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Optimum design of stayed columns with split-up cross arm
Advances in Engineering Software 36 (2005) 614–625 www.elsevier.com/locate/advengsoft
Optimum design of stayed columns with split-up cross arm Jan Van Steirteghema,*, Willy P. De Wildea, Philippe Samynb,c, Ben P. Verbeecka, Franc¸ois Wattela a
Mechanics of Materials and Constructions, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium b Department of Architecture, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium c Samyn and Partners, Steenweg op Waterloo 1537, B-1150 Brussels, Belgium Available online 11 April 2005
Abstract The columns considered in this paper consist of a central mast; a split-up cross arm hinged or fixed to the central mast and optimally stressed stays. The buckling load of these columns is calculated and the influence of the geometrical parameters is examined. The indicator of volume, W, is proposed as a measure of the efficiency of different structures that are designed to transmit the same load P. This paper demonstrates that the stayed columns with a split-up cross arm have a number of advantages with respect to the stayed columns examined previously. It further reveals that both the sections of the stays and the cross arm length influence the buckling load and the efficiency decisively. Compared to other columns with one cross arm, an efficiency increase of more than 20% can be achieved. The efficiency is defined as the minimal volume of material needed to sustain a given load P. The theory of morphological indicators is used to propose a design procedure for stayed columns. This procedure takes constraints (such as local buckling, yielding, choice of the morphology.) into consideration and proposes the most efficient column. Furthermore, it takes imperfection into account as prescribed by the construction code (Eurocode 3) [Eurocode Steel/3. Belgian Standard NBN ENV 1993-1-1: 1992/AX (1992)]. q 2005 Elsevier Ltd. All rights reserved. Keywords: Stayed column; Split-up cross arm; Indicator of volume; Efficiency; Imperfections; Design procedure; Comparison
1. Indicator of volume The indicator of volume W is a design tool developed by Samyn [2] allowing the optimization of structures for a chosen criteria, in case volume of the material, at an early stage and using only a very limited number of parameters. When designing stayed columns, W can be used to estimate the efficiency of the chosen morphology compared to others designed to transmit the same load. Given a structure characterized by its morphology and proportions presenting a volume of material V when loaded by a force system of which the resultant is F, main dimension L and composed of a material with allowable stress s, then W can be defined as the volume of material the morphological identical and homothetical structure loaded with a unit load of 1 N, of which the main dimension is 1 m and composed of * Corresponding author. Tel.: C32 2 6292936; fax: C32 2 6292928. E-mail address: [email protected] (J. Van Steirteghem).
0965-9978/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2005.03.007
a material with a unit allowable stress of 1 Pa and written as: WZ
sV : FL
(1)
It is possible to show that for statically determinate, fully stressed structures, when neglecting second order and instability effects as well as the material of the connections (trusses, arches, beams, frames, masts, membranes), with main dimension L and loaded by a uniformly distributed load or a mobile point load, W is only a function of the geometrical slenderness L/H (L and H are the dimensions of the rectangle in which the structure is inscribed, LOH). It is possible to use W to study stayed columns, in the same way Shanley [7] used the structural index F/L2. The design parameters in the case of a column are the load F and the length L, through which the load has to be transmitted. For pin-ended columns the stress at the onset of buckling can easily be obtained: scr Z
p2 E : ðL=rÞ2
(2)
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615
Symbols C1 D Ec Eca Es F h hi hs Ic Ica k2 KE KG KG L Lb Pcr PE PL,cr q t
constant external diameter of the central mast modulus of elasticity of mast material modulus of elasticity of cross arm material modulus of elasticity of stay material resultant of the total forces on a structure external diameter of the central mast internal diameter of the central mast diameter of the stays moment of inertia of the central mast moment of inertia of the cross arm constant assembled elastic stiffness matrix assembled geometric stiffness matrix assembled geometric stiffness matrix for unit load length of the column buckling length of the column maximum buckling load of the stayed column euler load of the central mast critical local buckling load dimensionless number Wh2/I wall thickness of the cross section of the central mast
The term r2ZIc/Uc is a dimensional number as it is the square of a length and hence cannot be used as a shape parameter. It is, therefore, suitable to replace r2 by a nondimensional number depending only on the geometrical properties of the cross section. Since r2 has the dimension of an area it can be divided by the square of the diameter of the central mast h to obtain the shape parameter q, which is a measure for the buckling sensitivity of a section: qZ
Uc h2 h2 Z 2: Ic r
(3)
T Topt V W a b g l L L1 L2 LE Uc Uca Us r s scr x
pretension of the stays optimal pretension of the stays total volume of the structure indicator of volume ratio of total volume to volume of central mast opening angle of the cross arm multiplication factor eigenvalues reduced slenderness of stayed column that can sustain a load P first approximation of the reduced slenderness second approximation of the reduced slenderness reduced slenderness of simple column that can sustain a load P cross-sectional area of the mast cross-sectional area of the cross arm member cross-sectional area of the stay gyros radius I/A allowable stress of the material used stress in the central mast at buckling ratio of the distance between the split-up cross arm and L, a constant
constant material parameters and length, the allowable critical stress increases as q decreases. This means that the ratio h/t, for example for a tube of wall thickness t, should be as large as possible, however limited by local buckling (‘crushing’, ‘crippling’, etc.) of the tin-walled section. Other parameters come into play when stayed columns are studied, the most important being the overall geometric slenderness L/H, the cross section of the stays Us, the cross section of the cross arm Uca and, for stayed columns with split-up cross arm, the opening angle of the cross arm b.
Eq. (2) can now be rewritten as: p2 Eh2 scr Z 2 L q
2. Problem statement (4)
The indicator of volume of a simple column, which fails due to buckling, can be established using Eq. (1): 2 sV 1 s L s q Z WZ Z ; (5) Pcr L p2 E h scr from which we obtain: scr Z p2
E 1 h2 s Z : 2 s q L W
(6)
W controls the allowable stress and therefore the structural efficiency of the column. It is observed in Eq. (6) that, for
For any compression member which is designed to transmit a given load F, the maximum obtainable stress depends most certainly on material properties, but also on geometric shape. W, being independent of the material used, allows comparison between different morphologies [2]. The indicator of volume contains, for a given cross-section, all the necessary information needed for the minimum volume design of an entire family of stayed columns. The indicator of volume is dimensionless and a function of a very limited number of parameters, which makes it suitable as a preliminary design tool. For large values of L/h, the stress at failure of simple columns is well below the elastic limit of the materials used and thus more efficient
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The efficiency of the stayed columns can be determined in two ways. The first one considers a constant volume of material and evaluates the maximum buckling load; the second one considers a given load and evaluates the minimal volume of material necessary to sustain it, which is the designer issue, as the load is usually known. This research indicates that for large values of L/h considerable weight reduction can be obtained.
H
L
h
Fig. 1. 2D drawing of stayed column with split-up cross arm.
structural concepts should be developed. Morphologies with one or several cross arm, hinged or fixed along the central column, have been proposed as efficient systems [3,8]. In this paper a column with one split-up cross arm (Fig. 1) hinged or fixed to the central column is investigated. This morphology introduces a rotational constraint at mid height due to the opening of the cross arm. It will be compared with a single cross arm stayed column in which the cross arm is fixed to the central mast in order to have some rotational constraint, proposed by Smith et al. [3] All stayed columns allow a given central mast, subjected to a compressive load, to operate at a higher stress level. Hence, a considerable decrease in weight of the central mast is possible but additional weight is added by the cables/stays and cross arms. Improvement is achieved only when the total volume is (considerably) less than that of a simple column capable of transmitting the same load F. The columns are studied using a finite element based eigenvalue analysis [4]. In order to propose a useful design procedure, the parameters that influence decisively the minimum weight of the stayed column have to be examined. The study is limited to hollow square or circular central masts. The required parameters are the tube diameter h, wall thickness t, the section of the stays Us, the section of the cross arm Uca, the pretension T, the opening angle b of the split-up cross arm and the geometrical slenderness of the stayed column L/H. Furthermore, the imperfections (eccentricities, residual stresses, elasto-plastic deformation) of the cross section are taken into consideration as prescribed by Eurocode 3 [1].
