Dynamic stability of multi-span frames subjected to periodic loading

Journal of Constructional Steel Research 70 (2012) 65–70

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Journal of Constructional Steel Research

Dynamic stability of multi-span frames subjected to periodic loading Gürkan Şakar a,⁎, Hasan Öztürk b, Mustafa Sabuncu b a b

Department of Mechanical Engineering, Atatürk University, 25240 Erzurum, Turkey Department of Mechanical Engineering, Dokuz Eylul University, 35100 Izmir, Turkey

a r t i c l e

i n f o

Article history: Received 16 July 2010 Accepted 9 October 2011 Available online 17 November 2011 Keywords: Stability Multi-span frame Vibration Buckling Finite element method

a b s t r a c t In this study, in-plane dynamic stability analysis of multi-span frames, which are composed of columns and beams and subjected to periodic loading, is investigated by using the Finite Element Method. Periodic loading is considered to be applied to each column member as an axial load and no loading is applied on the beams. The effects of beam to column length ratio, beam to column cross-section ratio and beam to column moment of area ratio are investigated for in-plane frames. In addition, the effect of the number of span and static and dynamic load parameters on the free vibration, buckling and dynamic stability analysis is also investigated. The results are compared with the recent literature in terms of the fundamental natural frequency and critical buckling load. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Various types of frames especially the civil-engineering type, are widely used in structural configurations such as buildings and bridges. These frames are subjected to concentrated and distributed static or dynamic loads which may cause static and dynamic instability. Many investigations about the vibration and buckling (static stability) characteristics of frames of various types have been carried out. The dynamic stability of mechanical systems represents a specific stability of motion according to Bolotin's definition [1]. When Bolotin's approach [1] is examined, three stages, vibration analysis, static stability (buckling analysis), and dynamic stability analysis, are seen to be included in the equation of dynamic stability. Therefore these three stages are studied in this paper. A numerical computer method using planar flexural finite line element for the determination of buckling loads of beams, shafts and frames, supported by rigid or elastic bearings has been studied by Bagci [2] and the buckling loads and the corresponding mode vectors were determined by the solution of a linear set of eigenvalue equations of elastic stability. Zaslavsky [3] has investigated the elastic stability of portal frames consisting of hinging and/or fixed columns and hinged girder with overhang under the action of a moving load. Simitses and Vlahinos [4] have presented a kinematically non-linear analysis of a two-bar frame connected by a flexible joint which is modeled by a rotational spring connecting one member to the other, subjected to eccentric concentrated loads. Syngellakis and Kameshki [5] have studied elastic critical loads for plane frames by

⁎ Corresponding author. E-mail address: [email protected] (G. Şakar). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.10.009

using the transfer matrix method. Bayo and Loureiro [6] have presented a direct one-step method, which is based on a non-linear analysis of the structure starting from an initial deformation state that includes the initial imperfections of the elements, for the buckling analysis of steel frame structures. Xu and Liu [7] have developed a practical method for the stability analysis of semi-braced steel frames with the effect of semi-rigid behavior of beam-to-column connections being taken into account. A simplified procedure for determining approximate values for the buckling loads of both regular and irregular frames was developed by Girgin et al. [8]; where the procedure utilizes lateral load analysis of frames and yields errors on the order of 5%, which may be considered suitable for design purposes. Tong and Xing [9] have studied the instability of braced frames by geometric and nonlinear material analysis accounting for residual stresses, initial sway imperfection and member initial bow. In this study, the change of buckling mode with increasing bracing stiffness was analyzed and the relationship between the ultimate load capacity and

Fig. 1. Multi-span frame structure subjected to periodic loading.

