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Slope-deflection equations for stability and second-order analysis ofTimoshenho beam–column structures with semi-rigid connections

J. Dario Aristizabal-Ochoa∗

School of Mines, National University, A.A. 75267, Medellin, Colombia

Received 24 December 2007; received in revised form 28 January 2008; accepted 12 February 2008

Abstract

A new set of slope-deflection equations for Timoshenko beam–columns of symmetrical cross section with semi-rigid connections that includethe combined effects of shear and bending deformations, and second-order axial load effects are developed in a classical manner. The proposedmethod that also includes the effects of the shear component of the applied axial force as the member deflects laterally (Haringx Model) has thefollowing advantages: (1) it can be utilized in the stability and second-order analyses of framed structures made up of Timoshenko beam–columnswith rigid, semi-rigid, and simple end connections; (2) the effects of semi-rigid connections are condensed into the slope-deflection equations fortension or compression axial loads without introducing additional degrees of freedom and equations; (3) it is more accurate than any other methodavailable and capable of capturing the phenomena of buckling under axial tension forces; and (4) it is powerful, practical, versatile and easy toteach. Analytical studies indicate that shear deformations increase the lateral deflections and reduce the critical axial loads of framed structuresmade of members with low shear stiffness. The effects of shear deformations must be considered in the analysis of beam–columns with relativelylow effective shear areas (like laced columns, columns with batten plates or with perforated cover plates, and columns with open webs) or withlow shear stiffness (like elastomeric bearing and short columns made of laminated composites with low shear modulus G when compared to theirelastic modulus E) making the shear stiffness GAs of the same order of magnitude as EI/L2. The shear effects are also of great importance in thestatic, stability and dynamic behavior of laminated elastomeric bearings used for seismic isolation of buildings. Four comprehensive examples areincluded that show the effectiveness of the proposed method and equations.c© 2008 Elsevier Ltd. All rights reserved.

Keywords: Beams; Beam–columns; Large deflections; Nonlinear analysis; Semi-rigid connections; Second-order analysis; Stability; Structural analysis; Timoshenkobeam

1. Introduction

The slope-deflection method represents a turning point inthe evolution and development of the matrix stiffness methodas it is known today (Samuelsson and Zienkiewicz [1]). It waspresented in 1915 by Wilson and Maney [2] in a Bulletin fromthe University of Illinois at Urbana-Champaign as a generalmethod to be used in the analysis of beam structures with rigid-joints subjected to transverse loads.

The slope-deflection method may be used to analyze alltypes of statically indeterminate beams and frames. Theclassic slope-deflection equations are derived by means ofthe moment–area theorems considering deformation caused by

∗ Tel.: +57 42686218; fax: +57 44255152.E-mail address: jdaristi2@yahoo.com.

bending moment only and neglecting those caused by shearand axial forces. Basically, a number of simultaneous equationsare formed with the unknowns taken as the angular rotationsand displacements of each joint. Once these equations havebeen solved, the moments at all joints may be determined.The method is simple to explain and apply since it isbased on the equilibrium of the joints and members. Theclassic slope-deflection method is generally taught in theintroductory structural analysis courses (Norris and Wilbur [3],Kassimili [4]) and used in the structural design (Salmon andJohnson [5]) because it provides a clear perspective and acomplete understanding of how the internal moments and thecorresponding deformations are interrelated, both of which areessential in structural engineering.

However, advances in composite materials of high resiliencecapacities and low shear stiffness as well as the need for lighter

0141-0296/$ - see front matter c© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2008.02.007

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Nomenclature

As Effective shear area of the beam–column crosssection;

E Young’s modulus of the material;G Shear modulus of the material;L Length of the beam–column AB;I Principal moment of inertia of the beam–column

about its axis of bending;Ma and Mb Bending moments (clockwise +) at ends A

and B, respectively;P Applied axial load at ends A and B (+

compression, − tension);Pcr Critical axial load;Pe π2EI/L2

= Euler load;Ra and Rb Stiffness indexes of the flexural connection at

A and B, respectively;u(x) Lateral deflection of the beam–column center

line;β =

11+P/(G As )

Shear reduction factor;∆ Sway of end B with respect to end A;κa and κb Flexural stiffness of the end connections at A

and B, respectively;ρa and ρb Fixity factors at A and B of column AB,

respectively;ψ(x) Rotation of the cross section due to bending alone

as shown by Fig. 1c;ψa′ and ψb′ Bending rotations of cross sections at ends A′

and B′ with respect to cord A′B′, respectively;

φ =

√∣∣P/ (βE I/L2)∣∣ Stability function in the plane of

bending;θa and θb Rotations of ends A and B due to bending with

respect to the vertical axis, respectively [θa =

ψa′ +Maκa

and θb = ψb′ +Mbκb

];

Γ =12(E I/L2)

G AsBending to shear coefficient.

and stronger beams and columns have created a great interest inthe shear effects and second-order analysis of framed structures.For instance, elastomeric seismic isolators and membersmade of light polymer materials may undergo extremelylarge deflections under combined axial and transverse loadswithout exceeding their elastic limit. The slope-deflectionequations for Timoshenko beams including the effects of sheardeformations and transverse loads were developed by Bryanand Baile [6]. Previously, Lin, Glauser and Johnson [7] haddeveloped the slope-deflection equations for laced and battenedbeam–columns including the effects of shear deformations,axial load and end rigid stay plates. On the other hand, thestability and second-order analysis of slender beam–columnsand of framed structures with semi-rigid connections has beeninvestigated by Aristizabal-Ochoa [8,9] using the classicalstability functions. However, the stability of framed structuresusing the classic slope-deflection method including thecombined effects of shear and bending deformations, second-

Fig. 1. Beam–column under end moments with semi-rigid bending connectionsat ends A and B: (a) Structural model; (b) Degrees of freedom, forces andmoments in the plane of bending; (c) Rotations at a cross section and at ends Aand B.

order P–∆ effects, and semi-rigid end connections is notknown yet.

