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Nonlinear stability analysis of steel cooling towers considering imperfection sensitivity
Thin–Walled Structures 146 (2020) 106448
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Nonlinear stability analysis of steel cooling towers considering imperfection sensitivity Jie Wu a, *, Junying Zhu a, You Dong b, Qilin Zhang a a b
College of Civil Engineering, Tongji University, 1239 Siping Road, Shanghai, 200092, PR China Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Steel cooling tower Imperfection sensitivity Stability analysis Nonlinearity Buckling mode
There is a lack of investigation on the nonlinear stability analysis of large steel hyperbolic cooling towers considering imperfection sensitivity. In this paper, nonlinear stability analysis of 150m-height steel hyperbolic cooling towers was assessed. Models with five structural systems were established, including two types of reticulated shells (i.e., single-layer and double-layer shells) and three forms of girds (i.e., triangular grid, rect angular grid and square pyramid grid). Additionally, geometrically and material nonlinear stability analyses for more than 220 cases were conducted considering various distributions and amplitudes of imperfections. The results showed that the five hyperbolic steel cooling towers are of relatively low imperfection sensitivity, which is different from most other thin-walled shells, and the imperfection sensitivity of rectangular grid is high, while triangle grid and square pyramid grid are of low imperfection sensitivity. In addition, structures with doublelayer reticulated shells are more sensitive to imperfection than those with single-layer ones. It is recom mended that the design of steel cooling towers can give priority to the scheme of single-layer reticulated shell with triangular grid. Furthermore, the imperfection amplitude of H/300 could reasonably represents the most unfavorable instability state for this type of structures.
1. Introduction Considering the increasing energy demand and environmental issues around the world, there is a strong need for reliable, economic and environmental friendly energy technologies. Thermal power plays an important role in meeting such energy demand. Cooling towers could reduce the temperature of spent steam in thermal power plant [1]. In general, the higher the tower, the better the cooling effectiveness and power generation efficiency, thus cooling towers should be well designed to meet the growing requirements. It has been 100 years since the first construction of cooling tower. In 1915, the world’s first cooling tower, a 45 m high reinforced concrete structure was built in Netherlands. In 1964, the first 100 m high cooling tower was built by Hitkamp Inc. In Germany. With the increasing demand for energy, the scale of cooling towers is becoming larger and larger. In 2001, a super-large cooling tower with a 200 m height was built at the RWE power plant in Niederaussem, Germany [2]. Over the past five decades, research on cooling towers has focused on the inelastic behavior [3–5], collapse mechanism [6,7], wind loads [8], wind induced responses based on wind tunnel tests [9–13] or field measurement [14], and
dynamic performances [15,16] of reinforced concrete cooling towers. At present, most of cooling towers are still reinforced concrete structures and they have several shortcomings, such as excessive self-weight and poor seismic resistance [17]. With advantages such as light-weight structure, small air resistance together with simple and quick con struction, steel reticulated shells have been more and more popular in structural design of cooling towers. However, the relevant studies on steel cooling towers are still scarce. In the 1980s, the Soviet Union and Iran established a 120 m high steel reticulated cooling tower [18]. Later, Saeed [19] conducted numerical simulation of dynamic response, sta bility and collapse for this type of cooling towers. In the 21st century, several studies have been conducted to aid the design and construction of steel cooling towers. For instance, Cholnoky et al. [20] conducted experimental and numerical studies on joints of steel cooling towers and developed an automatic 3D-CAD structural design tool. The effects of stiffening rings on a steel hyperbolic cooling tower was examined, and it was observed that the use of stiffening rings made the tower lighter [17]. Ma et al. [21] studied the static behaviors of a single-layer reticulated shell system under different load combinations and obtained the opti mum grid form and grid size of the single-layer reticulated shell system
* Corresponding author. E-mail addresses: [email protected] (J. Wu), [email protected] (J. Zhu), [email protected] (Y. Dong), [email protected] (Q. Zhang). https://doi.org/10.1016/j.tws.2019.106448 Received 16 April 2019; Received in revised form 25 September 2019; Accepted 2 October 2019 Available online 14 October 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.
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eigenvalue imperfection mode method accordingly. This paper took the N-order (e.g., N equals 20) eigenmodes of structures as imperfections, and the load factors were calculated respectively. The minimum load factor reflected the stability bearing capacity of the structure. In addition, Hühne et al. [40] proposed the single perturbation load approach (SPLA) to produce a single dimple-shape imperfection which has been widely used by many researches to investigate the imperfection sensitivity of shell structures. Based on a finite number of single dimple-shape imperfections, the worst multiple perturbation load approach (WMPLA) was proposed by Hao et al. [41]. In this method, multiple dimple-shape imperfections are applied on the shell surface, and then, the most detrimental combination of these dimple-shape im perfections can be determined by optimization methods. Besides, a hybrid framework for reliability-based design optimization of imperfect stiffened shells was investigated by considering various uncertain fac tors, such as manufacturing tolerance, material properties and envi ronment aspects [42]. The above studies mainly focused on cylindrical and spherical shells which are continuous thin-walled structures. Though there have been some researches on the stability of steel reticulated shells, such as overall stability [43–48] and member buckling [49,50], these studies mainly focused on the single-layer reticulated shell structures. With respect to cooling towers, meridional geometric imperfections in con crete cooling towers have gained attention from researchers in early studies. Gupta and Al-Dabbagh [51] found a significant increase in the circumferential and meridional bending moment when applying meridional geometric imperfections to a cooling tower, and a formula was proposed for calculation of maximum allowable geometric imper fections. Later, a more accurate formula was proposed by Alexandridis and Gardner [52], considering the weakening of the circumferential stiffness due to meridional geometric imperfections. With respect to the meridional imperfections, it was found by Jullien et al. [53] that self-weight plays an important role in the generation of meridional cracking. In follow-up studies, more accurate numerical methods and simulation models have been developed. For instance, it was found that the forced displacement and the annular ribs could have a large effect on the global stability of cooling towers [54] and the theoretical imper fection distribution pattern turned out to be unreliable [55]. In summary, several studies have been conducted on the mechanical properties of cooling towers and the stability of steel reticulated shell structures. However, studies on the stability of cooling towers consid ering the initial geometric imperfections are still lacking. The current research status can be summarized as follows: The effects of various imperfections on concrete cooling towers have been extensively studied. With respect to steel cooling towers, a comprehensive investigation on imperfection sensitivity is still needed. Steel cooling towers are gener ally annular vertical hyperbolic reticulated shells, to the best knowledge of the authors, there are no studies on the imperfection sensitivity for this type of reticulated shell structures. In this paper, a 150 m high hyperbolic cooling tower was taken as an example (see Fig. 1), and finite element analyses of the structure were conducted using ANSYS software, considering both geometric nonline arity and material nonlinearity. Different structural systems (e.g., singlelayer reticulated shell, double-layer reticulated shell, and three different grid forms) were conducted. Additionally, the effects of imperfection distribution and amplitude on the stability of hyperbolic steel cooling tower was analyzed. With comprehensive analysis of imperfection sensitivity and selection of structural forms for steel cooling towers, the study can provide a reference for the design of hyperbolic steel cooling towers in the future.