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66

Table 1 Comparison of results for the fundamental frequency parameter in the case of clamped and pinned single-span frame. Boundary conditions

E1I1/E2I2

m1/m2

L1/L2

Laurai et al. [14]

Present Results

0.25

0.25

1.5

1.5

6

6

0.25

0.25

1.5

1.5

6

6

1.5 3 6 1.5 3 6 1.5 3 6 1.5 3 6 1.5 3 6 1.5 3 6

2.561 3.337 4.078 3.935 4.628 5.071 4.044 4.571 4.976 1.237 1.583 1.893 1.743 2.052 2.244 1.507 1.848 2.103

2.559 3.327 4.029 3.933 4.620 5.030 4.043 4.567 4.950 1.236 1.577 1.867 1.742 2.047 2.219 1.507 1.845 2.084

the bracing rigidity was developed [10]. Xu and Wang have proposed an approach for evaluating the elastic buckling loads for multi-storey unbraced steel frames subjected to variable loading or nonproportional loading. Trahair [11] has studied the method of designing steel beams, columns, and plane frames against out-of-plane failure, which uses the results of an elastic flexural–torsional buckling analysis in the strength rules of design codes, called as design by buckling analysis (DBA). There are also various studies related to the vibration of the frames. A general digital computer method based on a Sturm sequence procedure has been described by Gupta [12] for determining the natural frequencies and associated modes of undamped free

Table 2 Comparison of results for the fundamental buckling load parameter in the case of clamped and pinned single-span frame. Boundary conditions

Timoshenko and Gere [24]

Present results

7.34

7.37

1.82

Two-span frame

E1I1/E2I2

m1/m2

L1/L2

Buckling load parameter for clamped ends

Buckling load parameter for pinned ends

0.25

0.25

1.5

1.5

6

6

1.5 3 6 1.5 3 6 1.5 3 6

9.355 9.585 9.651 7.603 8.516 9.056 5.126 6.391 7.556

2.338 2.396 2.412 1.880 2.123 2.263 1.115 1.528 1.867

vibration of frames and other structures whose stiffness and mass matrices are in a band form. Karabalis and Beskos [13] have investigated the static, dynamic and stability analysis of linear elastic plane structures consisting of beams with constant width and variable depth by using the finite element method. Laurai and Filipich [14,15] have investigated the vibrating frames carrying concentrated masses, and in-plane vibrations of portal frames with end supports, elastically restrained against rotation and translation by using analytical and numerical method. The natural frequencies and mode shapes for the in-plane vibration of triangular closed and planar frames (portal, H and T frames) have been studied by using the Rayleigh–Ritz method [16, 17]. A known fact is that a beam subjected to in-plane forces generally experiences forced in-plane vibration and when exciting frequencies overlap with the natural frequencies of the beam, resonance will occur. Apart from this phenomenon, for certain values of exciting frequencies, an entirely different type of resonance (called as parametric resonance) will also occur in the transverse direction and beam is said to be dynamically unstable. This type of resonance is also called parametric resonance. Thomas and Abbas [18] have examined the

Table 5 Buckling load parameters for various parameters for three-span frame. Three-span frame

E1I1/E2I2 m1/m2 L1/L2

Buckling load Buckling load parameter for parameter for clamped ends pinned ends

0.25

0.25

1.5

1.5

6

6

9.431 9.624 9.687 7.831 8.682 9.161 5.351 6.645 7.788

1.82

Table 3 Buckling load parameters for various parameters for single-span frame. Single-span frame

Table 4 Buckling load parameters for various parameters for two-span frame.

E1I1/E2I2

m1/m2

L1/L2

Buckling load parameter for clamped ends

Buckling load parameter for pinned ends

0.25

0.25

1.5

1.5

6

6

1.5 3 6 1.5 3 6 1.5 3 6

9.331 9.546 9.512 7.371 8.391 8.902 4.727 6.012 7.255

2.332 2.386 2.377 1.819 2.092 2.224 0.975 1.416 1.786

1.5 3 6 1.5 3 6 1.5 3 6

2.357 2.406 2.421 1.943 2.167 2.289 1.194 1.606 1.931

Table 6 Buckling load parameters for various parameters for four-span frame. Four-span frame

E1I1/E2I2 m1/m2 L1/L2 Buckling load Buckling load parameter for parameter for clamped ends pinned ends 0.25

0.25

1.5

1.5

6

6

1.5 3 6 1.5 3 6 1.5 3 6

9.463 9.646 9.709 7.950 8.763 9.224 5.486 6.791 7.916

2.365 2.411 2.427 1.974 2.188 2.305 1.240 1.649 1.965

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67

Table 7 Fundamental frequency parameters for various parameters for two-span frame. Two-span frame

