follows:
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Ma = −
3ρa [(1 − ρb ) + 3ρb tan (φ/2) /φ] (θ − θo ) EhI βφ 2 − (1 − ρa ) (1 − ρb ) βφ 2 + 3ρb (1 − βφ/ tan φ) M (1 − ρa ) (1 − ρb ) βφ 2 + 3 (ρa + ρb − 2ρa ρb ) (1 − βφ/ tan φ) + 9ρa ρb β [tan (φ/2) / (βφ/2) − 1] Box I.
Mb = −
3ρb [(1 − ρa ) + 3ρa tan (φ/2) /φ] (θ − θo ) EhI βφ 2 + 3ρb (1 − ρa ) (1 − βφ/ sin φ) M (1 − ρa ) (1 − ρb ) βφ 2 + 3 (ρa + ρb − 2ρa ρb ) (1 − βφ/ tan φ) + 9ρa ρb β [tan (φ/2) / (βφ/2) − 1] Box II.
3ρa [(1 − ρb ) + 3ρb tanh (φ/2) /φ] (θ − θo ) EhI βφ 2 − (1 − ρa ) (1 − ρb ) βφ 2 − 3ρb (1 − βφ/ tanh φ) M Ma = − (1 − ρa ) (1 − ρb ) βφ 2 + 3 (ρa + ρb − 2ρa ρb ) (1 − βφ/ tanh φ) + 9ρa ρb β [tanh (φ/2) / (βφ/2) − 1] Box III.
Mb =
3ρb [(1 − ρa ) + 3ρa tanh (φ/2) /φ] (θ − θo ) EhI βφ 2 − 3ρb (1 − ρa ) (1 − βφ/ sinh φ) M − (1 − ρa ) (1 − ρb ) βφ 2 + 3 (ρa + ρb − 2ρa ρb ) (1 − βφ/ tanh φ) + 9ρa ρb [tanh (φ/2) / (βφ/2) − 1] Box IV.
(a) For Pt ≥ 0 (compression) and β > 0 or Pt < 0 (tension) and β < 0: See equations in Boxes I and II. 2−β φ H Z = βφ (Ma2 + Mb2 ) cos φ + sin φ sin φ 2−β + 2βφ Ma Mb sin φ × (1 + φ/ tan φ) − 2(Ma + Mb )2 .
(5)
(b) For Pt < 0 (tension) and β > 0: See equations in Boxes III and IV. φ 2−β H Z = βφ(Ma2 + Mb2 ) cosh φ + sinh φ sinh φ 2−β + 2βφ Ma Mb sinh φ × (1 + φ/ tanh φ) − 2(Ma + Mb ) (6) q where: φ = |Pt / β E I / h 2 | and β = 1/[1 + Pt /(G As )]. Although in Eqs. (2) and (4) the lateral bracing is assumed to be a linear spring (linear with respect to the horizontal deflection of end A) on vertical rollers, so that vertical forces are not induced at any stage of lateral deflections, a nonlinear bracing with different end conditions can be used in the proposed model, as long as its induced vertical and horizontal forces are known as a function of the vertical and horizontal deflections of end A. Equations given in Boxes I–IV and Eqs. (5), (6) for Ma , Mb and Hz are derived in Appendix A. Eq. (2) could be solved for θ in two ways: a simplified one, by neglecting any change in the length of the cord AB (i.e., by making h r / h = 1 at all times); or in a complete manner, using Eq. (4) which includes any change in the ratio h r / h. Eq. (2) or (4) must be solved for θ or every load increment or for any new 2
problem with a given set of input values P, Q, M, θo , E A, E I , G As , h, and S∆ . Four examples that follow show the effectiveness, simplicity and accuracy of the proposed method. Example 1 is on the large deflection analyses of a cantilever beam–column under separate concentrated horizontal and vertical forces at the top. Example 2 shows the effects of an initial shape of a simply supported beam–column that consists of two identical straightline segments on its large deflection elastic response under eccentric axial loads at the ends. Example 3 addresses the large deflection analyses of a beam–column with both ends restrained against rotation and with sidesway uninhibited subjected to axial and lateral forces. Example 4 shows the application of the proposed method to the stability and postbuckling behavior of cantilever columns subjected to gravity and fixed nonconservative axial loads. 4. Verification and comprehensive examples 4.1. Example 1: Analysis of an out-of-plumb cantilever column Determine both the large-deflection behavior and stability of the cantilever column subjected to concentrated loads P and Q at the top end A as shown in Fig. 2(a). Assume that the member is initially out-of-plumb θo and with properties ρb , G, E, A, As , I and h. Also assume that the applied loads Q and P are always horizontal and vertical, respectively. Include in the analysis the effects of the axial strain Pt /E A on the cord length h r . Solution. In this example M = Ma = S∆ = ρa = 0; then from Eq. (4): Mb Pt Hz + {P sin θ + Q cos θ } 1 − − =0 (7a) h E A 4Pt2 h 2 where Pt = P cos θ − Q sin θ. For Pt ≥ 0 (compression) and β > 0 or Pt < 0 (tension) and β < 0
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Fig. 2. Example 1. Cantilever column semi-rigidly connected at the base B subjected to loads P and Q at the free end: (a) applied loads and properties; and (b) deflected shape.
3ρb (θ − θo ) EhI βφ 2 ; and (1 − ρb )βφ 2 + 3ρb (1 − βφ/ tan φ) 2−β φ 2 H Z = Mb βφ cos φ + −2 . sin φ sin φ
Mb = −
For Pt < 0 (tension) and β > 0 3ρb (θ − θo ) EhI βφ 2 ; and −(1 − ρb )βφ 2 + 3ρb (1 − βφ/ tanh φ) 2−β φ H Z = Mb2 βφ cosh φ + −2 . sinh φ sinh φ
Mb =
Substituting these expressions into Eq. (7a) (taking into p account that φ = |Pt /(β E I / h 2 )| and β = 1/[1 + Pt /(As G)]), and including the axial strain effects (Pt /E A), θ can be obtained from Eq. (7a). The following iterative process is suggested to determine the values of θ, ∆v , and ∆h for a given value of the applied loads P and Q: (1) Knowing the value of the applied loads P and Q and the input data θo , G, E, A, As , I and h, assume a trial value θ . (2) Calculate the values of Pt , β = 1/[1 + Pt /(As G)], φ = p |Pt /(β E I / h 2 )|, Mb and Hz . (3) Check if Eq. (7a) is fulfilled, otherwise select a new value for θ and return to step 2. Use the bisection method to determine the value of θ for the given applied loads P and Q and input data θo , G, E, A, As , I and h. (4) The vertical and horizontal deflections at the tip of the beam are approximately as follows: Hz Pt − h sin θ (8a) ∆h = 1 − E A 4Pt2 h 2
Fig. 3. Example 1. Transversal and vertical deflections of a perfectly clamped cantilever beam–column (ρb = 1) subjected to transverse load Q and P = 2E I /8h 2 with: (a) As G/(E I / h 2 ) = ∞; (b) As G/(E I / h 2 ) = 10; (c) As G/(E I / h 2 ) = 5.
∆h . (8b) tan θ Fig. 3(a)–(c) show the variations of ∆h / h and ∆v / h against the lateral load parameter Q/(E I / h 2 ) for three different values of G As /(E I / h 2 ) as indicated in the figures for a particular case of a cantilever column subjected to a constant vertical compressive load P of magnitude Pe /8 = π 2 E I /(8h 2 ) assuming that S∆ = θo = 0, ρb = 1 and E A = ∞. The results obtained with the proposed iterative method are compared to those obtained using Eqs. (14) and (15) for a perfectly clamped–clamped beam–column described in Example 3 (substituting h for 2h and with the corresponding values of ∆v and ∆h divided by two. These results are indicated in Fig. 3(a)–(c) as “Small Deflection Theory”). Notice that ∆v = h −
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Fig. 4. Example 1. Large-deflection stability of a cantilever column (with ρb = 1 and θo = 0) subjected to axial load P at A. Effects of shear deformations on the axial and lateral deflections at the free end for four different values of As G/(E I / h 2 ).
