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Analysis of guyed masts by the stability functions based on the Timoshenko beam-column
Engineering Structures 152 (2017) 597–606
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Analysis of guyed masts by the stability functions based on the Timoshenko beam-column Pablo M. Páez ⇑, Beradi Sensale Facultad de Ingeniería, Universidad de la República, Julio Herrera y Reissig 565, Montevideo, Uruguay
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 July 2017 Revised 16 August 2017 Accepted 16 September 2017
A new formulation for the analysis of guyed towers in which the mast is modeled as a continuous equivalent beam-column on non-linear elastic supports, is proposed. Because second-order effects must not be neglected and given the importance of consider the shear deformation in the analysis of this type of structures, the method proposes the calculation of the second-order deformation using the stability functions based on the Timoshenko beam column. The equivalent beam-column properties of the mast are calculated depending on the pattern construction of the tower and the guys are replaced by non-linear elastic supports. Based on the catenary configuration of the cable, the spring constant is obtained from the secant modulus of the cable. In order to validate the proposed method, a comparative study is carried out analyzing a guyed mast using the proposed method and a numerical model through the use of finite elements. As a main conclusions we will mention that, the proposed method are sufficiently accurate compared to the finite element method, confirming the validity of the hypotheses adopted in the development of the method and; the reduction of the computational effort, since the structure does not need to be discretized into a large number of elements for the convergence. Ó 2017 Elsevier Ltd. All rights reserved.
Keywords: Guyed towers Stability functions Second order analysis Geometric non-linearity
1. Introduction Radio and television communications, as well as cellular telephony, are possible in the modern world thanks to structures that support equipment transmitting signals from one place to another. Due to the great resistance in relation to the material consumption and permeability, steel trussed towers are structures in wide use in support of communication systems. These structures are usually slender and light elements, mostly located in exposed places, so environmental loads prevail in the design. Two types of trussed towers are used according to their structural behavior: the self-supporting and the guyed mast. When large heights are required and conditions at the installation site permit, guyed masts, because of their steel economy relative to the self-supporting ones, are the ones usually used. However, the presence of the guys confers a complex structural behavior on the environmental loads. For the analysis of guyed masts several methods can be applied. The TIA 222-G standard specifies three types of analysis [1]. The first one consists of an analysis where the tower is modeled as an equivalent beam-column supported by cables represented either ⇑ Corresponding author. E-mail addresses: (B. Sensale).
[email protected]
(P.M.
https://doi.org/10.1016/j.engstruct.2017.09.036 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
Páez),
[email protected]
as cable elements or as non-linear elastic supports. The second one consists of an elastic three-dimensional truss model where the mast is modeled as a space framework whose bars are connected by ideal frictionless joints and consequently can only produce axial force, and finally, an elastic three-dimensional model can be performed where some of the members of the mast are modeled as elements that can produce both bending moments and axial forces, and others as elements that can only produce axial force. In the last two methods of analysis, the guys are modeled as cable elements. The complexity in the analysis of the guyed towers lies in the inherent non-linearity that these structures present: the nonlinearity of the guys and the geometric non-linearity. The slenderness of these towers makes them susceptible to the buckling phenomenon; in other words, second-order effects must not be neglected. The horizontal actions cause displacements that increase the level of tension in the guys and in turn the forces of compression on the tower. These compression forces tend to increase displacement and so on, a phenomenon commonly referred to as the P-D effect. Among the methods of the first type, based on the beamcolumn model, different researchers have made different proposals for more than fifty years. Cohen and Perrin [2] proposed a two-dimensional analysis using a beam-column model on elastic
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supports, considering only the flexural stiffness of the beam. In the same sense, Ezra [3] proposed a two-dimensional analysis based on the beam-column model on non-linear elastic supports and considering the torsion of the structure by torsion springs. Basically, the analysis consists of calculating the displacements of the structure in iterative form from an arbitrary set of initial displacements. Kalha [4] proposes an approximate method for the analysis of guyed masts using an equivalent beam-column for the tower model and cable elements for the guys model. Although the method takes into account the geometric coupling among the different degrees of freedom and the second-order effects, it is difficult to apply because it requires a high degree of computational programming. Wahba et al. [5] analyzed three different models for the guyed masts, one of which consisted of modeling the tower as an equivalent beam-column and non-linear cable elements for the guys. Their results validated the conclusions of the previous studies by Kahla and confirmed the non-linear geometric response of these structures to wind loads. Margariti and Gantes [6] propose an approximate method based on the classic expressions of buckling for the calculation of the elastic critical load in guyed masts and pylons of cable-stayed bridges. This method is based on a single span beam-column model, considering only the flexural stiffness of the mast or pylon. The non-linearity of the guys is taken into account by the equivalent modulus of elasticity obtained from the tangent modulus of the cables. Williamson and Margolin [7] studied the effect of shear force in the design of guyed masts. For this, they used an equivalent beam-column model on elastic support, replacing the trussed web system of each face by a fictitious solid web of the same shear stiffness but having zero flexural stiffness. Their results showed the importance of considering the effects of shear deformation in the analysis of guyed masts. Based on the equivalent beam-column method, Páez and Sensale [8] propose an analytical method for the calculation of guyed masts using the stability functions based on the Euler-Bernoulli theory. The guys are replaced by non-linear elastic supports and the effect of the shear deformations are taken into account by a reduced second order moment obtained from the Timoshenko beam theory. Among the different methods of analysis, the most commonly used today are based on finite elements. These methods are very accurate; however, for convergence of the solutions, it is necessary to discretize the structure into a large number of elements, and requires a great effort of computer programming [4,5]. On the other hand, analytical methods require complex analyses, and although they allow us to obtain solutions whose results are close to those obtained by the use of finite elements, in general they present significant differences [2,3,6,7]. It is for this reason that in this work it is proposed to develop a new analytical method for the analysis of guyed masts based on the equivalent beam-column model. In this sence, the method maintains the rigor of the analytical methods but their application is simple. In other words, sufficiently accurate solutions are obtained for the design of guyed towers in a reasonable time and with a reasonable effort. The proposed method is motivated by the work of Páez and Sensale [8]. The main difference between this method and that of Páez and Sensale is essentially in the way the two methods compute the second-order deformation. The proposed method considers the non-linearity of the cables and the second-order effects. For this purpose, the tower is modeled as a continuous equivalent beam-column, on non-linear elastic supports, whose axial stiffness, flexural stiffness and shear stiffness are calculated depending on the pattern construction of the tower [9]. The guys are replaced by non-linear elastic supports whose spring constant is obtained from the secant modulus of elasticity of the cables [10]. The method proposes the calculation of the second-order deformation using the stability functions based on the Timoshenko beam-column [11–13].
One of the main advantages of the method is the reduction of the computational effort, since the structure does not need to be discretized into a large number of elements for the convergence and hence the method can be more useful in the pre-design stages. Another advantage of the method is that the effects of shear deformation are taken into account directly from the geometrical properties of the tower by the use of stability functions based on the Timoshenko beam-column, without the need for mathematical artifice to consider such effects. In addition, the proposed analytical method uses the basic and more general concepts from the point of view of structural engineering, this means that the engineer can quickly visualize the parameters that influence the design. In order to validate the proposed method, a comparative study is carried out, analyzing a guyed mast using the proposed method and a numerical model through the use of finite elements by SAP 2000 software [14]. 2. The guyed towers The towers are constructed from a series of vertical bars, generally denominated as legs, horizontal bars and diagonal bars, thus forming a space truss. The frequently used tower cross-section is the equilateral triangular, although square sections may be used in some cases. For smaller structures, the structural elements of the lattice are materialized by solid circular steel bars, while for larger sized structures circular steel tubes, angles profiles, and steel cold-formed profiles are used. The guys provide side support to the towers. In general, they are placed uniformly distributed on the height and are prestressed. In the case of towers of triangular section, three cables are placed at each level of guys and in the case of a square section four cables are placed per level. The guys are anchored to the ground in such a way that the angles that form the planes constituted by each cable and the tower are equal. The typical inclinations used for the cables, measured as the angle formed by the cable with the horizontal, are of the order of 40–60°, being able to reach smaller inclinations for the cables of the first level. The initial stress of pretensioning of the cables is one of the main parameters that affect the stability of this type of structure [15]. This is imposed during the construction of the mast and must be checked periodically. The TIA 222-G standard recommends average values of 10% of the cable breaking stress, with a range of variation between 7% and 15% of the breaking stress, since within this range the effects of vibration and the aeroelastic instability can be neglected. For its part, the EN 1993-1-11 standard provides somewhat more general guidelines, allowing initial stresses of up to 45% of the breaking stress. However, it requires the verification of vibration and the aeroelastic instability [16]. This work will focus on towers of equilateral triangular crosssection, although most of the reasonings presented here can extend to towers of square cross-section. 3. Model of the tower: Equivalent beam-column properties The model of the tower as an equivalent beam-column is based on the following assumptions: the material of the beam presents a linear elastic behavior, the deformations of the beam are small, and let us assume that, under the action of the bending moment and the shear force, the perpendicular plane sections to the longitudinal axis of the beam before it is deformed, remain plane but not necessarily perpendicular to the deformed longitudinal beam axis. These hypotheses correspond to the Timoshenko beam theory. Fig. 1 shows the typical lacing configuration of the trussed web of one of the faces of the tower. For each of these patterns, the geometrical properties of the equivalent beam-column can be
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Fig. 1. Typical lacing configurations of the trussed web of one of the faces of the tower.
