ESTABILIDAD Y ARRIOSTRAMIENTO MÍNIMO LATERAL PARA COLUMNAS ESCALONADAS CON CONEXIONES SEMI-RÍGIDAS INCLUYENDO LOS EFECTOS DE CORTANTE: II) VERIFICACIÓN Y EJEMPLOS

J. DARIO ARISTIZABAL-OCHOA
School of Mines, National University, Medellín, Colombia, jdaristi@unal.edu.co

Received for review January 26^th, 2012, accepted August 14^th, 2012, final version August, 15^th, 2012

ABSTRACT: The elastic stability analysis of single-stepped columns with sidesway totally inhibited, partially inhibited, and uninhibited subjected to concentrated axial loads located at the ends and at the intermediate joint including semi-rigid connections and shear force effects are presented using three different approaches in a companion paper. The first two approaches are those by Engesser and Haringx that include the shear component of the applied axial force proportional to the total slope (dy/dx) and to the angle of rotation of the cross section () along the member, respectively. The third approach is a simplified formulation based on the classical Euler theory that includes the effects of shear deformations but neglects the shear component of the applied axial force along the member. Definite criterion on minimum stiffness of lateral bracings for single-stepped columns is also presented. Five comprehensive examples are included that demonstrate the effectiveness of the proposed stability equations and minimum bracing stiffness criteria.

KEY WORDS: bracing, buckling, building codes, columns, construction, computer applications, frames, loads, semi-rigid connections, shear deformations, shoring, stability, stepped columns

PALABRAS CLAVE: arriostramiento, pandeo, códigos de construcción, columnas, construcción, aplicaciones informáticas, marcos, cargas, conexiones semi-rígidas, deformaciones por cortante, apuntalamiento, estabilidad, columnas con cambio de sección

1. INTRODUCTION

The elastic stability analysis and the corresponding equations for stepped columns subject to end and intermediate-axial loads including the effects of: 1) lateral and rotational restraints at the intermediate connection and column ends; and 2) shear deformations and shear force along the member caused by the applied axial loads are presented in a companion paper [1]. The main objective of this article is to present the use of the proposed stability equations and to demonstrate their effectiveness.

2. VERIFICATION STUDY AND EXAMPLES

2.1 Example 1. Consider the single-stepped column with hinged ends subjected to concentric axial loads P and aP at A and C shown in Fig. 1a (hinged at A and B: r_a = r_b = 0; and with rigid connection at C and no exterior rotational restraint: r_c = 1, K_c = 0).

Solution: Using Eq. (10) listed in the companion paper and assuming that r_a = r_b = 0, and k_c = 0, the eigenvalue equation becomes

For the particular case of ==1 (i.e., neglecting shear effects), Eq. (2) is identical to the solution reported by Timoshenko and Gere [1], Eq. (b), p.67.

An additional example was considered assuming that k_c = 3EI'/H (bending restraint provided by a transverse beam located at midspan with a hinged far end), h_t = h_b, I_t = I_b, b_t = b_b = 1, E_t = E_b = E, and a = 0, as shown in Fig. 1b. Then Eq. (2) becomes

Since u_t = u_b, Eq. (3) is reduced to Eq. (4)

Expression (4) is identical to the solution reported by Timoshenko and Gere ([2], bottom of p. 69). Note that in Timoshenko's notation, u_t = 2u₁, and u_b = 2u₂,.

Figure 2 shows the effects of shear deformations (using the Haringx approach) and semi-rigid connection at C (r_c) on the buckling loads of the column of Fig. 1(a) assuming that column segments are made of identical material, cross section, and equal length (h_t = h_b= L/2). Notice that: 1) shear deformations reduce the critical loads P and aP, while the effects of the semi-rigid connection at C are only significant when GA_s/(EI/L²) is very large and aP • 0; and 2) the proposed equation is capable of capturing the phenomena of buckling under tension when P or/and aP are negative

Using Eq. (10) listed in the companion paper and assuming that r_a = r_b = 0 and r_c = 1, the eigenvalue equation becomes

Results from Eq. (5) were tested against tabulated values presented by Timoshenko and Gere ([2], Table 2-6, p. 100) for the particular case of S_c = k_c = 0, b_t = b_b = 1 and E_t = E_b = E. The values of the reduced length L of the column calculated from Eq. (5) which are listed in Table 1 (denoted as mCal) are practically identical to those derived by Timoshenko and Gere [2] denoted as mTh) where (P_a+P_c)_cr = p²EI/L².