calculations based on small-deflection theory have a rather limited range of applicability, particularly for members made of high resilient materials like advanced composites (FRP). Fig. 4 shows the effects of the shear deformations on the large-deflection stability of a perfectly vertical cantilever column (with θo = 0 and ρb = 1) under vertical compressive load P. Notice that the deformation mode is in the range of 0 < ∆v / h < 1.5. Fig. 5(a)–(c) show the effects of shear deformations and factor of fixity at the base on the largedeflection stability of a perfectly vertical cantilever column (θo = 0) under vertical compressive load P. Fig. 6(a)–(c) show the effects of the factor of fixity at the base on the largedeflection behavior of a perfectly vertical cantilever column (θo = 0) subjected to tension load P and lateral load Q = 0.3π 2 E I / h 2 . Notice that the proposed method is capable of capturing the “phenomena” of tension buckling of Timoshenko beam–columns (Kelly [5] and Aristizabal-Ochoa [6,7]), as well as its postbuckling response. The axial shortening caused by P was neglected (i.e., Pt /E A = 0) in these analyses. Figs. 3(a)–(c), 4, 5(a)–(c), and 6(a)–(b) show that the largedeflection and stability analysis of a beam–column are affected by the end boundary conditions, the applied loads, the shear deformations along the member, and the shear force induced by the axial force as the member deforms. 4.2. Example 2: Large-deflection behavior of a simple supported beam–column with an initial imperfection subjected to an eccentric axial load Determine the large deflection-small strain nonlinear equation of a simple supported beam–column subjected to eccentric end axial loads as shown in Fig. 7(a). Assume that its initial shape consists of two identical straight-line segments (with properties G, E, A, As , I and length h) both making an angle θo with the cord. Bending occurs in the plane of the two segments about their cross-section major axis. Compare
the calculated results with those presented by Timoshenko and Gere [4] for small deflections and E A = G As = ∞. Include the effects of the axial strain Pt /E A on the cord length h r . Solution. Taking only one segment as shown in Fig. 7(b), then Q = S∆ = 0, M = Pe; ρa = 0 and ρb = 1; then from Eq. (4): (Ma + Mb ) Pt Hz + 1− − P sin θ = 0 (9) h E A 4Pt2 h 2 where Pt = P cos θ ≥ 0 (compression); Ma = Pe; Mb = θ +(1−βφ/ sin φ)(e/ h) − (θ −θo ) cos(1−βφ/ Ph; and tan φ) φ 2−β 2 2 H Z = βφ(Ma + Mb ) cos φ + sin φ sin φ 2−β + 2βφ Ma Mb sin φ × (1 + φ/ tan φ) − 2(Ma + Mb )2 . Substituting the expressions above for p Ma and Mb into Eq. (9) (taking into account that φ = |Pt /(β E I / h 2 )| and including the axial strain effects Pt /E A), θ can be obtained from Eq. (10): −(θ − θo ) cos θ +
1−cos φ sin φ βφ (e/ h)
(1 − βφ/ tan φ) Pt Hz + 1− − sin θ = 0. E A 4Pt2 h 2
(10)
Eq. (10) represents the large deflection-small strain behavior of the beam–column shown in Fig. 7(b) [after the expression for Hz is substituted in Eq. (10)]. With the solution for θ obtained from Eq. (10), the deflections ∆Lateral = [(1 − EPAt − Hz Pt 2 2 ) sin θ − sin θo ]h and ∆Axial = [cos θo − (1 − E A − 4Pt h Hz ) cos θ ]h 4Pt2 h 2
can be calculated directly.
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Fig. 5. Example 1. Large-deflection stability of a cantilever column subjected to axial compressive load P at A. Effects of shear deformations and fixity factor at the base on the axial and lateral deflections at the free end for three different values of As G/(E I / h 2 ).
Fig. 8(a)–(c) shows the effects of the shear stiffness on the large-deflection response of the column (for θo = 0.1, ρb = 1 and AE = ∞) under the end load P (compression) and
M = Pe for three different values of e/ h (0.05, 0.1, and 0.2). As expected, shear deformations and the axial load eccentricity increase both deflections.
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Fig. 6. Example 1. Large-deflection response of a cantilever beam–column subjected to tension load P and lateral load Q = 0.3Pe at the free end. Effects of shear deformations and fixity factor at the base on the axial and lateral deflections at the free end for two different values of As G/(E I / h 2 ).
β = 1), Eq. (10) is reduced to −(θ − θo ) +
or
1−cos φ sin φ φ (e/ h)
+θ =0 (1 − φ/ tan φ) 1 e sin φ θ= θo + (1 − cos φ) . cos φ φ h
(11) (12)
Expression (12) is identical to that obtained using the method presented by Timoshenko and Gere [4] for small deflections and strains using the classic stability functions that include bending deformations only.
Fig. 7. Example 2. Simple supported beam–column with an initial imperfection subjected to axial load P and end moment M = Pe.
Now, for small deflections and neglecting axial and shear deformations (i.e., small values of θ, E A = G As = ∞, and
4.3. Example 3: Large deflection analysis of a beam–column with sidesway uninhibited and both ends restrained Determine the large deflection-small strain nonlinear equations for the horizontal and vertical deflections of the beam–column (Fig. 9) restrained at both ends and with an
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Fig. 8. Example 2. Effects of shear deformations on the axial and lateral deflections at the free end of an out-of-plumb cantilever beam–column (θo = 0.1 and ρb = 1.0) for four different values of As G/(E I / h 2 ) and subjected to axial load P and moment M = Pe: (a) e/ h = 0.05; (b) e/ h = 0.1; and (c) e/ h = 0.2.
initial out-of-plumb angle θo . Assume that: (1) properties of the member are: ρa , ρb , G, E, A, As , I and h; and (2) the applied moment is M = Pe and the loads
P and Q are always vertical and horizontal, respectively. Include the effects of the axial strain Pt /E A on the cord length h r .
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Fig. 9. Example 3. Unbraced beam–column restrained at both ends. (a) Structural model with an initial out-of-plumb angle θo and applied loads P and Q. (b) Deflected shape and top end deflections.
Solution. In this example M = Pe and S∆ = 0, then from Eq. (4): Pt Hz (Ma + Mb ) + 1− − (P sin θ + Q cos θ ) = 0 h E A 4Pt2 h 2 (13) where Pt = P cos θ − Q sin θ. Substituting expressions for Ma , Mb , and Hz given by Boxes I–IV and Eqs. p (5), (6) into Eq. (13) (taking into account that M= Pe, φ = |Pt /(β E I / h 2 )| and β = 1/[1+ Pt /(As G)]) and including the axial strain effects Pt /E A), θ can be obtained using an iterative process similar to that described in Problem (1), and the corresponding values of ∆v and ∆h can be determined from Eqs. (8a) and (8b), respectively. When the beam–column is perfectly vertical (θo = 0), clamped at both ends (i.e., ρa = ρb = 1) and subjected to P and Q only, the solutions for ∆h , and ∆v proposed by Kelly [5] using small deflection theory are as follows: ∆h =
Qh βP
Q2 ∆v = 2β P 2
tan (φ/2) −β φ/2
4.4. Example 4: Large deflection and post bucking behavior of a beam–column subjected to a nonconservative force (Reut’s column) combined with a gravity load Determine both the large-deflection behavior and stability of the cantilever column (Fig. 10) subjected to two vertical concentrated loads: Po (gravity load) and P f (vertical force whose line of action always passes through point B). Assume that: (1) the member is initially out-of-plumb (θo ) with properties ρb , G, E, A, As , I and h; (2) bending occurs about one of its main axis; and (3) the applied loads Po and P f are always vertical and vary independently. Include in the analysis the effects of the axial strain Pt /E A on the cord length h r . Solution. In this example P = Po + P f ; Q = S∆ = 0, M = −(h r sin θ )P f ; ρa = 0 and ρb = 1; then from Eq. (4): (Ma + Mb ) Pt Hz P sin θ = 0 (16) + 1− − h E A 4Pt2 h 2 where Pt = P cos θ ≥ 0 (compression); Pf Hz Pt − Pt h tan θ Ma = M = − 1 − 2 2 E A 4Pt h Po + P f Mb = −
φβ cos φ − 2 sin φ + 2φ − β sin φ . (φ/ h)(1 + cos φ)
Fig. 10. Perfectly clamped cantilever column subjected to gravity load Po and a fixed force P f at the free end. (a) Applied loads and properties. (b) Deflected shape.
(14)
(15)
Notice that the cantilever beam–column of Problem (1) can be solved using the solution of this problem by simply making ρa = ρb = 0 and substituting h for 2h. The corresponding values of ∆v and ∆h must be divided by two.