The main objective of this publication is to present a newset of slope-deflection equations for the stability and second-order analysis of framed structures made of Timoshenkobeam–columns of symmetrical cross section with semi-rigidconnections under end loads including the combined effectsof: (1) bending and shear deformations; and (2) the shearcomponent of the applied axial forces (Haringx’s Model).The proposed method which is based on the “modified”stability functions for Timoshenko beam–columns with semi-rigid connections (Aristizabal-Ochoa [10,11]) has the followingadvantages: (1) the effects of semi-rigid connections arecondensed into the slope-deflection equations for tension orcompression axial loads without introducing additional degreesof freedom and equations; (2) it is more accurate than anyother method available and capable of capturing the phenomenaof buckling under axial tension forces of short columns likelaminated elastomeric bearings commonly used for seismicisolation of buildings; and (3) the method is powerful, practical,versatile and easy to teach. Four comprehensive examples areincluded that show the effectiveness of the proposed methodand corresponding equations.

2. Structural model

Assumptions. Consider a 2-D prismatic beam–column thatconnects points A and B as shown in Fig. 1a. The elementis made up of the beam–column itself A′B′, and the flexuralconnections AA′ and BB′ with bending stiffness κa and κb atends A and B, respectively. It is assumed that the beam–columnA′B′ of span L bends about the principal axis of its cross sectionwith a moment of inertia I , effective shear area As and: (1) ismade of a homogeneous linear elastic material with Young andshear moduli E and G, respectively; (2) its centroidal axis isa straight line; and (3) is loaded at extremes A and B with P(compressive axial load is +) along its centroidal axis.

The flexural connections have stiffnesses κa and κb (whoseunits are in force–distance/radian) in the plane of bending of

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the beam–column. The ratios Ra = κa/(EI/L) and Rb =

κb/(EI/L) are denoted as the stiffness indices of the flexuralconnections. 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 theproposed algorithm can be utilized in the inelastic analysis ofbeam–columns when the inelastic behavior is concentrated atthe connections. This can be carried out by updating the flexuralstiffness of the connections κa and κb for each load incrementin a linear-incremental fashion.

For convenience the following two parameters are intro-duced:

ρa =1

1 +3

Ra

; (1a)

and

ρb =1

1 +3

Rb

(1b)

where ρa and ρb are called the fixity factors. For hingedconnections, both the fixity factor ρ and the rigidity index Rare zero; but for rigid connections, the fixity factor is 1 and therigidity index is infinity. Since the fixity factor can only varyfrom 0 to 1 for elastic connections (while the rigidity index Rmay vary form 0 to ∞), it is more convenient to use in theelastic analysis of structures with semi-rigid connections.

3. Proposed slope-deflection equations

In order to investigate the lateral stability and second-orderanalysis of framed structures with semi-rigid connections, it isfirst necessary to establish the relationships between the endbending moments and the corresponding rotations for eachframe member and then apply the conditions of compatibilityof deformations at each joint. The application of classic slope-deflection method in the stability of plane frames (with rigidconnections and without transverse loads and shear deflectioneffects) is well explained by Salmon and Johnson ([5], pp.835–843). The proposed slope-deflection equations for planeframes with semi-rigid connections including shear deflectioneffects and without transverse loads along each member are asfollows:

Ma = SaaE I

L

(θa −

∆L

)+ Sab

E I

L

(θb −

∆L

)(2a)

Mb = SbaE I

L

(θa −

∆L

)+ Sbb

E I

L

(θb −

∆L

)(2b)

where θa and θb values are the rotations of ends A and Bmeasured with respect to the initial axis of the member, and∆/L the angle of rotation of the cord of the member AB. Bothrotations and end moments (Ma and Mb) are positive clockwiseas shown in Fig. 1. Saa , Sab, Sba and Sbb are denoted as thestiffness coefficients of the Timoshenko beam–column AB withsemi-rigid connections at both ends A and B. These coefficientsare listed below for easy reference.

First-order stiffness coefficients. For P = 0

Saa =3ρa(4 + ρbΓ )

(4 − ρaρb)+ (ρa + ρb + ρaρb)Γ(3)

Sab = Sba =3ρaρb(2 − Γ )

(4 − ρaρb)+ (ρa + ρb + ρaρb)Γ(4)

Sbb =3ρb(4 + ρaΓ )

(4 − ρaρb)+ (ρa + ρb + ρaρb)Γ. (5)

Second-order stiffness coefficients.

(a) For P > 0 (compression) or [1 + P/ (G As)] P > 0. SeeBox I.(b) For P < 0 (tension) and [1 + P/ (G As)] P < 0. SeeBox II.

In Boxes I and II Γ =12(E I/L2)

G As; φ =

√∣∣P/ (βE I/L2)∣∣;

and β =1

1+P/(G As ).

Notice that the proposed method and correspondingequations listed above are valid for beam–columns subjected totension and compression axial forces. When φ2 is negative (i.e.,[1 + P/ (G As)] P < 0), the following three exchanges weremade in the equations in Box I to obtain those in Box II: (1)sinφ for i sinhφ; (2) tanφ for i tanhφ; and (3) φ for iφ (where:i =

√−1). Eqs. (2a) and (2b) are derived in Appendix for quick

reference.To verify the coefficients given by Eqs. (3)–(5), the

numerators and denominators of expressions in Box I must bedifferentiated four times in accordance with L’Hospital’s ruleand then apply the P = 0 limit. However, since to carry outthis process is really tedious and cumbersome, the derivationsof Eqs. (3)–(5) are presented in the last part of the Appendix.

Four examples that follow show the effectiveness, simplicityand accuracy of the proposed slope-deflection equations listedabove. Examples 1 and 2 are on the stability analysis ofbeam–column structures subjected to concentrated forces atthe joints causing tension and/or compression in the members.Examples 3 and 4 show the application of the proposed methodto plane frames subjected to axial loads (compression and/ortension).