Figure 4 show a series of interaction diagrams of P/(EI/L²)-vs-aP/(EI/L²) for different values of S/(EI/L³) all obtained using Eq. (5) for the particular case of h_t = h_b = L/2, G_tA_st = G_bA_sb = ∞ and E_tI_t = E_bI_b = EI. Figure5 shows the variations of the compressive buckling load P_cr of a simply supported beam with the stiffness of the intermediate elastic support S located at midspan for different values of GA_s/P_e (notice that P_cr and S are normalized with respect to P_e = p²EI/L²).

Notice also that: 1) the values indicated in Fig. 5 corresponding to the curve with S = 0 are identical to those presented by Timoshenko and Gere ([2], p. 100 Table 2-6 for n = 1); 2) the buckling load capacity of the beam-column is affected significantly not only by the location and stiffness of the mid-support but also by the magnitude of the shear stiffness of the members as shown by the curves in Fig. 5 and 3) the curve corresponding to GA_s/P_e = ∞ in Fig. 5 shows that the compressive critical load increases in approximately the same proportion as S. This curve is identical to that presented by Timoshenko and Gere ([2], p. 73).

2.3 Example 3. The validity of Eq. (12) was tested against the solutions obtained utilizing the classical method of slope-deflection (Salmon and Johnson [3], p. 840) for a stepped column with hinged ends subjected to concentric axial loads P_a and P_c at A and C as shown in Fig. 6 (free at A: r_a = 0, S_a = 0; braced at C with no exterior rotational restraint: r_c = 1, K_c = 0; and hinged at B: r_b = 0).

Substituting into Eq. (12) r_a = r_b = 0, r_c = 1, and K_c = 0, the eigenvalue equation becomes

For the particular case: P_c = 0, h_t = h_b = h, E_tI_t = E_bI_b = EI, andb_t = b_b = 1, the stability equation is reduced to 2u_t = tanu_t, whose solution is u_t = 1.16556 or (P_a)_cr = p²EI/(2.6953h2). Using the slope-deflection method (Salmon and Johnson [3], pp. 840-842) a stability equation identical to (6b) can be obtained.

2.4 Example 4. Results from Eq. (13) were tested against tabulated solutions presented by Timoshenko and Gere ([2], Table 2-10, p. 115) for a stepped cantilever Euler-Bernoulli column (b_t = b_b = 1) shown in Fig. 7 free at A (r_a = S_a = 0), rigidly connected and unrestrained at C (r_c = 1, S_c = K_c = 0), and perfectly fixed at B (r_b = 1).

Table 2 shows that the values of m calculated using Eq. (13), which are listed with five significant figures, are practically identical to those by Timoshenko. Note that m is for the hinged-hinged column, and m/4 for the cantilever column, where m is used by Timoshenko in P_cr = mEI₂/(h₁+h₂)².

2.5 Example 5: Minimum lateral bracing for a bent-up frame

Utilizing the minimum bracing criteria and the steps described in the companion paper, determine the lateral bracings required to convert the bent frame shown in Fig. 8a into a braced frame. The bracings that need to be analyzed are: 1) S_a along the top level (i.e., frame braced at A and A' only); 2) S_c at the intermediate joint (i.e., frame braced at C and C' only); and 3) S_a and S_c along the top and intermediate joints (i.e., frame braced at A and C). Assume that r_b = r_c = 1 in both columns and neglect the effects of shear deformations (i.e., b = 1 in all members).

The first step is to find the fixity factors at the column ends for unbraced and braced conditions. Relationships between the rotational restraints and the fixity factors in framed structures are presented in the Appendix of the companion paper. For this particular frame, the fixity factors are as follows:

i) For unbraced conditions along AA', the frame would buckle in a anti-symmetric shape (Figs. 8b-c) with the beam providing rotational restraints at both ends A and A' of magnitude 6EI/L = 1.5EI/h (since L = 4h). Therefore, r_a = 1/(1+3/1.5) = 1/3, and r_b = r_c = 1.

ii) For braced conditions along AA', the frame would buckle in a symmetric shape (Fig. 8d) with the beam providing rotational restraints at both ends A and A' of magnitude 2EI/L = 0.5EI/h. Therefore, r_a = 1/(1+3/0.5) = 1/7, and r_b = r_c = 1.