Fig. 1. Saddle reticulated shell and hyperbolic cooling tower reticulated shell.
under different heights. With respect to wind loads, Ke et al. [22] studied the average wind load characteristics and wind-induced responses of a super-large straight-cone steel cooling tower. Cooling tower is a type of thin-walled structures, most of which have high imperfection sensitivity. For perfect structures, buckling load pre dicted by the classical bifurcation model is an upper limit. In practice, due to installation deviation during the construction of steel reticulated shell structures, initial geometric imperfections are inevitable, which can affect the stability of the structure to varying degrees. Imperfection sensitivity of thin-walled structures has been extensively investigated, with special emphasis on cylindrical shells [23,24], spherical shells [25, 26], lined pipes [26–29], and stiffed shells used in launch vehicles [30–34]. It is crucial to find a type of imperfection to predict the lower bounds of buckling loads. Three types of geometric imperfections were classified by Winterstetter and Schmidt [35]: realistic, worst and stim ulating geometric imperfections. Realistic imperfections can be measured and introduced into finite element (FE) models to perform geometrical and material nonlinear analysis. Unfortunately, this approach is usually not applicable in practical cases where the actual imperfect shape is unknown at the design stage. This worst geometric imperfections approach means to find the mathematically determined “worst possible” imperfection pattern. However, it is doubtful whether the approach can provide reasonable imperfection shapes and ampli tudes which are close enough to real imperfections occurring in practice [35,36]. Compared with the realistic and worst geometric imperfections, stimulating imperfections have been widely used in thin-walled shell buckling analysis, such as linear buckling mode imperfections (LBMI) which were recommended in European standard for steel shell struc tures [37] and Chinese code “JGJ7-2010, Technical specification for space frame structures” [38]. One important reason for widespread use of LBMI is that the imperfection shapes can be easily obtained by linear buckling analysis using common commercial FE packages. With respect to LBMI, the consistent imperfection mode method (CIMM) and the random imperfection mode method (RIMM) were widely employed in civil structures. CIMM was recommended for spatial truss structures by Chinese code “JGJ7-2010” [38], and it assumed that the first buckling mode is adopted as initial geometrical imperfection distribution, however, this assumption is inappropriate in many cases. In order to overcome the disadvantage of CIMM, RIMM was proposed based on reliability analysis (e.g., Monte-Carlo analysis), in which the initial geometric imperfection is treated as a multi-dimensional random variable. The advantage of RIMM is that it can take the uncertainties into consideration, but this method is time-consuming and requires large amount of computational effects. Combining the advantages and dis advantages of the two methods, Cai et al. [39] proposed the N-order
2. Structural modeling and analysis of steel cooling towers 2.1. Model description The finite element software ANSYS was employed to establish the 2
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considered within the stability analysis (GB/T 50102 2014) [56]. The surface of steel cooling towers is usually covered by color panels in engineering practice. Since the self-weight of color panels and roof purlins is 0.2kN/m2 approximately, the value of the dead load is taken as 0.2 kN/m2, in addition, the self-weight of members is auto-calculated. Wind load on the hyperbolic cooling tower was calculated according to the following equation [56]:
ωðz;θÞ ¼ βCg Cp ðθÞμz ω0
(1)
where ω(Z,θ) is the standard value of equivalent wind load applied on the outer surface of the cooling tower; β is the wind vibration coefficient; Cg is the tower interference coefficient; Cp(θ) is the average wind pressure distribution coefficient, as indicated in Fig. 5; θ is the wind direction angle; μz is the height variation factor of wind pressure; and ω0 is the basic wind pressure. In the analytical model, β ¼ 1.9, Cg ¼ 1.2, and ω0 ¼ 0.55 kPa [56]. Cp(θ) was calculated using the following equation: m X
Cp ðθÞ ¼
αk cos kθ
(2)
k¼0
where αk is coefficient which can be calculated based on the values of the Neiman curve K1.1 (as listed in Table 2) [56] and m is the total number of terms using in the Neiman curve. The first 10 terms were considered herein. During the operational condition of the cooling tower, the inside air flow could generate the internal wind suction load which can be computed as [56]:
Fig. 2. Geometry of the cooling tower.
models of different forms of steel cooling towers. The steel type is Q345, which yields at stress of 345 MPa. With respect to the single-layer reticulated shell models, the members were fixed at both ends and created using BEAM188 element which is in accordance with Timo shenko beam theory and includes shear-deformation effects. High-order shape function was used in BEAM188 and each element was simulated by a single unit. In double-layer reticulated shell models, joints between elements can be considered as pin connections and the elements only bear axial force, as stated in the design code “JGJ7-2010, Technical specification for space frame structures” [38], thus members were simulated by LINK180 element. The geometry of the steel cooling towers was designed based on the recommended value in the design code (GB/T 50102 2014) [56], as shown in Fig. 2. The ratio of height and bottom diameter is 1.20, the ratio of throat area and bottom area is 0.334, the ratio of throat height and structural height is 0.751, the diffusion angle on the top of tower is 11.5� , and the meridian dip at the bottom of tower is 26.9� . In order to investigate the stability of cooling towers with different structural systems, models with five different structure schemes were created, as shown in Fig. 3, respectively. The model parameters are shown in Fig. 4. As indicated in Fig. 4, the side length of the grid refers to the length of horizontal outer chord members at the throat of the tower. The thickness of the grid refers to the length of web members perpen dicular to the surface of the shell. Seven annular ribs were arranged in the single-layer reticulated shell models and all members consist of circular pipes. The parameters of the five structural systems are listed in Table 1. Based on fully stressed design (FSD) method, the finite element models were optimized by adjusting stress ratios of members to make sure that the structural weight of the five investigated structural forms is same. Thus, the influence of the structural weight on structural stability can be eliminated and the initial stability of the structural forms can be assessed. FSD is the most common approach in structural optimization areas, which assumes that each member reaches its stress limit at least in one of the load cases. Although the assumption is not always valid, the method usually can obtain a good solution after a few iterations. For detail of FSD, refer to [59,60].