3. Assumptions In the buckling analysis the axial deformation of the cross arm is not considered. However, when the magnitude of the pretension is calculated the deformation of the cross arm is taken into account. The connections between the cross arm and the central mast are considered as either ideal hinges or rigid. The connections between the stays and the cross arm as well as the mast are assumed to be ideal hinges. When the buckling load is calculated, the stayed column is assumed completely symmetrical and ideally centrally loaded. This implies that there is no initial eccentricity and crookedness. There will be no lateral deflection of the column prior to buckling. The imperfections of the column are accounted for when the design procedure is proposed. It is assumed that all the stays remain active until the maximum buckling load is reached. To achieve this goal the optimum pretension Topt is calculated and introduced in the stays [6]. The optimum pretension is defined as the pretension that becomes zero when the maximum buckling Pcr load is reached. Finally ‘perfect’ pin-ended supports are supposed.
4. Obtaining the buckling load The buckling load is obtained using a finite element method (FEM) based eigenvalue analysis discussed in detail by Temple [4]. The advantage of this approach is that it allows the study of all possible configurations, whereas analytical results are confined to a very small number of simple morphologies. The FEM approach can only provide numerical values to be rendered by curves for practical use. Fig. 2 illustrates the buckling modes for the different columns under consideration. The first buckling mode of the 4 columns are very similar (Fig. 2(a), (c), (e), and (g)). Buckling occurs due to exhaustion of the translational stiffness at mid height. The second buckling mode (Fig. 2(b), (d), (f), and (h)) of the 4 columns are distinct. In Fig. 2(b) and (f) it is observed that all the deformation energy is stored in the central mast. It is easily understood that the maximum buckling load, for the column with a split-up cross arm hinged to the central mast (Fig. 2(f)), will occur when, the ratio of the distance between the split-up cross arms and L is xZ1/3. For the two columns in which
J. Van Steirteghem et al. / Advances in Engineering Software 36 (2005) 614–625
hinged cross arm mode 1 symmetrical (a)
fixed cross arm
mode 2 asymmetrical (b)
mode 1 symmetrical (c)
mode 2 asymmetrical (d)
617
hinged cross arm
fixed cross arm
mode 1 symmetrical (e)
mode 1 symmetrical (g)
mode 2 symmetrical (f)
mode 2 symmetrical (h)
Fig. 2. Buckling modes for the different columns that are studied.
the cross arms are fixed to the central column (Fig. 2(d) and (h)) the stays supply a contribution to the deformation energy, hence the buckling load can further increase compared to Fig. 2(b) and (f), respectively.
A 4.9 m central mast is considered, the central mast is assumed to be a circular tube with an outside diameter h of 57.2 mm, an inside diameter hi of 44.5 mm and a weight of 7.9 kg/m. When the cross arm is fixed to the central mast, the same circular tube is initially selected for the cross arm; when hinged a bar with an outside diameter of 20 mm is initially selected, the stays are initially assumed to be steel wire ropes with a diameter hsZ11 m. They are then modified to cover the range 3%Uc/Uca%30 (Fig. 3) and 1%Uc/Us%10 (Fig. 4), respectively. The cross arm length is initially 0.82 m so as to obtain a slenderness L/H of 6. It is then modified to cover the range 1%L/H%10 (Fig. 5). When columns with split-up cross arm are studied an initial value of xZ0.3 is selected and then modified to cover the range 0.1%x%0.5 (Fig. 6). A value of 204 GPa is taken for the modulus of elasticity of the cross arm Eca and central
5. Numerical example In order to gain insight into the influence of different geometrical parameters (L/H,Uc/Us,Uc/Uca,x) on the buckling load of single split-up cross arm stayed column, the FEM approach is applied to a numerical example. In order to make comparison possible the column studied by Smith et al. [3] has been chosen. Furthermore, this example is also applied on the stayed column with a single cross arm hinged to the central mast, among others studied by Chu [5]. 14
Ec / Es = Eca / Es = 3.14 q = 9.96 12
L/H =4 Ωc / Ωs = 10.7
Pcr / PE
10
8
6
4
2 1
2
3
4
5
6
7
8
9
Ωc/Ωca Chu et al.
Smith et al.
split - up hinged
split - up fixed
Fig. 3. Effect of the cross arm properties on the buckling behavior for qZ9.96.
10
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14 Ec / Es = Eca / Es = 3.14 q = 9.96 12
L/H =6
Pcr / PE
10
8
6
4
2 1
2
3
4
5
6
8
7
9
10
Ωc/Ωca Chu et al.
Smith et al.
split - up hinged
split - up fixed
Fig. 4. Effect of stay size on buckling behavior for qZ9.96.
mast Ec, the stays are assumed to have a modulus of elasticity EsZ65 GPa. Figs. (3–6) illustrate the influence of the nondimensional geometrical parameters (L/H, Uc/Us, Uc/Uca, x) as a function of the nondimensional ratio Pcr to PE, PE being the Euler load of the central mast.
Fig. 3 illustrates clearly that for columns with hinged cross arm the cross arm(s) do not contribute to the deformation energy of the buckling mode associated with the maximum buckling load. The column studied by Chu [5] has a maximum buckling load of 4 times the Euler load, for a hinged split-up cross arm
14 Ec / Es = Eca / Es = 3.14 q = 9.96
12
Ωc / Ωs = 10.7
Pcr / PE
10
8
6
4
2 1
2
3
Chu et al.
4
Smith et al.
5 6 slenderness L/H
7
split - up hinged
Fig. 5. Effect of the slenderness on buckling behavior for qZ9.96.
8
9
split - up fixed
10
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14
12
Pcr / PE
10
8
6 Ec / Es = Eca / Es = 3.14 q = 9.96 4
L/H =4 Ωc / Ωs = 10.7
2 0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
X split - up fixed
split - up hinged
Fig. 6. Effect of opening angle for split-up cross arm.
the maximum buckling load is 9 times the Euler load. Furthermore, Fig. 3 illustrates that the column proposed by Smith et al. [3] and the column with fixed split-up cross arm are very sensible to a reduction of the cross sectional area of the cross arm. For large values of Uc/ Uca it is observed that results of the columns with rigid cross arm connection converges to results of the columns with hinged cross arm. The increase of the buckling load of the stayed column with hinged or fixed cross arm is respectively 28% and 71% higher than the buckling load associated with the column studied by Smith et al. [3]. In Fig. 4 one can see that for the column studied by Chu [5] the maximum buckling load is 4 times the Euler load. For the other columns it is clear that when the instability is governed by the translational stiffness (Mode I) the 3 columns exhibit a similar behaviour. For the fixed single cross arm stayed column, Mode II or double curvature buckling (Fig. 2(d)), becomes, at a certain ratio of Uc/Us, the controlling instability mode as opposed to the columns with split-up cross arm where single curvature buckling mode continues to occur and the buckling load can further increase. An increase of the buckling load up to 28% and 85% can be obtained for a hinged and fixed split-up cross arm, respectively compared to the column studied by Smith et al. [3]. Fig. 5 shows the influence of the slenderness L/H on the buckling load again it illustrates that the maximum buckling load of the columns studied by Chu [5] is 4 times the Euler load. Smith et al. [3] pointed out that as the cross arm member length increases (slenderness
decreases), the rotational constraint of the column at the cross arm is decreased and the translational restraint is increases, hence at a certain value of L/H mode II becomes the controlling mode and for further decrease of L/H the buckling load decreases. In the case of (a) split-up cross-arm(s). In the case of split-up a cross arm hinged to the central mast, the stays do not contribute to the rotational stiffness. Due to the geometrical disposition of the cross arm the stays reduce the buckling length of the central mast. The maximum buckling load will be obtained if x is 1/3 and the buckling mode will be one and a half sine. When the split-up cross arm is fixed to the central mast the buckling load can further increase due to the contribution of the stays. By using a split-up cross arm the buckling load may be increased significantly compared to the stayed column with one cross arm fixed to the central mast. If a hinged split-up cross arm is used the buckling load can be 28% higher, when the cross arm is fixed 85%. Fig. 6 illustrates that the maximum buckling load for a column with a split-up hinged cross arm will be maximized if xZ1/3, this corresponds with a buckling length of L/3 and a maximum buckling load of 9 times the Euler load. If the split-up cross arm ispffiffifixed ffi pthe ffiffiffi buckling load will be maximized for xZ 2=2C 2. Fig. 7 illustrates the theoretical buckling mode for a fixed split-up cross arm. The maximum theoretical buckling load, for infinitely stiff cross arm and stays, is 23.2 PE. However, it is reasonable to accept that the section of the cross arm will never exceed the section of the central mast, hence a buckling load of about 15 PE, can be achieved.