Table 1 Geometric properties of equivalent beam-column section according to the pattern of one face of the tower (adapted from [8]). Equivalent properties
Pattern 1
Pattern 2 and 3
Pattern 4
EA
3 E Am
3 E Am
3 E Am
E Iy , E Iz
1 2
G Ay , G Az
3 2
E Am a2
1 EAd w1
1 2
1
u þ tan EAh
E Am a2
3 2E
Ad w1
1 2
E Am a2
3 2E
Ad w1
Pattern 5 b 3 E Am þ A 1 2 b 2 E a Am þ A 3 E Ad w1
E and G are the modulus of elasticity and the shear modulus of the material. E A is the equivalent axial stiffness of the beam-column.E Iz , E Iy , G Az and G Ay are the flexural stiffness and shear stiffness of the equivalent beam-column with respect to coordinate axes z and y, respectively. Am, Ad and Ah are the cross-sections of the legs of the diagonal and horizontal bars that make up the lattice, respectively. a and b are the distance between the legs and the distance between the horizontal bars, respectively. u is the angle formed by the legs with the diagonal bars. 2 w1 ¼ sin u cos u b ¼ Ad Ah cos3 3u A Ah þ2Ad sin
u
obtained using the principle of virtual works. Table 1 shows these properties for the case of triangular cross-section towers [4,9]. 4. The guys model
2 1 2 1 d ðr1 r0 Þccb þc1 cb r1 sinhðccb c cb r1 Þr0 sinh ccb c cb r0 ¼ lcb 2Ecb ccb cosh 1=2ccb ccb r1 1 1 4Ecb ccb r0 sinhð1=2ccb ccb r1 Þ r sinh 1=2ccb ccb r1 1 0 1 þ 2Ecb ccb cosh 1=2ccb ccb r1 1 ð1Þ
4.1. The elastic cable: Secant modulus of elasticity Consider an inclined cable whose chord length is lcb and its horizontal projection ccb, subjected to the action of a chord force T0 and to a vertical dead load gcb (Fig. 2). Based on the catenary configuration of the cable, by applying a chord force T1, the cable elongation d can be expresed by Eq. (1) [10]:
where Ecb is the modulus of elasticity of the cable, r1 and r0 are the stresses in the cable due to the chord forces T1 and T0, respectively, and ccb is the density of the cable material. The non-linearity between the chord force and the elongation due to the sagging change under different load conditions can be
Fig. 2. Inclined cable subjected to a vertical dead load per unit length: relationship between the chord force and the chord length (adapted from [8]).
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Fig. 3. Secant modulus of elasticity vs. final cable stress, taking Ecb = 185 GPa and
r0 ¼ 0:1 f u ¼ 140 (adapted from [8]).
taken into account by the tangent and secant modulus of elasticity of the cable. In this way, the non-linear phenomenon can be treated as a linear one. However, the tangent modulus of elasticity should be used when the relationship between the stresses r1 and r0 is small; otherwise, the secant modulus of elasticity must be used. Its expression is given by Eq. (2) [10]:
Esec ¼
Dr r 1 r 0 ¼ ccb De d
ð2Þ
The graph of Fig. 3 shows the variation of the secant modulus of elasticity as a function of the final stress r1 for cable lengths of 60 m, 100 m and 160 m, whose modulus of elasticity is Ecb ¼ 185 GPa and whose initial stress is r0 ¼ 140 MPa, corresponding to 10% of the breaking stress. It can be seen that for final stresses of the order of 50% of the breaking stress, the ratio of the modulus of elasticity of the cable to the secant modulus of elasticity does not exceed 10% for a cable length of 160 m, or 4% for a cable length of 100 m. 4.2. The cable system: the equivalent spring stiffness Consider in the first instance the symmetrical plane system, in equilibrium, formed by the mast and two guys (Fig. 4). If we apply a horizontal force at the upper end of the mast, it will
Fig. 5. Chord force vs. elongation of the cable from the initial tensioning condition (adapted from [10]).
deform and its position can be described as a horizontal displacement u and a vertical one w, both components of the displacement being small. Since the mast displacement is composed of a rigid movement and by the flexion thereof, it can be assumed that the displacement w is an infinitesimal of the second order of u (w O u2 ) [17]. When the mast deforms, the cable to the left will lengthen while the one to the right will be shortened. If both cables are subjected to an initial chord force T0, the cable on the left will experience a rapid increase in tension while the cable on the right will undergo a rapid drop. Fig. 5 shows schematically the foregoing reasoning. Referring to Fig. 4, the elongation of the left cable d1 and the shortening of the right cable d2 as a function of the horizontal displacement u can be expressed by Eq. (3):
d1 ¼ d2 ¼
Fig. 4. Symmetrical plane system formed by the mast and two guys. Deformed configuration when applying a horizontal force F at the upper end of the mast (adapted form [17]).
ccb u lcb
ð3Þ
If the applied force F is of unit value and considering a linear relationship between stress and strain by using the secant modulus of elasticity, Eq. (2), the equivalent spring constant used to replace the cable system is Eq. (4):
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ccb u 2 lcb ccb u d2 ¼ lcb
d1 ¼
ð5Þ
Equating the horizontal forces, Eq. (6),
ðEsec;1 þ 2 Esec;2 Þ Acb u
c2cb 3
2 lcb
¼1
ð6Þ
Therefore, the equivalent elastic spring constant used to replace the cable system is determined by Eq. (7):
keq ¼ ðEsec;1 þ 2 Esec;2 Þ Acb
c2cb
ð7Þ
3
2 lcb
From Eqs. (1), (5) and (7), we can conclude that, under the assumptions made, the structure of Fig. 6 is more flexible in the positive direction of the U-axis than in the negative direction. 5. Proposed method Fig. 6. Three-dimensional system formed by the mast and three cables of equal chord length and equal inclination (adapted from [8]).
keq ¼ ðEsec;1 þ Esec;2 Þ Acb
c2cb 3 lcb
ð4Þ
where Acb is the cross-sectional area of the cables, Esec;1 and Esec;2 are the secant modulus of elasticity of the cable being stretched and the cable being shortened, respectively, and a the inclination of the cables with respect to the horizontal. Let us now consider the three-dimensional system constituted by the mast and three cables of equal chord length lcb, equal inclination a and such that the planes formed by each cable and the mast are equidistant to each other, to which a horizontal force F of unit value is applied in the upper end of the mast in the direction of the U-axis (Fig. 6). If the force is applied in the positive direction of the U-axis (as shown in the figure), the relationship between the elongation d1 of the cables to windward and the horizontal displacement u, as well as the relationship between the shortening d2 of the cable to leeward and the horizontal displacement can be expressed by Eq. (5) [8]:
Consider a guyed mast structure of height ht , with n levels of guys uniformly spaced by a distance hj. Without loss of generality, let us consider that the structure is fixed at the base, as shown in Fig. 7a. The tower will be subject to vertical distributed loads pz ðzÞ; being their self-weight; to vertical concentrated loads P z , due, for example, to the weight of the ancillaries, but in general they will be due to the lengthening and shortening of the cables and the wind load on them; and to horizontal concentrated and distributed loads Px and px ðzÞ, respectively, basically due to the action of the wind on the tower, the guys and the ancillaries. Taking into account the static equilibrium in the deformed position in node j, referring to Fig. 7b, we can write for the continuity bending moment, j-1, Eq. (8) [8]: M j1 ¼ M j þ
i¼n X i¼j
i¼n X i¼j
i¼n X Hi þ P x;i keq;i v i hi þ M e;i þ v j v j1
Esec;1 Esec;2 Pz0;i þ Pz;i þ 2 Esec;1 þ 2 Esec;2
i¼j
keq;i i
zcb;i zcb;i v i þ 3 T 0;i ccb;i ccb;i
Fig. 7. (a) External loads on the guyed mast. (b) Mathematical model of calculation, deformed position (adapted from [8]).
