Calculation of guyed masts in accordance with EN 1993-3-1 standard taking into account mast shaft geometrical imperfections

Engineering Structures 33 (2011) 2044–2048

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Calculation of guyed masts in accordance with EN 1993-3-1 standard taking into account mast shaft geometrical imperfections Monika Matuszkiewicz ∗ Koszalin University of Technology, Department of Steel Structures, Śniadeckich 2, 75-453 Koszalin, Poland

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Article history: Received 16 November 2010 Received in revised form 14 February 2011 Accepted 28 February 2011 Available online 27 March 2011 Keywords: Guyed mast Geometric imperfections Wind load Static analysis

abstract Selected problems concerning designing of guyed masts with lattice shaft in accordance with the ‘‘EN 1993-3-1: Design of steel structures. Part 3-1: Towers, masts and chimneys – Towers and masts’’ European standard have been described in this paper. The method of application of the mast shaft geometrical imperfections in calculations has been discussed. Based on the performed comparative analysis of a certain mast, the influence of such imperfections on the ultimate values of internal forces in the mast shaft has been demonstrated. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction According to the European standard [1] the guyed masts and chimneys are analysed with consideration of influence of deflections on the equilibrium conditions (according to the second order theory). However, in global analysis the perfect type structure is considered i.e. a structure without influence of initial geometrical imperfections of the mast shaft on the internal forces and deflection values. In Annex F to the above-named standard concerning mast structure execution, limitations regarding the maximum initial deflections of the mast shaft between two guy levels have been indicated, among other things. Those deflections should not exceed the L/1000 value (L—span length). In the case of mast structures featuring relatively small distances between adjacent guy fixing levels taking into account that the influence of the permissible assembly deviations in mast spans has no significant influence on the structural analysis results and can be neglected in calculations. However, in the case of considerable distance between two guy levels and significant normal forces in the mast shaft, the influence of the initial mast shaft imperfections may have a considerable impact on the internal force values. The simplest method of taking into account of the geometrical imperfections in the mast static calculations is replacement of the real curvilinear bars with straight ones but loaded with a certain substitute load, influence of which on the status of internal forces

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in the mast shaft is equivalent to the influence of initial span curvature. The best way is, in practise, to apply the geometrical imperfections described by a parabolic function because such curvature features a permanent value of the substitute load along the bar length Fig. 1 [2]. According to this assumption the substitute load value in a span can be defined by the following equation: q=

8Ne0 L2

,

(1)

where: N—mast span normal force, e0 —maximum imperfection amplitude value, L—span length. In the case of contra-positioned initial mast shaft curvatures the substitute load in particular spans should be applied with opposite senses (Fig. 2). Values of substitute load caused by the imperfections may constitute a significant fraction of the span wind load in the masts with slender shafts and considerable distances between the guy fixing levels. To show the influence of initial imperfections on the values of internal forces in mast shaft, static calculations of a certain antenna mast (Fig. 3) located at the Przysucha–Kozłowiec Radio and Television Broadcasting Centre were done. To simplify the matter only the mast structure without the aerials and other equipment was considered in the calculations. The calculations were accomplished using Mast1 software, described in [3], according to the second order theory. The nonlinear elastic analysis applied in this study pertained both to the guys and mast shaft.

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a

b

Fig. 1. Compression bar showing initial curvature: (a) bar scheme, (b) parabolic curvature substitute load.

a

b

Fig. 2. Substitute load with parabolic curvature caused by mast shaft geometrical imperfections.

The mast guys were considered in calculations as typical cables of nonlinear relation between force and elongation (shortening) of the cable chord. 2. Mast comparison analysis 2.1. Mast description The shaft of the mast featuring 204 m in height has been made as a steel trihedral space truss with side width a = 2.70 m. The leg and bracing members were made of steel tubular sections with yield strength fyk = 355 MPa. Pipes ∅193.7/10 mm were used for leg members and ∅88.9/5 mm for the mast face lacing. The mast has three guy fixing levels at the height of 67.175 m, 126.575 m and 176.075 m respectively. The guys are made of ∅36 mm steel ropes with T1 × 61-Z-II-g(7.66 cm2 ) structure where wire of ultimate strength fu = 1570 MPa and nominal break force 1195 kN was used. The applied values of initial guy forces for particular fixing levels are 40, 70 and 100 kN respectively. 2.2. Static calculations The mast static calculations were done taking into account the own weight of the structure as well as mast shaft and guys