ωi ¼ Cpi � qðHÞ
(3)
qðHÞ ¼ μH ⋅ Cg ⋅ω0
(4)
where q(H) is the design value of wind pressure at the top of the cooling tower; μH is the height variation factor of wind pressure at the top of the cooling tower; and Cpi is the internal suction coefficient. The nodal loads were obtained as the product of nodal influence areas and the wind pressure loads, which are shown in Eqs. (1) and (3), and each nodal load was perpendicular to the tower surface. In addition, the following aspect of pressure load is of pretty importance for shell elements near structural collapse. In this study, the stability analysis of steel hyperbolic cooling towers considering imperfection sensitivity by using beam elements was emphasized. Thus, the following aspect of pressure load was not considered. Furthermore, since the air inlet is hollow, the self-weight of the exterior panels, outside wind load and the inside wind suction load were applied above the air inlet, i.e., the part above the height of 26.9 m of the structure, as indicated in Fig. 6. Generally, there existed two ways of loading scenarios for the sta bility analysis of structures: D þ λ*(W) and λ*(D þ W) [57], where, λ is the load factor, D and W are the dead load and the wind load, respec tively. The former considered that the dead load was constant and only the wind load can be increased [21,58]. The latter deemed that λ was the buckling safety factor and the combination of dead load and wind load (or snow load [39]) was used. For different cases and research purposes, the relevant loading combination can be selected [57]. In this study, the load combination of dead load and wind load was adopted. Based on the Chinese code “JGJ7-2010, Technical specification for space frame structures” [38], the buckling safety factor λ (i.e., load factor) was applied to these two loads, in addition, the code recommends that the load factor should exceed 2.0 to ensure safety of the structure. As stated in the Chinese code “JGJ7-2010” [38], the load corre sponding to the first critical point during the nonlinear analysis could be regarded as the ultimate buckling load, at which local bucking may occur. There may exist other criteria for the determination of the ulti mate buckling load, for instance, for some specific cases, the first local buckling of the structure is still tolerable and the global buckling should be investigated. In this study, we adopted the recommendation from the
2.2. Loads The dead load, external wind load and internal wind suction load are 3
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Fig. 3. Configuration of cooling towers.
4
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Fig. 3. (continued).
Chinese code “JGJ7-2010” [38], in other words, we regarded the first buckling load encountered as the buckling load. The gap between the local and global buckling loads can be regarded as a “safety cushion”. This view was proposed by Gerasimidis et al. [24].
reticulated shells are much lower than those of double-layer ones. When considering material nonlinearity, the load factors (K2) of all the five models are approximately similar. The load factors (K2) of the double-layer reticulated shells are slightly smaller than those of the single-layer ones by GMNA. As listed in Table 3, due to the material
3. Imperfection sensitivity analysis of steel cooling tower 3.1. Nonlinear analysis without imperfections Without considering imperfections, buckling load factors of all models were calculated for the case with only geometrical nonlinearity and the case considering both geometrical and material nonlinearity. The effect of material nonlinearity on the structural stability can be assessed by comparing the variation of load factors. Newton-Raphson method was used to solve the nonlinear equations in the finite element analysis. In order to show the gap and differences between geometrical nonlinear analysis (GNA) and geometrical and material nonlinear analysis (GMNA), load-displacement curves of node 252 (see Fig. 7) in the S-T shell are shown in Fig. 8. The reason of selecting node 252 is that the maximal displacement occurs at this point by GNA. Table 3 shows the load factors of GNA and GMNA. As indicated in Fig. 8 and Table 3, there is a significant difference in load factors for the cases with and without considering material nonlinearity. Without considering material nonlinearity, the load factors (K1) of single-layer
Fig. 4. Explanation for grid size of the cooling tower.
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Table 1 Structural schemes of steel cooling towers. Structural systems
Type of reticulated shell
Form of girds
Side length of grids (m)
Thickness of grids (m)
Structural weight (t)
Stress ratio of members
Range of section sizes
S-T
Single-layer
Triangle
6
4
3462.3
<0.70
S-R
Single-layer
Rectangle
6
4
3437.3
<0.70
D-T
Double-layer
Triangle
4
4
3461.6
<0.90
D-R
Double-layer
Rectangle
4
4
3468.7
<0.92
D-S
Double-layer
Square pyramid
4
4
3487.7
<0.70
Φ159 � 6 – Φ900 � 18 Φ159 � 6 – Φ1000 � 12 Φ159 � 6 – Φ450 � 12 Φ159 � 6 – Φ400 � 12 Φ159 � 6 – Φ350 � 10
Note: S-T, S-R, D-T, D-R, and D-S refer to Fig. 3.
Fig. 5. Average wind pressure distribution coefficient of the cooling tower.
Table 2 Values for αk. α0 0.34387
α1
α2
α3
α4
0.40025
0.51139
0.41500
0.13856
α5 0.06904
α6
α7
α8
0.07317
0.01357
0.03466
Fig. 7. Position of node 252 in the S-T shell. Fig. 6. Area where loads (except self-weight of elements) are applied. 6
α9 0.00851
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Fig. 9. Load-displacement curves of the perfect and imperfect S-T shells by GNA and GMNA.
Fig. 8. Load-displacement curves of the perfect S-T shell. Table 3 Load factors by GNA and GMNA. Structural system
S-T
S-R
D-T
D-R
D-Q
Load factor: GNA (K1) Load factor: GMNA (K2) Variation: (K1–K2)/ K1 � 100%
13.166 4.556 65.4%
12.879 4.301 66.6%
63.360 4.079 93.6%
46.578 3.552 92.4%
67.874 3.722 94.5%
3.2. Effects of imperfection shapes In order to investigate the randomness of imperfection distributions, the first 20 eigenmode shapes were individually assigned as imperfec tions to the structure with imperfection amplitude H/300 (H refers to the height of the cooling towers, i.e., 150 m). The amplitude of imper fection of span/300 is recommended for spatial truss structures by Chinese code “JGJ7-2010” [38], which is widely used in China. The load factors of the five shells by GMNA with imperfection amplitude H/300 are listed in Table 4. The load-displacement curves of node 252 in the S-T shell by GNA and GMNA are shown in Fig. 9. To demonstrate the gap and differences between perfect and imperfection models, Fig. 10 shows the load-displacement curves of node 252 in the ST shell by only GMNA and the last points at the end of the curves correspond to the divergence of the calculation. The points corre sponding to the occurrence of the first yielding are also shown on the curves, which indicates that the occurrence time of the first yielding of imperfect shell is earlier than that of perfect shell. In Figs. 9 and 10, the 15th eigenmode shape with imperfection amplitude H/300 is assigned as imperfection to the imperfect structure. As shown in Fig. 10, the load-displacement curves are basically
Note: variation refers to the change of load factors before and after considering material nonlinearity.
nonlinearity, the load factors of S-T and D-T shells reduced by 65.4% and 93.6%, respectively. It is indicated that the influence of material nonlinearity on the stability of the double-layer reticulated shells is much higher than that of the single-layer ones. The above observations indicate that when considering geometric nonlinearity only, it may lead to incorrect conclusions of load factors associated with single-layer reticulated shells and double-layer ones. Therefore, both material and geometric nonlinearity were considered in the following analysis of imperfection sensitivity for hyperbolic steel cooling towers.