(θ − θo ) EhI βφ 2 + (1 − βφ/ sin φ) M ; (1 − βφ/ tan φ)
and 2−β φ cos φ + H Z = βφ(Ma2 + Mb2 ) sin φ sin φ 2−β + 2βφ Ma Mb (1 + φ/ tan φ)− 2(Ma + Mb )2 . sin φ Substituting the expressions above for p Ma and Mb into Eq. (16) (taking into account that φ = |Pt /(β E I / h 2 )| and including the axial strain effects Pt /E A), θ can be obtained
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from Eq. (17): θ − θo Pt Hz = 1− − tan θ E A 4Pt2 h 2 Pf βφ × 1− 1− (1 − 1/ cos φ) . (17) tan φ Po + P f Eq. (17) represents the large deflection-small strain behavior of the beam–column shown in Fig. 9 [after the expression for Hz is substituted in Eq. (17)]. With the solution for θ obtained from Eq. (17), the deflections at the top ∆h = [(1 − Hz Pt Pt 2 2 ) sin θ − sin θo ]h and ∆v = [cos θo − (1 − E A − EA − 4Pt h Hz ) cos θ ]h 4Pt2 h 2
can be calculated directly. Notice that when θ = θo = 0, Eq. (17) is reduced to the characteristic equation: cos φ = −P f /Po . This result is identical to Eq. (14c) reported by Bolotin [8] (p. 103) and Aristizabal-Ochoa [9] (p. 479) for a perfectly clamped cantilever column subjected to a force with a fixed line of action and gravity load. The four examples presented above show that the proposed method is very practical and versatile allowing both the small- as well as the large-deflection analysis of Timoshenko beam–columns subjected to end loads. 5. Summary and conclusions The nonlinear large deflection-small strain analysis and postbuckling behavior of Timoshenko beam–columns of symmetrical cross section with semi-rigid connections subjected to conservative and nonconservative end loads (forces and moments) including the combined effects of shear, axial and bending deformations, axial load eccentricities, lateral bracing and out-ofplumbness are developed in a simplified manner. A new set of stability functions based on the “modified shear equation” that include the effects of shear deformations and the shear component of the applied axial forces is derived. Also, an expression for the axial displacement δb caused by the “bowing” of the member subjected to end forces and moments with generalized end conditions is derived in a classic manner. The proposed method, although approximate, can also be used in the nonlinear large deflection analyses of Timoshenko beam–columns with rigid, semi-rigid, and simple connections subjected to tension forces. The analytical results indicate that the large deflectionsmall strain elastic response and postbuckling behavior of Timoshenko beam–columns of symmetrical cross section is not only affected by the magnitude of the axial load, the lateral drift restraints (i.e., bracing), the degree of fixity of the connections and its elastic properties, but also by the bowing effect caused by the flexural moments and shear deflections. Significant increases in the lateral and axial deflections are caused by the geometric nonlinear effects, particularly those induced by the shear deformations and by the shear component of the axial force. Also the buckling load of beam–columns is reduced by shear deformations and by the shear component of the axial force. These shear effects must be considered in the analysis of beam–columns with relatively low effective
shear areas (like in short laced columns, columns with batten plates or with perforated cover plates, and columns with open webs) or low shear stiffness (like elastomeric bearings and short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E making the shear stiffness G As of the same order of magnitude as E I / h 2 ). The shear effects are also of great importance in the tension and compression stability and dynamic behavior of laminated elastomeric bearings used for seismic isolation of buildings. The advantages of the proposed method are: (1) the effects of semi-rigid connections and shear deformations are condensed into a single nonlinear equation with a single unknown (for tension or compression axial loads) without introducing additional degrees of freedom and equations; (2) the proposed method, which is based on a new set of stability functions for Timoshenko beam–columns with semi-rigid connections, is very practical, versatile and more accurate than any other approximate method available; and (3) extension of the method to general elastic–plastic analysis requires the development and solution of incremental equations and the handling of the spread of plasticity throughout the volume of the members, including the interaction between normal and shear stresses. This is a complex extension and it is beyond the scope of this paper. Four comprehensive examples are included that show the effectiveness of the proposed method and equations. Acknowledgments The research presented in this paper was carried out at the National University of Colombia, School of Mines at Medellin. The author wants to express his appreciation to DIME for the financial support and to David Padilla-Llano and Johnny Moncada-Palacios, members of the Structural Stability Research Group of the National University of Colombia, for carrying out the analytical studies. Appendix A. Derivations of equations given in Boxes I–IV and Eqs. (5), (6) A.1. Expressions for Ma and Mb The stability analysis of a prismatic column including bending and shear deformations (Fig. 1(a)–(c)) is formulated using the “modified shear equation” proposed by Timoshenko and Gere [4] (p. 134). This approach has been utilized by Aristizabal-Ochoa [6,7] in the stability analysis of columns, and Kelly [5] in the analysis of Elastomeric Isolation Bearings. The governing equations are as follows: β E I u 00 (x) + Pu(x) = −Ma − (Ma + Mb + P∆) β E I ψ 00 (x) + Pψ(x) =
Ma + Mb + P∆ h
x h
(18a) (18b)
where u(x) = lateral deflection of the column center line; and ψ(x) = rotation of the cross section as shown by Fig. 1(c). The solutions to the second-order linear differential equations (18a) and (18b) are as follows:
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−
h
Ma =
1 Rb
+
2 βφ (θ − θo )βφ 2 (E I / h) + βφ + 1 − Rb tan φ tan(φ/2) βφ 1 1 + Ra + Rb 1 − tan φ + βφ/2 − 1 β
tan(φ/2) φ
βφ 2 Ra R b
i
M Ra
Box V.
− Mb =
h
1 Ra
+
βφ 2 Ra R b
βφ (θ − θo )βφ 2 (E I / h) + 1 − sin φ tan(φ/2) βφ 1 + Rb 1 − tan φ + βφ/2 − 1 β
tan(φ/2) φ
+
1 Ra
i
M Ra
Box VI.
h Ma =
1 Rb
+
2 φ (θ − θo )βφ 2 (E I / h) − βφ − 1 + Rb tanh φ tanh(φ/2) βφ 1 1 + Ra + Rb 1 − tan φ + − 1 β βφ/2
tanh(φ/2) φ
− Rφa Rb 2
i
M Ra
Box VII.
h Mb =
1 Ra
+
tanh(φ/2) φ
+ − Rβφ a Rb 2
1 Ra
βφ M (θ − θo )βφ 2 (E I / h) + 1 − sinh φ Ra tanh(φ/2) βφ + R1b 1 − tanh φ + βφ/2 − 1 β
i
Box VIII.
x u(x) = A cos φ h x x Ma + M b Ma + B sin φ + +∆ − h h P P x φ ψ(x) = C cos h x 1 Ma + M b + D sin φ + +∆ h h P
ψb =
but ∆ h = (θ − θo ) and (19b)
Ma + M b + ∆ tan(φ/2). P
Since u 0 = ψ + V /(As G) and V = Pψ − Ma +Mhb +P ∆ , the following expressions for ψa and ψb can be obtained: ψa =
(20b)
(19a)
where φ 2 = P/(β E I / h 2 ). The unknown coefficients A, B, C, and D can be obtained from the following boundary conditions: At x = 0: u = 0, ψ = ψa At x = h: u = ∆ and ψ = ψb where ψa and ψb = rotations of cross sections at A and B due to bending, respectively; ∆a and ∆b = lateral sway at A and B, respectively. ∆ = ∆a − ∆b = relative sidesway of column end A with respect to its bottom end B. +Mb Therefore: A = MPa ; B = MPa tan(φ/2) − MPasin φ ; C = M +M a b ψa − h1 + ∆ ; and P Ψa − Ψb cos φ D= − sin φ
Ma sin φ − βφ Mb sin φ − βφ cos φ ∆ + + 2 2 E I / h βφ sin φ E I/h h βφ sin φ
Ma sin φ − βφ cos φ Mb sin φ − βφ ∆ + + 2 2 E I/h E I / h βφ sin φ h βφ sin φ (20a)
Ma − M κa Ma sin φ − βφ cos φ Mb sin φ − βφ = + E I/h E I / h βφ 2 sin φ βφ 2 sin φ Mb θb0 = −(θ − θo ) − κb Ma sin φ − βφ Mb sin φ − βφ cos φ = + E I / h βφ 2 sin φ E I/h βφ 2 sin φ
θa0 = −(θ − θo ) −
(21a)
(21b)
where ψa and ψb = rotations of cross sections at A and B due to bending, respectively; θa0 and θb0 = rotations of ends A and B with respect to the p cord AB, respectively; and φ = |Pt /(β E I / h 2 )|. From Eqs. (21a) and (21b) the end moments Ma and Mb can be expressed in terms of (θ − θo ), M and the fixity ratios (Ra and Rb ) as shown in the equations given in Boxes V and VI, where β = 1/[1 + P/(As G)]; and As = effective shear area of the column. Similarly, when the axial load Pt is negative (tension) and β > 0: See equations in Boxes VII and VIII. Equations in Boxes√ VII and VIII were obtained by replacing φ or iφ (where i = −1) in equations in Boxes V and VI and also taking into account that sin(iφ) = i sinh φ, cos(iφ) = cosh φ, and tan(iφ) = i tanh φ. Equations in Boxes I, II, III and IV can be obtained once Ra = 3ρa /(1 − ρa ) and
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Rb = 3ρb /(1 − ρb ) are substituted into the equations given in Boxes V, VI, VII and VIII, respectively. The elastica is given by Eq. (19a) or x x Ma φ cos sin u(x) = φ + tan φ −1 P h 2 h x Ma + M b +∆ . (22a) × h P Similarly, when the axial load Pt is negative (tension) and β > 0: x x Ma φ u(x) = cosh sinh φ − tanh φ −1 P h 2 h x Ma + M b +∆ . (22b) + h P A.2. Expression for bowing factor Hz The axial displacement δb caused by the “bowing” of the beam–column given by Eq. (23) was derived by Kelly [5]: δb =
1 2
h
Z
2ψ
0
du − ψ 2 dx. dx
(23)
Substituting Eqs. (19a) and (19b) with ∆ = M = 0 into Eq. (23), carrying out the integration and after lots of algebra, the expression for the axial displacement δb caused by the “bowing” of the beam–column can be reduced to: h Hb2 Hb h ψa + ψb Hb tan(φ/2) δb = − + 2 P2 P 2 P βφ/2 φ + sin φ h Hb 2 + −1 4φ P β cos2 (φ/2) Hb h 2 −1 × (ψa + ψb ) + + P 8φ β
×
(2φ−sin 2φ)(ψa2 +ψb2 )−4(φ cos φ−sin φ)ψa ψb sin2 φ
(24)
b where Hb = − Ma +M . h After substituting Eq. (20) into Eq. (24), the expression for δb in terms of Ma and Mb is as follows: 1 δb = − 2(Ma + Mb )2 + βφ(Ma2 + Mb2 ) 4P 2 h 2−β φ × cos φ + sin φ sin φ 2−β φ + 2βφ Ma Mb 1+ . (25) sin φ tan φ
1003
Therefore: 2−β H Z = (4P 2 h)δb = βφ(Ma2 + Mb2 ) sin φ 2−β φ + 2βφ Ma Mb × cos φ + sin φ sin φ × (1 + φ/ tan φ) − 2(Ma + Mb )2 .
