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Approximate formulae for natural periods of plane steel frames C. Chrysanthakopoulos, N. Bazeos, D.E. Beskos ∗ Department of Civil Engineering, University of Patras, GR-26500 Patras, Greece Received 24 November 2004; accepted 28 September 2005

Abstract Approximate formulae for determining by hand with a high enough accuracy the first three natural periods of vibration of plane steel unbraced and braced frames are provided. These formulae are based on the modeling of a plane steel frame as an equivalent cantilever beam for which analytical expressions for the natural periods are available. Extensive parametric studies involving the finite element computation of the first three natural periods of 110 plane steel unbraced and braced frames are employed to establish correction factors for the equivalent beam modeling formulae which are functions of the number of stories and bays of the frame. The resulting corrected formulae permit a highly accurate determination of the first three natural periods of plane steel frames. c 2005 Elsevier Ltd. All rights reserved. Keywords: Natural periods; Plane steel frames; Braced frames; Unbraced frames; Multi-storey frames; Multi-bay frames; Equivalent flexural-shear beam

1. Introduction Seismic building codes, such as UBC [1] or EC 8 [2], provide analytical expressions for the computation of the design seismic acceleration in terms of the natural period of vibration of the structure. Thus, the design base shear can be computed easily by either multiplying this acceleration by the structural mass if only the first mode participates in the response, or by modal synthesis if the first few modes participate in the response. In the first case, a knowledge of the first (fundamental) period is necessary, while in the second case a knowledge of the first few natural periods of the structure is required. Lopez and Cruz [3] have established empirical formulae providing the required number of the first few modes necessary for obtaining the seismic response of building frames with a relative error of 5% and 10%, while seismic building codes [1,2] require the participation in the response of so many modes as to have at least a participation of 90% of the total structural mass. Seismic codes, such as UBC [1] or EC 8 [2], provide very simple but crude empirical formulae for the fundamental period of structures in terms of their material (steel or reinforced concrete), structural type (frame, shear wall, etc.) and height. ∗ Corresponding author. Tel.: +30 2610 996559; fax: +30 2610 996579.

E-mail address: [email protected] (D.E. Beskos). c 2005 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2005.09.005

Goel and Chopra [4], on the basis of experimental data gathered from eight earthquakes, were able to improve the accuracy of these formulae by modifying their coefficients. A more rational way of constructing formulae for the hand computation of natural periods of vibrations of tall plane wall-frame buildings is to establish relations to model those buildings as equivalent flexural-shear cantilever beams for which the natural periods are available in analytical form in standard structural dynamics texts. One can mention here, e.g., the works of Coull [5], Rosman [6], Rutenberg [7], Stafford Smith and Crowe [8], Li et al. [9] and Zalka [10]. The advantage of those formulae is that they enable the designer to rapidly compute natural periods by hand during the preliminary design stage. This advantage is not present in those methods of determining natural periods of large order plane or space frames and trusses by modeling them as continuous beams and employing the finite element method. In those methods the goal is to drastically reduce the computational work and not to provide simple formulae for the periods. For a comprehensive review on the subject, one can consult Noor [11]. In this work approximate formulae for determining by hand the first three natural periods of vibration of plane steel frames are presented. Formulae of high enough accuracy for both unbraced and braced frames are based on them being modeled as equivalent cantilever beams, in accordance with the approach of Stafford Smith and Crowe [8] for which analytic expressions for natural periods are available. These formulae are modified

C. Chrysanthakopoulos et al. / Journal of Constructional Steel Research 62 (2006) 592–604

by some correction factors, functions of the number of frame stories and bays, which are constructed with the aid of extensive parametric studies involving finite element computation of the first three natural periods of 110 plane steel unbraced and braced frames. Thus, the present work is analogous to that of Goel and Chopra [4] in the sense that, through data from numerical experiments, improvements in the accuracy of existing formulae are realized. In this work, frames of up to 15 stories are considered and thus the first three modes are enough to satisfy the 90% vibrating mass criterion of seismic codes [1, 2] as well as the empirical formulae of Lopez and Cruz [3]. 2. Free vibrations of a flexural-shear beam The free vibrations of a flexural-shear prismatic beam are governed by the equation [8] EI

2 ∂ 4v ∂ 2v 2∂ v − E I (αk) + m =0 ∂x4 ∂x2 ∂t 2

(1)

where E I and m are the flexural rigidity and mass per unit length, respectively, of the beam, v = v(x, t) is the lateral deflection of the beam, x and t denote axial coordinate and time, respectively, and αk expresses a shear rigidity to be defined explicitly later on. The above equation can be decomposed into the equivalent system of the two equations ∂ 4v f ∂ 2v f + m =0 ∂x4 ∂t 2 2 ∂ 2 vs f ∂ 4 vs f 2 ∂ vs f − E I (αk) + m =0 EI ∂x4 ∂x2 ∂t 2 EI

(2)

v1 = [(k 2 − 1)/k 2 ]v f ,

v2 = (1/k 2 )vs f

V (H, t) = 0,

∂v(0, t)/∂ x = 0 M(H, t) = 0

(4)