Table 4 Load factors of the five shells by GMNA with imperfection amplitude H/300. Mode order Perfect shells 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
S-T
S-R
D-T
Load factor
Variation
Load factor
Variation
Load factor
4.556 4.448 4.458 4.394 4.461 4.406 4.504 4.431 4.482 4.599 4.478 4.447 4.468 4.484 4.532 4.356 4.415 4.408 4.449 4.441 4.451
2.4% 2.2% 3.6% 2.1% 3.3% 1.1% 2.7% 1.6% 0.9% 1.7% 2.4% 1.9% 1.6% 0.5% 4.4% 3.1% 3.2% 2.3% 2.5% 2.3%
4.301 4.051 4.051 4.022 3.782 3.816 3.825 3.809 4.339 4.015 4.286 4.141 4.147 4.209 4.263 4.231 4.289 3.891 4.089 4.103 4.004
5.8% 5.8% 6.5% 12.1% 11.3% 11.1% 11.4% 0.9% 6.6% 0.3% 3.7% 3.6% 2.1% 0.9% 1.6% 0.3% 9.5% 4.9% 4.6% 6.9%
4.079 4.005 3.989 3.920 4.007 4.074 3.864 4.020 4.016 4.067 4.070 3.934 4.032 3.943 3.964 3.983 4.052 3.921 4.001 4.026 4.052
D-R Variation 1.8% 2.2% 3.9% 1.8% 0.1% 5.3% 1.4% 1.5% 0.3% 0.2% 3.6% 1.2% 3.3% 2.8% 2.4% 0.7% 3.9% 1.9% 1.3% 0.7%
D-Q
Load factor
Variation
Load factor
Variation
3.552 3.353 3.334 3.529 3.500 3.461 3.550 3.260 3.419 3.290 3.217 3.501 3.372 3.486 3.544 3.464 3.379 3.372 3.550 3.614 3.617
5.6% 6.1% 0.6% 1.5% 2.6% 0.1% 8.2% 3.7% 7.4% 9.4% 1.4% 5.1% 1.9% 0.2% 2.5% 4.9% 5.1% 0.1% 1.7% 1.8%
3.722 3.707 3.730 3.825 3.698 3.828 3.747 3.765 3.788 3.810 3.681 3.768 3.747 3.747 3.682 3.818 3.750 3.715 3.677 3.638 3.718
0.4% 0.2% 2.8% 0.6% 2.8% 0.7% 1.2% 1.8% 2.4% 1.1% 1.2% 0.7% 0.7% 1.1% 2.6% 0.8% 0.2% 1.2% 2.3% 0.1%
Note: variation refers to the change of load factors with and without considering imperfections, indicating the extent of favorable or unfavorable influence that imperfections have on the structure. It can be calculated by following equation: (K0-Ki)/Ki � 100%, where K0 is load factor for a perfect structure and Ki is load factor when taking the ith buckling mode as imperfection. The imperfection is unfavorable to the structure only if variation is negative. 7
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horizontal, thus the last points at the end of the curves can denote the load factors. Besides, in most cases the maximal displacements at the last load step are approximately 1.0 m which is not large, so the divergence of the incremental resolution of the equilibrium equation is mainly caused by excessive yielding of members. As indicated in Table 4, the load factors of the hyperbolic cooling tower shells do not increase with the increase of the eigenmode order. The correlation between load factors and modal order is not explicit. By taking different buckling modes as imperfections, some of load factors are smaller than the case when taking the first buckling mode as imperfection. The orders of buckling modes, which have larger effect than the first buckling mode, are listed in Table 5. As indicated, the first buckling mode is not necessarily the most unfavorable buckling mode for the hyperbolic steel cooling towers. For instance, with respect to the Model S-T, S-R and D-T, nearly half of the first 20 buckling modes are more much unfavorable than the first buckling mode. The above anal ysis demonstrates that CIMM is not applicable to hyperbolic steel cool ing tower. The maximum and minimum values of the load factors are shown in Fig. 11. The change of load factors before and after considering imper fections can be compared through the variation, which is listed in Table 6. As indicated in Fig. 11, load factors associated with all five structural systems change within a relatively small range, and hyperbolic steel cooling towers are of low imperfection sensitivity. For the researched five models, load factors of single-layer reticulated shells are higher than those of double-layer ones. Therefore, single-layer reticulated shells are more suitable for hyperbolic steel cooling towers; and among singlelayer reticulated shell models, load factors of shells with triangular grid are higher than those of shells with rectangular grid. In double-layer reticulated shell models, load factors of shells with triangular grid are generally higher than those of shells with rectangular grid and square pyramid grid. Thus, triangular grid has a better performance compared with other gird forms. According to Table 6, the absolute value of negative variation is taken as imperfection sensitivity. Stability performance of each grid form is evaluated by comparing their imperfection sensitivity accord ingly. The maximum imperfection sensitivity of shells with triangular grid is 5.3%, while for square pyramid grid that is merely 2.3%. The imperfection sensitivity of the both grid forms is at a relatively low level. Imperfection sensitivity of shells with rectangular grid reach at approximately 10% for both single-layer reticulated shell and doublelayer one. Therefore, grid forms can be sorted by imperfection sensi tivity as: rectangular grid > triangular grid > square pyramid grid. Rectangular grid is not recommended for hyperbolic steel cooling towers. Based on the above analysis, Model S-T is not sensitive to imper fections and has the highest load factor among all the five structural systems. Therefore, for hyperbolic steel cooling towers with height of approximately 150 m, the single-layer reticulated shell with triangular grid is recommended.
Fig. 10. Load-displacement curves of the perfect and imperfect S-T shells by GMNA. Table 5 Orders of buckling modes which are more unfavorable than the first buckling mode. Number
S-T
S-R
D-T
D-R
D-Q
Mode orders
3, 5, 7, 11, 15 16, 17, 19
2, 3, 4, 5, 6, 7 9, 17, 20
2, 7, 9, 10
11, 14, 19
Number of modes
8
9
2, 3, 6, 11, 13 14, 15, 17, 18 9
4
3
Fig. 11. Range of load factor for steel cooling towers with different struc tural systems. Table 6 Range of variation for load factors. Structural system
S-T
S-R
D-T
D-R
D-Q
Max of Load factor Variation Min of Load factor Variation
4.599 0.9% 4.356 4.4%
4.339 0.9% 3.782 12.1%
4.074 0.1% 3.864 5.3%
3.617 1.8% 3.217 9.4%
3.828 2.8% 3.638 2.3%
3.3. Effects of imperfection amplitude
Note: the meaning of variation refers to Table 4.