4. Verification and comprehensive examples

Example 1. Second-order analysis of a cantilever columnunder several loads at the top.

Determine the second-order load-deflection equation of thecantilever column shown in Fig. 2. Compare the resultingexpressions with those derived by Timoshenko and Gere ([12],pp. 5 and 13) for a perfectly clamped Bernoulli–Euler cantilevercolumn (i.e., ρ = β = 1).

Solution: From equilibrium (Fig. 2b):

M21 = M (6)

M12 + M + P∆ + QL = 0. (7)

Now using Eqs. (2a) and (2b)

M12 = S11E I

L

(θ1 −

∆L

)+ S12

E I

L

(θ2 −

∆L

)(8)

Please cite this article in press as: Dario Aristizabal-Ochoa J. Slope-deflection equations for stability and second-order analysis of Timoshenho beam–columnstructures with semi-rigid connections. Engineering Structures (2008), doi:10.1016/j.engstruct.2008.02.007

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Saa =

3ρa(1 − ρb)βφ2+ 9ρaρb

(1 −

βφtanφ

)(1 − ρa)(1 − ρb)βφ2 + 3(ρa + ρb − 2ρaρb)

(1 −

βφtanφ

)+ 9ρaρb

(tan(φ/2)φ/2 − β

)Sab = Sba =

9ρaρb

(βφ

sinφ − 1)

(1 − ρa)(1 − ρb)βφ2 + 3(ρa + ρb − 2ρaρb)(

1 −βφ

tanφ

)+ 9ρaρb

(tan(φ/2)φ/2 − β

)Sbb =

3ρb(1 − ρa)βφ2+ 9ρaρb

(1 −

βφtanφ

)(1 − ρa)(1 − ρb)βφ2 + 3(ρa + ρb − 2ρaρb)

(1 −

βφtanφ

)+ 9ρaρb

(tan(φ/2)φ/2 − β

)Box I.

Saa =

−3ρa(1 − ρb)βφ2+ 9ρaρb

(1 −

βφtanhφ

)−(1 − ρa)(1 − ρb)βφ2 + 3(ρa + ρb − 2ρaρb)

(1 −

βφtanhφ

)+ 9ρaρb

(tanh(φ/2)φ/2 − β

)Sab = Sba =

9ρaρb

(βφ

sinhφ − 1)

−(1 − ρa)(1 − ρb)βφ2 + 3(ρa + ρb − 2ρaρb)(

1 −βφ

tanhφ

)+ 9ρaρb

(tanh(φ/2)φ/2 − β

)Sbb =

−3ρb(1 − ρa)βφ2+ 9ρaρb

(1 −

βφtanhφ

)−(1 − ρa)(1 − ρb)βφ2 + 3(ρa + ρb − 2ρaρb)

(1 −

βφtanhφ

)+ 9ρaρb

(tanh(φ/2)φ/2 − β

)Box II.

Fig. 2. Example 1: Second-order analysis of a cantilever column with semi-rigid connections at the support subjected to top end loads: (a) Structural model;and (b) Deflected shape and end actions.

M21 = S21E I

L

(θ1 −

∆L

)+ S22

E I

L

(θ2 −

∆L

). (9)

In this example ρa = ρ and ρb = 1; then for P > 0(compression) and using the equations in Box I:

S11 =

3ρ(

1 −βφ

tanφ

)(1 − ρ)

(1 −

βφtanφ

)+ 3ρ

(tan(φ/2)φ/2 − β

) (10)

S12 = S21 =

3ρ(βφ

sinφ − 1)

(1 − ρ)(

1 −βφ

tanφ

)+ 3ρ

(tan(φ/2)φ/2 − β

) (11)

S22 =

(1 − ρ)βφ2+ 3ρ

(1 −

βφtanφ

)(1 − ρ)

(1 −

βφtanφ

)+ 3ρ

(tan(φ/2)φ/2 − β

) . (12)

Knowing that θ1 = 0 and using Eqs. (8) and (9) the followingmatrix equation can be obtained from Eqs. (6) and (7): S22

E I

L−(S12 + S22)

E I

L2

−(S21 + S22)E I

L2 (S11 + 2S12 + S22)E I

L3 − P/L

×

{θ2∆

}=

{MQ

}. (13)

From Eq. (13) the following expressions can be obtaineddirectly for a cantilever column subjected simultaneously to P ,Q and M at the top end:

θ2 =

[(1 +

S21

S22

)∆L

+M

S22

]L

E I(14a)

and

∆ =QL + M(1 + S12/S22)

E I/L(S11 − S212/S22)− P L

L . (14b)

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Fig. 3. Example 2: Stability analysis of a two span beam–column with an intermediate elastic support: (a) Structural model and deflected shape; and (b) End forcesand moments on elements 1 and 2.

Expression (14b) can be reduced to ∆ =QL3

E Iφ2

(tanφφ

− 1)

+

M L2

E I

(1−cosφφ2 cosφ

)which is identical to the superposition of Eqs.

(g) and (1-33) presented by Timoshenko and Gere ([12],pp. 5 and 14) for the particular case of a perfectly clampedBernoulli–Euler cantilever column (i.e., ρ = β = 1) subjectedto lateral load Q and overturning moment M , respectively. Thelarge deflection and post-buckling behavior of a Timoshenkocantilever column subjected to loadings at the top end is fullydiscussed by Aristizabal-Ochoa [11].

Example 2. Stability of a two-span beam–column with anintermediate elastic support.

Determine the buckling load of the two-span continuousbeam–column shown in Fig. 3a. Assume that members 1 and2 are initially horizontal with properties (GAs)1, (EI)1, l1 and(GAs)2, (EI)2, l2, respectively. Also assume that the appliedloads αP and P remain always horizontal and are applied atnodes 2 and 3, respectively. Include the effects of the transversestiffness S of the intermediate support 2. Compare the resultswith those presented by Timoshenko and Gere ([12], pp. 70and 99) when the member is made of two Euler–Bernoullibeam–columns.