(26)
Similarly, when the axial load Pt is negative (tension) and β > 0: 2−β φ 2 2 H Z = βφ(Ma + Mb ) cosh φ + sinh φ sinh φ 2−β + 2βφ Ma Mb sinh φ × (1 + φ/ tanh φ) − 2(Ma + Mb )2 .
(27)
Notice that the expression for the bowing factor Hz given by Eq. (26) is identical to that developed by Ekhande et al. [10] when β = 1 (i.e., when the effects of shear deformations and those of the shear component of the axial force are neglected). The expression for Hz developed by Ekhande et al. [10] was used by Aristizabal-Ochoa [1] in the nonlinear large-deflection small-strain analysis of slender beam–columns with semi-rigid connections. References [1] Aristizabal-Ochoa J Dario. Nonlinear large-deflection small-strain analysis of beam–columns under end loads. Journal of Structural Engineering ASCE 2001;127(1):92–6. [2] Aristizabal-Ochoa J Dario. Large-deflection stability of slender beam–columns with semi-rigid connections: The elastica approach. Journal of Engineering Mechanics ASCE 2004;274–82. [3] Torkamani MAM, Sonmez M, Cao J. Second-order elastic plane-frame analysis using finite-element method. Journal of Structural Engineering ASCE 1997;123(9):1225–35. [4] Timoshenko S, Gere J. Theory of elastic stability. 2nd ed. McGraw-Hill; 1961 [Chapter II]. [5] Kelly JM. Tension buckling in multilayer elastomeric bearings. Journal of Engineering Mechanics ASCE 2003;129(12):1363–8. [6] Aristizabal-Ochoa J Dario. Tension buckling in multilayer elastomeric bearings by James M. Kelly. Journal of Engineering Mechanics ASCE 2003;129(12):1363–8. Discussion: 131(1):106–8. [7] Aristizabal-Ochoa J Dario. Column stability and minimum lateral bracing: Effects of shear deformations. Journal of Engineering Mechanics ASCE 2004;130(10):1223–32. [8] Bolotin VV. Nonconservative problems of the theory of elastic stability. 60 Fifth Avenue, New York (NY): MacMillan; 1963. p. 103. [9] Aristizabal-Ochoa J Dario. Stability of beam–columns with semirigid connections under conservative and non-conservative end axial forces: Static method. Journal of Engineering Mechanics ASCE 2005;131(5): 473–84. [10] Ekhande SG, Selvappalam M, Madugula MKS. Stability functions for three-dimensional beam columns. Journal of Structural Engineering ASCE 1989;115(2):467–79.
Large deflection and postbuckling behavior of Timoshenko beam–columns with semi-rigid connections including shear and axial effects J. Dario Aristizabal-Ochoa ∗ School of Mines, National University, A.A. 75267, Medell´ın, Colombia Received 8 June 2006; received in revised form 24 July 2006; accepted 26 July 2006 Available online 20 September 2006
Abstract The nonlinear large deflection-small strain analysis and postbuckling behavior of Timoshenko beam–columns of symmetrical cross section with semi-rigid connections subjected to conservative and non-conservative end loads (forces and moments) including the combined effects of shear, axial and bending deformations, axial load eccentricities, lateral bracing and out-of-plumbness are developed in a simplified manner. A new set of stability functions based on the “modified shear equation” that includes the effects of shear deformations and the shear component of the applied axial forces is derived. Also, an expression for the axial displacement δb caused by the “bowing” of the beam–column subjected to end forces and moments with generalized end conditions is derived in a classic manner. The proposed method and corresponding nonlinear equations, although approximate, can be used in the tension and compression stability and nonlinear large deflection-small strain elastic analyses of Timoshenko beam–columns with rigid, semi-rigid, and simple connections. Analytical studies indicate that shear deformations increase the longitudinal and transverse deflections and reduce the buckling axial load capacities of beam–columns. The effects of shear deformations must be considered in the analysis of beam–columns with relatively low effective shear areas (like in short laced columns, columns with batten plates and with open webs) or low shear stiffness (like elastomeric bearings and short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E making the shear stiffness G As of the same order of magnitude as E I / h 2 ). The shear effects are also of great importance in the tension and compression stability and dynamic behavior of laminated elastomeric bearings used for seismic isolation of buildings. Four comprehensive examples are included that show the effectiveness of the proposed method and equations. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Beams; Beam–columns; Large deflections; P–∆ analysis; Nonlinear analysis; Semi-rigid connections; Stability; Structural analysis; Postbuckling behavior; Timoshenko beams
1. Introduction Advances in composite materials of high resilience capacities and low shear stiffness as well as the need for lighter and stronger structural members have created a great interest in their large-deflection analysis and postbuckling behavior. For instance, seismic isolators made of light polymer materials may undergo extremely large deflections under axial and transverse loads without exceeding their elastic limit. The nonlinear geometric elastic behavior of a slender beam–column with semi-rigid connections under conservative ∗ Corresponding address: National University of Colombia, Civil Engineering, Calle 9c # 15-165, Casa 2 Urbanizacion, Villaverde, Medell´ın, Antioquia, Colombia. Tel.: +57 42686218; fax: +57 44255152. E-mail address: [email protected].
c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.07.012
and nonconservative end loads has been investigated by Aristizabal-Ochoa [1,2] using the classical stability functions and the exact expression for the curvature in the differential equation of the deflection curve (i.e., the “Elastica” approach), respectively. Approximate methods based on the Finite Element Method (FEM) with large deflections and with or without large strains have been utilized by other researchers like Torkamani et al. [3] to solve the nonlinear geometric elastic behavior of slender beam–columns and frames with rigid connections. However, the exact solution for the large deflection behavior of Timohenko beam–columns under any type of end loads (conservative and nonconservative) including the combined effects of bending, shear and axial forces and their corresponding deformations is not known yet. The main objective of this publication is to present an approximate and practical method for the nonlinear large
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Nomenclature A cross-sectional area of the beam–column As effective shear area of the beam–column E and G elastic and shear moduli of the material, respectively Hz “bowing” bending function given by Eqs. (5) and (6) h initial length of the beam–column AB hr chord length of the beam–column AB after “bowing” caused by bending and shear actions I principal moment of inertia of the beam–column about its axis of bending M external bending moment applied at A Ma and Mb end moments of beam–column at A and B, respectively P applied vertical load at A (+ compression, − tension) Pe = π 2 E I / h 2 Euler load Pt axial load along the cord AB (+ compression, − tension) Q applied transverse load at A Ra and Rb stiffness indexes of the flexural connection at A and B, respectively ∆h and ∆v horizontal and vertical deflections of end A of beam–column AB, respectively κa and κb the flexural stiffness of the end connections at A and B, respectively ρa and ρb fixity factors at A and B of column AB, respectively θ angle of the cord AB with respect to the vertical axis (Fig. 1(a)) θo initial out-of-plumb angle with respect to the vertical axis (Fig. 1(a)) θa0 and θb0 end slopes of member AB measured with respect to its cord (Fig. 1(b)) u(x) lateral deflection of the column center line as shown by Fig. 2(c) ψ(x) rotation of the cross section as shown by Fig. 2(c) ∆a and ∆b lateral sway at A and B, respectively ∆ = ∆a − ∆b relative sidesway of column end A with respect to its bottom end B ψa and ψb rotations of cross sections at A and B due to bending, respectively deflection-small strain analysis and postbuckling behavior of Timoshenko beam–columns of symmetrical cross section with semi-rigid connections subjected to end loads and moments (conservative or nonconservative) including the combined effects of: (1) bending, axial and shear deformations; (2) the shear component of the applied axial forces (using the “modified shear equation” presented by Timoshenko and Gere [4]); (3) out-of-plumbness of the member’s longitudinal axis; and (4) axial load eccentricities. Also, an expression for the axial displacement δb caused by the “bowing” of the Timoshenko beam–column with generalized end conditions
subjected to end forces and moments is derived in a classic manner. The proposed method and corresponding nonlinear equations, although approximate, can be used in the tension and compression stability and postbuckling behavior as well as in the nonlinear large deflection-small strain elastic analyses of Timoshenko beam–columns with rigid, semi-rigid, and simple connections. The proposed method and derived equations are extensions of those previously proposed by AristizabalOchoa [1] for slender beam–columns. The proposed method and equations can also be utilized in the large deflectionsmall strain inelastic analysis and postbuckling behavior of Timoshenko beam–columns whose connections suffer from flexural degradation or, on the contrary, flexural stiffening. The advantages of the proposed method are: (1) the effects of semi-rigid connections are condensed into a single nonlinear equation with a single unknown (for tension or compression axial loads) without introducing additional degrees of freedom and equations; (2) the proposed method, which is based on the “modified shear equation” for Timoshenko beam–columns with semi-rigid connections, is more accurate than any other approximate method available and capable of capturing the phenomena of tension buckling in members with low shear stiffness; (4) the method can be incorporated into the large deflection elastic analysis and postbuckling behavior of Timoshenko beam–columns without major difficulties; and (5) extension of the method to general elastic–plastic analysis requires the development and solution of incremental equations and the handling of the spread of plasticity throughout the volume of the members, including the interaction between normal and shear stresses. This is a complex extension and it is beyond the scope of this paper. Four comprehensive examples are included that show the effectiveness of the proposed method and equations. 2. Structural model 2.1. Assumptions Consider a 2-D prismatic beam–column that connects points A and B as shown in Fig. 1(a). The element is made up of the beam–column itself AB and the flexural connections κa and κb at the top and bottom ends, respectively. It is assumed that the member AB of span h bends about the principal axis of its cross section with a moment of inertia I , gross sectional area A, and effective shear area As and: (1) is made of a homogeneous linear elastic material with Young and shear moduli E and G, respectively; (2) its centroidal axis is a straight line with an initial out-of-plumbness θo with respect to the vertical axis; and (3) is loaded at extreme A with P (vertical load), Q (transverse load), and M (M = Pe + M 0 , where M 0 is an additional moment applied at A). Each one of these three loads can vary independently and are applied in the plane of bending. The lateral bracing at A is provided by a linear displacement spring of magnitude S∆ with one extreme connected to end A of the beam–column and the other end on vertical rollers that slide on a frictionless rigid vertical wall as shown in Fig. 1(a). The dimensions of S∆ are in force/distance. The flexural
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993
Fig. 1. Model of column with sidesway partially inhibited and with rotational and lateral end restraints. (a) Structural model with an initial out-of-plumb angle θo ; (b) end moments, forces, rotations and deflection moments in the plane of bending; and (c) internal forces and external loads on deformed element.