Yamada et al. [47] and Guo [48] found that the load factors of single-layer latticed shells do not necessarily decrease with the increase of imperfection value. As for the researched hyperbolic steel cooling
Table 7 Summary of the most unfavorable buckling modes of each structural system. Structural system
S-T
Modal order Load factor
3 4.394
S-R 15 4.356
4 3.782
D-T 7 3.809
3 3.920
8
D-R 6 3.864
7 3.260
D-Q 10 3.217
10 3.681
19 3.638
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Fig. 12. Buckling modes of cooling towers.
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Fig. 12. (continued).
towers, there may be an imperfection, which is less than H/300 and is more disadvantageous to the structures. Therefore, the rationality of H/300 needs to be verified. Based on the data in Table 4, the two most unfavorable buckling modes and their corresponding load factors are listed in Table 7. The corresponding buckling modes are shown in Fig. 12, which were taken as imperfections in the following calculation. Imperfection amplitudes of � H/1000, � H/700, � H/500 and � H/300 were taken into consideration. For a certain buckling mode, the imperfection was applied positively or negatively. As shown in Fig. 13, when the value of imperfection amplitude is negative, the offset direction of each node is reversed, i.e., the imperfection is reversely applied. The calculation re sults are shown in Table 8 and Fig. 14. As indicated in Fig. 14, the effects of imperfection amplitudes on the structural stability can be described by “symmetrical imperfection sensitivity (SIS)” and “asymmetric imperfection sensitivity (ASIS)”. SIS means that the load-imperfection amplitude curve is symmetric, i.e., both positive and negative imperfections will cause decrease of load factors, as shown in Fig. 14(a, b, c). ASIS means that the load-
imperfection amplitude curve is asymmetric, e.g., positive imperfec tions cause decrease of load factors, but negative imperfections cause increase of load factors, as shown in the 7th mode curve in Fig. 14(d). The change pattern of stability with imperfection amplitudes is listed in Table 9. As indicated, in most cases, the buckling pattern of hyper bolic steel cooling towers is related with the SIS. Therefore, under normal circumstances, a larger value of the imperfection amplitude could result in a larger reduction in the structural stability. Hence, for hyperbolic steel cooling towers investigated in this paper, imperfection amplitude of H/300 can reasonably present the most unfavorable buckling state of the structure. 3.4. Effects of the structural weight and stress ratio The above analytical results in Section 3.1–3.3 are based on the structural schemes of cooling towers listed in Table 1, in which the five structures have the same level of weight. However, as listed in Table 1, the maximal stress ratio of members in the five researched shells has large differences under the same structural weight, e.g., the maximal 10
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Table 8 Load factors for different imperfection amplitudes. Structural system
S-T
S-R
D-T
Imperfection amplitude
3rd mode
15th mode
4th mode
7th mode
3rd mode
6th mode
7th mode
10th mode
10th mode
19th mode
-H/300 - H/500 - H/700 - H/1000 0 H/1000 H/700 H/500 H/300 The most Unfavorable amplitude
4.466 4.535 4.550 4.553 4.556 4.548 4.540 4.523 4.394 H/300
4.417 4.521 4.536 4.543 4.556 4.553 4.546 4.527 4.356 H/300
3.779 4.038 4.107 4.173 4.301 4.173 4.107 4.040 3.782 -H/300
3.809 4.080 4.161 4.221 4.301 4.221 4.153 4.080 3.809 �H/300
3.920 3.919 3.957 4.033 4.079 4.040 3.993 3.946 3.920 �H/300
3.908 3.909 3.931 3.945 4.079 3.989 3.962 3.926 3.864 H/300
3.707 3.640 3.591 3.564 3.552 3.540 3.516 3.408 3.260 H/300
3.217 3.409 3.476 3.492 3.552 3.492 3.476 3.409 3.217 �H/300
3.702 3.727 3.724 3.738 3.722 3.736 3.729 3.727 3.681 H/300
3.860 3.799 3.773 3.750 3.722 3.697 3.677 3.662 3.638 H/300
stress ratio of S-T and D-T shells is 0.7 and 0.9, respectively. The section sizes of double-layer shells are much smaller than those of single-layer shells, which may lead to the load factors of double-layer shells are smaller than those of single-layer ones. In order to evaluate the effects of structural weight and stress ratio of members on load factors, two structural systems (i.e., S-T and D-T) with three allowable stress ratio (i. e., 0.7, 0.8, and 0.9) were considered, as listed in Table 10. The corre lation between load factors and structural weight under different allowable stress ratio of members is listed in Table 11. Fig. 15 shows the load factors with structural weight of S-T and D-T shells with and without imperfections. As indicated in Table 11 and Fig. 15, single-layer reticulated (S-T) shell has higher load factors than double-layer reticu lated (D-T) shell with the similar weight, i.e., the weight of S-T shell is lighter than that of D-T shell with the same stress ratio. In addition, Fig. 16 shows the load factors with stress ratio of S-T and D-T shells with and without imperfection, and indicates that D-T shell has higher load factors than S-T shell with the same stress ratio. Furthermore, Fig. 17 shows the ratio of load factor to weight with stress ratio of members. As indicated in Table 11 and Fig. 17, S-T shell has higher load factors per weight than D-T shell, thus single-layer reticulated shell is recom mended considering costs of the structures. Note: the 1st column lists the allowable stress ratio of members, the 2nd column lists the load factor of perfect structure, the 3rd column lists the minimal load factor with imperfect amplitude of H/300, the 4th column lists the structural weight, and the 5th column lists the ratio of load factor with perfect structure to weight.
D-R
D-Q
4. Conclusions The increasing demand for resources results in a need for super-large cooling towers and there is a trend of development of steel structure considering its superior performance in the large-span spatial structures. In order to ensure structural performance, the nonlinear stability and the selection of structural systems are of great importance. In this paper, the nonlinear stability analysis was conducted on a 150 m high hyperbolic steel cooling tower considering different structural systems. Some con clusions of this study can be summarized as follows. (1) Material nonlinearity should be taken into account in nonlinear stability analysis of hyperbolic steel cooling towers and could have a significant influence on the stability analysis. The effects of material nonlinearity on the stability of the double-layer reticulated shells are much higher than those of the single-layer ones. (2) For the hyperbolic steel cooling tower shells, the first-order buckling mode is not necessarily the most unfavorable mode. It is recommended that more forms of imperfection (not only the first buckling mode) should be taken into consideration in nonlinear stability analysis. (3) The correlation of the stability factors and the modal orders is not explicit. The load factors vary in a small range with different imperfection distributions. For the researched five models, hy perbolic steel cooling towers are of relatively low imperfection sensitivity, which is different from most other thin-walled shells.