wind load for two characteristic wind action directions W1 and W2 (Fig. 4). Additionally, for comparison purposes, calculations with consideration of the substitute mast shaft load caused by geometrical imperfections equal to the admissible assembly deviations in accordance with [1], in the form of the contrapositioned shaft initial curvatures, were accomplished. The mast is located in the first wind load zone at the territory of Poland. In the calculations performed according to [1] the mast was qualified to the second reliability class so, values of the partial factors for actions are γG = 1.1, γQ = 1.4 respectively. According to [4] it has been assumed that the prime value of the wind basic velocity pressure for the first zone is 300 Pa. Numerical calculations have been performed substituting the latticed mast shaft with a solid bar of relevant geometric characteristics taking into account the bending, shear and torsional flexibility. In the calculations made as per [1] the real dynamic wind action on the mast can usually be substituted by a quasi-equivalent static load composed of the constant mean wind loading (Fig. 5(a)) and a number of the so-called patch loads acting only on some mast shaft fragments (Fig. 5(b)). The calculation method as per [1] applied for the mean wind loading and patch loads acting on the mast structure elements was described in [5]. In order to determine the dynamic response of a mast to wind loading, one should consider, in accordance with [1], a series of static patch load patterns in combination with the mean load. The patch load effect shall be determined as a difference between the patch load combination effect with the mean load and the effect of just mean load. Internal force (or deflection) increments Sp caused by the patch loads are calculated with consideration of the vector summation in accordance with [3] formula.

  N − Sp =  (Si − Sm )2 ,

(2)

i=1

where: Sm —internal forces caused by the mean wind load, Si — internal forces caused by the mean load increased by i—patch load, N—number of patch load schemes required in the calculations. Then the extreme internal forces are calculated from the following formula: S = Sm ± Sp .

(3)

One can easily find out that the Sp value calculated from formula (2) is in general considerably higher than the maximum Si –Sm difference. In the event of taking into account the initial mast shaft imperfections, the calculation procedure is slightly more complex. For each combination, based on the obtained normal force values

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Fig. 3. Mast scheme. Table 1 Values of substitute loads caused by mast shaft imperfections. Mast span number

1

2

3

Span length Li (m)

67.175

59.400

49.500

0.119 0.118 0.125 0.121 0.124 0.123 0.122 0.120 0.120 0.118 0.121 0.123 0.125 0.122 0.123 0.122 0.121 0.120

0.100 0.100 0.102 0.098 0.103 0.099 0.103 0.099 0.101 0.099 0.101 0.099 0.103 0.100 0.103 0.100 0.103 0.100

0.069 0.066 0.070 0.070 0.070 0.070 0.071 0.070 0.070 0.068 0.069 0.067 0.070 0.067 0.070 0.067 0.071 0.069

(a) Own weight + mean wind load (b) combination (a) + wind sectional load 1 (c) combination (a) + wind sectional load 2 (d) combination (a) + wind sectional load 3 Substitute load q (kN/m)

Load combinations

(e) combination (a) + wind sectional load 4 (f) combination (a) + wind sectional load 5 (g) combination (a) + wind sectional load 6 (h) combination (a) + wind sectional load 7 (i) combination (a) + wind sectional load 8

in particular mast spans, values of the equivalent load from imperfections have been calculated (using formula (1)) and then the calculations have been repeated summing up the wind load and substitute load as per the diagram shown in Fig. 2(a) or (b).

The calculated values of the equivalent load caused by mast shaft imperfections have been listed in Table 1; the higher values pertain to the W1 wind load direction and the lower values to the W2 direction.