(5)

where V and M denote shear force and bending moment, respectively. Thus, Eq. (2), describing flexural free vibrations, yields the eigenvalue equation [12] 1 + cos λ f H cosh λ f H = 0

(6)

(7)

where λ4f = ω2f m/E I

λ22 = λ21 + (αk)2 ,

λ21 λ22 = λ4s f = ωs2f m/E I

(10)

with ωs f being the natural frequency of coupled flexural-shear vibrations. According to Rutenberg [7] and the Southwell–Dunkerley approximation, one can finally obtain the natural periods T of free vibrations governed by Eq. (1) in the form T = (2π/λ2 ) m/E I (11) where 1 ≈ λ4

k2 − 1 k2

1 + λ4f

1 k2

1 . λ4s f

(12)

Approximate, yet reasonably accurate, solutions of Eq. (9) have the form [8] kα H < 6 (13) (λs f H )2 ∼ = (λ f H )2[1 + (kα H /λ f H )]1/2, 2 ∼ (λs f H ) = (n − 0.5)π(1 + kα H ), kα H ≥ 6. (14)

Plane orthogonal, braced or unbraced frames, shear walls or coupled frame-wall systems fixed on the ground can be modeled as equivalent flexural-shear cantilever beams. This equivalence can be established by expressing E I and αk of Eq. (1) in terms of the geometrical and material parameters of the frame or the frame-wall system. Following [8] one has that, for α = [G A/E I ]1/2,

(8)

k = [1 + (E I /E Ac2 )]1/2

(15)

the equivalent flexural-shear beam can be established provided the three parameters E I , E Ac2 and G A can be expressed in terms of the properties of the frame or the frame-wall system. Thus, for an unbraced bay of a frame, E I and E Ac2 of the equivalent beam can be expressed as [8] EI =

n j =1

whose solution in terms of its eigenvalues λ f is [12] (λ f H )1 = 1.875, (λ f H )2 = 4.694 n = 3, 4, . . . (λ f H )n ∼ = (n − 0.5)π,

for which

3. Plane frame structures as equivalent beams

with v f denoting the deflection component due purely to flexural motion and the vs f deflection component due to a coupled shear-flexural motion. For a cantilever beam of length H with the fixed end at x = 0 and the free end at x = H , the boundary conditions read [12] v(0, t) = 0,

with ω f being the natural frequency of flexural vibration. Eq. (3) on the other hand, describing coupled flexural-shear free vibrations, yields the eigenvalue equation [7] 2 λ1 2 λ2 2+ cos λ1 H cosh λ1 H + λ2 λ1 λ2 λ1 + − (9) sin λ1 H sinh λ2 H = 0 λ1 λ2

(3)

for which v = v1 + v2

593

(E I ) j ,

E Ac2 =

n

(E Ac2) j

(16)

j =1

where (E I ) j is the flexural rigidity of the j th vertical member (column or wall) of the system, (E A) j the axial rigidity of the j th vertical member, (c) j the distance of the j th column to the center of the area of the vertical members of the lateral load resisting frame, and n the total number of vertical members of the frame. On the other hand, G A of the equivalent beam can

594

C. Chrysanthakopoulos et al. / Journal of Constructional Steel Research 62 (2006) 592–604 Table 1 Section properties of frames with constant sections per height

Fig. 1. Geometrical parameters (a) and types (b, c, d, e) of braced parts of frames.

be computed from [8]

j =1

(17)

i=1

where H j is the height of floor j , L i the i th bay length, Ibi and Ici are beam and column second moments of area of bay i , respectively, and n and m denote the total number of columns and beams, respectively. For a braced bay of a frame, E Ac2 is given again by (16)2 , while E I is given by [13] E I = E A f h 2 /2

(18)

where A f is the cross-sectional area of the column and h the distance between the centroids of the two columns (Fig. 1a). On the other hand, G A receives different expressions for different types of bracing. Thus, following [8,13], one has that G A = n E Ad sh 2 /d 3

(19)

for x or double-diagonal (n = 2) and single-diagonal (n = 1) bracing (Fig. 1b, c), 2d 3 h GA = sE (20) + 4 Av h 2 Ad for K bracing (Fig. 1d), and d3 m s 2 (1 − 2m)2 GA = sE + + 2 Av 12h Iv 2m 2 Ad

(21)

for knee bracing (Fig. 1e). In the above expressions, m is the horizontal distance of the brace top connection from the column (Fig. 1e), s and h the dimensions of the composite brace elements (Fig. 1a), while Ad and Av denote the sectional areas of the bracing element and column, respectively (Fig. 1a). For a framed structure consisting of b braced and u unbraced bays, the parameters E I , E Ac2 and G A of the equivalent flexural-shear beam are obtained using EI =

u b (E I )i + (E I ) j i=1

j =1

Beams

1 Bay

4 Storey 7 Storey 10 Storey

HEB 280 HEB 320 HEB 340

IPE 360 IPE 400 IPE 450

2 Bay

4 Storey 7 Storey 10 Storey 15 Storey

HEB 280 HEB 360 HEB 400 HEB 450

IPE 360 IPE 450 IPE 450 IPE 450

3 Bay

4 Storey 7 Storey 10 Storey 15 Storey

HEB 280 HEB 360 HEB 400 HEB 450

IPE 360 IPE 450 IPE 450 IPE 450

G A = (G A)u +

b (G A) j

(23)

j =1

1 1 + m G A = 12E/H n Icj /H j /L (Ibi i )