Fig. 13. Imperfection applied positively (left) or negatively (right). 11
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Thin-Walled Structures 146 (2020) 106448
Fig. 14. Relationship between load factor and imperfection amplitude.
12
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Table 9 Stability with imperfection amplitudes. Structural system
S-T S-R D-T D-R D-S
Mode order
3 15 4 7 3 6 7 10 10 19
Change in stability after applying imperfections Imperfection applied positively
Imperfection applied reversely
Decrease Decrease Decrease Decrease Decrease Decrease Decrease Decrease Slightly decrease Decrease
Decrease Decrease Decrease Decrease Decrease Decrease Increase Decrease Slightly decrease Increase
Imperfection sensitivity pattern
SIS SIS SIS SIS SIS SIS ASIS SIS SIS ASIS
Fig. 15. Load factors with structural weight of models S-T and D-T with and without imperfection.
(4) Under the premise that the weight of each structural system is at the same level, considering the effects of imperfections, the load factors of shells with rectangular grid can be reduced by 12.1%, of which the imperfection sensitivity is relatively high. When it comes to triangular grid and square pyramid grid, the largest reductions are 5.3% and 2.4%, respectively, the imperfection sensitivity of which is relatively low. According to the load fac tors, single-layer reticulated shells are better than double-layer ones, and the triangular grid is better than the other two grid forms. The load factors of single-layer reticulated shell with triangular grid range from 4.36 to 4.60. (5) With the same level weight, the load factors of single-layer reticulated shells are higher than those of double-layer ones, while the former is lower than the latter with the same allowable stress ratio of members. Furthermore, the load factors per weight of single-layer reticulated shells are higher than those of doublelayer ones. Considering load factors, imperfection sensitivity and costs of each structural system, it is recommended that the design of steel cooling towers could give priority to the scheme of singlelayer reticulated shell with triangular grid.
Fig. 16. Load factors with stress ratio of models S-T and D-T with and without imperfection.
Table 10 Structure schemes of cooling towers under different stress ratio. Structural system
Type of reticulated shell
Form of girds
Side length of grids (m)
Thickness of grids (m)
Structural weight (t)
Stress ratio of members
Range of section size
S-T-0.7
Single-layer
Triangle
6
4
3462.3
<0.70
S-T-0.8
Single-layer
Triangle
6
4
3164.9
<0.80
S-T-0.9
Single-layer
Triangle
6
4
2937.4
<0.70
D-T-0.7
Double-layer
Triangle
4
4
3987.7
<0.70
D-T-0.8
Double-layer
Triangle
4
4
3739.1
<0.80
D-T-0.9
Double-layer
Triangle
4
4
3461.6
<0.90
Φ159 � 6 – Φ900 � 18 Φ159 � 6 – Φ800 � 16 Φ159 � 6 – Φ800 � 12 Φ159 � 6 – Φ550 � 12 Φ159 � 6 – Φ500 � 12 Φ159 � 6 – Φ450 � 12
Table 11 Correlation between load factors and structural weight under different allowable stress ratios of members. Stress ratio 0.7 0.8 0.9
S-T
D-T
Load factor (LF)
LF with imperfection
Weight (t)
LF/Weight ( � 10 3/t)
LF
LF with imperfection
Weight (t)
LF/Weight ( � 10 3/t)
4.556 4.027 3.677
4.356 3.769 3.476
3462.3 3164.9 2937.4
1.316 1.273 1.252
4.746 4.464 4.079
4.427 4.022 3.864
3987.7 3739.1 3461.6
1.190 1.194 1.178
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Fig. 17. Ratio of load factor to weight with stress ratio of members.
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Full length article
Nonlinear stability analysis of steel cooling towers considering imperfection sensitivity Jie Wu a, *, Junying Zhu a, You Dong b, Qilin Zhang a a b
College of Civil Engineering, Tongji University, 1239 Siping Road, Shanghai, 200092, PR China Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Steel cooling tower Imperfection sensitivity Stability analysis Nonlinearity Buckling mode
There is a lack of investigation on the nonlinear stability analysis of large steel hyperbolic cooling towers considering imperfection sensitivity. In this paper, nonlinear stability analysis of 150m-height steel hyperbolic cooling towers was assessed. Models with five structural systems were established, including two types of reticulated shells (i.e., single-layer and double-layer shells) and three forms of girds (i.e., triangular grid, rect angular grid and square pyramid grid). Additionally, geometrically and material nonlinear stability analyses for more than 220 cases were conducted considering various distributions and amplitudes of imperfections. The results showed that the five hyperbolic steel cooling towers are of relatively low imperfection sensitivity, which is different from most other thin-walled shells, and the imperfection sensitivity of rectangular grid is high, while triangle grid and square pyramid grid are of low imperfection sensitivity. In addition, structures with doublelayer reticulated shells are more sensitive to imperfection than those with single-layer ones. It is recom mended that the design of steel cooling towers can give priority to the scheme of single-layer reticulated shell with triangular grid. Furthermore, the imperfection amplitude of H/300 could reasonably represents the most unfavorable instability state for this type of structures.