connectors have stiffnesses κa and κb (whose dimensions are in force × distance/radian) in the plane of bending of the beam–column. The ratios Ra = κa /(E I / h) and Rb = κb /(E I / h) are denoted as the stiffness indices of the flexural connections. These indices vary from zero (i.e., Ra = Rb = 0) for simple connections (i.e., pinned) to infinity (i.e., Ra = Rb = ∞) for fully restrained connections (i.e., rigid). Notice that the proposed algorithm can be utilized in the inelastic analysis of beam–columns when the inelastic behavior is concentrated at the connections. This can be carried out by updating the flexural stiffness of the connections κa and κb for each load increment in a linear-incremental fashion or including the corresponding moment–rotation relationship of the connection as shown by Aristizabal-Ochoa [1,2]. For convenience the following two parameters are introduced: ρa =
1 1+
3 Ra
and
ρb =
1 1+
3 Rb
(1a–b)
where ρa and ρb are called the fixity factors. For perfectly hinged connections, both the fixity factor ρ and the rigidity index R are zero; but for rigid connections (i.e., perfectly clamped), the fixity factor is 1 and the rigidity index is infinity. Since the fixity factor can only vary from 0 to 1 for elastic connections (while the rigidity index R may vary from 0 to ∞), it is more convenient to use in the elastic analysis of structures with semi-rigid connections. 3. Proposed method and equations The large deflection-small strain equations are developed in the plane of bending about one of the principal axes. The proposed method consists in solving simultaneously the
nonlinear equations (2) and (3). Ma + M b + P sin θ + Q − S∆ h h hr hr sin θ − sin θo cos θ =0 × h h Pt Hz hr / h = 1 − − E A 4Pt2 h 2
(2) (3)
where: hr Pt = P cos θ + S∆ h sin θ − sin θo − Q sin θ. h By substituting Eq. (3) into Eq. (2), only the following nonlinear equation has to be solved: Ma + M b + P sin θ + Q − S∆ h h Pt Hz × 1− − sin θ − sin θo cos θ E A 4Pt2 h 2 Pt Hz − = 0. (4) × 1− E A 4Pt2 h 2 Eq. (2) represents the condition of moment equilibrium of the beam–column shown in Fig. 1(b) about B, and Eq. (3) the geometric condition of the actual distance between the two extremes of the beam–column (expression that includes the effects of the axial strain and the “bowing” caused by its bending and shear deformations). Notice that the axial load on the beam–column Pt is taken as the sum of the actual components of the vertical and horizontal forces at A along its chord. The magnitudes of the end moments Ma and Mb and the “bowing” factor Hz depend on the actual sign of Pt , as follows:
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J.D. Aristizabal-Ochoa / Engineering Structures 29 (2007) 991–1003
Ma = −
3ρa [(1 − ρb ) + 3ρb tan (φ/2) /φ] (θ − θo ) EhI βφ 2 − (1 − ρa ) (1 − ρb ) βφ 2 + 3ρb (1 − βφ/ tan φ) M (1 − ρa ) (1 − ρb ) βφ 2 + 3 (ρa + ρb − 2ρa ρb ) (1 − βφ/ tan φ) + 9ρa ρb β [tan (φ/2) / (βφ/2) − 1] Box I.
Mb = −
3ρb [(1 − ρa ) + 3ρa tan (φ/2) /φ] (θ − θo ) EhI βφ 2 + 3ρb (1 − ρa ) (1 − βφ/ sin φ) M (1 − ρa ) (1 − ρb ) βφ 2 + 3 (ρa + ρb − 2ρa ρb ) (1 − βφ/ tan φ) + 9ρa ρb β [tan (φ/2) / (βφ/2) − 1] Box II.
3ρa [(1 − ρb ) + 3ρb tanh (φ/2) /φ] (θ − θo ) EhI βφ 2 − (1 − ρa ) (1 − ρb ) βφ 2 − 3ρb (1 − βφ/ tanh φ) M Ma = − (1 − ρa ) (1 − ρb ) βφ 2 + 3 (ρa + ρb − 2ρa ρb ) (1 − βφ/ tanh φ) + 9ρa ρb β [tanh (φ/2) / (βφ/2) − 1] Box III.
Mb =
3ρb [(1 − ρa ) + 3ρa tanh (φ/2) /φ] (θ − θo ) EhI βφ 2 − 3ρb (1 − ρa ) (1 − βφ/ sinh φ) M − (1 − ρa ) (1 − ρb ) βφ 2 + 3 (ρa + ρb − 2ρa ρb ) (1 − βφ/ tanh φ) + 9ρa ρb [tanh (φ/2) / (βφ/2) − 1] Box IV.
(a) For Pt ≥ 0 (compression) and β > 0 or Pt < 0 (tension) and β < 0: See equations in Boxes I and II. 2−β φ H Z = βφ (Ma2 + Mb2 ) cos φ + sin φ sin φ 2−β + 2βφ Ma Mb sin φ × (1 + φ/ tan φ) − 2(Ma + Mb )2 .