Columns

(22)

in which (E I ) j is given by (18), (G A) j by (19)–(21), (E I )i and (G A)u by (16)1 and (17), respectively, and E Ac2 by (16)2 with the braced bays accounted for in the calculation of c j by replacing each braced bay by an equivalent column with the values E I and G A of the given braced bay. 4. Parametric studies and comparisons In order to check the validity of the simple approximate formulae for natural period determination of steel frames presented in the previous section, the natural periods of a large number of steel plane frames were computed using these formulae and compared against those obtained using a finite element analysis considered to be the ‘exact’ values. In addition, the same periods were also compared against those obtained using the crude approximate formulae of codes. 4.1. Properties of analyzed frames The analyzed plane steel frames are divided into two major categories: those that retain their member properties constant along the height of the building and those that do not. Furthermore, there are five subcategories depending on the type of brace that each frame incorporates. As a result, there are unbraced frames and frames with single, double, K or knee bracing. In each subcategory, there are frames composed of 1–3 bays and 4–15 stories. The total number of plane steel frames analyzed is 110. Some of these frames are depicted in Figs. 2 and 3. The length of one bay is 4 m and the story height is 3 m. In knee bracing, the m value (Fig. 1e) is equal to 1.6 m. The section properties of columns and beams are summarized in Tables 1 and 2. The diagonal brace in every case is of the tubular cross section D127.4. Young’s modulus of elasticity E is 200 GPa, the Poisson ratio v is 0.3, and mass the per unit volume is 795 kg/m3 . The mass of each frame comes from the G + 0.3Q combination, where the dead load G = 36 kN/m and the live load Q = 10 kN/m.

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595

Fig. 2. Unbraced frames.

Fig. 3. Double-diagonal braced frames.

4.2. Number of evaluated periods

According to them, for buildings below 20 stories

Lopez and Cruz [3] have determined the number of modes that are necessary in a modal superposition method so that the error in forces and deformations does not exceed 5%.

NM = 3, if T1 /T ∗ < 1.5 1 NM = (T1 /T ∗ − 1.5) + 3, 2

if T1 /T ∗ > 1.5

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for braced steel frames with H measured in m. The current Greek seismic code E.A.K. [15] proposes the formula H H (27) T1 = 0.09 √ H + ρL L where H is the height in m, L the width in m, and ρ the ratio of the wall surface to that of all the vertical elements. 4.4. Comparison of results and error study Fig. 4. Modeling of a frame structure as an equivalent cantilever.

in which NM is the number of modes, T1 is the first natural period of the structure, and T ∗ is the period at the knee of the spectrum. It can easily be seen that the minimum number of modes that must be taken into account are 3. For the set of analyzed steel frames, NM could reach 4, depending on the parameter T ∗ , which is not a variable in this study. However, modal analyses on all the selected frames led to the conclusion that with 3 modes the modal mass participation ratio exceeded 90% of the total mass of the structure. As a result, all the calculations, comparisons and corrections refer to the first three natural periods. 4.3. Analysis methods All the plane steel frames are analyzed by the finite element method. Once the finite element model of every frame is created, a modal analysis is performed using SAP 2000 [14] and the first three natural periods are calculated. The calculated natural periods are considered to be the exact and correct values, so they serve as control variables to check the periods obtained from the equivalent cantilever formulae, as well as the formulae of the various codes. The equivalent cantilever modeling considers the entire structure as a single cantilever (Fig. 4). The properties E I , E Ac2 and G A of the equivalent cantilever are derived from the equations of Section 3. For the calculation of λs f , Eq. (14) is used because preliminary results showed that it is more accurate compared with Eq. (13) in predicting the first periods. For reasons of completeness, the first natural periods of vibration of the considered plane steel frames are compared against those obtained by the very simple but crude formulae of some codes. Thus, according to UBC [1] T1 = 0.035H

3/4

(24)

where the height H of the frame is measured in feet (1 ft = 0.3048 m), while according to EC 8 [2] T1 = 0.085H 3/4

(25)

for unbraced frames and T1 = 0.075H 3/4

(26)

The values of the first three natural periods of the 110 plane steel frames considered here, as obtained by the equivalent cantilever formulae and the finite element method together with the values of the fundamental periods as obtained by the seismic codes formulae, were tabulated and compared. The relative % errors were computed on the basis of the relation ε(%) = (TFEM − T )100/TFEM