1. Introduction Considering the increasing energy demand and environmental issues around the world, there is a strong need for reliable, economic and environmental friendly energy technologies. Thermal power plays an important role in meeting such energy demand. Cooling towers could reduce the temperature of spent steam in thermal power plant [1]. In general, the higher the tower, the better the cooling effectiveness and power generation efficiency, thus cooling towers should be well designed to meet the growing requirements. It has been 100 years since the first construction of cooling tower. In 1915, the world’s first cooling tower, a 45 m high reinforced concrete structure was built in Netherlands. In 1964, the first 100 m high cooling tower was built by Hitkamp Inc. In Germany. With the increasing demand for energy, the scale of cooling towers is becoming larger and larger. In 2001, a super-large cooling tower with a 200 m height was built at the RWE power plant in Niederaussem, Germany [2]. Over the past five decades, research on cooling towers has focused on the inelastic behavior [3–5], collapse mechanism [6,7], wind loads [8], wind induced responses based on wind tunnel tests [9–13] or field measurement [14], and
dynamic performances [15,16] of reinforced concrete cooling towers. At present, most of cooling towers are still reinforced concrete structures and they have several shortcomings, such as excessive self-weight and poor seismic resistance [17]. With advantages such as light-weight structure, small air resistance together with simple and quick con struction, steel reticulated shells have been more and more popular in structural design of cooling towers. However, the relevant studies on steel cooling towers are still scarce. In the 1980s, the Soviet Union and Iran established a 120 m high steel reticulated cooling tower [18]. Later, Saeed [19] conducted numerical simulation of dynamic response, sta bility and collapse for this type of cooling towers. In the 21st century, several studies have been conducted to aid the design and construction of steel cooling towers. For instance, Cholnoky et al. [20] conducted experimental and numerical studies on joints of steel cooling towers and developed an automatic 3D-CAD structural design tool. The effects of stiffening rings on a steel hyperbolic cooling tower was examined, and it was observed that the use of stiffening rings made the tower lighter [17]. Ma et al. [21] studied the static behaviors of a single-layer reticulated shell system under different load combinations and obtained the opti mum grid form and grid size of the single-layer reticulated shell system
* Corresponding author. E-mail addresses: [email protected] (J. Wu), [email protected] (J. Zhu), [email protected] (Y. Dong), [email protected] (Q. Zhang). https://doi.org/10.1016/j.tws.2019.106448 Received 16 April 2019; Received in revised form 25 September 2019; Accepted 2 October 2019 Available online 14 October 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.
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eigenvalue imperfection mode method accordingly. This paper took the N-order (e.g., N equals 20) eigenmodes of structures as imperfections, and the load factors were calculated respectively. The minimum load factor reflected the stability bearing capacity of the structure. In addition, Hühne et al. [40] proposed the single perturbation load approach (SPLA) to produce a single dimple-shape imperfection which has been widely used by many researches to investigate the imperfection sensitivity of shell structures. Based on a finite number of single dimple-shape imperfections, the worst multiple perturbation load approach (WMPLA) was proposed by Hao et al. [41]. In this method, multiple dimple-shape imperfections are applied on the shell surface, and then, the most detrimental combination of these dimple-shape im perfections can be determined by optimization methods. Besides, a hybrid framework for reliability-based design optimization of imperfect stiffened shells was investigated by considering various uncertain fac tors, such as manufacturing tolerance, material properties and envi ronment aspects [42]. The above studies mainly focused on cylindrical and spherical shells which are continuous thin-walled structures. Though there have been some researches on the stability of steel reticulated shells, such as overall stability [43–48] and member buckling [49,50], these studies mainly focused on the single-layer reticulated shell structures. With respect to cooling towers, meridional geometric imperfections in con crete cooling towers have gained attention from researchers in early studies. Gupta and Al-Dabbagh [51] found a significant increase in the circumferential and meridional bending moment when applying meridional geometric imperfections to a cooling tower, and a formula was proposed for calculation of maximum allowable geometric imper fections. Later, a more accurate formula was proposed by Alexandridis and Gardner [52], considering the weakening of the circumferential stiffness due to meridional geometric imperfections. With respect to the meridional imperfections, it was found by Jullien et al. [53] that self-weight plays an important role in the generation of meridional cracking. In follow-up studies, more accurate numerical methods and simulation models have been developed. For instance, it was found that the forced displacement and the annular ribs could have a large effect on the global stability of cooling towers [54] and the theoretical imper fection distribution pattern turned out to be unreliable [55]. In summary, several studies have been conducted on the mechanical properties of cooling towers and the stability of steel reticulated shell structures. However, studies on the stability of cooling towers consid ering the initial geometric imperfections are still lacking. The current research status can be summarized as follows: The effects of various imperfections on concrete cooling towers have been extensively studied. With respect to steel cooling towers, a comprehensive investigation on imperfection sensitivity is still needed. Steel cooling towers are gener ally annular vertical hyperbolic reticulated shells, to the best knowledge of the authors, there are no studies on the imperfection sensitivity for this type of reticulated shell structures. In this paper, a 150 m high hyperbolic cooling tower was taken as an example (see Fig. 1), and finite element analyses of the structure were conducted using ANSYS software, considering both geometric nonline arity and material nonlinearity. Different structural systems (e.g., singlelayer reticulated shell, double-layer reticulated shell, and three different grid forms) were conducted. Additionally, the effects of imperfection distribution and amplitude on the stability of hyperbolic steel cooling tower was analyzed. With comprehensive analysis of imperfection sensitivity and selection of structural forms for steel cooling towers, the study can provide a reference for the design of hyperbolic steel cooling towers in the future.
Fig. 1. Saddle reticulated shell and hyperbolic cooling tower reticulated shell.
under different heights. With respect to wind loads, Ke et al. [22] studied the average wind load characteristics and wind-induced responses of a super-large straight-cone steel cooling tower. Cooling tower is a type of thin-walled structures, most of which have high imperfection sensitivity. For perfect structures, buckling load pre dicted by the classical bifurcation model is an upper limit. In practice, due to installation deviation during the construction of steel reticulated shell structures, initial geometric imperfections are inevitable, which can affect the stability of the structure to varying degrees. Imperfection sensitivity of thin-walled structures has been extensively investigated, with special emphasis on cylindrical shells [23,24], spherical shells [25, 26], lined pipes [26–29], and stiffed shells used in launch vehicles [30–34]. It is crucial to find a type of imperfection to predict the lower bounds of buckling loads. Three types of geometric imperfections were classified by Winterstetter and Schmidt [35]: realistic, worst and stim ulating geometric imperfections. Realistic imperfections can be measured and introduced into finite element (FE) models to perform geometrical and material nonlinear analysis. Unfortunately, this approach is usually not applicable in practical cases where the actual imperfect shape is unknown at the design stage. This worst geometric imperfections approach means to find the mathematically determined “worst possible” imperfection pattern. However, it is doubtful whether the approach can provide reasonable imperfection shapes and ampli tudes which are close enough to real imperfections occurring in practice [35,36]. Compared with the realistic and worst geometric imperfections, stimulating imperfections have been widely used in thin-walled shell buckling analysis, such as linear buckling mode imperfections (LBMI) which were recommended in European standard for steel shell struc tures [37] and Chinese code “JGJ7-2010, Technical specification for space frame structures” [38]. One important reason for widespread use of LBMI is that the imperfection shapes can be easily obtained by linear buckling analysis using common commercial FE packages. With respect to LBMI, the consistent imperfection mode method (CIMM) and the random imperfection mode method (RIMM) were widely employed in civil structures. CIMM was recommended for spatial truss structures by Chinese code “JGJ7-2010” [38], and it assumed that the first buckling mode is adopted as initial geometrical imperfection distribution, however, this assumption is inappropriate in many cases. In order to overcome the disadvantage of CIMM, RIMM was proposed based on reliability analysis (e.g., Monte-Carlo analysis), in which the initial geometric imperfection is treated as a multi-dimensional random variable. The advantage of RIMM is that it can take the uncertainties into consideration, but this method is time-consuming and requires large amount of computational effects. Combining the advantages and dis advantages of the two methods, Cai et al. [39] proposed the N-order
2. Structural modeling and analysis of steel cooling towers 2.1. Model description The finite element software ANSYS was employed to establish the 2
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considered within the stability analysis (GB/T 50102 2014) [56]. The surface of steel cooling towers is usually covered by color panels in engineering practice. Since the self-weight of color panels and roof purlins is 0.2kN/m2 approximately, the value of the dead load is taken as 0.2 kN/m2, in addition, the self-weight of members is auto-calculated. Wind load on the hyperbolic cooling tower was calculated according to the following equation [56]:
ωðz;θÞ ¼ βCg Cp ðθÞμz ω0
(1)
where ω(Z,θ) is the standard value of equivalent wind load applied on the outer surface of the cooling tower; β is the wind vibration coefficient; Cg is the tower interference coefficient; Cp(θ) is the average wind pressure distribution coefficient, as indicated in Fig. 5; θ is the wind direction angle; μz is the height variation factor of wind pressure; and ω0 is the basic wind pressure. In the analytical model, β ¼ 1.9, Cg ¼ 1.2, and ω0 ¼ 0.55 kPa [56]. Cp(θ) was calculated using the following equation: m X
Cp ðθÞ ¼
αk cos kθ
(2)
k¼0
where αk is coefficient which can be calculated based on the values of the Neiman curve K1.1 (as listed in Table 2) [56] and m is the total number of terms using in the Neiman curve. The first 10 terms were considered herein. During the operational condition of the cooling tower, the inside air flow could generate the internal wind suction load which can be computed as [56]:
Fig. 2. Geometry of the cooling tower.