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7. Efficiency of stayed columns ≈ L / 3.41
Lb ≈ L / 4.82
≈ L / 2.41
Lb ≈ L / 4.82
≈ L / 3.41
Lb ≈ L / 4.82
Fig. 7. Theoretical buckling mode of a column with split-up cross arm.
6. Pretension In order to remain effective the stays cannot become slack before the buckling load has been attained. The pretension that becomes zero at the onset of buckling has been defined by Hafez as the optimal pretension, Topt [6]: ‘the initial pretension in the stays that disappears completely just after the load in the column reaches its maximum buckling load’. As shown by Hafez [6] it is possible to calculate the optimal pretension of the stays as a function of the geometrical and material parameters of the stayed column. Topt Z C1 Pcr :
(7)
C1 is a function of the geometrical and material properties of the stayed column. " 1 C ðH=LÞ2 Ac Ec x2 ð1 C ðH=xLÞ2 Þ3=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C1 Z C 2 As Es 1 K x 2 1 Kx 1 C ðH=LÞ2 #K1 A E 4 : ð8Þ ! c c C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aca Eca 1 C ðH=LÞ2 If the numerical example discussed earlier is taken and a variable slenderness is considered (Fig. 5), it is possible to plot the diagram Topt/TminZf(L/H) (Fig. 8). The pretension is only necessary to guarantee that the stays do not become slack prior to buckling and as a consequence the ratio Topt/Tmin is proportional to the ratio Pcr/PE. Tmin being the ‘minimal initial pretension in the stays that remain effective until the Euler load has been reached. At a load larger than the Euler load, the stays become slack.’ [6]
Figs. (3–7) show the influence of Uc/Uca, Uc/Us, L/H and x on the buckling load of the stayed columns. However, they do not give insight into the efficiency of the column. Smith has defined the ‘relative efficiency’ as the ratio Pcr to the total weight. For reasons of dimensional similarity, the nondimensional indicator of volume, W, is preferred to compare structural morphologies. Four columns are selected as a basis of comparison: the single cross arm hinged to the central mast (Fig. 9(a)), fixed to the central mast (Fig. 9(b)) the stayed column with splitup cross arm hinged to the central mast (Fig. 9(c)) and finally the stayed column with split-up cross arm fixed to the central mast (Fig. 9(d)). The simple column is taken as reference column. Two basic assumptions are possible when investigating the efficiency of stayed columns. Either a constant volume of material is assumed and the maximum buckling load is evaluated or a constant load is supposed and the minimal volume of material is determined. Since the latter is the designer issue, it is selected. The columns investigated in the numerical example were selected in order to compare different morphologies. The search algorithm finds the best foursome (Uc/Uca, Uc/Us, L/H, x) that minimizes W for different values of L/h. The length of the mast takes values between 1.14 m and 11.4 m, which covers the range 20%L/h%200; the stay diameter hs, values between 5.2 mm and 16.3 mm, which covers the range 5%Uc/Us%50, the cross arm length may vary so that L/H covers the range 5%L/H%50, the section of the cross arm takes values between 25.7 cm2 and 0.79 cm2, which covers the range 0.4%Uc/Uca%13 and finally for columns with split-up cross arm x can vary between 0.1 and 0.5. However, when the rotational stiffness is guaranteed by the cross arm, the external diameter of the cross arm cannot be superior to the external diameter of the central mast. The simple (Euler) column bearing a load FZPcr is used for reference. The expression of the indicator of volume W can easily be obtained starting from the definition of W: WZ
sV : FL
(9)
Replacing V by Uc times the length L and F by the buckling load Pcr, Eq. (10) is obtained: WZ
sUc L : p Ec Ic L=L2 2
(10)
Eq. (10) can be rearranged and written as a function of the reduced slenderness of the simple column LE as defined by Eurocode 3 [1]. s Uc h2c L 2 WZ 2 Z L2E : (11) h p Ec Ic
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14 Ec / Es = Eca / Es = 3.14 q = 9.96
12
Ac / As = 10.7
Topt / Pmin
10
8
6
4
2 1
2
3
4
5
6
8
7
9
10
slenderness L/H Chu et al.
Smith et al.
split - up hinged
split - up fixed
Fig. 8. Effect of the slenderness on the optimal pretension.
The buckling load of a stayed column is equal to g times the buckling load of the central mast, hence there exists a relation between the reduced slenderness of the stayed column and the simple column that can sustain the same load: pffiffiffi L ¼ 4 gLE : (12)
load of the stayed column with split-up cross arm fixed to the central mast is significantly higher, the efficiency is the same as the hinged split-up cross arm due to the volume of material needed to guarantee the bending stiffness of the cross arm.
The indicator of volume of the stayed column can be expressed in terms of the reduced slenderness of the simple column that sustains the same load: a W Z pffiffiffi L2E : g
(13)
a being the ratio of the total volume of the stayed column to the volume of the central mast. Fig. 10 illustrates that the efficiency can be increased significantly when stayed columns are used. For values of the reduced slenderness LE % 1, stays and cross arm do not increase the efficiency as yielding of the central mast governs the failure mode of the column. For higher values of the slenderness the efficiency of the different morphologies increases rapidly compared with the Euler column. For example, for a reduced slenderness of 3, the stayed column with one cross arm hinged to the central mast needs approximately 49% less volume of material, when the cross arm is fixed 52%, for split-up cross arm, hinged or fixed to the central mast, this gain of material is about 62%. This means that the column with split-up cross arm can be up to 23% more efficient than the column proposed by Smith et al. [3] when assuming a constant load. Furthermore, it is interesting to mention that although the maximum buckling
(a)
(b)
(c)
Fig. 9. Different columns under investigation.
(d)
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10 WE = 9
9
W = Indicator of volume
8 7 6 5
WE = 4.7 = 54%WE
4
WE = 4.4 = 49%WE
3
WE = 3.4 = 38%WE
2 1 0 1.5
1
2
3
2.5
3.5
4
Le Chu et al.
Smith et al.
split - up hinged
split - up fixed
Euler
Fig. 10. The efficiency of stayed columns when considering a constant load.
It should be noted that when a constant load is considered, the reduced slenderness of the stayed column and the simple column are considerably different. Fig. 11 confirms what has been discussed earlier. The difference in slenderness of the central mast of the stayed column and the simple column is considerable. For
example, a stayed column with fixed split-up cross arm and reduced slenderness of LZ 4:5 can sustain the same load as a simple column with a slenderness of LE Z 2:5. Fig. 11 further reveals that the slenderness of the central mast of a fixed split-up cross arm is larger than when a hinged split-up cross arm is used.