(28)

where TFEM is the “exact” value of the period as computed by the finite element code of SAP 2000 [14] and T is the corresponding value as obtained by any other method. Tables 3 and 4 present results for unbraced and x-braced frames, respectively, with constant cross-sections along the height, while Tables 5 and 6 present corresponding results for the case of cross-sections varying with height. Additional results for the other three types of bracing for both constant and variable cross-sections with height can be found in Chrysanthakopoulos [16]. It is observed that for unbraced frames the error is quite low. The prediction of the first period is very good and its error ranges between 10% and 20%. The results for the second and third natural periods are similar. On the contrary, for braced frames the relative error for the first period increases drastically up to 50% (double-diagonal and K bracing) and 60% (single-diagonal and knee bracing). Minimum errors stay above 20% in all cases. The estimation of the second and third periods is even worse, with deviations reaching 90%. Generally speaking, the relative error increases as frame height decreases for the same type of diagonal bracing. An increase in the number of bays slightly improves the estimated periods for a given plane frame. 5. Correction factors 5.1. Establishment of correction factors From the error analysis of the previous section, it becomes necessary to improve the prediction of natural periods using the equivalent cantilever method with the appropriate introduction of correction factors in order to bring the error down to acceptable levels. The fact that, for the braced frames, the error increases significantly in comparison with the unbraced frames leads to the conclusion that the flexural-shear component of the equivalent cantilever response requires adjustment. The correction in the form of a correction factor is applied to the obtained eigenvalues λ or λs f of the equivalent cantilever.

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Table 2 Section properties of frames with variable sections per height Columns

Beams

1 Bay

4 Storey 7 Storey 10 Storey

HEB280, HEB260 HEB 320, HEB 280, HEB 260 HEB 340, HEB 320, HEB 280, HEB 260

IPE 360, IPE 330 IPE 400, IPE 360, IPE 330 IPE 450, IPE 400, IPE 360, IPE 330

2 Bay

4 Storey 7 Storey 10 Storey 15 Storey

HEB280, HEB260 HEB 360, HEB 340, HEB 320 HEB 400, HEB 360, HEB 340, HEB 320 HEB 450, HEB 400, HEB 360, HEB 340, HEB 320

IPE 360, IPE 330 IPE 450, IPE 400, IPE 360 IPE 450, IPE 400, IPE 360 IPE 450, IPE 400, IPE 360

3 Bay

4 Storey 7 Storey 10 Storey 15 Storey

HEB280, HEB260 HEB 360, HEB 340, HEB 320 HEB 400, HEB 360, HEB 340, HEB 320 HEB 450, HEB 400, HEB 360, HEB 340, HEB 320

IPE 360, IPE 330 IPE 450, IPE 400, IPE 360 IPE 450, IPE 400, IPE 360 IPE 450, IPE 400, IPE 360

Table 3 Comparison of the first three natural periods of unbraced frames with constant sections per height Unbraced frames

Equivalent cantilever T1 T2

T3

Relative error (%) Equivalent cantilever T1 T2

T3

EAK T1

UBC T1

EC 8 T1

1 Bay

1 2 3

4 Storey 7 Storey 10 Storey

0.6405 1.0286 1.4054

0.2099 0.3255 0.4169

0.1255 0.1933 0.2437

16.19 12.82 11.36

9.78 11.71 15.77

−0.76 2.79 8.58

29.34 19.91 14.85

28.02 29.06 31.02

28.29 29.33 31.28

2 Bay

4 5 6 7

4 Storey 7 Storey 10 Storey 15 Storey

0.6647 0.8717 1.2664 2.0114

0.2199 0.2816 0.3982 0.5990

0.1317 0.1679 0.2360 0.3506

18.78 15.87 13.22 10.54

13.30 14.85 14.54 15.53

5.98 8.70 8.44 9.20

53.34 35.51 34.59 36.32

32.78 19.23 25.05 34.07

33.03 19.53 25.34 34.32

3 Bay

8 9 10 11

4 Storey 7 Storey 10 Storey 15 Storey

0.6765 0.8735 1.2468 1.9215

0.2245 0.2858 0.4012 0.5967

0.1346 0.1709 0.2390 0.3527

18.74 16.54 14.13 11.84

14.21 15.06 14.31 14.82

8.13 9.82 9.22 9.67

62.55 47.87 46.32 46.36

33.93 20.03 24.67 31.98

34.18 20.33 24.95 32.24

Table 4 Comparison of the first three natural periods of braced frames with constant sections per height Braced frames