ð8Þ
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Fig. 8. Deformed position of the continuous beam-column on discrete elastic supports for the bars between the nodes j 1, j and j + 1. Free-body diagrams to apply the slopedeflection equations based on the beam-column of Timoshenko (adapted from [8]).
where zcb;j is the height corresponding to the level j of guys, v j is the value of the lateral deflection for said level of guys, keq;j is the value of the equivalent spring constant for the level j of guys, ccb;j is the horizontal distance between the guys’ anchor on the ground and the tower for the level j of guys, Hj is the horizontal resultant force R zcb;j þhj =2 between two levels of guys, obtained as Hj ¼ zcb;j hj =2 px ðzÞdz; and Pz0;j is the vertical resultant force due to the uniformly distributed vertical load between two levels of guys, obtained as R zcb;j þhj =2 Pz0;j ¼ zcb;j hj =2 pz ðzÞdz, P z;j is the vertical force at the level j of guys due to the action of the wind on these, and Me;j is the bending moment due to the cables’ eccentricity at the level j of guys and is given by Eq. (9):
M e;j ¼
! pffiffiffi pffiffiffi a Acb;j zcb;j ccb;j v j 3 3 Esec;1 þ Esec;2 3 6 3 lcb:j
ð9Þ
j
5.1. Slope-deflection equations based on the Timoshenko beamcolumn: ‘‘the modified approach” Two different formulations have been proposed to analyze the influence of shear force on the elastic critical buckling load and the second-order effect in beam-columns, one performed by Engesser and another by Haringx [18]. The Haringx model has been applied and discussed by Timoshenko and Gere and is known as ‘‘the modified approach” [11].
According to the hypotheses proposed by Haringx, we can deduce the slope-deflection equations that allow us to consider the bending deformations and the shear force deformations; its expression for the bar j between nodes j-1 and j (Fig. 8) is given by Eq. (10) [12]:
M j1;j M j;j1
v j v j1 EI ¼ C hj1 þ S hj ðC þ SÞ M F;j h j h j v j v j1 EI ¼ S hj1 þ C hj ðC þ SÞ þ M F;j h h j j
ð10Þ where Mj1;j and Mj;j1 are the fixed end moments for the bar j; C j and Sj are the stability functions based on the Timoshenko beamcolumn for the bar j, Eq. (11):
Cj ¼ Sj ¼
1 wj bj cot bj 2 tanðbj =2Þ bj
wj wj bj cosec bj 1 2tanðbj =2Þ bj
ð11Þ
wj
bj is the stability function in the plane bending and its expression is given by Eq. (12):
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi u
u Pj h2j
bj ¼ t
ðw E IÞj
ð12Þ
P.M. Páez, B. Sensale / Engineering Structures 152 (2017) 597–606
wj is the shear reduction factor, Eq. (13):
wj ¼ 1 þ
Pj Gj Xred;j
1 ð13Þ
Xred is the effective shear area of the beam-column cross-section, namely, Xred ¼ A (Table 1); Pj is the compression force in the bar j, it is deduced from the equilibrium equations and their expression is given by Eq. (14) [8]: Pj ¼
i¼n X ðEsec;1 Esec;2 Þ zcb;i zcb;i P 0;i þ P z;i þ 2 v i þ 3 T 0;i keq;i ðEsec;1 þ 2 Esec;2 Þ ccb;i ccb;i i¼j ð14Þ
The subscripts for the expressions in parentheses indicate the indicates that the modulus corresponding bar; for example EI h j of elasticity, the second-order moment and the span length are those corresponding to the bar j. M F;j is the fixed end moment for a beam subjected to a uniformly distributed load in its plane bending and an axial compressive force. Its expression depends on the support conditions; for the case of a beam fixed at both ends, its expression is given by Eq. (15) [13]: 2
M F;j ¼
px;j hj 12
12 b2j
!
1 wj
!"
1
bj 2 tan bj =2
#
ð15Þ
603
For a beam-column with n spans, from Eq. (16) a 2n-1 2n-1 system of equations is formed whose unknowns are the rotations and the lateral deflection of the nodes h1 hj hn1 and v 1 v j v n , respectively, since, on the one hand, the upper end of the tower is considered as pinned, implying that hn can be written as a function of hn1 ; and on the other hand, when the lower end is fixed h0 ¼ v 0 ¼ 0 and when it is pinned v 0 ¼ 0 and h0 can be written as a function of h1 . 5.3. Obtaining the lateral displacement: second order analysis, iterative method Under the action of the external forces, the proposed method assumes the initial shape of the elastic curve correspond to a parabolic function and such that the maximum value of the elastic curve is less than an assumed value v m;i , with v m;i ¼ 0:001 ht . The equivalent spring constants at each of the levels are calculated from the assumed lateral displacements. In this way the relationship between the compressive force in the bar j and the compression force in the end bar n can be reasonably estimated. Let f be the ratio factor between these forces, this is P j ¼ f P n , in this way the relationship between the buckling parameters can be written as Eq. (18):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE I Þn hj bn bj ¼ f ðE I Þj hn
ð18Þ
5.2. Analysis of the structure Consider a continuous equivalent beam-column on discrete elastic supports, for which both the sectional and material properties are known, subjected to the action of external loads. Taking into account the static equilibrium in the deformed position for the elements between the nodes j-1, j and j+1 (Fig. 8) and according to the slope-deflection equations based on the Timoshenko beamcolumn with axial compression force (Eq. (10)), we obtain the Equations (16): a2j;1 hjþ1 þ a2j;2 v jþ1 þ a2j;3 hj þ a2j;4 v j þ a2j;5 hj1 þ a2j;6 v j1 ¼ Me;j þ MF;jþ1 MF;j a2j1;1 hjþ1 þ a2j1;2 v jþ1 þ a2j1;3 hj þ a2j1;4 v j þ a2j1;5 hj1 þ a2j1;6 v j1 ¼ Hj þ Px;j
ð16Þ
The coefficients aij in Eq. (16) depend on the geometrical properties of the beam, the material properties, the stability functions, the compressive forces and the equivalent spring constants. Their expressions are given by Eq. (17):
a2j;1 ¼
EI S jþ1 h
a2j;2 ¼ a2j1;1 ¼ EI ðC þ SÞ h2 jþ1 EI a2j;3 ¼ EI C þ C h h jþ1 j EI EI a2j;4 ¼ a2j1;3 ¼ ðC þ SÞ ðC þ SÞ 2 2 h h j jþ1 EI S a2j;5 ¼ h j EI a2j;6 ¼ a2j1;5 ¼ ðC þ SÞ 2 h j EI Pjþ1 a2j1;2 ¼ 2 3 ðC þ SÞ þ hjþ1 h jþ1 EI EI Pjþ1 Pj a2j1;4 ¼ 2 3 ðC þ SÞ þ 2 3 ðC þ SÞ þ keq;j hjþ1 hj h h jþ1 j EI Pj a2j1;6 ¼ 2 3 ðC þ SÞ þ hj h j ð17Þ
The steps of the iterative procedure to obtain the lateral displacement are [8]: 1. The lateral displacements of the tower are calculated, that is vj for j = 1 to n. The proposed method assumes that the initial shape of the elastic curve is a parabolic function. 2. With the values of lateral displacements calculated in Step 1, the modulus of elasticity Esec;1 and Esec;2 are obtained for each level of guys. 3. According to Eq. (16), the new lateral displacement of the tower under the action of the acting loads is obtained by solving the 2n-1 2n-1 system of equations. 4. If for each of the guys levels the difference between the displacement obtained in Step 3 and the displacement of Step 1 is less than a predetermined value, the process is finished. Otherwise, with the displacement obtained in Step 3, the process is repeated until convergence. 5. If the difference between the displacements at each level of guys cannot be made less than a predetermined value, the structure is unstable under the action of the acting loads. The rigidity of the beam-column and/or the rigidity of the spring constants should be modified. Referring to Fig. 8, for the bar j and for a section at a distance x from the end j-1, the bending moment in said section is given by Eq. (19):
!2 2 px;j hj 1 Mj ðxÞ ¼ bj wj ! " # 1 cosðbj Þ sinðbj =hj xÞ cosðbj =hj xÞ þ 1 sinðbj Þ
1 Mj1;j cosðbj Þ M j;j1 sinðbj =hj xÞ sinðbj Þ
þ M j1;j cosðbj =hj xÞ ð19Þ
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6. Case study: comparison between the finite elements method and the proposed method In order to validate the proposed method (PM), a guyed mast structure of 120 m height are analyzed. For this, one model is designed using the Finite Element Method (FEM) through the use of the SAP 2000 program. In this model (FEM: 3D truss), the structure is modeled as a three-dimensional truss where the bars work mainly in axial force and the cables are modeled as cable elements. The tower has pinned base connections and the second-order effects are considered from a non-linear P-D