(5)
(b) For Pt < 0 (tension) and β > 0: See equations in Boxes III and IV. φ 2−β H Z = βφ(Ma2 + Mb2 ) cosh φ + sinh φ sinh φ 2−β + 2βφ Ma Mb sinh φ × (1 + φ/ tanh φ) − 2(Ma + Mb ) (6) q where: φ = |Pt / β E I / h 2 | and β = 1/[1 + Pt /(G As )]. Although in Eqs. (2) and (4) the lateral bracing is assumed to be a linear spring (linear with respect to the horizontal deflection of end A) on vertical rollers, so that vertical forces are not induced at any stage of lateral deflections, a nonlinear bracing with different end conditions can be used in the proposed model, as long as its induced vertical and horizontal forces are known as a function of the vertical and horizontal deflections of end A. Equations given in Boxes I–IV and Eqs. (5), (6) for Ma , Mb and Hz are derived in Appendix A. Eq. (2) could be solved for θ in two ways: a simplified one, by neglecting any change in the length of the cord AB (i.e., by making h r / h = 1 at all times); or in a complete manner, using Eq. (4) which includes any change in the ratio h r / h. Eq. (2) or (4) must be solved for θ or every load increment or for any new 2
problem with a given set of input values P, Q, M, θo , E A, E I , G As , h, and S∆ . Four examples that follow show the effectiveness, simplicity and accuracy of the proposed method. Example 1 is on the large deflection analyses of a cantilever beam–column under separate concentrated horizontal and vertical forces at the top. Example 2 shows the effects of an initial shape of a simply supported beam–column that consists of two identical straightline segments on its large deflection elastic response under eccentric axial loads at the ends. Example 3 addresses the large deflection analyses of a beam–column with both ends restrained against rotation and with sidesway uninhibited subjected to axial and lateral forces. Example 4 shows the application of the proposed method to the stability and postbuckling behavior of cantilever columns subjected to gravity and fixed nonconservative axial loads. 4. Verification and comprehensive examples 4.1. Example 1: Analysis of an out-of-plumb cantilever column Determine both the large-deflection behavior and stability of the cantilever column subjected to concentrated loads P and Q at the top end A as shown in Fig. 2(a). Assume that the member is initially out-of-plumb θo and with properties ρb , G, E, A, As , I and h. Also assume that the applied loads Q and P are always horizontal and vertical, respectively. Include in the analysis the effects of the axial strain Pt /E A on the cord length h r . Solution. In this example M = Ma = S∆ = ρa = 0; then from Eq. (4): Mb Pt Hz + {P sin θ + Q cos θ } 1 − − =0 (7a) h E A 4Pt2 h 2 where Pt = P cos θ − Q sin θ. For Pt ≥ 0 (compression) and β > 0 or Pt < 0 (tension) and β < 0
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Fig. 2. Example 1. Cantilever column semi-rigidly connected at the base B subjected to loads P and Q at the free end: (a) applied loads and properties; and (b) deflected shape.
3ρb (θ − θo ) EhI βφ 2 ; and (1 − ρb )βφ 2 + 3ρb (1 − βφ/ tan φ) 2−β φ 2 H Z = Mb βφ cos φ + −2 . sin φ sin φ
Mb = −
For Pt < 0 (tension) and β > 0 3ρb (θ − θo ) EhI βφ 2 ; and −(1 − ρb )βφ 2 + 3ρb (1 − βφ/ tanh φ) 2−β φ H Z = Mb2 βφ cosh φ + −2 . sinh φ sinh φ
Mb =
Substituting these expressions into Eq. (7a) (taking into p account that φ = |Pt /(β E I / h 2 )| and β = 1/[1 + Pt /(As G)]), and including the axial strain effects (Pt /E A), θ can be obtained from Eq. (7a). The following iterative process is suggested to determine the values of θ, ∆v , and ∆h for a given value of the applied loads P and Q: (1) Knowing the value of the applied loads P and Q and the input data θo , G, E, A, As , I and h, assume a trial value θ . (2) Calculate the values of Pt , β = 1/[1 + Pt /(As G)], φ = p |Pt /(β E I / h 2 )|, Mb and Hz . (3) Check if Eq. (7a) is fulfilled, otherwise select a new value for θ and return to step 2. Use the bisection method to determine the value of θ for the given applied loads P and Q and input data θo , G, E, A, As , I and h. (4) The vertical and horizontal deflections at the tip of the beam are approximately as follows: Hz Pt − h sin θ (8a) ∆h = 1 − E A 4Pt2 h 2
Fig. 3. Example 1. Transversal and vertical deflections of a perfectly clamped cantilever beam–column (ρb = 1) subjected to transverse load Q and P = 2E I /8h 2 with: (a) As G/(E I / h 2 ) = ∞; (b) As G/(E I / h 2 ) = 10; (c) As G/(E I / h 2 ) = 5.
∆h . (8b) tan θ Fig. 3(a)–(c) show the variations of ∆h / h and ∆v / h against the lateral load parameter Q/(E I / h 2 ) for three different values of G As /(E I / h 2 ) as indicated in the figures for a particular case of a cantilever column subjected to a constant vertical compressive load P of magnitude Pe /8 = π 2 E I /(8h 2 ) assuming that S∆ = θo = 0, ρb = 1 and E A = ∞. The results obtained with the proposed iterative method are compared to those obtained using Eqs. (14) and (15) for a perfectly clamped–clamped beam–column described in Example 3 (substituting h for 2h and with the corresponding values of ∆v and ∆h divided by two. These results are indicated in Fig. 3(a)–(c) as “Small Deflection Theory”). Notice that ∆v = h −
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Fig. 4. Example 1. Large-deflection stability of a cantilever column (with ρb = 1 and θo = 0) subjected to axial load P at A. Effects of shear deformations on the axial and lateral deflections at the free end for four different values of As G/(E I / h 2 ).
calculations based on small-deflection theory have a rather limited range of applicability, particularly for members made of high resilient materials like advanced composites (FRP). Fig. 4 shows the effects of the shear deformations on the large-deflection stability of a perfectly vertical cantilever column (with θo = 0 and ρb = 1) under vertical compressive load P. Notice that the deformation mode is in the range of 0 < ∆v / h < 1.5. Fig. 5(a)–(c) show the effects of shear deformations and factor of fixity at the base on the largedeflection stability of a perfectly vertical cantilever column (θo = 0) under vertical compressive load P. Fig. 6(a)–(c) show the effects of the factor of fixity at the base on the largedeflection behavior of a perfectly vertical cantilever column (θo = 0) subjected to tension load P and lateral load Q = 0.3π 2 E I / h 2 . Notice that the proposed method is capable of capturing the “phenomena” of tension buckling of Timoshenko beam–columns (Kelly [5] and Aristizabal-Ochoa [6,7]), as well as its postbuckling response. The axial shortening caused by P was neglected (i.e., Pt /E A = 0) in these analyses. Figs. 3(a)–(c), 4, 5(a)–(c), and 6(a)–(b) show that the largedeflection and stability analysis of a beam–column are affected by the end boundary conditions, the applied loads, the shear deformations along the member, and the shear force induced by the axial force as the member deforms. 4.2. Example 2: Large-deflection behavior of a simple supported beam–column with an initial imperfection subjected to an eccentric axial load Determine the large deflection-small strain nonlinear equation of a simple supported beam–column subjected to eccentric end axial loads as shown in Fig. 7(a). Assume that its initial shape consists of two identical straight-line segments (with properties G, E, A, As , I and length h) both making an angle θo with the cord. Bending occurs in the plane of the two segments about their cross-section major axis. Compare
the calculated results with those presented by Timoshenko and Gere [4] for small deflections and E A = G As = ∞. Include the effects of the axial strain Pt /E A on the cord length h r . Solution. Taking only one segment as shown in Fig. 7(b), then Q = S∆ = 0, M = Pe; ρa = 0 and ρb = 1; then from Eq. (4): (Ma + Mb ) Pt Hz + 1− − P sin θ = 0 (9) h E A 4Pt2 h 2 where Pt = P cos θ ≥ 0 (compression); Ma = Pe; Mb = θ +(1−βφ/ sin φ)(e/ h) − (θ −θo ) cos(1−βφ/ Ph; and tan φ) φ 2−β 2 2 H Z = βφ(Ma + Mb ) cos φ + sin φ sin φ 2−β + 2βφ Ma Mb sin φ × (1 + φ/ tan φ) − 2(Ma + Mb )2 . Substituting the expressions above for p Ma and Mb into Eq. (9) (taking into account that φ = |Pt /(β E I / h 2 )| and including the axial strain effects Pt /E A), θ can be obtained from Eq. (10): −(θ − θo ) cos θ +
1−cos φ sin φ βφ (e/ h)
(1 − βφ/ tan φ) Pt Hz + 1− − sin θ = 0. E A 4Pt2 h 2
(10)
Eq. (10) represents the large deflection-small strain behavior of the beam–column shown in Fig. 7(b) [after the expression for Hz is substituted in Eq. (10)]. With the solution for θ obtained from Eq. (10), the deflections ∆Lateral = [(1 − EPAt − Hz Pt 2 2 ) sin θ − sin θo ]h and ∆Axial = [cos θo − (1 − E A − 4Pt h Hz ) cos θ ]h 4Pt2 h 2
can be calculated directly.
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Fig. 5. Example 1. Large-deflection stability of a cantilever column subjected to axial compressive load P at A. Effects of shear deformations and fixity factor at the base on the axial and lateral deflections at the free end for three different values of As G/(E I / h 2 ).
Fig. 8(a)–(c) shows the effects of the shear stiffness on the large-deflection response of the column (for θo = 0.1, ρb = 1 and AE = ∞) under the end load P (compression) and
M = Pe for three different values of e/ h (0.05, 0.1, and 0.2). As expected, shear deformations and the axial load eccentricity increase both deflections.
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Fig. 6. Example 1. Large-deflection response of a cantilever beam–column subjected to tension load P and lateral load Q = 0.3Pe at the free end. Effects of shear deformations and fixity factor at the base on the axial and lateral deflections at the free end for two different values of As G/(E I / h 2 ).
β = 1), Eq. (10) is reduced to −(θ − θo ) +
or
1−cos φ sin φ φ (e/ h)
+θ =0 (1 − φ/ tan φ) 1 e sin φ θ= θo + (1 − cos φ) . cos φ φ h
(11) (12)
Expression (12) is identical to that obtained using the method presented by Timoshenko and Gere [4] for small deflections and strains using the classic stability functions that include bending deformations only.