7
6
5
Λ
4
3
2
1
0 0
0.5
1
1.5
2
2.5
3
3.5
4
ΛE Chu et al.
Smith et al.
split - up hinged
split - up fixed
Euler
Fig. 11. Comparison of the slenderness of the stayed column and the simple column for a constant load.
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It will be shown how the imperfections can be accounted for in an early design stage.
8. Imperfections Until now, perfect columns have been discussed. Clearly, for a real (stayed) columns imperfections are unavoidable. Initial crookedness is always present and is amplified in the presence of axial loading. Therefore, if the buckling load of a column is to be predicted, it is necessary to account for these imperfections. Eurocode 3 [1], design of steel structures, proposes 4 buckling curves (a, b, c, d). The a-curve corresponds to sections that are almost insensitive to buckling; the d-curve corresponds to the sections with the most pronounced tendency to buckle. In Fig. 12 next to the 4 buckling curves of the Eurocode 3, the curve of the perfect column and an approximation for the a and d curves are plotted [2]. For ‘perfect’ columns the relation between L and scr/s is given by: scr 1 Z 2: s L
(14)
The a-curve (for LR 1) can be approximated by: scr 1 Z ; s 0:5 C L2
(15)
and the d-curve by: scr 1 Z s 1 C L2
(16)
9. Design procedure It is now possible to propose a design procedure. From what has been discussed previously, it is possible to determine the relationship between the slenderness of the central mast L/h, the overall slenderness L/H, the section of the stays Us, the section of the cross arm Uca and x for a load Fand length L. The applied load F and the length L of the column are two known design parameters. The designer will impose limits on the overall slenderness L/H, the slenderness of the central mast L/h and in the case of a split-up cross arm also x. Obviously, F has to be inferior to the buckling load of the stayed column, that is: F% Pcr Z g
p2 EIc : L2
(17)
Tables can be drawn up that determine the column that satisfies Eq. (17) and minimizes the value of W [9]. For the intervals imposed by the designer these tables provide the values of Us, Uca and x. These tables take constraints into consideration: for example it can be demonstrated that the lightest columns always coincides with the smallest value L/h and q; still, this cannot be done endlessly because of local buckling. The critical
1 0.9
1 Λ2
a b
0.8
c 0.7
1 0.5+Λ2
d
σcr/σ
0.6 0.5
1 1+Λ2
0.4 0.3 0.2 0.1 0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
Λ Fig. 12. Four buckling curves (a,b,c,d) of Eurocode 3 and the proposed approximation.
3.5
4.0
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‘local’ buckling load has been defined by Shanley [7]: PL;cr Z k2
Et Et Z k2 : D h
(18)
Here k2 is a constant with a theoretical value of 1.212, but tests have indicated that much smaller values are to be considered at failure. Typically k2 is taken equal to 0.4 to take imperfections into account. Eq. (18) and the limits imposed on L/h will yield the section needed to sustain a load F. Other considered boundary conditions are yielding of the stays and buckling and yielding of the cross arm. At this point a first design of the stayed column is available. Yet, the inevitable imperfections were not accounted for. To include these in the design, Eq. (12) is used to calculate a first approximation of the reduced slenderness L1 . Furthermore, the a-curve is considered, hence an improved approximation of the reduced slenderness can be obtained: rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s L qc L2 Z Z g K 0:5: (19) p E h 2 The stays and the cross arm do not need to be changed because taking the values from the first approximation always results in a safe design. Furthermore, their contribution to the total weight is small. Finally, it is possible to calculate the magnitude of the pretension by using Eqs. (7 and 8). The successive steps discussed in this paragraph have been used to develop a computer program that designs these columns for given boundary conditions. The user provides the load F, the length L of the column, and the limits of L/H and L/h. The design of the column, including the section of the central mast, cross arm and stays as well as the value of L/H, L/h and the pretension in the stays are returned by the program taking local buckling and yielding into consideration. To calculate the buckling load of a given configuration a basic eigenvalue–eigenmode calculation is performed [4]. To optimize the column a constrained nonlinear optimization algorithm is used. The results found by the program agree very closely with those found by an exhaustive numerical analysis of a set of columns.
10. Conclusions The buckling load of a stayed column may be increased many times by reinforcing it with a system of pretensioned and split-up hinged or fixed cross arm members. The magnitude of the externally applied load on a stayed column depends not only on the critical buckling load but also on the initial pretension in the stays. The magnitude of the pretension should be equal to the optimal pretension [6].
The rotational stiffness of the split-up cross arm stayed columns is very large, therefore, the mode of instability is almost always a symmetrical (single curvature) deflected shape. This coincides with the exhaustion of the translational stiffness of the column at mid height. For a hinged split-up cross arm the buckling load may be up to 9 times the Euler load of the central mast. The maximum ‘theoretical’ buckling load of the fixed split-up cross arm is 23.2 times the Euler load. The buckling load is influenced by changes in the cross arm length, the stay diameter, cross arm properties and the opening angle of the split-up cross arm. This change is substantial for small values of both cross arm length and stay diameter and decreases for larger values. The influence of the cross arm properties is negligible if the cross arm is hinged to the central mast, if the cross arm is fixed the influence is substantial due to the fact that the cross arm contributes to the rotational stiffness. The numerical buckling solutions obtained can be used to determine, for stayed columns, the ratios L/H, Uc/Us, Uc/Uca and x that maximizes the efficiency of the column, based on the indicator of volume W. The morphological indicators in general and the indicator of volume in particular are powerful tools to compare and design structures. Two basic assumptions are possible. A constant volume can be considered and the load evaluated or a constant load can be considered and the volume evaluated. For design reasons the second approach seems more suitable. Compared to other stayed columns with one cross arm, their efficiency is increased up to 23% when a split-up cross arm is used. Finally, an easy-to-use procedure is proposed to design stayed columns. This paper has dealt with the stability of a stayed column with hinged or fixed split-up cross arm. The conclusions reached herein are considered to be useful in designing other morphologies of stayed columns.
Acknowledgements This research is supported by the Institute of Innovation through Science and Technology in Flanders. (www.iwt.be)
References [1] Eurocode Steel/3. Belgian Standard NBN ENV 1993-1-1: 1992/AX, 1992. [2] Samyn P. Etude de la morphologie des structures, a` l’aide des indicateurs de volume et de de´placement. Brussels: Acade´mie royale de belgique; 2004. ISBN 2–8031–0201–3. [3] Smith RJ, McCaffrey GT, Ellis JS. Buckling of a single cross arm stayed column. J Struct Div ASCE 1975;101(ST. 1):249–68. [4] Temple MC. Buckling of stayed columns. J Struct Div ASCE 1977; 103(ST. 4):839–85.
J. Van Steirteghem et al. / Advances in Engineering Software 36 (2005) 614–625 [5] Chu KH, Berge SS. Analysis and design of struts, with tension ties. J Struct Div ASCE 1963;89(ST. 1):127–63. [6] Hafez HH, Temple MC, Ellis JS. Pretensioning of single cross arm stayed columns. J Struct Div ASCE 1979;103(ST. 2):359–75. [7] Shanley FR. Weight–strength analysis of aircraft structures. 2nd ed. New York: Dover Publications, Inc.; 1960.
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[8] Van Steirteghem J, Buyl C, De Wilde WP, Samyn Ph. Morphological optimization of 2D-stayed columns. Proceedings of OPTI 2003, 261– 270, May 2003, Detroit, USA. [9] Wattel F, Van Steirteghem J, De Wilde P, Samyn Ph. Buckling of stayed columns, Master Thesis, Vrije Universiteit Brussel, Brussels, Belgium, Sept. 2003.