analysis (Fig. 9). The results will be compared with those obtained by the PM. The tower has a triangular cross-section of 1.2 m width. The bars are materialized by circular steel tubes with modulus of elasticity Es = 200 GPa. The legs (vertical bars) are 73.00 mm outside diameter and 5.20 mm thick and the horizontal bars and diagonals are 42.16 mm outside diameter and 3.42 mm thick. The inclination of the diagonals is 45°. The guys are EHS steel with an ultimate tensile strength of 1400 MPa and modulus of elasticity Ecb = 185 GPa. They are placed with spacings of 12 m in height. The guys whose anchorage levels to the tower are +12 m, +24 m and +36 m are anchored to the ground at a distance of 23 m from the tower axis and are 6.35 mm nominal diameter (Acb = 24.632 mm2). The guys whose anchorage levels to the tower are +48 m, +60 m and +72 m are anchored to the ground at a distance of 46 m from the tower axis and are 8.00 mm nominal diameter (Acb = 38.511 mm2). The remaining cables, whose anchorage levels to the tower are +84 m, +96 m, +108 m and +120 m, are anchored to the ground at a distance of 69 m from the tower axis and their nominal diameters are 8.00 mm. The wind loads on the structure are calculated from the guidelines established by the TIA 222-G standard and a basic wind speed of 43.4 m/s is considered. The graphs of Figs. 10–12 show the lateral deflection, the axial forces diagram and the bending moment diagram obtained from the PM and the finite element FEM: 3D truss models in the case that the tower base connection is pinned.
6.1. Analysis, comparison and discussion of results
Fig. 9. Three-dimensional model of the guyed tower of 120 m height in FEM. Initial configuration: pretensioning of cables at 10% of the breaking strength.
In relation to the maximum displacements of the tower, the difference between the PM and FEM: 3D truss models is of the order of 0.86%. The difference between the maximum positive bending moments is of the order of 1.5%, whereas between the maximum negative bending moments is of the order of 12.8%. In relation to axial force, only small differences are observed.
Fig. 10. Lateral deflection vs. tower height for the PM and for the FEM: 3D truss models when the base connection of the tower is pinned.
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Fig. 11. Axial force diagram for the PM and FEM: 3D truss models when the base connection of the tower is pinned.
Fig. 12. Bending moment diagram for the PM and FEM: 3D truss models when the base connection of the tower is pinned.
7. Conclusions In this work an approximate method has been proposed for the calculation of the deflection produced by the second-order effects in guyed masts of triangular cross-section. The method is based on the stability functions for the Timoshenko beam-column. The geometrical properties for the equivalent beam-column are obtained according to the lattice pattern of each of the faces of the tower. The guys are modeled as equivalent spring constants from the secant modulus of elasticity of the cables, and the model takes into account the eccentricities of the cables to the anchor points to the tower. The proposed method has been numerically validated through a case study comparing it with finite element methods. From the results presented, the following conclusions can be deduced: 1. Only small differences were observed in the calculation of the effects produced by the second-order deformations of the structure between the PM and the finite element methods. In other words, the values of the stresses and displacements of the
structure obtained by the proposed method are sufficiently accurate compared to those obtained by the finite element method, confirming the validity of the hypotheses adopted in the development of the method. 2. The PM maintains the complexity and mathematical rigor of analytical methods, but it has been conceived as a simpler application method. It uses basic and more general concepts from the point of view of structural engineering, which means that the engineer can quickly visualize the parameters that influence the design. 3. The proposed method considers the effects of shear deformation and second-order effects directly from the stability functions, which makes it possible to work with greater accuracy, without the need for mathematical artífice, to consider the effect of the shear deformation, or to make approximations and/or divisions of the elements between each guys levels. 4. It is very simple to program, even compared with the use of spreadsheets, as it does not need to discretize the structure into a large number of elements for convergence. This is why the method can be very useful in the preliminary design stages,
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enabling a significant saving of time. In other words, the finite element method does not present greater advantages in the preliminary design stage than the proposed method. 5. Although this work has been focused on triangular crosssection towers, with an arrangement of three cables per guys level, the proposed method can be extended to other types of cable arrangement and to other cross-section forms. Therefore, the proposed method, initially developed for the analysis of guyed towers, can be extended to the analysis of cable-stayed bridge pylons.
References [1] TIA 222-G. Structural Standard for Antenna: Supporting Structures and Antennas, Telecommunications Industry Association, Arlington VA; 2006. [2] Cohen E, Perrin H. Design of multi-level guyed towers: structural analysis. J Struct Div ASCE 1957;83:1–29. [3] Ezra GO. Analysis of high guyed towers. J Struct Div ASCE 1966;92:169–98. [4] Kalha NB. Equivalent beam-column analysis of guyed towers. Comput Struct 1995;55(4):631–45. [5] Wahba Y, Madugula M, Monforton G. Evaluations of non-linear analysis of guyed antenna towers. Comput Struct 1998;68:207–12.
[6] Margariti G, Gantes G. Linear and nonlinear buckling response and imperfection sensitivity of cable-stayed mast and pylon. Struct Eng Int 2015;25(1):43–9. [7] Williamson RA, Margolin MN. Shear effects in design of guyed towers. J Struct Div ASCE 1966;92:213–33. [8] Páez PM, Sensale B. Calculating guyed towers without using finite elements. Hormigón y acero 2017. https://doi.org/10.1016/j.hya.2017.05.012. [9] Romstad KM, Chiesa M. Approximate analysis of guyed towers. ASCE; 1977. [10] Gimsing NJ, Georgakis CT. Cable supported bridges: concept and design. 3rd ed. J. Wiley and Sons; 2012. [11] Timoshenko S, Gere J. Theory of elastic stability. 2nd ed. McGraw-Hill; 1961. [12] Dario Aristizabal-Ochoa J. Slope-deflection equations for stability and secondorder analysis of Timoshenko beam-column structures with semi-rigid connections. Eng Struct 2008;30(9):2517–27. [13] Lin FJ, Glauser EC, Johnston BJ. Behavior of laced and battened structural members. J Struct Div ASCE 1960;123(5):1377–401. [14] Computers and Structures, Inc. SAP2000. Analysis Reference Manual. Berkeley California, USA; 2011. [15] Carrasco A, Parnás V, Rodriguez P. Guy tension influence on the structural behavior of a Guyed Mast. J Int Assoc Shell Spatial Struct 2012;53(2):111–6. [16] UNE-EN 1993-3-1: 2013. Eurocode 3: Design of steel structures – Part 3–1: Tower, masts and chimneys – tower and masts. CEN; 2013. [17] Irvine HM, O’Sullivan MJ. Elastic stability of simple guyed towers. Appl Ocean Res 1979;1(4):203–7. [18] Ziegler H. Arguments for and against Engesser’s buckling formulas. Ing Arch 1982;52:105–13.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Analysis of guyed masts by the stability functions based on the Timoshenko beam-column Pablo M. Páez ⇑, Beradi Sensale Facultad de Ingeniería, Universidad de la República, Julio Herrera y Reissig 565, Montevideo, Uruguay
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 11 July 2017 Revised 16 August 2017 Accepted 16 September 2017
A new formulation for the analysis of guyed towers in which the mast is modeled as a continuous equivalent beam-column on non-linear elastic supports, is proposed. Because second-order effects must not be neglected and given the importance of consider the shear deformation in the analysis of this type of structures, the method proposes the calculation of the second-order deformation using the stability functions based on the Timoshenko beam column. The equivalent beam-column properties of the mast are calculated depending on the pattern construction of the tower and the guys are replaced by non-linear elastic supports. Based on the catenary configuration of the cable, the spring constant is obtained from the secant modulus of the cable. In order to validate the proposed method, a comparative study is carried out analyzing a guyed mast using the proposed method and a numerical model through the use of finite elements. As a main conclusions we will mention that, the proposed method are sufficiently accurate compared to the finite element method, confirming the validity of the hypotheses adopted in the development of the method and; the reduction of the computational effort, since the structure does not need to be discretized into a large number of elements for the convergence. Ó 2017 Elsevier Ltd. All rights reserved.