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Table 2 Values of internal forces in mast shaft for W1 wind direction action. Span number Span beginning 1

Span middle Span end Span beginning

Fig. 4. The most unfavourable mast wind load directions. 2

Tables 2 and 3 indicate the ultimate values of mast shaft internal forces obtained from numerical calculations: a – without consideration of the initial imperfections, b – with consideration of the mast shaft initial imperfections in accordance with Fig. 2(a) or (b). This study does not contain the obtained values of forces in guys because the differences obtained for particular calculations (mast shaft with and without imperfections) appeared to be insignificant (the highest ones did not exceed 2%). Based on the results of numerical calculations one can say that the differences in internal force values calculated for the perfect and imperfect type structures pertained mostly to all bending moments acting in the middle of mast shaft spans. In the case of the structures featuring initial imperfections, the bending moment values increased compared with the perfect type structures by 14.2% for the bottom span, 14.4%—for the middle one and 5.1% for the top one. The initial mast shaft curvatures have no practical impact on the normal force values. The highest difference did not exceed 1%.

Span middle Span end Span beginning

3

Span middle Span end

Span number Span beginning 1

Span middle Span end Span beginning

2

Span middle Span end

a

M (kNm)

V (kN)

−1161.0 −1162.1 −1088.3 −1089.4 −1020.3 −1019.5 −866.2 −866.4 −773.1 −772.9 −728.2 −728.5 −513.1 −514.1 −443.2 −443.9 −397.1 −396.2

0 0 563.88 643.78 −252.72 −269.91 −437.09 −439.83 420.56 481.13 −341.96 −371.91 −446.61 −464.24 314.97 328.95 −338.23 −346.41

27.2 31.0 −9.8 −10.3 −39.1 −35.0 38.2 35.1 −10.9 −11.8 −30.4 −27.7 30.5 32.5 −9.1 −9.1 −30.8 −29.5

Table 3 Values of internal forces in mast shaft for W2 wind direction action.

3. Remarks and final conclusions Selected issues regarding the static analysis of guyed masts with lattice shaft have been presented in this study. Based on the comparative analysis of a mast structure subjected to wind load and, additionally, to substitute loads caused by geometrical imperfections as per [1] quite substantial increases of the bending

a b a b a b a b a b a b a b a b a b

N (kN)

Span beginning 3

Span middle Span end

a b a b a b a b a b a b a b a b a b

b

Fig. 5. Wind load schemes for the analysed mast: (a) mean load, (b) patch loads.

N (kN)

M (kNm)

V (kN)

−1139.7 −1140.8 −1066.8 −1068.2 −999.1 −1000.2 −825.4 −825.4 −732.4 −732.3 −687.6 −687.7 −484.8 −486.1 −414.8 −416.0 −368.8 −367.7

0 0 −682.94 −765.85 −270.92 −294.55 285.95 289.56 −546.36 −603.28 378.67 414.21 487.64 512.28 −395.28 −405.60 344.42 351.51

−30.8 −34.6 8.4 8.9 36.5 40.5 −35.6 −38.6 16.0 16.7 32.7 30.3 −28.4 −30.4 11.8 11.7 33.3 32.3

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moment values in the mast shaft compared with the analysed perfect type structure have been shown. These remarks pertain to the lowest, reliable mast spans because in those spans high normal force values occur. Values of admissible initial deflection indicated in Annex F to the [1] standard are clearly not equivalent to and do not substitute any geometrical as well as structural imperfections but the [1] standard does not comprise any guidelines regarding consideration of those imperfections in calculations. Initial mast shaft imperfections can, of course, be different than the limits for the tolerances of initial deflections. It has been demonstrated in this study that taking into account even such imperfection values as L/1000 may have a considerable influence on the mast shaft internal force values. Therefore, it seems appropriate to take into account, as in the case of uniform

built-up columns calculated in accordance with EN 1993-1-1, the initial imperfections during mast calculations, the magnitude of which should be determined during further detailed studies. References [1] EN 1993-3-1. Eurocode 3: design of steel structures. Part 3-1: towers, masts and chimneys—towers and masts. [2] Pałkowski Sz. On numerical analysis of masts using theory of the second order. Inżynieria i Budownictwo 2002;8:436–8 [in Polish]. [3] Pałkowski Sz. Steel structures. Some issues of calculating and designing. Warszawa: PWN; 2009 [in Polish]. [4] EN 1991-1-4. Eurocode 1: actions on structures. Part 1–4: general actions—wind actions. [5] Matuszkiewicz M. Computation lattice guyed masts according to PN-EN 19933-1. Inżynieria i Budownictwo 2010;4:194–9 [in Polish].