Optimum design of stayed columns with split-up cross arm Jan Van Steirteghema,*, Willy P. De Wildea, Philippe Samynb,c, Ben P. Verbeecka, Franc¸ois Wattela a
Mechanics of Materials and Constructions, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium b Department of Architecture, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium c Samyn and Partners, Steenweg op Waterloo 1537, B-1150 Brussels, Belgium Available online 11 April 2005
Abstract The columns considered in this paper consist of a central mast; a split-up cross arm hinged or fixed to the central mast and optimally stressed stays. The buckling load of these columns is calculated and the influence of the geometrical parameters is examined. The indicator of volume, W, is proposed as a measure of the efficiency of different structures that are designed to transmit the same load P. This paper demonstrates that the stayed columns with a split-up cross arm have a number of advantages with respect to the stayed columns examined previously. It further reveals that both the sections of the stays and the cross arm length influence the buckling load and the efficiency decisively. Compared to other columns with one cross arm, an efficiency increase of more than 20% can be achieved. The efficiency is defined as the minimal volume of material needed to sustain a given load P. The theory of morphological indicators is used to propose a design procedure for stayed columns. This procedure takes constraints (such as local buckling, yielding, choice of the morphology.) into consideration and proposes the most efficient column. Furthermore, it takes imperfection into account as prescribed by the construction code (Eurocode 3) [Eurocode Steel/3. Belgian Standard NBN ENV 1993-1-1: 1992/AX (1992)]. q 2005 Elsevier Ltd. All rights reserved. Keywords: Stayed column; Split-up cross arm; Indicator of volume; Efficiency; Imperfections; Design procedure; Comparison
1. Indicator of volume The indicator of volume W is a design tool developed by Samyn [2] allowing the optimization of structures for a chosen criteria, in case volume of the material, at an early stage and using only a very limited number of parameters. When designing stayed columns, W can be used to estimate the efficiency of the chosen morphology compared to others designed to transmit the same load. Given a structure characterized by its morphology and proportions presenting a volume of material V when loaded by a force system of which the resultant is F, main dimension L and composed of a material with allowable stress s, then W can be defined as the volume of material the morphological identical and homothetical structure loaded with a unit load of 1 N, of which the main dimension is 1 m and composed of * Corresponding author. Tel.: C32 2 6292936; fax: C32 2 6292928. E-mail address: [email protected] (J. Van Steirteghem).
0965-9978/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2005.03.007
a material with a unit allowable stress of 1 Pa and written as: WZ
sV : FL
(1)
It is possible to show that for statically determinate, fully stressed structures, when neglecting second order and instability effects as well as the material of the connections (trusses, arches, beams, frames, masts, membranes), with main dimension L and loaded by a uniformly distributed load or a mobile point load, W is only a function of the geometrical slenderness L/H (L and H are the dimensions of the rectangle in which the structure is inscribed, LOH). It is possible to use W to study stayed columns, in the same way Shanley [7] used the structural index F/L2. The design parameters in the case of a column are the load F and the length L, through which the load has to be transmitted. For pin-ended columns the stress at the onset of buckling can easily be obtained: scr Z
p2 E : ðL=rÞ2
(2)
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615
Symbols C1 D Ec Eca Es F h hi hs Ic Ica k2 KE KG KG L Lb Pcr PE PL,cr q t
constant external diameter of the central mast modulus of elasticity of mast material modulus of elasticity of cross arm material modulus of elasticity of stay material resultant of the total forces on a structure external diameter of the central mast internal diameter of the central mast diameter of the stays moment of inertia of the central mast moment of inertia of the cross arm constant assembled elastic stiffness matrix assembled geometric stiffness matrix assembled geometric stiffness matrix for unit load length of the column buckling length of the column maximum buckling load of the stayed column euler load of the central mast critical local buckling load dimensionless number Wh2/I wall thickness of the cross section of the central mast
The term r2ZIc/Uc is a dimensional number as it is the square of a length and hence cannot be used as a shape parameter. It is, therefore, suitable to replace r2 by a nondimensional number depending only on the geometrical properties of the cross section. Since r2 has the dimension of an area it can be divided by the square of the diameter of the central mast h to obtain the shape parameter q, which is a measure for the buckling sensitivity of a section: qZ
Uc h2 h2 Z 2: Ic r
(3)
T Topt V W a b g l L L1 L2 LE Uc Uca Us r s scr x
pretension of the stays optimal pretension of the stays total volume of the structure indicator of volume ratio of total volume to volume of central mast opening angle of the cross arm multiplication factor eigenvalues reduced slenderness of stayed column that can sustain a load P first approximation of the reduced slenderness second approximation of the reduced slenderness reduced slenderness of simple column that can sustain a load P cross-sectional area of the mast cross-sectional area of the cross arm member cross-sectional area of the stay gyros radius I/A allowable stress of the material used stress in the central mast at buckling ratio of the distance between the split-up cross arm and L, a constant
constant material parameters and length, the allowable critical stress increases as q decreases. This means that the ratio h/t, for example for a tube of wall thickness t, should be as large as possible, however limited by local buckling (‘crushing’, ‘crippling’, etc.) of the tin-walled section. Other parameters come into play when stayed columns are studied, the most important being the overall geometric slenderness L/H, the cross section of the stays Us, the cross section of the cross arm Uca and, for stayed columns with split-up cross arm, the opening angle of the cross arm b.
Eq. (2) can now be rewritten as: p2 Eh2 scr Z 2 L q
2. Problem statement (4)
The indicator of volume of a simple column, which fails due to buckling, can be established using Eq. (1): 2 sV 1 s L s q Z WZ Z ; (5) Pcr L p2 E h scr from which we obtain: scr Z p2
E 1 h2 s Z : 2 s q L W
(6)
W controls the allowable stress and therefore the structural efficiency of the column. It is observed in Eq. (6) that, for
For any compression member which is designed to transmit a given load F, the maximum obtainable stress depends most certainly on material properties, but also on geometric shape. W, being independent of the material used, allows comparison between different morphologies [2]. The indicator of volume contains, for a given cross-section, all the necessary information needed for the minimum volume design of an entire family of stayed columns. The indicator of volume is dimensionless and a function of a very limited number of parameters, which makes it suitable as a preliminary design tool. For large values of L/h, the stress at failure of simple columns is well below the elastic limit of the materials used and thus more efficient
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The efficiency of the stayed columns can be determined in two ways. The first one considers a constant volume of material and evaluates the maximum buckling load; the second one considers a given load and evaluates the minimal volume of material necessary to sustain it, which is the designer issue, as the load is usually known. This research indicates that for large values of L/h considerable weight reduction can be obtained.
H
L
h
Fig. 1. 2D drawing of stayed column with split-up cross arm.
structural concepts should be developed. Morphologies with one or several cross arm, hinged or fixed along the central column, have been proposed as efficient systems [3,8]. In this paper a column with one split-up cross arm (Fig. 1) hinged or fixed to the central column is investigated. This morphology introduces a rotational constraint at mid height due to the opening of the cross arm. It will be compared with a single cross arm stayed column in which the cross arm is fixed to the central mast in order to have some rotational constraint, proposed by Smith et al. [3] All stayed columns allow a given central mast, subjected to a compressive load, to operate at a higher stress level. Hence, a considerable decrease in weight of the central mast is possible but additional weight is added by the cables/stays and cross arms. Improvement is achieved only when the total volume is (considerably) less than that of a simple column capable of transmitting the same load F. The columns are studied using a finite element based eigenvalue analysis [4]. In order to propose a useful design procedure, the parameters that influence decisively the minimum weight of the stayed column have to be examined. The study is limited to hollow square or circular central masts. The required parameters are the tube diameter h, wall thickness t, the section of the stays Us, the section of the cross arm Uca, the pretension T, the opening angle b of the split-up cross arm and the geometrical slenderness of the stayed column L/H. Furthermore, the imperfections (eccentricities, residual stresses, elasto-plastic deformation) of the cross section are taken into consideration as prescribed by Eurocode 3 [1].