Keywords: Guyed towers Stability functions Second order analysis Geometric non-linearity
1. Introduction Radio and television communications, as well as cellular telephony, are possible in the modern world thanks to structures that support equipment transmitting signals from one place to another. Due to the great resistance in relation to the material consumption and permeability, steel trussed towers are structures in wide use in support of communication systems. These structures are usually slender and light elements, mostly located in exposed places, so environmental loads prevail in the design. Two types of trussed towers are used according to their structural behavior: the self-supporting and the guyed mast. When large heights are required and conditions at the installation site permit, guyed masts, because of their steel economy relative to the self-supporting ones, are the ones usually used. However, the presence of the guys confers a complex structural behavior on the environmental loads. For the analysis of guyed masts several methods can be applied. The TIA 222-G standard specifies three types of analysis [1]. The first one consists of an analysis where the tower is modeled as an equivalent beam-column supported by cables represented either ⇑ Corresponding author. E-mail addresses: (B. Sensale).
[email protected]
(P.M.
https://doi.org/10.1016/j.engstruct.2017.09.036 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
Páez),
[email protected]
as cable elements or as non-linear elastic supports. The second one consists of an elastic three-dimensional truss model where the mast is modeled as a space framework whose bars are connected by ideal frictionless joints and consequently can only produce axial force, and finally, an elastic three-dimensional model can be performed where some of the members of the mast are modeled as elements that can produce both bending moments and axial forces, and others as elements that can only produce axial force. In the last two methods of analysis, the guys are modeled as cable elements. The complexity in the analysis of the guyed towers lies in the inherent non-linearity that these structures present: the nonlinearity of the guys and the geometric non-linearity. The slenderness of these towers makes them susceptible to the buckling phenomenon; in other words, second-order effects must not be neglected. The horizontal actions cause displacements that increase the level of tension in the guys and in turn the forces of compression on the tower. These compression forces tend to increase displacement and so on, a phenomenon commonly referred to as the P-D effect. Among the methods of the first type, based on the beamcolumn model, different researchers have made different proposals for more than fifty years. Cohen and Perrin [2] proposed a two-dimensional analysis using a beam-column model on elastic
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supports, considering only the flexural stiffness of the beam. In the same sense, Ezra [3] proposed a two-dimensional analysis based on the beam-column model on non-linear elastic supports and considering the torsion of the structure by torsion springs. Basically, the analysis consists of calculating the displacements of the structure in iterative form from an arbitrary set of initial displacements. Kalha [4] proposes an approximate method for the analysis of guyed masts using an equivalent beam-column for the tower model and cable elements for the guys model. Although the method takes into account the geometric coupling among the different degrees of freedom and the second-order effects, it is difficult to apply because it requires a high degree of computational programming. Wahba et al. [5] analyzed three different models for the guyed masts, one of which consisted of modeling the tower as an equivalent beam-column and non-linear cable elements for the guys. Their results validated the conclusions of the previous studies by Kahla and confirmed the non-linear geometric response of these structures to wind loads. Margariti and Gantes [6] propose an approximate method based on the classic expressions of buckling for the calculation of the elastic critical load in guyed masts and pylons of cable-stayed bridges. This method is based on a single span beam-column model, considering only the flexural stiffness of the mast or pylon. The non-linearity of the guys is taken into account by the equivalent modulus of elasticity obtained from the tangent modulus of the cables. Williamson and Margolin [7] studied the effect of shear force in the design of guyed masts. For this, they used an equivalent beam-column model on elastic support, replacing the trussed web system of each face by a fictitious solid web of the same shear stiffness but having zero flexural stiffness. Their results showed the importance of considering the effects of shear deformation in the analysis of guyed masts. Based on the equivalent beam-column method, Páez and Sensale [8] propose an analytical method for the calculation of guyed masts using the stability functions based on the Euler-Bernoulli theory. The guys are replaced by non-linear elastic supports and the effect of the shear deformations are taken into account by a reduced second order moment obtained from the Timoshenko beam theory. Among the different methods of analysis, the most commonly used today are based on finite elements. These methods are very accurate; however, for convergence of the solutions, it is necessary to discretize the structure into a large number of elements, and requires a great effort of computer programming [4,5]. On the other hand, analytical methods require complex analyses, and although they allow us to obtain solutions whose results are close to those obtained by the use of finite elements, in general they present significant differences [2,3,6,7]. It is for this reason that in this work it is proposed to develop a new analytical method for the analysis of guyed masts based on the equivalent beam-column model. In this sence, the method maintains the rigor of the analytical methods but their application is simple. In other words, sufficiently accurate solutions are obtained for the design of guyed towers in a reasonable time and with a reasonable effort. The proposed method is motivated by the work of Páez and Sensale [8]. The main difference between this method and that of Páez and Sensale is essentially in the way the two methods compute the second-order deformation. The proposed method considers the non-linearity of the cables and the second-order effects. For this purpose, the tower is modeled as a continuous equivalent beam-column, on non-linear elastic supports, whose axial stiffness, flexural stiffness and shear stiffness are calculated depending on the pattern construction of the tower [9]. The guys are replaced by non-linear elastic supports whose spring constant is obtained from the secant modulus of elasticity of the cables [10]. The method proposes the calculation of the second-order deformation using the stability functions based on the Timoshenko beam-column [11–13].
One of the main advantages of the method is the reduction of the computational effort, since the structure does not need to be discretized into a large number of elements for the convergence and hence the method can be more useful in the pre-design stages. Another advantage of the method is that the effects of shear deformation are taken into account directly from the geometrical properties of the tower by the use of stability functions based on the Timoshenko beam-column, without the need for mathematical artifice to consider such effects. In addition, the proposed analytical method uses the basic and more general concepts from the point of view of structural engineering, this means that the engineer can quickly visualize the parameters that influence the design. In order to validate the proposed method, a comparative study is carried out, analyzing a guyed mast using the proposed method and a numerical model through the use of finite elements by SAP 2000 software [14]. 2. The guyed towers The towers are constructed from a series of vertical bars, generally denominated as legs, horizontal bars and diagonal bars, thus forming a space truss. The frequently used tower cross-section is the equilateral triangular, although square sections may be used in some cases. For smaller structures, the structural elements of the lattice are materialized by solid circular steel bars, while for larger sized structures circular steel tubes, angles profiles, and steel cold-formed profiles are used. The guys provide side support to the towers. In general, they are placed uniformly distributed on the height and are prestressed. In the case of towers of triangular section, three cables are placed at each level of guys and in the case of a square section four cables are placed per level. The guys are anchored to the ground in such a way that the angles that form the planes constituted by each cable and the tower are equal. The typical inclinations used for the cables, measured as the angle formed by the cable with the horizontal, are of the order of 40–60°, being able to reach smaller inclinations for the cables of the first level. The initial stress of pretensioning of the cables is one of the main parameters that affect the stability of this type of structure [15]. This is imposed during the construction of the mast and must be checked periodically. The TIA 222-G standard recommends average values of 10% of the cable breaking stress, with a range of variation between 7% and 15% of the breaking stress, since within this range the effects of vibration and the aeroelastic instability can be neglected. For its part, the EN 1993-1-11 standard provides somewhat more general guidelines, allowing initial stresses of up to 45% of the breaking stress. However, it requires the verification of vibration and the aeroelastic instability [16]. This work will focus on towers of equilateral triangular crosssection, although most of the reasonings presented here can extend to towers of square cross-section. 3. Model of the tower: Equivalent beam-column properties The model of the tower as an equivalent beam-column is based on the following assumptions: the material of the beam presents a linear elastic behavior, the deformations of the beam are small, and let us assume that, under the action of the bending moment and the shear force, the perpendicular plane sections to the longitudinal axis of the beam before it is deformed, remain plane but not necessarily perpendicular to the deformed longitudinal beam axis. These hypotheses correspond to the Timoshenko beam theory. Fig. 1 shows the typical lacing configuration of the trussed web of one of the faces of the tower. For each of these patterns, the geometrical properties of the equivalent beam-column can be
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Fig. 1. Typical lacing configurations of the trussed web of one of the faces of the tower.