Solution: From moment equilibrium of beam segments 1 and2 (Fig. 3b) the following two equations are obtained:

M21 + (P + αP)∆ − V1l1 = 0 (15)

M23 − P∆ + V2l2 = 0. (16)

From moment equilibrium of joint 2:

M21 + M23 = 0. (17)

Knowing that V1 +V2 = S∆ and using Eq. (17), then from Eqs.(15) and (16) the following equation is obtained:

M21

(1 +

l1l2

)+

[(1 +

l1l2

+ α

)P − Sl1

]∆ = 0. (18)

In this example ρa = 0 and ρb = 1 for element 1, and ρa = 1and ρb = 0 for element 2. Then calling ϕ the bending rotation

of node 2 with respect to the horizontal axis and using Eq. (2):

M21 = (Sbb)1E I

l1

(ϕ −

∆l1

)and

M23 = (Saa)2E I

l2

(ϕ +

∆l2

)where for P > 0 and P(1 + α) > 0 and using the equations in

Box I: (Sbb)1 =β1φ

21

1−β1φ1tanφ1

and (Saa)2 =β2φ

22

1−β2φ2tanφ2

.

Now, Eqs. (17), (18) can be presented in matrix form as inBox III.

By making the determinant of the 2 × 2 matrix of equationgiven in Box III the characteristic equation can be determinedand the buckling loads can be calculated. For instance for theparticular case of EI1 = EI2 and l1 = l2 = L/2 and S = 0, thecharacteristic equation in Box III is reduced to:(

1 −β1φ1

tanφ1

)(1 +

β2φ22

β1φ21

)

+

(1 −

β2φ2

tanφ2

)(1 +

β1φ21

β2φ22

)= 4. (19)

Eq. (19) yields identical results to those obtained byTimoshenko and Gere ([12], p. 99, Eqs. 2–42) whenthe member is a simply supported member made of anEuler–Bernoulli beam–column (i.e., β1 = β2 = 1) withcompressive loads at mid span (node 2) and at end 3.

Another interesting case is when the value of S becomesvery large making ∆ ≈ 0, and then the characteristic equationis reduced to the first term of the equation given in Box III asfollows:

(Sbb)1E I1

l1+ (Saa)2

E I2

l2= 0

or

β2φ22

β1φ21

tanφ2

tanφ1

(tanφ1 − β1φ1

tanφ2 − β2φ2

)= −

l2l1

E I1

E I2. (20)

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ARTICLE IN PRESS6 J. Dario Aristizabal-Ochoa / Engineering Structures ( ) –

(Sbb)1E I1

l1+ (Saa)2

E I2

l2−(Sbb)1

E I1

l21

+ (Saa)2E I2

l22

(Sbb)1E I1

l1

L

l2−(Sbb)1

E I1

l21

L

l2+ β1φ

21

E I1

l21

+ β2φ22

E I2

l22

l1l2

+ Sl1

{ϕ∆}

= 0

Box III.

Fig. 4. Interaction diagrams P/(EI/L2) versus αP/(EI/L2) for the two-spanbeam–column of Example 2 (for the particular case of l1 = l2 = L/2,G1 As1 = G2 As2 = ∞ and E1 I1 = E2 I2 = EI) with an intermediate elasticsupport with S/(EI/L3) = ∞, 100, and 0.

Fig. 5. Variations of the critical Loads of a simple supported beam–columnwith the stiffness of the intermediate elastic support S (where Pe = π2EI/L2).

Eq. (20) yields identical results to those obtained byTimoshenko and Gere ([12], p. 67, Eq. (b)) for a two-spanEuler–Bernoulli beam–column (i.e., β1 = β2 = 1) withcompressive loads located at nodes 1 and 3.

Fig. 4 shows a series of interaction diagrams of P/(EI/L2)

versus αP/(EI/L2) for different values of S/(EI/L3) allobtained using the equation given in Box III for the particularcase of l1 = l2 = L/2, (GAs)1 = (GAs)2 = ∞ and(EI)1 = (EI)2 = EI. Fig. 5 shows the variations of the bucklingload Pcr of a simply supported beam with the stiffness of theintermediate elastic support S located at midspan for differentvalues of GAs/Pe (notice that Pcr and S are normalized withrespect to Pe = π2EI/L2).

Notice that: (1) the values indicated in Fig. 5 correspondingto curve with S = 0 are identical to those presented byTimoshenko and Gere ([12], p. 100, Tables 2–6 for n = 1);

Fig. 6. Example 3: Stability analysis of a double symmetrical frame with beam-to-column connection of stiffness κ . (After Timoshenko & Gere [12], p. 34.)

(2) buckling load capacity of the beam–column is affectedsignificantly not only by the location and stiffness of the mid-support but also by the magnitude of the shear stiffness ofthe members as shown by the curves in Fig. 5; and (3) thecurve corresponding to GAs/Pe = ∞ in Fig. 5 shows that thecompressive critical load increases in approximately the sameproportion as S. This curve is identical to that presented byTimoshenko and Gere ([12], pp. 73).

Example 3. Stability analysis of a double symmetrical framewith semi-rigid connections.

Determine the buckling load of the double symmetricalframe with semi-rigid beam-to-column connections shownin Fig. 6. The horizontal members are connected to thevertical columns at both ends with semi-rigid connectionsof stiffness κ (or fixity factor ρ). Assume that the verticaland horizontal members are initially straight each withthe following set of properties (G1, E1, A1, As1, I1, h) and(G2, E2, A2, As2, I2, b), respectively. Also assume that theapplied loads P and Q are always vertical and horizontal,respectively. Include the effects of the end connections.Compare the results with those presented by Timoshenko andGere ([12], p. 62) when the beam-to-column connections arerigid (i.e. ρ = 1). Study the stability of the frame under tensionand compression forces.