Equivalent cantilever T1 T2

T3

Relative error (%) Equivalent cantilever T1 T2

T3

EAK T1

UBC T1

EC 8 T1

1 Bay

12 13 14

4 Storey 7 Storey 10 Storey

0.1333 0.3701 0.7348

0.0213 0.0591 0.1173

0.0076 0.0211 0.0419

54.58 35.90 24.24

77.47 64.83 53.77

86.16 76.06 66.76

−84.05 −63.66 −39.18

−87.49 −44.96 −12.75

−64.81 −27.42 0.89

2 Bay

15 16 17 18

4 Storey 7 Storey 10 Storey 15 Storey

0.1729 0.3684 0.6233 1.1618

0.0527 0.1063 0.1694 0.2883

0.031 0.0617 0.0966 0.1591

54.94 43.13 37.05 30.85

58.03 47.24 42.70 39.06

58.56 44.01 37.94 33.60

0.49 −3.14 3.60 14.77

−43.37 −29.18 −10.45 11.77

−26.02 −13.56 2.91 22.44

3 Bay

19 20 21 22

4 Storey 7 Storey 10 Storey 15 Storey

0.2106 0.4263 0.6951 1.2328

0.0669 0.1308 0.2047 0.3383

0.0398 0.0771 0.1194 0.1935

52.11 38.66 31.78 24.52

53.56 41.11 35.94 32.01

53.49 38.07 31.55 26.78

29.10 21.50 23.50 28.42

−25.11 −20.43 −7.34 9.24

−9.97 −5.86 5.64 20.22

The correction factor must be applicable to all frames. Thus, it must depend on some characteristic parameters of the frame. Parameters like these are the numbers of floors, bays, section properties and even α and k of the equivalent cantilever. The following general linear form for the correction factor (C) is adopted C = c1 + n · c2 + (m − 1) · c3

(29)

where c1 , c2 and c3 are constants, n is the number of floors, and

m is the number of bays. The determination of the constants c1 , c2 and c3 has been made so that (i) Correction factor C depends only on the number of natural periods and not on the type of the brace. (ii) Constants c2 and c3 are independent of the bracing and of the number of bays. (iii) Eq. (29) can be used in every case of frame, brace type and natural period.

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Table 5 Comparison of the first three natural periods of unbraced frames with variable sections per height Unbraced frames

Equivalent cantilever T1 T2

T3

Relative error (%) Equivalent cantilever T1 T2

T3

EAK T1

UBC T1

EC 8 T1

1 Bay

1 2 3

4 Storey 7 Storey 10 Storey

0.6856 1.1574 1.6380

0.2251 0.3687 0.4976

0.1346 0.2192 0.2927

13.17 7.60 4.69

10.66 13.44 14.86

−1.30 6.66 10.36

31.60 24.56 21.45

30.32 33.18 36.36

30.59 33.43 36.60

2 Bay

4 5 6 7

4 Storey 7 Storey 10 Storey 15 Storey

0.7118 0.9788 1.3953 2.1452

0.2356 0.3180 0.4417 0.6418

0.1412 0.1898 0.2622 0.3760

15.33 9.90 7.85 6.82

14.32 14.69 15.20 16.44

5.57 8.13 10.20 13.41

54.58 38.49 36.96 37.81

34.57 22.96 27.77 35.61

34.81 23.25 28.04 35.85

3 Bay

8 9 10 11

4 Storey 7 Storey 10 Storey 15 Storey

0.7246 0.9824 1.3796 2.0582

0.2406 0.3225 0.4458 0.6413

0.1442 0.1929 0.2659 0.3794

15.75 10.48 8.68 8.14

15.27 15.08 15.24 16.15

7.73 9.38 11.03 13.97

63.75 50.28 48.41 47.82

36.04 23.72 27.61 33.84

36.28 24.01 27.88 34.09

Table 6 Comparison of the first three natural periods of braced frames with variable sections per height Braced frames

Equivalent cantilever T1 T2

T3

Relative error (%) Equivalent cantilever T1 T2

T3

EAK T1

UBC T1

EC 8 T1

1 Bay

12 13 14

4 Storey 7 Storey 10 Storey

0.1365 0.3952 0.7826

0.0218 0.0631 0.1249

0.0078 0.0225 0.0446

53.50 31.78 19.56

77.09 63.25 51.80

85.96 75.03 65.77

−83.92 −63.13 −38.76

−87.36 −44.48 −12.42

−64.70 −27.01 1.18

2 Bay

15 16 17 18

4 Storey 7 Storey 10 Storey 15 Storey

0.1742 0.3700 0.6339 1.1953

0.0529 0.1061 0.1699 0.2895

0.0311 0.0615 0.0964 0.1581

54.90 44.18 37.57 30.18

58.56 49.20 44.94 41.30

59.14 46.01 40.44 36.93

1.13 −0.82 5.98 16.36

−42.44 −26.28 −7.72 13.41

−25.21 −11.01 5.31 23.88

3 Bay

19 20 21 22

4 Storey 7 Storey 10 Storey 15 Storey

0.2121 0.4295 0.7065 1.2622

0.0673 0.1314 0.2063 0.3403

0.0400 0.0774 0.1201 0.1936

52.26 40.03 32.66 24.30

54.39 43.86 39.04 35.00

54.17 40.45 34.61 30.77

29.81 23.82 25.71 29.88

−23.84 −16.86 −4.25 11.10

−8.86 −2.73 8.36 21.85

Thus, Eq. (29) for braced frames becomes CBR = c1 + 0.02 · n + 0.03 · (m − 1) ≤ Cmax

(30)

with Cmax = α1 + α2 · (m − 1) ≤ 1.0

(31)

while for unbraced frames it reduces to a single constant CUBR = c1 .