Table 1 Geometric properties of equivalent beam-column section according to the pattern of one face of the tower (adapted from [8]). Equivalent properties
Pattern 1
Pattern 2 and 3
Pattern 4
EA
3 E Am
3 E Am
3 E Am
E Iy , E Iz
1 2
G Ay , G Az
3 2
E Am a2
1 EAd w1
1 2
1
u þ tan EAh
E Am a2
3 2E
Ad w1
1 2
E Am a2
3 2E
Ad w1
Pattern 5 b 3 E Am þ A 1 2 b 2 E a Am þ A 3 E Ad w1
E and G are the modulus of elasticity and the shear modulus of the material. E A is the equivalent axial stiffness of the beam-column.E Iz , E Iy , G Az and G Ay are the flexural stiffness and shear stiffness of the equivalent beam-column with respect to coordinate axes z and y, respectively. Am, Ad and Ah are the cross-sections of the legs of the diagonal and horizontal bars that make up the lattice, respectively. a and b are the distance between the legs and the distance between the horizontal bars, respectively. u is the angle formed by the legs with the diagonal bars. 2 w1 ¼ sin u cos u b ¼ Ad Ah cos3 3u A Ah þ2Ad sin
u
obtained using the principle of virtual works. Table 1 shows these properties for the case of triangular cross-section towers [4,9]. 4. The guys model
2 1 2 1 d ðr1 r0 Þccb þc1 cb r1 sinhðccb c cb r1 Þr0 sinh ccb c cb r0 ¼ lcb 2Ecb ccb cosh 1=2ccb ccb r1 1 1 4Ecb ccb r0 sinhð1=2ccb ccb r1 Þ r sinh 1=2ccb ccb r1 1 0 1 þ 2Ecb ccb cosh 1=2ccb ccb r1 1 ð1Þ
4.1. The elastic cable: Secant modulus of elasticity Consider an inclined cable whose chord length is lcb and its horizontal projection ccb, subjected to the action of a chord force T0 and to a vertical dead load gcb (Fig. 2). Based on the catenary configuration of the cable, by applying a chord force T1, the cable elongation d can be expresed by Eq. (1) [10]:
where Ecb is the modulus of elasticity of the cable, r1 and r0 are the stresses in the cable due to the chord forces T1 and T0, respectively, and ccb is the density of the cable material. The non-linearity between the chord force and the elongation due to the sagging change under different load conditions can be
Fig. 2. Inclined cable subjected to a vertical dead load per unit length: relationship between the chord force and the chord length (adapted from [8]).
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Fig. 3. Secant modulus of elasticity vs. final cable stress, taking Ecb = 185 GPa and
r0 ¼ 0:1 f u ¼ 140 (adapted from [8]).
taken into account by the tangent and secant modulus of elasticity of the cable. In this way, the non-linear phenomenon can be treated as a linear one. However, the tangent modulus of elasticity should be used when the relationship between the stresses r1 and r0 is small; otherwise, the secant modulus of elasticity must be used. Its expression is given by Eq. (2) [10]:
Esec ¼
Dr r 1 r 0 ¼ ccb De d
ð2Þ
The graph of Fig. 3 shows the variation of the secant modulus of elasticity as a function of the final stress r1 for cable lengths of 60 m, 100 m and 160 m, whose modulus of elasticity is Ecb ¼ 185 GPa and whose initial stress is r0 ¼ 140 MPa, corresponding to 10% of the breaking stress. It can be seen that for final stresses of the order of 50% of the breaking stress, the ratio of the modulus of elasticity of the cable to the secant modulus of elasticity does not exceed 10% for a cable length of 160 m, or 4% for a cable length of 100 m. 4.2. The cable system: the equivalent spring stiffness Consider in the first instance the symmetrical plane system, in equilibrium, formed by the mast and two guys (Fig. 4). If we apply a horizontal force at the upper end of the mast, it will
Fig. 5. Chord force vs. elongation of the cable from the initial tensioning condition (adapted from [10]).
deform and its position can be described as a horizontal displacement u and a vertical one w, both components of the displacement being small. Since the mast displacement is composed of a rigid movement and by the flexion thereof, it can be assumed that the displacement w is an infinitesimal of the second order of u (w O u2 ) [17]. When the mast deforms, the cable to the left will lengthen while the one to the right will be shortened. If both cables are subjected to an initial chord force T0, the cable on the left will experience a rapid increase in tension while the cable on the right will undergo a rapid drop. Fig. 5 shows schematically the foregoing reasoning. Referring to Fig. 4, the elongation of the left cable d1 and the shortening of the right cable d2 as a function of the horizontal displacement u can be expressed by Eq. (3):
d1 ¼ d2 ¼
Fig. 4. Symmetrical plane system formed by the mast and two guys. Deformed configuration when applying a horizontal force F at the upper end of the mast (adapted form [17]).
ccb u lcb
ð3Þ
If the applied force F is of unit value and considering a linear relationship between stress and strain by using the secant modulus of elasticity, Eq. (2), the equivalent spring constant used to replace the cable system is Eq. (4):
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ccb u 2 lcb ccb u d2 ¼ lcb
d1 ¼
ð5Þ
Equating the horizontal forces, Eq. (6),
ðEsec;1 þ 2 Esec;2 Þ Acb u
c2cb 3
2 lcb
¼1
ð6Þ
Therefore, the equivalent elastic spring constant used to replace the cable system is determined by Eq. (7):
keq ¼ ðEsec;1 þ 2 Esec;2 Þ Acb
c2cb
ð7Þ
3
2 lcb
From Eqs. (1), (5) and (7), we can conclude that, under the assumptions made, the structure of Fig. 6 is more flexible in the positive direction of the U-axis than in the negative direction. 5. Proposed method Fig. 6. Three-dimensional system formed by the mast and three cables of equal chord length and equal inclination (adapted from [8]).
keq ¼ ðEsec;1 þ Esec;2 Þ Acb
c2cb 3 lcb
ð4Þ
where Acb is the cross-sectional area of the cables, Esec;1 and Esec;2 are the secant modulus of elasticity of the cable being stretched and the cable being shortened, respectively, and a the inclination of the cables with respect to the horizontal. Let us now consider the three-dimensional system constituted by the mast and three cables of equal chord length lcb, equal inclination a and such that the planes formed by each cable and the mast are equidistant to each other, to which a horizontal force F of unit value is applied in the upper end of the mast in the direction of the U-axis (Fig. 6). If the force is applied in the positive direction of the U-axis (as shown in the figure), the relationship between the elongation d1 of the cables to windward and the horizontal displacement u, as well as the relationship between the shortening d2 of the cable to leeward and the horizontal displacement can be expressed by Eq. (5) [8]:
Consider a guyed mast structure of height ht , with n levels of guys uniformly spaced by a distance hj. Without loss of generality, let us consider that the structure is fixed at the base, as shown in Fig. 7a. The tower will be subject to vertical distributed loads pz ðzÞ; being their self-weight; to vertical concentrated loads P z , due, for example, to the weight of the ancillaries, but in general they will be due to the lengthening and shortening of the cables and the wind load on them; and to horizontal concentrated and distributed loads Px and px ðzÞ, respectively, basically due to the action of the wind on the tower, the guys and the ancillaries. Taking into account the static equilibrium in the deformed position in node j, referring to Fig. 7b, we can write for the continuity bending moment, j-1, Eq. (8) [8]: M j1 ¼ M j þ
i¼n X i¼j
i¼n X i¼j
i¼n X Hi þ P x;i keq;i v i hi þ M e;i þ v j v j1
Esec;1 Esec;2 Pz0;i þ Pz;i þ 2 Esec;1 þ 2 Esec;2
i¼j
keq;i i
zcb;i zcb;i v i þ 3 T 0;i ccb;i ccb;i
Fig. 7. (a) External loads on the guyed mast. (b) Mathematical model of calculation, deformed position (adapted from [8]).