Solution: In this particular frame ρa = ρb = 1 for both thecolumns, and ρa = ρb = ρ for both the beams. Now, frommoment equilibrium of joint 1:

(M12)col. + (M14)beam = 0. (21)

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ARTICLE IN PRESSJ. Dario Aristizabal-Ochoa / Engineering Structures ( ) – 7

Due to symmetry in the buckling mode, (i.e., θ2 = θ4 =

−θ1 and consequently a single unknown rotation and no swaybetween the ends of the members), then using Eqs. (2a) and(2b):

(M12)col. =E1 I1

hθ1 [(Saa)1 − (Sab)1] (22)

and

(M14)beam =E2 I2

bθ1 [(Saa)2 − (Sab)2] (23)

where: (Saa)1 =1−

β1φ1tanφ1

β1−tan(φ1/2)φ1/2

; (Sab)1 =

β1φ1sinφ1

−1

β1−tan(φ1/2)φ1/2

;

(Saa)2 =

3ρ(1 − ρ)β2φ22 + 9ρ2

(1 −

β2φ2tanφ2

)(1 − ρ)2β2φ

22 + 6ρ(1 − ρ)

(1 −

β2φ2tanφ2

)+ 9ρ2

(tan(φ2/2)φ2/2

− β2

) ;

and

(Sab)2 =

9ρ2(β2φ2sinφ2

− 1)

(1 − ρ)2β2φ22 + 6ρ(1 − ρ)

(1 −

β2φ2tanφ2

)+ 9ρ2

(tan(φ2/2)φ2/2

− β2

) .Substituting expressions (22) and (23) into the moment

equilibrium equation (21), Eq. (21′) is obtained:[E1 I1

E2 I2

b

h{(Saa)1 − (Sab)1} + (Saa)2 − (Sab)2

]θ1 = 0. (21′)

By making the left-hand term of Eq. (21′) equal to zero andsubstituting the four expressions for Saa and Sab listed above,the following characteristic equation is finally obtained:

E1 I1E2 I2

b

h

β1φ1sinφ1

(1 + cosφ1)− 2

β1 −tan(φ1/2)φ1/2

+

ρ(1 − ρ)β2φ22 + 9ρ2

(1 −

β2φ2tanφ2

)− 9ρ2

(β2φ2sinφ2

− 1)

(1 − ρ)2β2φ22 + 6ρ(1 − ρ)

(1 −

β2φ2tanφ2

)+ 9ρ2

(tan(φ2/2)φ2/2

− β2

) = 0.

(24)

The characteristic equation (24) can be reduced further totan(φ1/2)φ1/2

= −bh

E1 I1E2 I2

tanφ2/2φ2/2

for a frame with Bernoulli–Eulercolumns and beams with ρ = 1 which is identical to Eqs. (2)–(25) presented by Timoshenko and Gere ([12], p. 63).

Fig. 7a–c show a series of interaction diagrams ofP/(EI/L2) versus Q/(EI/L2) for different values ρ andGAs/(EI/L2) all obtained using Eq. (24) for the particular caseof h = b = L , G1 As1 = G2 As2 = GAs and E1 I1 = E2 I2 =

EI. Notice that: (1) buckling load capacities are affectedsignificantly not only by the stiffness of beam-to-columnconnections but also by the magnitude of the shear stiffnessof the members; and (2) buckling under axial tension forcesis a mirror image of buckling under axial compression forcesfor short members with low shear stiffness like multilayerelastomeric bearings as claimed by Kelly [13]. Further detailsof buckling of beam–columns under axial tension are given byAristizabal-Ochoa [10,11,14] and Kelly [13].

Fig. 7. Interaction diagrams P/(EI/L2)-versus −Q/(EI/L2) for a doublesymmetrical frame of Example 3 with different values of GAs/(EI/L2) andbeam-to-column fixity factor ρ := (a) 1; (b) 0.6; and (c) 0.2.

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ARTICLE IN PRESS8 J. Dario Aristizabal-Ochoa / Engineering Structures ( ) –

Fig. 8. Example 4: Stability analysis of an unsymmetrical frame with lateral sway partially inhibited: (a) Structural model and loads; and (b) Forces on eachdeflected member.

Example 4. Stability analysis of an unsymmetrical plane frameunder axial loads with lateral sway partially inhibited.

Determine the buckling loads of the plane frame withlateral sway partially inhibited as shown in Fig. 8a.The flexural stiffness κ of beam-to-column and thoseof the column-to-foundation semi-rigid connections areindicated in Fig. 8a. Assume that the vertical and horizontalmembers are initially straight each with the followingset of properties [(EI)1, (GAs)1, h1], [(EI)2, (GAs)2, h2] and[(EI)3, (GAs)3, L], respectively. Also assume that the mode ofbuckling is as shown in Fig. 8a and that the applied loads P andαP are always vertical. Include the effects of the stiffness S ofthe lateral bracing at node 2. Compare the results with thosepresented by Salmon and Johnson [5] for a symmetrical framemade of Euler–Bernoulli members when the beam-to-columnconnections are rigid (i.e., ρ1 = ρ2 = 1) and the bases of thecolumns are perfectly hinged (i.e., ρ3 = ρ4 = 0) and α = 1.Discuss also the stability of the frame under tension forces.