(32)

The corrected eigenvalue is given by λc = λ · Ci

(33)

or λs f,c = λs f · Ci

(34)

where Ci is the appropriate correction factor. The coefficients α1 , α2 and c1 of the above equations are given for various frame cases in Table 7. 5.2. Validation of correction factors After applying the correction factors, a recalculation of the deviations between the exact periods (SAP 2000) and the

predicted ones (equivalent cantilever with corrections) was performed. Eq. (28) was used on this occasion as well. Observed relative errors after the correction are very low (Figs. 5–10). For unbraced frames, improvement is little since the predicted periods were quite close to the exact ones even before the correction. Independent of bracing or not, errors remain below 15% for the first natural period. This is valid for both constant and variable section frames. The second natural period displays deviations ranging from −5% to 15%, while for the third natural period the deviation remains under 20%. Only for braced frames with 1 bay and 4 stories, the error ranges between 30% and 40%. In general, great improvement is achieved in all periods and in all frames when correction factors are applied. The first natural period is best estimated in every case. The occurrence of negative values in deviation is not alarming. Negative relative errors mean that the predicted periods from the equivalent cantilever are lower than the exact SAP 2000 values thus being less conservative. This is something one wants to avoid. Nevertheless, the obtained negative relative errors are kept very low (below 5%), which is practically a match to the exact natural period.

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Table 7 Parameters a1 , a2 and c1 for correction factor equations Bracing c1

T1 a1

a2

c1

T2 a1

a2

c1

T3 a1

a2

Apply to

Constant sections

No

0.94

–

–

0.94

–

–

0.99

–

–

λs f

Variable sections

No

0.98

–

–

0.95

–

–

1.00

–

–

λs f

Constant sections

Double Single K Knee

0.67 0.62 0.67 0.62

1.00 1.00 1.00 0.85

− − − −

0.48 0.43 0.44 0.40

0.90 0.85 0.65 0.60

− − − −

0.39 0.34 0.35 0.30

0.85 0.80 0.55 0.50

− − − −

λ λ λ λ

Variable sections

Double Single K Knee

0.71 0.65 0.70 0.66

1.00 1.00 1.00 0.90

– − − −

0.51 0.43 0.46 0.41

0.90 0.85 0.65 0.65

− − − −

0.39 0.39 0.35 0.30

0.85 0.80 0.55 0.50

− − − −

λ λ λ λ

0.55

0.75 0.75 0.80 0.80

0.10 0.10 0.05 0.05

0.56

0.75 0.75 0.75 0.75

0.05 0.05 0.10 0.05

0.58

0.80 0.80 0.80 0.80

0.05 0.05 0.05 0.05

λs f λs f λs f λs f

1 Bay

More than 1 bay

Double Single K Knee

Fig. 5. First period relative error after correction for frames with constant section along the height of the structure.

Fig. 6. Second period relative error after correction for frames with constant section along the height of the structure.

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Fig. 7. Third period relative error after correction for frames with constant section along the height of the structure.

Fig. 8. First period relative error after correction for frames with variable section along the height of the structure.

Fig. 9. Second period relative error after correction for frames with variable section along the height of the structure.

C. Chrysanthakopoulos et al. / Journal of Constructional Steel Research 62 (2006) 592–604

601

Fig. 10. Third period relative error after correction for frames with variable section along the height of the structure.

Parameters α and k can be calculated with the aid of Eq. (15) and yield GA EI = 0.6552/m k = 1+ = 1.0004. (38) α= EI E Ac2 Thus kα H equals 7.8655. The total mass of the frame per meter of height is M 20 235 = = 1686 kg/m. (39) H 12 Eq. (7) provides the values of λ f H directly for the first three modes. Eq. (14) is then used to calculate λs f . Thus, one has 2

λs f · H 1 = (1 − 0.5) π (1 + k · α · H ) m=

Fig. 11. Plane steel frame of examples.

6. Examples of using corrected formulae Consider the plane steel frame of Fig. 11 with and without the x-bracing system and with constant and variable sections along its height. The following three cases are presented to illustrate the method in a detailed manner for the convenience of the reader or the user.

= 1.5708 · (1 + 7.8655) ⇔ λs f1 = 0.3110 2

λs f · H 2 = (2 − 0.5) π (1 + k · α · H ) = 4.7124 · (1 + 7.8655) ⇔ λs f2 = 0.5386 2

λs f · H 3 = (3 − 0.5) π (1 + k · α · H )

(40)

= 7.8540 · (1 + 7.8655) ⇔ λs f3 = 0.6954. 6.1. Frame without bracing Due to the symmetry of the frame, the center of the area of the vertical members is easily found to be at the center of the frame, that is, at x = 6 m. The term E Ac2 is computed with the aid of Eq. (16)2 and yields E Ac2 = E [2 · 0.0131 · 62 + 2 · 0.0131 · 22 ] = 1.0480E. (35) The term E I is computed with the aid of Eq. (16)1 and yields E I = 4 · 0.0001927 · E = 0.0008 · E = 154 160 kN m2 . (36) The term G A is computed with the aid of Eq. (17) and yields

GA = 3

12 · 200 000 000 1 0.0001927 ·4 3

+

1 0.0001627 ·3 4

= 66 186.3 kN.

(37)

Having determined λs f and λ f for the first three natural periods, Eq. (12) is finally used to obtain the “total” eigenvalues λ as λ1 = 0.3101,

λ2 = 0.5385,

λ3 = 0.6955.