Fundamental E1I1/E2I2 m1/m2 L1/L2 Fundamental frequency parameter frequency parameter for pinned ends for clamped ends 0.25

0.25

1.5

1.5

6

6

1.5 3 6 1.5 3 6 1.5 3 6

2.280 3.027 3.775 3.747 4.477 4.958 4.056 4.604 4.990

1.109 1.448 1.768 1.682 2.00 2.201 1.572 1.893 2.119

dynamic stability of beams under periodical loading using the FEM and Bolotin's approach. Saito and Otomi [19] have studied the stability and vibration of a beam, which is elastically supported at both ends and carrying a mass subjected to periodical load, using the Laplace transform. Park [20] has examined the dynamic stability of a free–free beam under follower load by the FEM-Hamilton principle. Sinha [21] has determined the dynamic stability of simply supported beams, under periodical axial loading, using the Galerkin Method. Brisseghella et al. [22] have investigated the dynamic stability of elastic structures using the FEM. The dynamic stability of plane elastic frames subjected to a vertical foundation motion of the random input stationary, ergodic type with zero mean value has been investigated by Zingone and Muscolino [23]. As it is mentioned in the above literature review, there are many investigations about the vibration and buckling (static stability) characteristics of frames of various types. However, dynamic stability, which is the analysis of instability regions, has not been studied widely prior to this study. In addition, in-plane dynamic stability analysis of multi-span frames, which are composed of columns and beams

Table 8 Fundamental frequency parameters for various parameters for three-span frame. Three-span frame

E1I1/E2I2 m1/m2 L1/L2 Fundamental frequency parameter for clamped ends

Fundamental frequency parameter for pinned ends

0.25

0.25

1.5

1.5

6

6

1.064 1.396 1.721 1.670 1.991 2.196 1.607 1.927 2.147

1.5 3 6 1.5 3 6 1.5 3 6

2.182 2.908 3.661 3.686 4.431 4.929 4.067 4.632 5.022

Fig. 2. Effect of L1/L2 ratios on the first dynamic instability regions of single-span frame. E1I1/E2I2 =0.25, m1/m2 = 0.25, α =0, (—— L1/L2 =1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/L2 = 3).

and subjected to periodic loading, has also not been studied prior to this paper. Within this context, the objective of this paper is to study the effects of beam to column length ratio, beam to column cross-section ratio, beam to column moment of area ratio, number of span, and static and dynamic load parameters on the free vibration, buckling and dynamic stability analysis, and to provide information for the future researches. 2. The finite element formulation Fig. 1 shows the multi-span frame subjected to periodic loading. The beam that forms the frame is assumed to be a Euler beam and is modeled by using the FEM. The beam element has 2 nodes with three degrees of freedom in each node. After applying the standard procedure for a finite element analysis for a beam element, elastic


stiffness matrix k
e ; mass matrix ½ m e  and geometric matrix k
ge ; can be obtained. Transforming these matrices in terms of the reference coordinates a set of matrices below is obtained: h h h ih T ke  ¼ T k¯ e T ¼ 0

h h h ih T ¯ me  ¼ T m e T ¼ 0

h h h i T kge  ¼ T k¯ ge ½T ¼ 0

ð1Þ

Table 9 Fundamental frequency parameters for various parameters for four-span frame. Four-span frame

E1I1/E2I2 m1/m2 L1/L2 Fundamental frequency parameter for clamped ends

Fundamental frequency parameter for pinned ends

0.25

0.25

1.5

1.5

6

6

1.039 1.368 1.695 1.661 1.984 2.193 1.626 1.946 2.162

1.5 3 6 1.5 3 6 1.5 3 6

2.128 2.844 3.598 3.650 4.402 4.913 4.072 4.648 5.039

Fig. 3. Effect of L1/L2 ratios on the first dynamic instability regions of four-span frame. E1I1/E2I2 = 0.25, m1/m2 = 0.25, α = 0, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/ L2 = 3).