Solution: In this particular frame there are three unknownsθ1, θ2 and ∆ as shown in Fig. 8a. Now, from momentequilibrium of joints 1 and 2:

M13 + M12 = 0 (25)

M24 + M21 = 0. (26)

The third equation is obtained by applying equilibrium toeach member (see Fig. 8b):

Column 1: M31 + M13 + P∆V1h1 = 0Column 2: M42 + M24 + αP∆V1h1 = 0Beam: V1+V2 − S∆ = 0.From these three conditions of equilibrium and Eqs. (25) and

(26), Eq. (27) is obtained:

M31 + M13

h1+

M42 + M24

h2

+

[(1h1

+α

h2

)P − S

]∆ = 0. (27)

Now using Eqs. (2a) and (2b):

M31 =E I1

h1θ1(Sab)1 −

E I1

h21

[(Saa)1 + (Sab)1] ∆ (28)

M13 =E I1

h1θ1(Sbb)1 −

E I1

h21

[(Sbb)1 + (Sab)1] ∆ (29)

M42 =E I2

h2θ2(Sab)2 −

E I2

h22

[(Saa)2 + (Sab)2] ∆ (30)

M24 =E I2

h2θ2(Sbb)2 −

E I2

h22

[(Sbb)2 + (Sab)2] ∆ (31)

M12 =E I3

Lθ1(Saa)3 +

E I3

Lθ2(Sab)3 (32)

M21 =E I3

Lθ1(Sab)3 +

E I3

Lθ2(Sbb)3. (33)

Substituting (28)–(33) into Eqs. (25)–(27) the 3 × 3homogeneous system given in Box IV is obtained:

In Box IV:

(Saa)1 =

3ρ3

(β1φ1tanφ1

− 1)

(1 − ρ3)(

1 −β1φ1tanφ1

)+ 3ρ3

(tan(φ1/2)φ1/2

− β1

) ;

(Sab)1 =

3ρ3

(β1φ1sinφ1

− 1)

(1 − ρ3)(

1 −β1φ1tanφ1

)+ 3ρ3

(tan(φ1/2)φ1/2

− β1

) ;

(Sbb)1 =

(1 − ρ3)β1φ21 + 3ρ3

(1 −

β1φ1tanφ1

)(1 − ρ3)

(1 −

β1φ1tanφ1

)+ 3ρ3

(tan(φ1/2)φ1/2

− β1

) ;

(Saa)2 =

3ρ4

(β2φ2tanφ2

− 1)

(1 − ρ4)(

1 −β2φ2tanφ2

)+ 3ρ4

(tan(φ2/2)φ2/2

− β2

) ;

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ARTICLE IN PRESSJ. Dario Aristizabal-Ochoa / Engineering Structures ( ) – 9

(Sbb)1E I1

h1+ (Saa)3

E I3

L(Sab)3

E I3

L−[(Sab)1 + (Sbb)1

] E I1

h21

(Sab)3E I3

L(Sbb)2

E I2

h2+ (Sbb)3

E I3

L−[(Sab)2 + (Sbb)2

] E I2

h22

−[(Sab)1 + (Sbb)1

] E I1

h21

−[(Sab)2 + (Sbb)2

] E I2

h22

S −

(1

h1+α

h2

)P +

[(Sab)1 + 2(Sab)1 + (Sbb)1

] E I1

h1+[

(Sab)2 + 2(Sab)2 + (Sbb)2] E I2

h2

θ1θ2∆

= 0

Box IV.

(Sab)2 =

3ρ4

(β2φ2sinφ2

− 1)

(1 − ρ4)(

1 −β2φ2tanφ2

)+ 3ρ3

(tan(φ2/2)φ2/2

− β2

) ;

(Sbb)2 =

(1 − ρ4)β2φ22 + 3ρ4

(1 −

β2φ2tanφ2

)(1 − ρ4)

(1 −

β2φ2tanφ2

)+ 3ρ4

(tan(φ2/2)φ2/2

− β2

) ;

(Saa)3 =3ρ1(4 + Γρ2)

(4 − ρ1ρ2)+ Γ (ρ1 + ρ2 + ρ1ρ2);

(Sab)3 =3ρ1ρ2(2 − Γ )

(4 − ρ1ρ2)+ Γ (ρ1 + ρ2 + ρ1ρ2); and

(Sbb)3 =3ρ2(4 + Γρ1)

(4 − ρ1ρ2)+ Γ (ρ1 + ρ2 + ρ1ρ2).

By making the determinant of the 3 × 3 matrix of equation inBox IV equal to zero the characteristic equation is determinedand then the buckling loads can be calculated.

For the particular case of a symmetrical unbraced framewith Bernoulli–Euler columns and beam with EI1 = EI2,β1 = β2 = β3 = 1, h1 = h2 = h, α = 1, ρ1 = ρ2 = 1,ρ3 = ρ4 = 0, S = Γ = 0, and assuming that θ1 = θ2 thecharacteristic equation in Box IV can be reduced to:

φ1 tanφ1 = 6E I3/L

E I1/h. (34)

Another interesting case is when the value of S becomes verylarge making ∆ ≈ 0 (i.e. for a frame with sidesway inhibited),then the characteristic equation is reduced to the first two rowsand columns of equation in Box IV:∣∣∣∣∣∣∣(Sbb)1

E I1

h1+ (Saa)3

E I3

L(Sab)3

E I3

L

(Sab)3E I3

L(Sbb)2

E I2

h2+ (Saa)3

E I3

L

∣∣∣∣∣∣∣ = 0.

(35)

For the particular case of a braced symmetrical frame withBernoulli–Euler columns and beam with EI1 = EI2, β1 = β2 =

β3 = 1, h1 = h2 = h, α = 1, ρ1 = ρ2 = 1, ρ3 = ρ4 = 0,Γ = 0 and assuming that θ1 = −θ2 the characteristic equation(35) can be further reduced to:

φ21 sinφ1

sinφ1 − φ1 cosφ1= −2

E I3/L

E I1/h. (36)

Eqs. (34) and (36) are identical to Eqs. (14-2-30) and (14-2-18) presented by Salmon and Johnson ([5], pp. 900–912) forunbraced and braced symmetrical frames, respectively. It must

Fig. 9. Comparison of results for a symmetrical frame with lateral swayinhibited using the proposed method and a simplified formula after Salmonand Johnson [5].

be emphasized that Eq. (34) is only capable of capturing theanti-symmetric modes of buckling of this unbraced symmetricframe (since it was assumed that θ1 = θ2). Likewise, Eq. (36)is only capable of capturing the symmetric modes of bucklingof this braced symmetric frame (since it was assumed thatθ1 = −θ2). Fig. 9 shows the first three eigen-values obtainedusing the characteristic equation in Box IV and the first twoobtained using Eq. (34). Therefore, the proposed method ismore powerful, since it is capable of capturing all modes ofbuckling including those under axial tension forces (as it wasshown in Example 3).