(41)

Thus, the initial natural periods according to Eq. (11) are T1 = 0.6765 s,

T2 = 0.2245 s,

T3 = 0.1346 s.

(42)

The above values of periods should be corrected through appropriate correction factors. Using Table 7 for the case of a frame without bracing and constant sections, one can obtain the coefficient c1 . Then Eq. (32) yields CUBR,1 = 0.94,

CUBR,2 = 0.94,

CUBR,3 = 0.99.

(43)

Use of Eq. (33) enables one to compute the corrected “total” eigenvalues λ of the flexure-shear cantilever, which read λ1 = 0.2902,

λ2 = 0.5062,

λ3 = 0.6885.

(44)

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C. Chrysanthakopoulos et al. / Journal of Constructional Steel Research 62 (2006) 592–604

Thus, the corrected natural periods on account of Eq. (11) are T1 = 0.7728 s,

T2 = 0.2540 s,

T3 = 0.1373 s.

(45)

For comparison purposes, the “exact” values of these periods, as obtained by finite elements, read T1 = 0.8325 s,

T2 = 0.2621 s,

T3 = 0.1465 s

(46)

and the values of the first period, as determined by the codes, read UBC : T1 = 0.5501 s, EC 8 : T1 = 0.5481 s,

λ1 = 0.16275,

Due to the symmetry of the frame, the center of the area of the vertical members is found easily to be at the center of the frame, that is, at x = 6 m. The term E Ac2 is computed with the aid of Eq. (16)2 and yields E Ac2 = E 0.0131 · 62 + 6 · 0.0131 · 02 + 0.0131 · 62 = 0.9432E.

(47)

The term E I is computed with the aid of Eq. (16)1 and (22) and yields E I = (E I )eq + E · I1 + E · I3 = 0.1048 · E (48)

To determine G A, one first computes G A from Eq. (19) as

12 · 200 000 000 1 0.0001927 0.0001927 + 3 3

+

1

0.0001627 0.0001627 + 4 4

= 277 312.8 kN.

T1 = 0.2106 s,

T2 = 0.0669 s,

T3 = 0.0398 s.

CBR,1 = 0.55 + 0.02 · 4 + 0.03 · (3 − 1) ≤ 0.75 + 0.10 · (3 − 1) ⇔ CBR,1 = 0.69 CBR,2 = 0.56 + 0.02 · 4 + 0.03 · (3 − 1)

(55)

CBR,3 = 0.58 + 0.02 · 4 + 0.03 · (3 − 1) ≤ 0.80 + 0.05 · (3 − 1) ⇔ CBR,3 = 0.72.

(49)

λ2 = 0.203249,

λ3 = 0.270189.(56)

Thus, the corrected natural periods, on account of Eq. (11), are T1 = 0.4216 s,

T2 = 0.1350 s,

T3 = 0.0764 s.

Thus kα H becomes equal to 1.4525. The total mass of the frame per meter of height is

6.3. Frame with bracing and variable sections (51)

Eq. (7) provides the values of λ f H directly for the first three modes. Eq. (14) is then used to calculate λs f . Thus,

(57)

For comparison purposes, the “exact” values of these periods, as obtained by finite elements, read

EAK : T1 = 0.2357 s.

20 235 M = = 1686 kg/m. H 12

(54)

The above values of periods should be corrected through appropriate correction factors. Using Table 7 for the case of a frame with more than 1 bay and double-diagonal bracing, one can obtain the coefficients α1 , α2 and c1 . Then Eqs. (30) and (31) yield, for the first three modes, the values

Parameters α and k can be calculated with the aid of Eq. (15) and yield GA α= = 0.11481/m EI (50) EI k = 1+ = 1.05429. E Ac2

m=

(53)

Thus, the initial natural periods according to Eq. (11) are

λ1 = 0.115019,

and then, from Eq. (23), evaluates G A as

3

λ3 = 0.37452.

Use of Eq. (33) enables one to compute the corrected “total” eigenvalues λ of the flexure-shear cantilever, which read

2 · 3 · 200 000 000 · 42 · 0.001546 = 237 465.6 kN 53

λ2 = 0.28870,

≤ 0.75 + 0.05 · (3 − 1) ⇔ CBR,2 = 0.70

+ 2 · 0.0001927 · E = 21 037 080 kN m2 .

G A = 237465.6 +

(52)

Having determined λs f and λ f for the first three natural periods, Eq. (12) is finally used to obtain the “total” eigenvalues λ as

EAK : T1 = 0.3118 s.

6.2. Frame with bracing

GA =

one has 2

λs f · H 1 = (1 − 0.5) π (1 + k · α · H )

= 1.5708 · (1 + 1.4525) ⇔ λs f · H 1 = 1.9628 2

λs f · H 2 = (2 − 0.5) π (1 + k · α · H )

= 4.7124 · (1 + 1.4525) ⇔ λs f · H 2 = 3.3996 2

λs f · H 3 = (3 − 0.5) π (1 + k · α · H )

= 7.8540 · (1 + 1.4525) ⇔ λs f · H 3 = 4.3889.