Fig. 7. Example 2. Simple supported beam–column with an initial imperfection subjected to axial load P and end moment M = Pe.
Now, for small deflections and neglecting axial and shear deformations (i.e., small values of θ, E A = G As = ∞, and
4.3. Example 3: Large deflection analysis of a beam–column with sidesway uninhibited and both ends restrained Determine the large deflection-small strain nonlinear equations for the horizontal and vertical deflections of the beam–column (Fig. 9) restrained at both ends and with an
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Fig. 8. Example 2. Effects of shear deformations on the axial and lateral deflections at the free end of an out-of-plumb cantilever beam–column (θo = 0.1 and ρb = 1.0) for four different values of As G/(E I / h 2 ) and subjected to axial load P and moment M = Pe: (a) e/ h = 0.05; (b) e/ h = 0.1; and (c) e/ h = 0.2.
initial out-of-plumb angle θo . Assume that: (1) properties of the member are: ρa , ρb , G, E, A, As , I and h; and (2) the applied moment is M = Pe and the loads
P and Q are always vertical and horizontal, respectively. Include the effects of the axial strain Pt /E A on the cord length h r .
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Fig. 9. Example 3. Unbraced beam–column restrained at both ends. (a) Structural model with an initial out-of-plumb angle θo and applied loads P and Q. (b) Deflected shape and top end deflections.
Solution. In this example M = Pe and S∆ = 0, then from Eq. (4): Pt Hz (Ma + Mb ) + 1− − (P sin θ + Q cos θ ) = 0 h E A 4Pt2 h 2 (13) where Pt = P cos θ − Q sin θ. Substituting expressions for Ma , Mb , and Hz given by Boxes I–IV and Eqs. p (5), (6) into Eq. (13) (taking into account that M= Pe, φ = |Pt /(β E I / h 2 )| and β = 1/[1+ Pt /(As G)]) and including the axial strain effects Pt /E A), θ can be obtained using an iterative process similar to that described in Problem (1), and the corresponding values of ∆v and ∆h can be determined from Eqs. (8a) and (8b), respectively. When the beam–column is perfectly vertical (θo = 0), clamped at both ends (i.e., ρa = ρb = 1) and subjected to P and Q only, the solutions for ∆h , and ∆v proposed by Kelly [5] using small deflection theory are as follows: ∆h =
Qh βP
Q2 ∆v = 2β P 2
tan (φ/2) −β φ/2
4.4. Example 4: Large deflection and post bucking behavior of a beam–column subjected to a nonconservative force (Reut’s column) combined with a gravity load Determine both the large-deflection behavior and stability of the cantilever column (Fig. 10) subjected to two vertical concentrated loads: Po (gravity load) and P f (vertical force whose line of action always passes through point B). Assume that: (1) the member is initially out-of-plumb (θo ) with properties ρb , G, E, A, As , I and h; (2) bending occurs about one of its main axis; and (3) the applied loads Po and P f are always vertical and vary independently. Include in the analysis the effects of the axial strain Pt /E A on the cord length h r . Solution. In this example P = Po + P f ; Q = S∆ = 0, M = −(h r sin θ )P f ; ρa = 0 and ρb = 1; then from Eq. (4): (Ma + Mb ) Pt Hz P sin θ = 0 (16) + 1− − h E A 4Pt2 h 2 where Pt = P cos θ ≥ 0 (compression); Pf Hz Pt − Pt h tan θ Ma = M = − 1 − 2 2 E A 4Pt h Po + P f Mb = −
φβ cos φ − 2 sin φ + 2φ − β sin φ . (φ/ h)(1 + cos φ)
Fig. 10. Perfectly clamped cantilever column subjected to gravity load Po and a fixed force P f at the free end. (a) Applied loads and properties. (b) Deflected shape.
(14)
(15)
Notice that the cantilever beam–column of Problem (1) can be solved using the solution of this problem by simply making ρa = ρb = 0 and substituting h for 2h. The corresponding values of ∆v and ∆h must be divided by two.
(θ − θo ) EhI βφ 2 + (1 − βφ/ sin φ) M ; (1 − βφ/ tan φ)
and 2−β φ cos φ + H Z = βφ(Ma2 + Mb2 ) sin φ sin φ 2−β + 2βφ Ma Mb (1 + φ/ tan φ)− 2(Ma + Mb )2 . sin φ Substituting the expressions above for p Ma and Mb into Eq. (16) (taking into account that φ = |Pt /(β E I / h 2 )| and including the axial strain effects Pt /E A), θ can be obtained
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from Eq. (17): θ − θo Pt Hz = 1− − tan θ E A 4Pt2 h 2 Pf βφ × 1− 1− (1 − 1/ cos φ) . (17) tan φ Po + P f Eq. (17) represents the large deflection-small strain behavior of the beam–column shown in Fig. 9 [after the expression for Hz is substituted in Eq. (17)]. With the solution for θ obtained from Eq. (17), the deflections at the top ∆h = [(1 − Hz Pt Pt 2 2 ) sin θ − sin θo ]h and ∆v = [cos θo − (1 − E A − EA − 4Pt h Hz ) cos θ ]h 4Pt2 h 2
can be calculated directly. Notice that when θ = θo = 0, Eq. (17) is reduced to the characteristic equation: cos φ = −P f /Po . This result is identical to Eq. (14c) reported by Bolotin [8] (p. 103) and Aristizabal-Ochoa [9] (p. 479) for a perfectly clamped cantilever column subjected to a force with a fixed line of action and gravity load. The four examples presented above show that the proposed method is very practical and versatile allowing both the small- as well as the large-deflection analysis of Timoshenko beam–columns subjected to end loads. 5. Summary and conclusions The nonlinear large deflection-small strain analysis and postbuckling behavior of Timoshenko beam–columns of symmetrical cross section with semi-rigid connections subjected to conservative and nonconservative end loads (forces and moments) including the combined effects of shear, axial and bending deformations, axial load eccentricities, lateral bracing and out-ofplumbness are developed in a simplified manner. A new set of stability functions based on the “modified shear equation” that include the effects of shear deformations and the shear component of the applied axial forces is derived. Also, an expression for the axial displacement δb caused by the “bowing” of the member subjected to end forces and moments with generalized end conditions is derived in a classic manner. The proposed method, although approximate, can also be used in the nonlinear large deflection analyses of Timoshenko beam–columns with rigid, semi-rigid, and simple connections subjected to tension forces. The analytical results indicate that the large deflectionsmall strain elastic response and postbuckling behavior of Timoshenko beam–columns of symmetrical cross section is not only affected by the magnitude of the axial load, the lateral drift restraints (i.e., bracing), the degree of fixity of the connections and its elastic properties, but also by the bowing effect caused by the flexural moments and shear deflections. Significant increases in the lateral and axial deflections are caused by the geometric nonlinear effects, particularly those induced by the shear deformations and by the shear component of the axial force. Also the buckling load of beam–columns is reduced by shear deformations and by the shear component of the axial force. These shear effects must be considered in the analysis of beam–columns with relatively low effective
shear areas (like in short laced columns, columns with batten plates or with perforated cover plates, and columns with open webs) or low shear stiffness (like elastomeric bearings and short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E making the shear stiffness G As of the same order of magnitude as E I / h 2 ). The shear effects are also of great importance in the tension and compression stability and dynamic behavior of laminated elastomeric bearings used for seismic isolation of buildings. The advantages of the proposed method are: (1) the effects of semi-rigid connections and shear deformations are condensed into a single nonlinear equation with a single unknown (for tension or compression axial loads) without introducing additional degrees of freedom and equations; (2) the proposed method, which is based on a new set of stability functions for Timoshenko beam–columns with semi-rigid connections, is very practical, versatile and more accurate than any other approximate method available; and (3) extension of the method to general elastic–plastic analysis requires the development and solution of incremental equations and the handling of the spread of plasticity throughout the volume of the members, including the interaction between normal and shear stresses. This is a complex extension and it is beyond the scope of this paper. Four comprehensive examples are included that show the effectiveness of the proposed method and equations. Acknowledgments The research presented in this paper was carried out at the National University of Colombia, School of Mines at Medellin. The author wants to express his appreciation to DIME for the financial support and to David Padilla-Llano and Johnny Moncada-Palacios, members of the Structural Stability Research Group of the National University of Colombia, for carrying out the analytical studies. Appendix A. Derivations of equations given in Boxes I–IV and Eqs. (5), (6) A.1. Expressions for Ma and Mb The stability analysis of a prismatic column including bending and shear deformations (Fig. 1(a)–(c)) is formulated using the “modified shear equation” proposed by Timoshenko and Gere [4] (p. 134). This approach has been utilized by Aristizabal-Ochoa [6,7] in the stability analysis of columns, and Kelly [5] in the analysis of Elastomeric Isolation Bearings. The governing equations are as follows: β E I u 00 (x) + Pu(x) = −Ma − (Ma + Mb + P∆) β E I ψ 00 (x) + Pψ(x) =
Ma + Mb + P∆ h
x h
(18a) (18b)
where u(x) = lateral deflection of the column center line; and ψ(x) = rotation of the cross section as shown by Fig. 1(c). The solutions to the second-order linear differential equations (18a) and (18b) are as follows:
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−
h
Ma =
1 Rb
+
2 βφ (θ − θo )βφ 2 (E I / h) + βφ + 1 − Rb tan φ tan(φ/2) βφ 1 1 + Ra + Rb 1 − tan φ + βφ/2 − 1 β
tan(φ/2) φ
βφ 2 Ra R b
i
M Ra
Box V.