ð8Þ
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Fig. 8. Deformed position of the continuous beam-column on discrete elastic supports for the bars between the nodes j 1, j and j + 1. Free-body diagrams to apply the slopedeflection equations based on the beam-column of Timoshenko (adapted from [8]).
where zcb;j is the height corresponding to the level j of guys, v j is the value of the lateral deflection for said level of guys, keq;j is the value of the equivalent spring constant for the level j of guys, ccb;j is the horizontal distance between the guys’ anchor on the ground and the tower for the level j of guys, Hj is the horizontal resultant force R zcb;j þhj =2 between two levels of guys, obtained as Hj ¼ zcb;j hj =2 px ðzÞdz; and Pz0;j is the vertical resultant force due to the uniformly distributed vertical load between two levels of guys, obtained as R zcb;j þhj =2 Pz0;j ¼ zcb;j hj =2 pz ðzÞdz, P z;j is the vertical force at the level j of guys due to the action of the wind on these, and Me;j is the bending moment due to the cables’ eccentricity at the level j of guys and is given by Eq. (9):
M e;j ¼
! pffiffiffi pffiffiffi a Acb;j zcb;j ccb;j v j 3 3 Esec;1 þ Esec;2 3 6 3 lcb:j
ð9Þ
j
5.1. Slope-deflection equations based on the Timoshenko beamcolumn: ‘‘the modified approach” Two different formulations have been proposed to analyze the influence of shear force on the elastic critical buckling load and the second-order effect in beam-columns, one performed by Engesser and another by Haringx [18]. The Haringx model has been applied and discussed by Timoshenko and Gere and is known as ‘‘the modified approach” [11].
According to the hypotheses proposed by Haringx, we can deduce the slope-deflection equations that allow us to consider the bending deformations and the shear force deformations; its expression for the bar j between nodes j-1 and j (Fig. 8) is given by Eq. (10) [12]:
M j1;j M j;j1
v j v j1 EI ¼ C hj1 þ S hj ðC þ SÞ M F;j h j h j v j v j1 EI ¼ S hj1 þ C hj ðC þ SÞ þ M F;j h h j j
ð10Þ where Mj1;j and Mj;j1 are the fixed end moments for the bar j; C j and Sj are the stability functions based on the Timoshenko beamcolumn for the bar j, Eq. (11):
Cj ¼ Sj ¼
1 wj bj cot bj 2 tanðbj =2Þ bj
wj wj bj cosec bj 1 2tanðbj =2Þ bj
ð11Þ
wj
bj is the stability function in the plane bending and its expression is given by Eq. (12):
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi u
u Pj h2j
bj ¼ t
ðw E IÞj
ð12Þ
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wj is the shear reduction factor, Eq. (13):
wj ¼ 1 þ
Pj Gj Xred;j
1 ð13Þ
Xred is the effective shear area of the beam-column cross-section, namely, Xred ¼ A (Table 1); Pj is the compression force in the bar j, it is deduced from the equilibrium equations and their expression is given by Eq. (14) [8]: Pj ¼
i¼n X ðEsec;1 Esec;2 Þ zcb;i zcb;i P 0;i þ P z;i þ 2 v i þ 3 T 0;i keq;i ðEsec;1 þ 2 Esec;2 Þ ccb;i ccb;i i¼j ð14Þ
The subscripts for the expressions in parentheses indicate the indicates that the modulus corresponding bar; for example EI h j of elasticity, the second-order moment and the span length are those corresponding to the bar j. M F;j is the fixed end moment for a beam subjected to a uniformly distributed load in its plane bending and an axial compressive force. Its expression depends on the support conditions; for the case of a beam fixed at both ends, its expression is given by Eq. (15) [13]: 2
M F;j ¼
px;j hj 12
12 b2j
!
1 wj
!"
1
bj 2 tan bj =2
#
ð15Þ
603
For a beam-column with n spans, from Eq. (16) a 2n-1 2n-1 system of equations is formed whose unknowns are the rotations and the lateral deflection of the nodes h1 hj hn1 and v 1 v j v n , respectively, since, on the one hand, the upper end of the tower is considered as pinned, implying that hn can be written as a function of hn1 ; and on the other hand, when the lower end is fixed h0 ¼ v 0 ¼ 0 and when it is pinned v 0 ¼ 0 and h0 can be written as a function of h1 . 5.3. Obtaining the lateral displacement: second order analysis, iterative method Under the action of the external forces, the proposed method assumes the initial shape of the elastic curve correspond to a parabolic function and such that the maximum value of the elastic curve is less than an assumed value v m;i , with v m;i ¼ 0:001 ht . The equivalent spring constants at each of the levels are calculated from the assumed lateral displacements. In this way the relationship between the compressive force in the bar j and the compression force in the end bar n can be reasonably estimated. Let f be the ratio factor between these forces, this is P j ¼ f P n , in this way the relationship between the buckling parameters can be written as Eq. (18):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE I Þn hj bn bj ¼ f ðE I Þj hn
ð18Þ
5.2. Analysis of the structure Consider a continuous equivalent beam-column on discrete elastic supports, for which both the sectional and material properties are known, subjected to the action of external loads. Taking into account the static equilibrium in the deformed position for the elements between the nodes j-1, j and j+1 (Fig. 8) and according to the slope-deflection equations based on the Timoshenko beamcolumn with axial compression force (Eq. (10)), we obtain the Equations (16): a2j;1 hjþ1 þ a2j;2 v jþ1 þ a2j;3 hj þ a2j;4 v j þ a2j;5 hj1 þ a2j;6 v j1 ¼ Me;j þ MF;jþ1 MF;j a2j1;1 hjþ1 þ a2j1;2 v jþ1 þ a2j1;3 hj þ a2j1;4 v j þ a2j1;5 hj1 þ a2j1;6 v j1 ¼ Hj þ Px;j
ð16Þ
The coefficients aij in Eq. (16) depend on the geometrical properties of the beam, the material properties, the stability functions, the compressive forces and the equivalent spring constants. Their expressions are given by Eq. (17):
a2j;1 ¼
EI S jþ1 h
a2j;2 ¼ a2j1;1 ¼ EI ðC þ SÞ h2 jþ1 EI a2j;3 ¼ EI C þ C h h jþ1 j EI EI a2j;4 ¼ a2j1;3 ¼ ðC þ SÞ ðC þ SÞ 2 2 h h j jþ1 EI S a2j;5 ¼ h j EI a2j;6 ¼ a2j1;5 ¼ ðC þ SÞ 2 h j EI Pjþ1 a2j1;2 ¼ 2 3 ðC þ SÞ þ hjþ1 h jþ1 EI EI Pjþ1 Pj a2j1;4 ¼ 2 3 ðC þ SÞ þ 2 3 ðC þ SÞ þ keq;j hjþ1 hj h h jþ1 j EI Pj a2j1;6 ¼ 2 3 ðC þ SÞ þ hj h j ð17Þ
The steps of the iterative procedure to obtain the lateral displacement are [8]: 1. The lateral displacements of the tower are calculated, that is vj for j = 1 to n. The proposed method assumes that the initial shape of the elastic curve is a parabolic function. 2. With the values of lateral displacements calculated in Step 1, the modulus of elasticity Esec;1 and Esec;2 are obtained for each level of guys. 3. According to Eq. (16), the new lateral displacement of the tower under the action of the acting loads is obtained by solving the 2n-1 2n-1 system of equations. 4. If for each of the guys levels the difference between the displacement obtained in Step 3 and the displacement of Step 1 is less than a predetermined value, the process is finished. Otherwise, with the displacement obtained in Step 3, the process is repeated until convergence. 5. If the difference between the displacements at each level of guys cannot be made less than a predetermined value, the structure is unstable under the action of the acting loads. The rigidity of the beam-column and/or the rigidity of the spring constants should be modified. Referring to Fig. 8, for the bar j and for a section at a distance x from the end j-1, the bending moment in said section is given by Eq. (19):
!2 2 px;j hj 1 Mj ðxÞ ¼ bj wj ! " # 1 cosðbj Þ sinðbj =hj xÞ cosðbj =hj xÞ þ 1 sinðbj Þ
1 Mj1;j cosðbj Þ M j;j1 sinðbj =hj xÞ sinðbj Þ
þ M j1;j cosðbj =hj xÞ ð19Þ
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6. Case study: comparison between the finite elements method and the proposed method In order to validate the proposed method (PM), a guyed mast structure of 120 m height are analyzed. For this, one model is designed using the Finite Element Method (FEM) through the use of the SAP 2000 program. In this model (FEM: 3D truss), the structure is modeled as a three-dimensional truss where the bars work mainly in axial force and the cables are modeled as cable elements. The tower has pinned base connections and the second-order effects are considered from a non-linear P-D