T1 = 0.4397 s,

T2 = 0.1441 s,

T3 = 0.0855 s

and the values of the first period, as determined by the codes, read UBC : T1 = 0.5501 s,

EC 8 : T1 = 0.4836 s,

Consider the plane steel frame of Fig. 11 with x-bracing and variable sections according to Table 2. Following similar procedures, as in Section 6.2, one can compute E Ac2 , G A and E I for the first and second floors, which have identical sections,

C. Chrysanthakopoulos et al. / Journal of Constructional Steel Research 62 (2006) 592–604

and obtain

603

account of Eq. (11) and obtain

(E Ac2 )I = 0.9432E (E I )I = 21 037 080 kN m2

(58) (59)

(G A)I = 277 312.8 kN.

(60)

Parameters αI and kI can be calculated with the aid of Eq. (15) and yield αI = 0.11481/m,

kI = 1.05429.

(61)

For the third and fourth floors, which have identical sections, the terms E Ac2 , E I and G A are computed and yield the values (E Ac2 )II = 0.8496E (E I )II = 18 939 680 kN m2

(62) (63)

(G A)II = 267 045 kN.

(64)

T1 = 0.4279 s,

T2 = 0.1370 s,

T3 = 0.0775 s.

(73)

For comparison purposes, the “exact” values of the periods, as obtained by finite elements, are T1 = 0.4442 s,

T2 = 0.1475 s,

T3 = 0.08723 s (74)

and the values of the first period as determined by the codes are UBC : T1 = 0.5501 s, EAK : T1 = 0.2357 s.

EC 8 : T1 = 0.4836 s,

(75)

7. Conclusions On the basis of the preceding discussion, one can draw the following conclusions:

Eq. (7) provides the values of λ f H directly for the first three modes. Eq. (13) is then used, since kα Hoλ = 1.4774, to calculate λs f . Thus, one has

(1) The proposed method of modeling a plane steel frame (braced or unbraced) as an equivalent cantilever in conjunction with the use of certain correction factors constitutes a simple, sufficiently conservative and accurate way of predicting, by hand calculation, the first three natural periods of that frame. (2) The first and second periods are determined with a relative error below 15%, while the third period is determined with a relative error below 20%. (3) The method is limited to braced frames with only one braced bay. Further studies and some modifications of the correction factors should be made if a frame consists of more than one braced bay. (4) The height of the story and the span of the bay play little part in the accuracy of the prediction, provided one is kept away from extreme cases.

λs f1 = 0.1644,

Acknowledgement

Then parameters αII and kII can be calculated and yield αII = 0.11875/m,

kII = 1.05426.

(65)

Thus, for the entire frame, the parameters α, k and E I are computed as αI · HI + αII · HII = 0.11678/m HI + HII kI · HI + kII · HII = 1.05428/m k= HI + HII (E I )I · HI + (E I )II · HII = 19 988 400. EI = HI + HII α=

λs f2 = 0.2847,

λs f3 = 0.3676.

(66) (67) (68)

(69)

Having obtained λs f and λ f for the first three natural periods, Eq. (12) is finally used to obtain the “total” eigenvalues λ as λ1 = 0.16347,

λ2 = 0.29011,

λ3 = 0.37639.

(70) References

Finally, the initial natural periods according to Eq. (11) are T1 = 0.2139 s,

T2 = 0.0679 s,

T3 = 0.0404 s.

(71)

The above values of periods should be corrected through appropriate correction factors. Using Table 7 for the case of a frame with more than one bay and double-diagonal bracing, the coefficients α1 , α2 and c1 are obtained. Then Eqs. (30) and (31) yield, for the first three modes, the values CBR,1 = 0.55 + 0.02 · 4 + 0.03 · (3 − 1) ≤ 0.75 + 0.10 · (3 − 1) = 0.69 CBR,2 = 0.56 + 0.02 · 4 + 0.03 · (3 − 1) ≤ 0.75 + 0.05 · (3 − 1) = 0.70 CBR,3 = 0.58 + 0.02 · 4 + 0.03 · (3 − 1)

The authors would like to thank Miss Maria Dimitriadi for her excellent typing of the manuscript.

(72)

≤ 0.75 + 0.05 · (3 − 1) = 0.72. Using Eq. (33), one can compute the corrected “total” eigenvalues λ and finally the corrected natural periods on

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[11] Noor AK. Continuum modeling for repetitive lattice structures. Applied Mechanics Reviews 1988;41:285–96. [12] Graff K. Wave motion in elastic solids. Columbus (Ohio): Ohio State University Press; 1975. [13] EC 3 (Eurocode 3). Design of steel structures. Part 1.1. Brussels: CEN (European Committee for Standardization); 1992. [14] SAP 2000. Three dimensional static and dynamic finite element analysis

and design of structures, basic analysis reference. Berkeley (California): Computers and Structures Inc.; May 2000. [15] E.A.K. Greek Earthquake Resistant Design Code. Earthquake design and protection organization (O.A.. .), Athens; September 1999. [16] Chrysanthakopoulos C. Approximate formulae for computing natural periods of steel frames and simple bridges. M.S. thesis. Patras: (Greece); Department of Civil Engineering, University of Patras. 2004 [in Greek].

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