5. Summary and conclusions

A new set of slope-deflections equations that include theeffects of shear deformations and the shear component of theapplied axial forces (Haringx Model) are derived and thenutilized in the stability and second-order analysis of framedstructures made of Timoshenko beam–columns with rigid,semi-rigid, and simple connections. The proposed method isbased on the “modified” stability functions for beam–columnswith semi-rigid connections (Aristizabal-Ochoa [10,11]). Thevalidity and effectiveness of the proposed equations are verifiedagainst four well-documented solutions on elastic stability andsecond-order analyses of beam–columns and plane frames.

The advantages of the proposed method are: (1) the effectsof semi-rigid connections are condensed into two nonlinear

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ARTICLE IN PRESS10 J. Dario Aristizabal-Ochoa / Engineering Structures ( ) –

equations with three unknowns (the end flexural rotations andthe lateral sway between the ends of the member) for tension orcompression axial loads without introducing additional degreesof freedom and equations; (2) more accurate than any othermethod available and capable of capturing the phenomenaof buckling under axial tension forces. This is particularlyimportant in short columns like laminated elastomeric bearingscommonly used for seismic isolation of buildings; and (3) themethod is powerful, practical, versatile, and easy to teach.

Analytical studies indicate that shear deformations increasethe lateral deflections of framed structures, and reduce theircritical axial loads. The effects of shear deformations andsecond-order P∆ effects must be considered in the analysisof beam–columns with relatively low effective shear areas(like short laced columns, columns with batten plates or withperforated cover plates, and columns with open webs) or lowshear modulus G (like elastomeric bearing and short columnsmade of laminated composites) resulting in members with shearstiffness GAs of the same order of magnitude as EI/L2. Theshear effects are also of great importance in the static, stabilityand dynamic behavior of laminated elastomeric bearings usedfor seismic isolation of buildings.

Acknowledgments

The research presented in this paper was carried out atthe National University of Colombia, School of Mines atMedellin. The author wants to express his appreciation to theDepartment of Civil Engineering, School of Mines (DIME)for the financial support and encouragement, and also to JhonMonsalve-Cano, Jaime Hernandez-Urrea, Camilo Builes-Mejiaand David Padilla-Llano members of the Structural StabilityGroup of the National University of Colombia, for preparingthe figures of all four examples.

Appendix. Derivation of proposed slope-deflection equa-tion (2a) and (2b)

The stability analysis of a prismatic column includingbending and shear deformations (Fig. 1a–c) is formulated usingthe “modified” approach proposed by Haringx and discussed byTimoshenko and Gere ([12], page 134). This approach has beenutilized by Kelly [13] in the analysis of Elastomeric IsolationBearings and by Aristizabal-Ochoa [10,11,14] in the stabilityanalysis of columns, multicolumn systems and frames.

Knowing that the governing equation of bending for thebeam–column shown in Fig. 1b is:

M = −E I∂ψ

∂x(37)

and using Haringx’s approach:

V = G Asγ − Pψ (38)

γ =∂u

∂x− ψ. (39)

On using expression (39) suggested by Haringx, thegoverning Eqs. (37) and (38) for bending moment and

transverse shear force of the beam–column shown in Fig. 1b[for which M = Ma − (Ma + Mb + P∆) x

L − Pu and V =

−Pψ + (Ma + Mb + P∆)/L] become:

βE I∂2u

∂x2 + Pu = −Ma + (Ma + Mb + P∆)x

L(40)

βE I∂2ψ

∂x2 + Pψ =Ma + Mb + P∆

L(41)

where: u = Lateral deflection along the beam–column centerline; andψ = Rotation of the cross section along the beam–column

due to bending alone as shown in Fig. 1c.γ = Shear deformation of the cross section along the

beam–column as shown in Fig. 1c.The solutions to the second-order linear differential

equations (40) and (41) are as follows:

u(x) = A cos( x

Lφ)

+ B sin( x

Lφ) x

L

(Ma + Mb + P∆

P

)−

Ma

P(42)

ψ. (x. ) = C cos( x

Lφ)

+ D sin( x

Lφ)

+1L

(Ma + Mb + P∆

P

). (43)

The unknown coefficients A, B, C , and D can be obtained fromthe following boundary conditions:

At A′ (x = 0): u = 0, ψ = ψa′

At B′ (x = h) : u = ∆ and ψ = ψb′ where: ψa′ andψb′ = rotations of cross sections at ends A′ and B′ with respectto the vertical axis due to bending, respectively;

Therefore: A =MaP ; B =

MaP tan(φ/2) −

Ma+MbP sinφ ; C =

ψa′ −1L

(Ma+Mb

P + ∆)

; and

D =Ψa′ − Ψb′ cosφ

sinφ−

(Ma + Mb

P+ ∆

)tan(φ/2).

Since u′= ψ + V/(As G) and V = Pψ −

Ma+Mb+P∆L , the

following expressions for ψa′ and ψb′ can be obtained:

ψa′ =sinφ − βφ cosφ

βφ2 sinφ

Ma

E I/L+

sinφ − βφ

βφ2 sinφ

Mb

E I/L+

∆L

(44a)

ψb′ =sinφ − βφ

βφ2 sinφ

Ma

E I/L+

sinφ − βφ cosφ

βφ2 sinφ

Mb

E I/L+

∆L.