analysis (Fig. 9). The results will be compared with those obtained by the PM. The tower has a triangular cross-section of 1.2 m width. The bars are materialized by circular steel tubes with modulus of elasticity Es = 200 GPa. The legs (vertical bars) are 73.00 mm outside diameter and 5.20 mm thick and the horizontal bars and diagonals are 42.16 mm outside diameter and 3.42 mm thick. The inclination of the diagonals is 45°. The guys are EHS steel with an ultimate tensile strength of 1400 MPa and modulus of elasticity Ecb = 185 GPa. They are placed with spacings of 12 m in height. The guys whose anchorage levels to the tower are +12 m, +24 m and +36 m are anchored to the ground at a distance of 23 m from the tower axis and are 6.35 mm nominal diameter (Acb = 24.632 mm2). The guys whose anchorage levels to the tower are +48 m, +60 m and +72 m are anchored to the ground at a distance of 46 m from the tower axis and are 8.00 mm nominal diameter (Acb = 38.511 mm2). The remaining cables, whose anchorage levels to the tower are +84 m, +96 m, +108 m and +120 m, are anchored to the ground at a distance of 69 m from the tower axis and their nominal diameters are 8.00 mm. The wind loads on the structure are calculated from the guidelines established by the TIA 222-G standard and a basic wind speed of 43.4 m/s is considered. The graphs of Figs. 10–12 show the lateral deflection, the axial forces diagram and the bending moment diagram obtained from the PM and the finite element FEM: 3D truss models in the case that the tower base connection is pinned.
6.1. Analysis, comparison and discussion of results
Fig. 9. Three-dimensional model of the guyed tower of 120 m height in FEM. Initial configuration: pretensioning of cables at 10% of the breaking strength.
In relation to the maximum displacements of the tower, the difference between the PM and FEM: 3D truss models is of the order of 0.86%. The difference between the maximum positive bending moments is of the order of 1.5%, whereas between the maximum negative bending moments is of the order of 12.8%. In relation to axial force, only small differences are observed.
Fig. 10. Lateral deflection vs. tower height for the PM and for the FEM: 3D truss models when the base connection of the tower is pinned.
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Fig. 11. Axial force diagram for the PM and FEM: 3D truss models when the base connection of the tower is pinned.
Fig. 12. Bending moment diagram for the PM and FEM: 3D truss models when the base connection of the tower is pinned.
7. Conclusions In this work an approximate method has been proposed for the calculation of the deflection produced by the second-order effects in guyed masts of triangular cross-section. The method is based on the stability functions for the Timoshenko beam-column. The geometrical properties for the equivalent beam-column are obtained according to the lattice pattern of each of the faces of the tower. The guys are modeled as equivalent spring constants from the secant modulus of elasticity of the cables, and the model takes into account the eccentricities of the cables to the anchor points to the tower. The proposed method has been numerically validated through a case study comparing it with finite element methods. From the results presented, the following conclusions can be deduced: 1. Only small differences were observed in the calculation of the effects produced by the second-order deformations of the structure between the PM and the finite element methods. In other words, the values of the stresses and displacements of the
structure obtained by the proposed method are sufficiently accurate compared to those obtained by the finite element method, confirming the validity of the hypotheses adopted in the development of the method. 2. The PM maintains the complexity and mathematical rigor of analytical methods, but it has been conceived as a simpler application method. It uses basic and more general concepts from the point of view of structural engineering, which means that the engineer can quickly visualize the parameters that influence the design. 3. The proposed method considers the effects of shear deformation and second-order effects directly from the stability functions, which makes it possible to work with greater accuracy, without the need for mathematical artífice, to consider the effect of the shear deformation, or to make approximations and/or divisions of the elements between each guys levels. 4. It is very simple to program, even compared with the use of spreadsheets, as it does not need to discretize the structure into a large number of elements for convergence. This is why the method can be very useful in the preliminary design stages,
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enabling a significant saving of time. In other words, the finite element method does not present greater advantages in the preliminary design stage than the proposed method. 5. Although this work has been focused on triangular crosssection towers, with an arrangement of three cables per guys level, the proposed method can be extended to other types of cable arrangement and to other cross-section forms. Therefore, the proposed method, initially developed for the analysis of guyed towers, can be extended to the analysis of cable-stayed bridge pylons.
References [1] TIA 222-G. Structural Standard for Antenna: Supporting Structures and Antennas, Telecommunications Industry Association, Arlington VA; 2006. [2] Cohen E, Perrin H. Design of multi-level guyed towers: structural analysis. J Struct Div ASCE 1957;83:1–29. [3] Ezra GO. Analysis of high guyed towers. J Struct Div ASCE 1966;92:169–98. [4] Kalha NB. Equivalent beam-column analysis of guyed towers. Comput Struct 1995;55(4):631–45. [5] Wahba Y, Madugula M, Monforton G. Evaluations of non-linear analysis of guyed antenna towers. Comput Struct 1998;68:207–12.
[6] Margariti G, Gantes G. Linear and nonlinear buckling response and imperfection sensitivity of cable-stayed mast and pylon. Struct Eng Int 2015;25(1):43–9. [7] Williamson RA, Margolin MN. Shear effects in design of guyed towers. J Struct Div ASCE 1966;92:213–33. [8] Páez PM, Sensale B. Calculating guyed towers without using finite elements. Hormigón y acero 2017. https://doi.org/10.1016/j.hya.2017.05.012. [9] Romstad KM, Chiesa M. Approximate analysis of guyed towers. ASCE; 1977. [10] Gimsing NJ, Georgakis CT. Cable supported bridges: concept and design. 3rd ed. J. Wiley and Sons; 2012. [11] Timoshenko S, Gere J. Theory of elastic stability. 2nd ed. McGraw-Hill; 1961. [12] Dario Aristizabal-Ochoa J. Slope-deflection equations for stability and secondorder analysis of Timoshenko beam-column structures with semi-rigid connections. Eng Struct 2008;30(9):2517–27. [13] Lin FJ, Glauser EC, Johnston BJ. Behavior of laced and battened structural members. J Struct Div ASCE 1960;123(5):1377–401. [14] Computers and Structures, Inc. SAP2000. Analysis Reference Manual. Berkeley California, USA; 2011. [15] Carrasco A, Parnás V, Rodriguez P. Guy tension influence on the structural behavior of a Guyed Mast. J Int Assoc Shell Spatial Struct 2012;53(2):111–6. [16] UNE-EN 1993-3-1: 2013. Eurocode 3: Design of steel structures – Part 3–1: Tower, masts and chimneys – tower and masts. CEN; 2013. [17] Irvine HM, O’Sullivan MJ. Elastic stability of simple guyed towers. Appl Ocean Res 1979;1(4):203–7. [18] Ziegler H. Arguments for and against Engesser’s buckling formulas. Ing Arch 1982;52:105–13.