G. Şakar et al. / Journal of Constructional Steel Research 70 (2012) 65–70

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Fig. 4. Effect of L1/L2 ratios on the first dynamic instability regions of single-span frame. E1I1/E2I2 = 1, m1/m2 = 1, α = 0, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/L2 = 3).

where ½T is the transformation matrix. Langrange's equation of motion for the completed structure in the matrix form is given as, h

hh i i h ii Me fq€ g þ K e −PðtÞ K ge fqg ¼ 0

ð2Þ


where ½K e  ; K ge  and ½Me  are global elastic stiffness, global geometrical stiffness and global mass matrices of completed structure respectively. The periodic load P ðtÞ ¼ P o þ P t cosΩt with Ω is the disturbing frequency which is considered to apply to each column member as an axial load. There are no axial periodic forces on the beams. The static and time dependent components of the load can be represented as a fraction of the fundamental static buckling load Pcr. Therefore substituting P ðtÞ ¼ αP cr þ βP cr cosΩt in Eq. (2) gives: h

i

Me fq€ g þ

hh

i  h ii K e − αP cr þ βP cr cosΩt K ge fqg ¼ 0:

Fig. 6. Effect of L1/L2 ratios on the first dynamic instability regions of single-span frame. E1I1/E2I2 = 1.5, m1/m2 = 1.5, α = 0, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/ L2 = 3).

the dynamic instability can be determined from the equation given below, " h

# i   h i Ω2 K e − α  1=2β P cr K ge − ½M e  fqg ¼ 0: 4

ð4Þ

This equation represents the solution of the three related problems: (i) Free vibration with α ¼ 0, β ¼ 0 and ω ¼ Ω=2 the natural frequency hh

i h ii 2 K e −ω Me fqg ¼ 0

ð5Þ

ð3Þ (ii) Static stability with α ¼ 1, β ¼ 0 and Ω ¼ 0

Eq. (3) represents a system of second order differential equation with periodic coefficients of the Mathieu–Hill type. From the theory of linear equations with periodic coefficients, the boundaries between stable and unstable solutions of Eq. (3) are formed by periodic solutions of period T and 2T, where T ¼ 2π=Ω. It has been shown by Bolotin [1] that solutions with period 2T carry great practical importance. For the first approximation, the boundaries of the principal regions of

Fig. 5. Effect of L1/L2 ratios on the first dynamic instability regions of four-span frame. E1I1/E2I2 = 1, m1/m2 = 1, α = 0, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/L2 = 3).

hh

h ii i K e −P cr K ge fqg ¼ 0

ð6Þ

Fig. 7. Effect of L1/L2 ratios on the first dynamic instability regions of four-span frame. E1I1/E2I2 = 1.5, m1/m2 = 1.5, α = 0, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/ L2 = 3).

G. Şakar et al. / Journal of Constructional Steel Research 70 (2012) 65–70

Fig. 8. Effect of L1/L2 ratios on the first dynamic instability regions of single-span frame. E1I1/E2I2 = 0.25, m1/m2 = 0.25, α = 0.2, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/ L2 = 3).

(iii) Dynamic stability when all terms are present " h

# i h i Ω2 h i K e −ðα  1=2βÞP cr K ge − M e fqg ¼ 0: 4

ð7Þ

3. Result and discussion In this study, in-plane free vibration, static and dynamic stability analyses subjected to periodic loads of multi-span frame are investigated by using the finite element method. Periodic loading is considered to be applied to each column members as an axial load while there is no loading on the beams. Material properties such as modulus of elasticity, dimensions etc. of the frame for these analyses and the results obtained from the present study are shown as parametrical values for generalization. 3.1. Buckling and free vibration analyses In the present study, parametrical comparison for the natural frequency and buckling load of multi-span frames is performed and compared with the results obtained by Laurai et al. [14] and

Fig. 9. Effect of L1/L2 ratios on the first dynamic instability regions of four-span frame. E1I1/E2I2 = 0.25, m1/m2 = 0.25, α = 0.2, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/ L2 = 3).

69

Fig. 10. Effect of L1/L2 ratios on the first dynamic instability regions of single-span frame. E1I1/E2I2 = 1, m1/m2 = 1, α = 0.2, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·L1/L2 = 3).