(44b)

However, the rotations at ends A and B are θa = ψa′ +Maκa

and

θb = ψb′ +Mbκb

, respectively.Therefore:

θa −∆L

=sinφ − βφ cosφ

βφ2 sinφ

Ma

E I/L

+sinφ − βφ

βφ2 sinφ

Mb

E I/L+

Ma

κa(45a)

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ARTICLE IN PRESSJ. Dario Aristizabal-Ochoa / Engineering Structures ( ) – 11

θa −∆L

=Ma

E I/L

sinφ − βφ

βφ2 sinφ

+Mb

E I/L

sinφ − βφ cosφ

βφ2 sinφ+

Mb

κb(45b)

where: ∆ = relative sidesway of end B with respect to end A;andθa and θb= rotations of ends A and B due to bending with

respect to the vertical axis, respectively.Notice that Eqs. (44a) and (44b) are identical to

those derived by Salmon and Johnson ([6], p. 902) forBernoulli–Euler columns (i.e. when β = 1). Now, expressing(45a) and (45b) in matrix form, then by inverting the matrixof coefficients and using expressions (1a) and (1b) [or Ra =

3ρa/(1 − ρa) and Rb = 3ρb/(1 − ρb)], Eqs. (2a) and (2b) canbe obtained.

A.1. Derivations of Eqs. (3)–(5)

The governing Eqs. (40) and (41) when P = 0 are reducedto:

EI∂2u

∂x2 = −Ma + (Ma + Mb)x

L(46a)

EI∂2ψ

∂x2 =Ma + Mb

L. (46b)

Eqs. (46a) and (46b) are now much simpler to solve. Afterapplying the boundary conditions [i.e., u = 0, ψ = ψa′ atA′ (x = 0); and u = ∆ and ψ = ψb′ at B′ (x = L)], theirsolutions are:

u(x) =∆L

x +L

6E I(2Ma − Mb) x −

Ma

2E Ix2

+1

6E I L(Ma + Mb) x3 (47a)

ψ(x) =∆L

+ (Ma + Mb)1

G As+

L

6E I(2Ma − Mb)

−Ma

E Ix +

12E I L

(Ma + Mb) x2. (47b)

Knowing that ψ(x) = u′(x) − V/(G As) and V = −(Ma +

Mb)/L , the following expressions for ψa′ and ψb′ can beobtained from Eq. (47b):

ψa′ =

(13

+E I/L2

G As

)Ma

E I/L

+

(−

16

+E I/L2

G As

)Mb

E I/L+

∆L

(48a)

ψb′ =

(−

16

+E I/L2

G As

)Ma

E I/L

+

(13

+E I/L2

G As

)Mb

E I/L+

∆L. (48b)

As explained above, the rotations at ends A and B are:θa = ψa′ +

Maκa

and θb = ψb′ +Mbκb

, respectively. Therefore:

θa −∆h

=

(13

+E I/h2

G As

)Ma

E I/h

+

(−

16

+E I/h2

G As

)Mb

E I/h+

Ma

κa(49a)

θa −∆h

=

(−

16

+E I/h2

G As

)Ma

E I/h

+

(13

+E I/h2

G As

)Mb

E I/h+

Mb

κb. (49b)

Expressing (49a) and (49b) in the matrix form, then by invertingthe matrix of coefficients and using Ra = 3ρa/(1 − ρa)

and Rb = 3ρb/(1 − ρb), Eqs. (2a)–(2b) and (3)–(4) can beobtained.

References

[1] Samuelsson A, Zienkiewicz OC. History of the stiffness method.International Journal for Numerical Methods in Engineering 2006;67:149–57.

[2] Wilson WM, Maney GA. Slope-deflection method. University of IllinoisEngineering Experiment Station, Bulletin 80, 1915.

[3] Norris CH, Wilbur JB. Elementary structural analysis. 2nd ed. McGraw-Hill Book Co; 1960.

[4] Kassimali A. Structural analysis. 2nd ed. Thomson-Engineering; 1998.[5] Salmon CG, Johnson JE. Steel structures: Design and behavior. 4th ed.

Harper Collins College Publishers; 1996 [Chapter 14].[6] Bryant RH, Baile OC. Slope deflection analysis including transverse

shear. Journal of the Structural Division 1977;103(2):443–6.[7] Lin FJ, Glauser EC, Johnston BJ. Behavior of laced and battened

structural members. Journal of Structural Engineering, ASCE 1960;123(5):1377–401.

[8] Dario Aristizabal-Ochoa J. First- and second-order stiffness matrices andload vector of beam–columns with semirigid connections. Journal ofStructural Engineering, ASCE 1997;123(5):669–78.

[9] Dario Aristizabal-Ochoa J. Story stability of braced, partially braced, andunbraced frames: Classical approach. Journal of Structural Engineering,ASCE 1997;123(6):799–807.

[10] Dario Aristizabal-Ochoa J. Column stability and minimum lateral bracing:effects of shear deformations. Journal of Engineering Mechanics 2004;130(10):1223–32.

[11] Dario Aristizabal-Ochoa J. Large deflection and postbuckling behavior ofTimoshenko beam–columns with semirigid connections including shearand axial effects. Journal of Engineering Structures 2007;29(6):991–1003.

[12] Timoshenko S, Gere J. Theory of elastic stability. 2nd ed. McGraw-Hill;1961 [Chapter 2].

[13] Kelly JM. Tension buckling in multilayer elastomeric bearings. Journalof Engineering Mechanics 2003;129(12):1363–8. Discussion by DarioAristizabal-Ochoa J. 2005; 131 (1): 106–108.

[14] Dario Aristizabal-Ochoa J. Tension and compression stability and second-order analyses of three-dimensional multicolumn systems: Effects ofshear deformations. ASCE Journal of Engineering Mechanics 2007;133(1):106–16.

Please cite this article in press as: Dario Aristizabal-Ochoa J. Slope-deflection equations for stability and second-order analysis of Timoshenho beam–columnstructures with semi-rigid connections. Engineering Structures (2008), doi:10.1016/j.engstruct.2008.02.007