− Mb =
h
1 Ra
+
βφ 2 Ra R b
βφ (θ − θo )βφ 2 (E I / h) + 1 − sin φ tan(φ/2) βφ 1 + Rb 1 − tan φ + βφ/2 − 1 β
tan(φ/2) φ
+
1 Ra
i
M Ra
Box VI.
h Ma =
1 Rb
+
2 φ (θ − θo )βφ 2 (E I / h) − βφ − 1 + Rb tanh φ tanh(φ/2) βφ 1 1 + Ra + Rb 1 − tan φ + − 1 β βφ/2
tanh(φ/2) φ
− Rφa Rb 2
i
M Ra
Box VII.
h Mb =
1 Ra
+
tanh(φ/2) φ
+ − Rβφ a Rb 2
1 Ra
βφ M (θ − θo )βφ 2 (E I / h) + 1 − sinh φ Ra tanh(φ/2) βφ + R1b 1 − tanh φ + βφ/2 − 1 β
i
Box VIII.
x u(x) = A cos φ h x x Ma + M b Ma + B sin φ + +∆ − h h P P x φ ψ(x) = C cos h x 1 Ma + M b + D sin φ + +∆ h h P
ψb =
but ∆ h = (θ − θo ) and (19b)
Ma + M b + ∆ tan(φ/2). P
Since u 0 = ψ + V /(As G) and V = Pψ − Ma +Mhb +P ∆ , the following expressions for ψa and ψb can be obtained: ψa =
(20b)
(19a)
where φ 2 = P/(β E I / h 2 ). The unknown coefficients A, B, C, and D can be obtained from the following boundary conditions: At x = 0: u = 0, ψ = ψa At x = h: u = ∆ and ψ = ψb where ψa and ψb = rotations of cross sections at A and B due to bending, respectively; ∆a and ∆b = lateral sway at A and B, respectively. ∆ = ∆a − ∆b = relative sidesway of column end A with respect to its bottom end B. +Mb Therefore: A = MPa ; B = MPa tan(φ/2) − MPasin φ ; C = M +M a b ψa − h1 + ∆ ; and P Ψa − Ψb cos φ D= − sin φ
Ma sin φ − βφ Mb sin φ − βφ cos φ ∆ + + 2 2 E I / h βφ sin φ E I/h h βφ sin φ
Ma sin φ − βφ cos φ Mb sin φ − βφ ∆ + + 2 2 E I/h E I / h βφ sin φ h βφ sin φ (20a)
Ma − M κa Ma sin φ − βφ cos φ Mb sin φ − βφ = + E I/h E I / h βφ 2 sin φ βφ 2 sin φ Mb θb0 = −(θ − θo ) − κb Ma sin φ − βφ Mb sin φ − βφ cos φ = + E I / h βφ 2 sin φ E I/h βφ 2 sin φ
θa0 = −(θ − θo ) −
(21a)
(21b)
where ψa and ψb = rotations of cross sections at A and B due to bending, respectively; θa0 and θb0 = rotations of ends A and B with respect to the p cord AB, respectively; and φ = |Pt /(β E I / h 2 )|. From Eqs. (21a) and (21b) the end moments Ma and Mb can be expressed in terms of (θ − θo ), M and the fixity ratios (Ra and Rb ) as shown in the equations given in Boxes V and VI, where β = 1/[1 + P/(As G)]; and As = effective shear area of the column. Similarly, when the axial load Pt is negative (tension) and β > 0: See equations in Boxes VII and VIII. Equations in Boxes√ VII and VIII were obtained by replacing φ or iφ (where i = −1) in equations in Boxes V and VI and also taking into account that sin(iφ) = i sinh φ, cos(iφ) = cosh φ, and tan(iφ) = i tanh φ. Equations in Boxes I, II, III and IV can be obtained once Ra = 3ρa /(1 − ρa ) and
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Rb = 3ρb /(1 − ρb ) are substituted into the equations given in Boxes V, VI, VII and VIII, respectively. The elastica is given by Eq. (19a) or x x Ma φ cos sin u(x) = φ + tan φ −1 P h 2 h x Ma + M b +∆ . (22a) × h P Similarly, when the axial load Pt is negative (tension) and β > 0: x x Ma φ u(x) = cosh sinh φ − tanh φ −1 P h 2 h x Ma + M b +∆ . (22b) + h P A.2. Expression for bowing factor Hz The axial displacement δb caused by the “bowing” of the beam–column given by Eq. (23) was derived by Kelly [5]: δb =
1 2
h
Z
2ψ
0
du − ψ 2 dx. dx
(23)
Substituting Eqs. (19a) and (19b) with ∆ = M = 0 into Eq. (23), carrying out the integration and after lots of algebra, the expression for the axial displacement δb caused by the “bowing” of the beam–column can be reduced to: h Hb2 Hb h ψa + ψb Hb tan(φ/2) δb = − + 2 P2 P 2 P βφ/2 φ + sin φ h Hb 2 + −1 4φ P β cos2 (φ/2) Hb h 2 −1 × (ψa + ψb ) + + P 8φ β
×
(2φ−sin 2φ)(ψa2 +ψb2 )−4(φ cos φ−sin φ)ψa ψb sin2 φ
(24)
b where Hb = − Ma +M . h After substituting Eq. (20) into Eq. (24), the expression for δb in terms of Ma and Mb is as follows: 1 δb = − 2(Ma + Mb )2 + βφ(Ma2 + Mb2 ) 4P 2 h 2−β φ × cos φ + sin φ sin φ 2−β φ + 2βφ Ma Mb 1+ . (25) sin φ tan φ
1003
Therefore: 2−β H Z = (4P 2 h)δb = βφ(Ma2 + Mb2 ) sin φ 2−β φ + 2βφ Ma Mb × cos φ + sin φ sin φ × (1 + φ/ tan φ) − 2(Ma + Mb )2 .
(26)
Similarly, when the axial load Pt is negative (tension) and β > 0: 2−β φ 2 2 H Z = βφ(Ma + Mb ) cosh φ + sinh φ sinh φ 2−β + 2βφ Ma Mb sinh φ × (1 + φ/ tanh φ) − 2(Ma + Mb )2 .
(27)
Notice that the expression for the bowing factor Hz given by Eq. (26) is identical to that developed by Ekhande et al. [10] when β = 1 (i.e., when the effects of shear deformations and those of the shear component of the axial force are neglected). The expression for Hz developed by Ekhande et al. [10] was used by Aristizabal-Ochoa [1] in the nonlinear large-deflection small-strain analysis of slender beam–columns with semi-rigid connections. References [1] Aristizabal-Ochoa J Dario. Nonlinear large-deflection small-strain analysis of beam–columns under end loads. Journal of Structural Engineering ASCE 2001;127(1):92–6. [2] Aristizabal-Ochoa J Dario. Large-deflection stability of slender beam–columns with semi-rigid connections: The elastica approach. Journal of Engineering Mechanics ASCE 2004;274–82. [3] Torkamani MAM, Sonmez M, Cao J. Second-order elastic plane-frame analysis using finite-element method. Journal of Structural Engineering ASCE 1997;123(9):1225–35. [4] Timoshenko S, Gere J. Theory of elastic stability. 2nd ed. McGraw-Hill; 1961 [Chapter II]. [5] Kelly JM. Tension buckling in multilayer elastomeric bearings. Journal of Engineering Mechanics ASCE 2003;129(12):1363–8. [6] Aristizabal-Ochoa J Dario. Tension buckling in multilayer elastomeric bearings by James M. Kelly. Journal of Engineering Mechanics ASCE 2003;129(12):1363–8. Discussion: 131(1):106–8. [7] Aristizabal-Ochoa J Dario. Column stability and minimum lateral bracing: Effects of shear deformations. Journal of Engineering Mechanics ASCE 2004;130(10):1223–32. [8] Bolotin VV. Nonconservative problems of the theory of elastic stability. 60 Fifth Avenue, New York (NY): MacMillan; 1963. p. 103. [9] Aristizabal-Ochoa J Dario. Stability of beam–columns with semirigid connections under conservative and non-conservative end axial forces: Static method. Journal of Engineering Mechanics ASCE 2005;131(5): 473–84. [10] Ekhande SG, Selvappalam M, Madugula MKS. Stability functions for three-dimensional beam columns. Journal of Structural Engineering ASCE 1989;115(2):467–79.