Timeshenko and Gere [24], respectively. Tables 1 and 2 show the comparison of results for the fundamental frequency and buckling load coefficients in the case of clamped and pinned single-span frame. As seen from Tables 1 and 2, comparison shows that agreement between the related results is very good. Tables 3–6 show the non-dimension buckling load for various parameters for the multi-span frame. When the number of spans of the frame structure increases, the buckling load parameter also increases. Increase of E1I1/E2I2 and m1/m2 ratios reduces the buckling load parameter. In addition, as can be seen from Table 3, when the L1/L2 ratio increases the buckling load parameter also increases; except for the case of E1I1/E2I2 = 0.25 and m1/m2 = 0.25 and L1/L2 = 6 parameters of single-span frame. However, the increment rate of buckling load parameter in this case rises with increasing E1I1/E2I2 and m1/ m2 ratios. The effect of various parameters mentioned above on the buckling load parameter is valid for both clamped ends and pinned end conditions. Consequently as can be seen from Tables 3–6, the buckling load parameter for the frames with clamped ends is larger than those of the frame with pinned ends. Tables 1, 7–9 show the natural frequency parameters for various parameters for the multi-span frame. When the number of spans of frame structure increases for both boundary conditions, the fundamental frequency parameter decreases except for the case of E1I1/ E2I2 = 6 and m1/m2 = 6. For clamped ends boundary condition, the fundamental frequency parameter generally increases with the

Fig. 11. Effect of L1/L2 ratios on the first dynamic instability regions of four-span frame. E1I1/E2I2 = 1, m1/m2 = 1, α = 0.2, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/L2 = 3).

70

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origin and gets narrower with increasing number of spans of the frame structure. When E1I1/E2I2 and m1/m2 ratios increase for all the frames, the first dynamic unstable regions widen and shift them towards higher Ω=p ratios, as seen Figs. 2–7. In this case, the maximum narrowing occurs at L1/L2 = 1 ratio. Figs. 8–13 show effect of the static load parameter (α) on the first dynamic unstable region. As seen in the figures, the first dynamic unstable regions widen and shift to the origin by increasing the static load parameter. 4. Conclusion

Fig. 12. Effect of L1/L2 ratios on the first dynamic instability regions of single-span frame. E1I1/E2I2 = 1.5, m1/m2 = 1.5, α = 0.2, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·L1/L2 = 3).

In this paper, the static and the dynamic stability of multi-span frames are studied. When the number of spans of the frame structure increases, the buckling load parameter also increases and on the other hand, the first dynamic unstable region moves towards the origin and gets narrower. Finally, by changing the L1/L2, E1I1/E2I2, and m1/m2 ratios, number of span of the frame, boundary conditions, and static and dynamic load parameters, the static and the dynamic stability of the multi-span frames may be conserved. References

increase of E1I1/E2I2 and m1/m2 ratios except for single-span frame with E1I1/E2I2 = m1/m2 = 6, L1/L2 = 3 and 6. The fundamental frequency parameters of all the frames for pinned end boundary conditions have a similar increase as for the frames with clamped ends boundary condition. In this case, the frequency parameter shows a drop trend after E1I1/E2I2 = m1/m2 = 1.5 values. However when L1/ L2 ratio increases, the fundamental frequency also increases. The fundamental frequency parameter for the frames with clamped ends is larger than those of the frame pinned ends, as expected. 3.2. Dynamic stability analysis Figs. 2–13 show the effects of beam to column length ratio, beam to column cross-section ratio, beam to column moment of area ratio, static and dynamic load parameters on the dynamic stability analysis for the first unstable region for only single and four span frames. The forcing (disturbing) frequency (Ω) to the first natural frequency (p) of single-span frame (E1I1/E2I2 = 1 and m1/m2 = 1 and L1/L2 = 1) ratios, which construct the boundaries of unstable regions versus the dynamic load parameter (β) is plotted. In addition, all columns of the frames are considered clamped ends. When L1/L2 ratio increases, the first dynamic unstable region widens and shifts to higher T ¼ Ω=p ratios for all the frames. On the other hand, the first dynamic unstable region moves towards the

Fig. 13. Effect of L1/L2 ratios on the first dynamic instability regions of four-span frame. E1I1/E2I2 = 1.5, m1/m2 = 1.5, α = 0.2, (—— L1/L2 = 1, - - - - - - L1/L2 = 1.5, -·-·-·- L1/ L2 = 3).

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