3. Assumptions In the buckling analysis the axial deformation of the cross arm is not considered. However, when the magnitude of the pretension is calculated the deformation of the cross arm is taken into account. The connections between the cross arm and the central mast are considered as either ideal hinges or rigid. The connections between the stays and the cross arm as well as the mast are assumed to be ideal hinges. When the buckling load is calculated, the stayed column is assumed completely symmetrical and ideally centrally loaded. This implies that there is no initial eccentricity and crookedness. There will be no lateral deflection of the column prior to buckling. The imperfections of the column are accounted for when the design procedure is proposed. It is assumed that all the stays remain active until the maximum buckling load is reached. To achieve this goal the optimum pretension Topt is calculated and introduced in the stays [6]. The optimum pretension is defined as the pretension that becomes zero when the maximum buckling Pcr load is reached. Finally ‘perfect’ pin-ended supports are supposed.
4. Obtaining the buckling load The buckling load is obtained using a finite element method (FEM) based eigenvalue analysis discussed in detail by Temple [4]. The advantage of this approach is that it allows the study of all possible configurations, whereas analytical results are confined to a very small number of simple morphologies. The FEM approach can only provide numerical values to be rendered by curves for practical use. Fig. 2 illustrates the buckling modes for the different columns under consideration. The first buckling mode of the 4 columns are very similar (Fig. 2(a), (c), (e), and (g)). Buckling occurs due to exhaustion of the translational stiffness at mid height. The second buckling mode (Fig. 2(b), (d), (f), and (h)) of the 4 columns are distinct. In Fig. 2(b) and (f) it is observed that all the deformation energy is stored in the central mast. It is easily understood that the maximum buckling load, for the column with a split-up cross arm hinged to the central mast (Fig. 2(f)), will occur when, the ratio of the distance between the split-up cross arms and L is xZ1/3. For the two columns in which
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hinged cross arm mode 1 symmetrical (a)
fixed cross arm
mode 2 asymmetrical (b)
mode 1 symmetrical (c)
mode 2 asymmetrical (d)
617
hinged cross arm
fixed cross arm
mode 1 symmetrical (e)
mode 1 symmetrical (g)
mode 2 symmetrical (f)
mode 2 symmetrical (h)
Fig. 2. Buckling modes for the different columns that are studied.
the cross arms are fixed to the central column (Fig. 2(d) and (h)) the stays supply a contribution to the deformation energy, hence the buckling load can further increase compared to Fig. 2(b) and (f), respectively.
A 4.9 m central mast is considered, the central mast is assumed to be a circular tube with an outside diameter h of 57.2 mm, an inside diameter hi of 44.5 mm and a weight of 7.9 kg/m. When the cross arm is fixed to the central mast, the same circular tube is initially selected for the cross arm; when hinged a bar with an outside diameter of 20 mm is initially selected, the stays are initially assumed to be steel wire ropes with a diameter hsZ11 m. They are then modified to cover the range 3%Uc/Uca%30 (Fig. 3) and 1%Uc/Us%10 (Fig. 4), respectively. The cross arm length is initially 0.82 m so as to obtain a slenderness L/H of 6. It is then modified to cover the range 1%L/H%10 (Fig. 5). When columns with split-up cross arm are studied an initial value of xZ0.3 is selected and then modified to cover the range 0.1%x%0.5 (Fig. 6). A value of 204 GPa is taken for the modulus of elasticity of the cross arm Eca and central
5. Numerical example In order to gain insight into the influence of different geometrical parameters (L/H,Uc/Us,Uc/Uca,x) on the buckling load of single split-up cross arm stayed column, the FEM approach is applied to a numerical example. In order to make comparison possible the column studied by Smith et al. [3] has been chosen. Furthermore, this example is also applied on the stayed column with a single cross arm hinged to the central mast, among others studied by Chu [5]. 14
Ec / Es = Eca / Es = 3.14 q = 9.96 12
L/H =4 Ωc / Ωs = 10.7
Pcr / PE
10
8
6
4
2 1
2
3
4
5
6
7
8
9
Ωc/Ωca Chu et al.
Smith et al.
split - up hinged
split - up fixed
Fig. 3. Effect of the cross arm properties on the buckling behavior for qZ9.96.
10
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14 Ec / Es = Eca / Es = 3.14 q = 9.96 12
L/H =6
Pcr / PE
10
8
6
4
2 1
2
3
4
5
6
8
7
9
10
Ωc/Ωca Chu et al.
Smith et al.
split - up hinged
split - up fixed
Fig. 4. Effect of stay size on buckling behavior for qZ9.96.
mast Ec, the stays are assumed to have a modulus of elasticity EsZ65 GPa. Figs. (3–6) illustrate the influence of the nondimensional geometrical parameters (L/H, Uc/Us, Uc/Uca, x) as a function of the nondimensional ratio Pcr to PE, PE being the Euler load of the central mast.
Fig. 3 illustrates clearly that for columns with hinged cross arm the cross arm(s) do not contribute to the deformation energy of the buckling mode associated with the maximum buckling load. The column studied by Chu [5] has a maximum buckling load of 4 times the Euler load, for a hinged split-up cross arm
14 Ec / Es = Eca / Es = 3.14 q = 9.96
12
Ωc / Ωs = 10.7
Pcr / PE
10
8
6
4
2 1
2
3
Chu et al.
4
Smith et al.
5 6 slenderness L/H
7
split - up hinged
Fig. 5. Effect of the slenderness on buckling behavior for qZ9.96.
8
9
split - up fixed
10
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619
14
12
Pcr / PE
10
8
6 Ec / Es = Eca / Es = 3.14 q = 9.96 4
L/H =4 Ωc / Ωs = 10.7
2 0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
X split - up fixed
split - up hinged
Fig. 6. Effect of opening angle for split-up cross arm.
the maximum buckling load is 9 times the Euler load. Furthermore, Fig. 3 illustrates that the column proposed by Smith et al. [3] and the column with fixed split-up cross arm are very sensible to a reduction of the cross sectional area of the cross arm. For large values of Uc/ Uca it is observed that results of the columns with rigid cross arm connection converges to results of the columns with hinged cross arm. The increase of the buckling load of the stayed column with hinged or fixed cross arm is respectively 28% and 71% higher than the buckling load associated with the column studied by Smith et al. [3]. In Fig. 4 one can see that for the column studied by Chu [5] the maximum buckling load is 4 times the Euler load. For the other columns it is clear that when the instability is governed by the translational stiffness (Mode I) the 3 columns exhibit a similar behaviour. For the fixed single cross arm stayed column, Mode II or double curvature buckling (Fig. 2(d)), becomes, at a certain ratio of Uc/Us, the controlling instability mode as opposed to the columns with split-up cross arm where single curvature buckling mode continues to occur and the buckling load can further increase. An increase of the buckling load up to 28% and 85% can be obtained for a hinged and fixed split-up cross arm, respectively compared to the column studied by Smith et al. [3]. Fig. 5 shows the influence of the slenderness L/H on the buckling load again it illustrates that the maximum buckling load of the columns studied by Chu [5] is 4 times the Euler load. Smith et al. [3] pointed out that as the cross arm member length increases (slenderness
decreases), the rotational constraint of the column at the cross arm is decreased and the translational restraint is increases, hence at a certain value of L/H mode II becomes the controlling mode and for further decrease of L/H the buckling load decreases. In the case of (a) split-up cross-arm(s). In the case of split-up a cross arm hinged to the central mast, the stays do not contribute to the rotational stiffness. Due to the geometrical disposition of the cross arm the stays reduce the buckling length of the central mast. The maximum buckling load will be obtained if x is 1/3 and the buckling mode will be one and a half sine. When the split-up cross arm is fixed to the central mast the buckling load can further increase due to the contribution of the stays. By using a split-up cross arm the buckling load may be increased significantly compared to the stayed column with one cross arm fixed to the central mast. If a hinged split-up cross arm is used the buckling load can be 28% higher, when the cross arm is fixed 85%. Fig. 6 illustrates that the maximum buckling load for a column with a split-up hinged cross arm will be maximized if xZ1/3, this corresponds with a buckling length of L/3 and a maximum buckling load of 9 times the Euler load. If the split-up cross arm ispffiffifixed ffi pthe ffiffiffi buckling load will be maximized for xZ 2=2C 2. Fig. 7 illustrates the theoretical buckling mode for a fixed split-up cross arm. The maximum theoretical buckling load, for infinitely stiff cross arm and stays, is 23.2 PE. However, it is reasonable to accept that the section of the cross arm will never exceed the section of the central mast, hence a buckling load of about 15 PE, can be achieved.