The required bracings for column AB for each one of the requested cases are calculated as follows:

1. Braced at top end A only. Taking into consideration that: I_t = I, h_t = h, I_b = 2I, h_t = 2h, P_a = P, P_c = 3P, r_a = 1/3, and r_b = r_c = 1, the solution for braced conditions obtained from Eq. (11) are: (P_a)_cr = 0.19072(p²EI/h²) and (P_a+P_c)_cr = 0.76288(p²EI/h²) (or u_t = p = 1.37198 and u_b = p = 3.88055). Now, the magnitude of S_a can obtained directly from Eq. (11) by substituting the values of u_t = 1.37198 and u_b = 3.88055, r_a = 1/3, and r_b = r_c = 1. The result is S_a = 3.3818EI/h³ per column.
2. Braced at the intermediate joint C only. Taking into consideration that: I_t = I, h_t = h, I_b = 2I, h_t = 2h, P_a = P, P_c = 3P, r_a = 1/3, and r_b = r_c = 1, the solution for braced conditions obtained from Eq. (12) are: (P_a)_cr = 0.20909(p²EI/h²) and (P_a+P_c)_cr = 0.83637(p²EI/h²) (or u_t = p = 1.43654 and u_b = p = 4.06315). Now, the magnitude of S_c can obtained directly from Eq. (12) by substituting the values of u_t = 1.43654, u_b = 4.06315, r_a = 1/3, and r_b = r_c = 1. The result is S_c = 2.77322EI/h³ per column.
3. Braced at end A and at the intermediate joint C simultaneously. Taking into consideration that: I_t = I, h_t = h, I_b = 2I, h_t = 2h, P_a = P, P_c = 3P, r_a = 1/7, and r_b =r_c = 1, the solution for braced conditions obtained from Eq. (10) are: (P_a)_cr = 0.329338(p²EI/h²) and (P_a+P_c)_cr = 1.31735(p²EI/h²) (or u_t = p = 1.802898 and u_b = p = 5.09936).

Now, the magnitude of S_a can obtained directly from the characteristic equation = 0 by deleting the second row and column (or ) and substituting the values of u_t = 1.802898, u_b = 5.09936, r_a= 1/7, and r_b =r_c = 1. The result is S_a = 47.3354EI/h³ per column.

Similarly, S_c can obtained directly from the characteristic equation = 0 by deleting the third row and column (or ) and substituting the values of u_t = 1.802898, u_b = 5.09936, r_a = 1/7, and r_b = r_c = 1. The result is S_c = 1.096677EI/h³ per column.

It is interesting to note that: 1) the trend for this particular frame is indicated by Eq. (18) as follows: (P_a+P_c)_cr = 1.31735(p²EI/h²) > 0.83637(p²EI/h²) > 0.76288(p²EI/h²) > 0.16074(p²EI/h²) (this last value corresponds to the unbraced frame); and 2) the required lateral bracing at A is relatively small compared to that required at the intermediate joint C.

3. SUMMARY AND CONCLUSIONS

A complete column classification and the corresponding stability equations for single-stepped columns with sidesway totally inhibited, partially inhibited, and uninhibited subjected to concentrated axial loads located at the ends and at the intermediate joint including the shear force effects are presented using three different approaches in a companion paper.

To understand the four-way classification of single-stepped columns and post shores, the corresponding stability equations, five examples are used in this paper and the results are compared to those using other methods. A verification study indicates that the calculated elastic buckling loads are theoretically exact.

Analytical studies indicate that: 1) the stability of a single-stepped column increases substantially with the magnitude of the lateral restraints and the fixity at the base and at the intermediate connection; 2) shear deformations and the shear forces induced by the applied axial loads reduce the buckling loads; 3) the degree of fixity at the top end has less influence on the overall stability of stepped columns and post shores in frames with sidesway inhibited than in frames with sidesway uninhibited; 4) the critical axial loads using the Engesser approach is lower than that using the Haringx and simplified Euler approaches; 5) the critical loads in tension and in compression predicted by the Haringx approach are very sensitive to the shear stiffness GA_s; 6) the Haringx approach is the only one among the three approaches capable of capturing the phenomena of tension buckling; and 7)as expected, the critical axial loads in compression are highly affected by the degree of flexural fixity at the supports, but the critical axial loads in tension are not affected as much. It is important to emphasize that shear effects are not of significance and can be neglected for slender columns of solid cross sections like rectangular and I-cross sections.

ACKNOWLEDGMENTS

The research presented in this paper was carried out at the National University of Colombia at Medellin. The author wants to express his appreciation to Jaime Hernandez-Urrea, currently a Ph.D. student at the University of Illinois at Champaign-Urbana and former member of the Structural Stability Group GES of the National University of Colombia at Medellin for helping and preparing the figures of all the examples.

NOTATION

REFERENCES

[1] Aristizabal, J. D., Stability and minimum lateral bracing for stepped columns with semi-rigid connections including shear effects: I) Theory, Revista DYNA, 174, 2012, pp. 96-106.         [ Links ]
[2] Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, 2nd Ed., McGraw-Hill Book Inc., New York, N.Y. 1961.         [ Links ]
[3] Salmon, C. G. and Johnson, J.E., Steel Structures: Design and Behavior, 4th Ed., HarperCollins College Publishers (Chapter 14), 1996.         [ Links ]