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7. Efficiency of stayed columns ≈ L / 3.41
Lb ≈ L / 4.82
≈ L / 2.41
Lb ≈ L / 4.82
≈ L / 3.41
Lb ≈ L / 4.82
Fig. 7. Theoretical buckling mode of a column with split-up cross arm.
6. Pretension In order to remain effective the stays cannot become slack before the buckling load has been attained. The pretension that becomes zero at the onset of buckling has been defined by Hafez as the optimal pretension, Topt [6]: ‘the initial pretension in the stays that disappears completely just after the load in the column reaches its maximum buckling load’. As shown by Hafez [6] it is possible to calculate the optimal pretension of the stays as a function of the geometrical and material parameters of the stayed column. Topt Z C1 Pcr :
(7)
C1 is a function of the geometrical and material properties of the stayed column. " 1 C ðH=LÞ2 Ac Ec x2 ð1 C ðH=xLÞ2 Þ3=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C1 Z C 2 As Es 1 K x 2 1 Kx 1 C ðH=LÞ2 #K1 A E 4 : ð8Þ ! c c C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aca Eca 1 C ðH=LÞ2 If the numerical example discussed earlier is taken and a variable slenderness is considered (Fig. 5), it is possible to plot the diagram Topt/TminZf(L/H) (Fig. 8). The pretension is only necessary to guarantee that the stays do not become slack prior to buckling and as a consequence the ratio Topt/Tmin is proportional to the ratio Pcr/PE. Tmin being the ‘minimal initial pretension in the stays that remain effective until the Euler load has been reached. At a load larger than the Euler load, the stays become slack.’ [6]
Figs. (3–7) show the influence of Uc/Uca, Uc/Us, L/H and x on the buckling load of the stayed columns. However, they do not give insight into the efficiency of the column. Smith has defined the ‘relative efficiency’ as the ratio Pcr to the total weight. For reasons of dimensional similarity, the nondimensional indicator of volume, W, is preferred to compare structural morphologies. Four columns are selected as a basis of comparison: the single cross arm hinged to the central mast (Fig. 9(a)), fixed to the central mast (Fig. 9(b)) the stayed column with splitup cross arm hinged to the central mast (Fig. 9(c)) and finally the stayed column with split-up cross arm fixed to the central mast (Fig. 9(d)). The simple column is taken as reference column. Two basic assumptions are possible when investigating the efficiency of stayed columns. Either a constant volume of material is assumed and the maximum buckling load is evaluated or a constant load is supposed and the minimal volume of material is determined. Since the latter is the designer issue, it is selected. The columns investigated in the numerical example were selected in order to compare different morphologies. The search algorithm finds the best foursome (Uc/Uca, Uc/Us, L/H, x) that minimizes W for different values of L/h. The length of the mast takes values between 1.14 m and 11.4 m, which covers the range 20%L/h%200; the stay diameter hs, values between 5.2 mm and 16.3 mm, which covers the range 5%Uc/Us%50, the cross arm length may vary so that L/H covers the range 5%L/H%50, the section of the cross arm takes values between 25.7 cm2 and 0.79 cm2, which covers the range 0.4%Uc/Uca%13 and finally for columns with split-up cross arm x can vary between 0.1 and 0.5. However, when the rotational stiffness is guaranteed by the cross arm, the external diameter of the cross arm cannot be superior to the external diameter of the central mast. The simple (Euler) column bearing a load FZPcr is used for reference. The expression of the indicator of volume W can easily be obtained starting from the definition of W: WZ
sV : FL
(9)
Replacing V by Uc times the length L and F by the buckling load Pcr, Eq. (10) is obtained: WZ
sUc L : p Ec Ic L=L2 2
(10)
Eq. (10) can be rearranged and written as a function of the reduced slenderness of the simple column LE as defined by Eurocode 3 [1]. s Uc h2c L 2 WZ 2 Z L2E : (11) h p Ec Ic
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621
14 Ec / Es = Eca / Es = 3.14 q = 9.96
12
Ac / As = 10.7
Topt / Pmin
10
8
6
4
2 1
2
3
4
5
6
8
7
9
10
slenderness L/H Chu et al.
Smith et al.
split - up hinged
split - up fixed
Fig. 8. Effect of the slenderness on the optimal pretension.
The buckling load of a stayed column is equal to g times the buckling load of the central mast, hence there exists a relation between the reduced slenderness of the stayed column and the simple column that can sustain the same load: pffiffiffi L ¼ 4 gLE : (12)
load of the stayed column with split-up cross arm fixed to the central mast is significantly higher, the efficiency is the same as the hinged split-up cross arm due to the volume of material needed to guarantee the bending stiffness of the cross arm.
The indicator of volume of the stayed column can be expressed in terms of the reduced slenderness of the simple column that sustains the same load: a W Z pffiffiffi L2E : g
(13)
a being the ratio of the total volume of the stayed column to the volume of the central mast. Fig. 10 illustrates that the efficiency can be increased significantly when stayed columns are used. For values of the reduced slenderness LE % 1, stays and cross arm do not increase the efficiency as yielding of the central mast governs the failure mode of the column. For higher values of the slenderness the efficiency of the different morphologies increases rapidly compared with the Euler column. For example, for a reduced slenderness of 3, the stayed column with one cross arm hinged to the central mast needs approximately 49% less volume of material, when the cross arm is fixed 52%, for split-up cross arm, hinged or fixed to the central mast, this gain of material is about 62%. This means that the column with split-up cross arm can be up to 23% more efficient than the column proposed by Smith et al. [3] when assuming a constant load. Furthermore, it is interesting to mention that although the maximum buckling
(a)
(b)
(c)
Fig. 9. Different columns under investigation.
(d)
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10 WE = 9
9
W = Indicator of volume
8 7 6 5
WE = 4.7 = 54%WE
4
WE = 4.4 = 49%WE
3
WE = 3.4 = 38%WE
2 1 0 1.5
1
2
3
2.5
3.5
4
Le Chu et al.
Smith et al.
split - up hinged
split - up fixed
Euler
Fig. 10. The efficiency of stayed columns when considering a constant load.
It should be noted that when a constant load is considered, the reduced slenderness of the stayed column and the simple column are considerably different. Fig. 11 confirms what has been discussed earlier. The difference in slenderness of the central mast of the stayed column and the simple column is considerable. For
example, a stayed column with fixed split-up cross arm and reduced slenderness of LZ 4:5 can sustain the same load as a simple column with a slenderness of LE Z 2:5. Fig. 11 further reveals that the slenderness of the central mast of a fixed split-up cross arm is larger than when a hinged split-up cross arm is used.
7
6
5
Λ
4
3
2
1
0 0
0.5
1
1.5
2
2.5
3
3.5
4
ΛE Chu et al.
Smith et al.
split - up hinged
split - up fixed
Euler
Fig. 11. Comparison of the slenderness of the stayed column and the simple column for a constant load.
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It will be shown how the imperfections can be accounted for in an early design stage.
8. Imperfections Until now, perfect columns have been discussed. Clearly, for a real (stayed) columns imperfections are unavoidable. Initial crookedness is always present and is amplified in the presence of axial loading. Therefore, if the buckling load of a column is to be predicted, it is necessary to account for these imperfections. Eurocode 3 [1], design of steel structures, proposes 4 buckling curves (a, b, c, d). The a-curve corresponds to sections that are almost insensitive to buckling; the d-curve corresponds to the sections with the most pronounced tendency to buckle. In Fig. 12 next to the 4 buckling curves of the Eurocode 3, the curve of the perfect column and an approximation for the a and d curves are plotted [2]. For ‘perfect’ columns the relation between L and scr/s is given by: scr 1 Z 2: s L
(14)
The a-curve (for LR 1) can be approximated by: scr 1 Z ; s 0:5 C L2
(15)
and the d-curve by: scr 1 Z s 1 C L2
(16)
9. Design procedure It is now possible to propose a design procedure. From what has been discussed previously, it is possible to determine the relationship between the slenderness of the central mast L/h, the overall slenderness L/H, the section of the stays Us, the section of the cross arm Uca and x for a load Fand length L. The applied load F and the length L of the column are two known design parameters. The designer will impose limits on the overall slenderness L/H, the slenderness of the central mast L/h and in the case of a split-up cross arm also x. Obviously, F has to be inferior to the buckling load of the stayed column, that is: F% Pcr Z g
p2 EIc : L2
(17)
Tables can be drawn up that determine the column that satisfies Eq. (17) and minimizes the value of W [9]. For the intervals imposed by the designer these tables provide the values of Us, Uca and x. These tables take constraints into consideration: for example it can be demonstrated that the lightest columns always coincides with the smallest value L/h and q; still, this cannot be done endlessly because of local buckling. The critical
1 0.9
1 Λ2
a b
0.8
c 0.7
1 0.5+Λ2
d
σcr/σ
0.6 0.5
1 1+Λ2
0.4 0.3 0.2 0.1 0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
Λ Fig. 12. Four buckling curves (a,b,c,d) of Eurocode 3 and the proposed approximation.
3.5
4.0
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‘local’ buckling load has been defined by Shanley [7]: PL;cr Z k2
Et Et Z k2 : D h
(18)
Here k2 is a constant with a theoretical value of 1.212, but tests have indicated that much smaller values are to be considered at failure. Typically k2 is taken equal to 0.4 to take imperfections into account. Eq. (18) and the limits imposed on L/h will yield the section needed to sustain a load F. Other considered boundary conditions are yielding of the stays and buckling and yielding of the cross arm. At this point a first design of the stayed column is available. Yet, the inevitable imperfections were not accounted for. To include these in the design, Eq. (12) is used to calculate a first approximation of the reduced slenderness L1 . Furthermore, the a-curve is considered, hence an improved approximation of the reduced slenderness can be obtained: rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s L qc L2 Z Z g K 0:5: (19) p E h 2 The stays and the cross arm do not need to be changed because taking the values from the first approximation always results in a safe design. Furthermore, their contribution to the total weight is small. Finally, it is possible to calculate the magnitude of the pretension by using Eqs. (7 and 8). The successive steps discussed in this paragraph have been used to develop a computer program that designs these columns for given boundary conditions. The user provides the load F, the length L of the column, and the limits of L/H and L/h. The design of the column, including the section of the central mast, cross arm and stays as well as the value of L/H, L/h and the pretension in the stays are returned by the program taking local buckling and yielding into consideration. To calculate the buckling load of a given configuration a basic eigenvalue–eigenmode calculation is performed [4]. To optimize the column a constrained nonlinear optimization algorithm is used. The results found by the program agree very closely with those found by an exhaustive numerical analysis of a set of columns.
10. Conclusions The buckling load of a stayed column may be increased many times by reinforcing it with a system of pretensioned and split-up hinged or fixed cross arm members. The magnitude of the externally applied load on a stayed column depends not only on the critical buckling load but also on the initial pretension in the stays. The magnitude of the pretension should be equal to the optimal pretension [6].
The rotational stiffness of the split-up cross arm stayed columns is very large, therefore, the mode of instability is almost always a symmetrical (single curvature) deflected shape. This coincides with the exhaustion of the translational stiffness of the column at mid height. For a hinged split-up cross arm the buckling load may be up to 9 times the Euler load of the central mast. The maximum ‘theoretical’ buckling load of the fixed split-up cross arm is 23.2 times the Euler load. The buckling load is influenced by changes in the cross arm length, the stay diameter, cross arm properties and the opening angle of the split-up cross arm. This change is substantial for small values of both cross arm length and stay diameter and decreases for larger values. The influence of the cross arm properties is negligible if the cross arm is hinged to the central mast, if the cross arm is fixed the influence is substantial due to the fact that the cross arm contributes to the rotational stiffness. The numerical buckling solutions obtained can be used to determine, for stayed columns, the ratios L/H, Uc/Us, Uc/Uca and x that maximizes the efficiency of the column, based on the indicator of volume W. The morphological indicators in general and the indicator of volume in particular are powerful tools to compare and design structures. Two basic assumptions are possible. A constant volume can be considered and the load evaluated or a constant load can be considered and the volume evaluated. For design reasons the second approach seems more suitable. Compared to other stayed columns with one cross arm, their efficiency is increased up to 23% when a split-up cross arm is used. Finally, an easy-to-use procedure is proposed to design stayed columns. This paper has dealt with the stability of a stayed column with hinged or fixed split-up cross arm. The conclusions reached herein are considered to be useful in designing other morphologies of stayed columns.
Acknowledgements This research is supported by the Institute of Innovation through Science and Technology in Flanders. (www.iwt.be)
References [1] Eurocode Steel/3. Belgian Standard NBN ENV 1993-1-1: 1992/AX, 1992. [2] Samyn P. Etude de la morphologie des structures, a` l’aide des indicateurs de volume et de de´placement. Brussels: Acade´mie royale de belgique; 2004. ISBN 2–8031–0201–3. [3] Smith RJ, McCaffrey GT, Ellis JS. Buckling of a single cross arm stayed column. J Struct Div ASCE 1975;101(ST. 1):249–68. [4] Temple MC. Buckling of stayed columns. J Struct Div ASCE 1977; 103(ST. 4):839–85.
J. Van Steirteghem et al. / Advances in Engineering Software 36 (2005) 614–625 [5] Chu KH, Berge SS. Analysis and design of struts, with tension ties. J Struct Div ASCE 1963;89(ST. 1):127–63. [6] Hafez HH, Temple MC, Ellis JS. Pretensioning of single cross arm stayed columns. J Struct Div ASCE 1979;103(ST. 2):359–75. [7] Shanley FR. Weight–strength analysis of aircraft structures. 2nd ed. New York: Dover Publications, Inc.; 1960.
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[8] Van Steirteghem J, Buyl C, De Wilde WP, Samyn Ph. Morphological optimization of 2D-stayed columns. Proceedings of OPTI 2003, 261– 270, May 2003, Detroit, USA. [9] Wattel F, Van Steirteghem J, De Wilde P, Samyn Ph. Buckling of stayed columns, Master Thesis, Vrije Universiteit Brussel, Brussels, Belgium, Sept. 2003.