models of different forms of steel cooling towers. The steel type is Q345, which yields at stress of 345 MPa. With respect to the single-layer reticulated shell models, the members were fixed at both ends and created using BEAM188 element which is in accordance with Timo shenko beam theory and includes shear-deformation effects. High-order shape function was used in BEAM188 and each element was simulated by a single unit. In double-layer reticulated shell models, joints between elements can be considered as pin connections and the elements only bear axial force, as stated in the design code “JGJ7-2010, Technical specification for space frame structures” [38], thus members were simulated by LINK180 element. The geometry of the steel cooling towers was designed based on the recommended value in the design code (GB/T 50102 2014) [56], as shown in Fig. 2. The ratio of height and bottom diameter is 1.20, the ratio of throat area and bottom area is 0.334, the ratio of throat height and structural height is 0.751, the diffusion angle on the top of tower is 11.5� , and the meridian dip at the bottom of tower is 26.9� . In order to investigate the stability of cooling towers with different structural systems, models with five different structure schemes were created, as shown in Fig. 3, respectively. The model parameters are shown in Fig. 4. As indicated in Fig. 4, the side length of the grid refers to the length of horizontal outer chord members at the throat of the tower. The thickness of the grid refers to the length of web members perpen dicular to the surface of the shell. Seven annular ribs were arranged in the single-layer reticulated shell models and all members consist of circular pipes. The parameters of the five structural systems are listed in Table 1. Based on fully stressed design (FSD) method, the finite element models were optimized by adjusting stress ratios of members to make sure that the structural weight of the five investigated structural forms is same. Thus, the influence of the structural weight on structural stability can be eliminated and the initial stability of the structural forms can be assessed. FSD is the most common approach in structural optimization areas, which assumes that each member reaches its stress limit at least in one of the load cases. Although the assumption is not always valid, the method usually can obtain a good solution after a few iterations. For detail of FSD, refer to [59,60].
ωi ¼ Cpi � qðHÞ
(3)
qðHÞ ¼ μH ⋅ Cg ⋅ω0
(4)
where q(H) is the design value of wind pressure at the top of the cooling tower; μH is the height variation factor of wind pressure at the top of the cooling tower; and Cpi is the internal suction coefficient. The nodal loads were obtained as the product of nodal influence areas and the wind pressure loads, which are shown in Eqs. (1) and (3), and each nodal load was perpendicular to the tower surface. In addition, the following aspect of pressure load is of pretty importance for shell elements near structural collapse. In this study, the stability analysis of steel hyperbolic cooling towers considering imperfection sensitivity by using beam elements was emphasized. Thus, the following aspect of pressure load was not considered. Furthermore, since the air inlet is hollow, the self-weight of the exterior panels, outside wind load and the inside wind suction load were applied above the air inlet, i.e., the part above the height of 26.9 m of the structure, as indicated in Fig. 6. Generally, there existed two ways of loading scenarios for the sta bility analysis of structures: D þ λ*(W) and λ*(D þ W) [57], where, λ is the load factor, D and W are the dead load and the wind load, respec tively. The former considered that the dead load was constant and only the wind load can be increased [21,58]. The latter deemed that λ was the buckling safety factor and the combination of dead load and wind load (or snow load [39]) was used. For different cases and research purposes, the relevant loading combination can be selected [57]. In this study, the load combination of dead load and wind load was adopted. Based on the Chinese code “JGJ7-2010, Technical specification for space frame structures” [38], the buckling safety factor λ (i.e., load factor) was applied to these two loads, in addition, the code recommends that the load factor should exceed 2.0 to ensure safety of the structure. As stated in the Chinese code “JGJ7-2010” [38], the load corre sponding to the first critical point during the nonlinear analysis could be regarded as the ultimate buckling load, at which local bucking may occur. There may exist other criteria for the determination of the ulti mate buckling load, for instance, for some specific cases, the first local buckling of the structure is still tolerable and the global buckling should be investigated. In this study, we adopted the recommendation from the
2.2. Loads The dead load, external wind load and internal wind suction load are 3
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Fig. 3. Configuration of cooling towers.
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Fig. 3. (continued).
Chinese code “JGJ7-2010” [38], in other words, we regarded the first buckling load encountered as the buckling load. The gap between the local and global buckling loads can be regarded as a “safety cushion”. This view was proposed by Gerasimidis et al. [24].
reticulated shells are much lower than those of double-layer ones. When considering material nonlinearity, the load factors (K2) of all the five models are approximately similar. The load factors (K2) of the double-layer reticulated shells are slightly smaller than those of the single-layer ones by GMNA. As listed in Table 3, due to the material
3. Imperfection sensitivity analysis of steel cooling tower 3.1. Nonlinear analysis without imperfections Without considering imperfections, buckling load factors of all models were calculated for the case with only geometrical nonlinearity and the case considering both geometrical and material nonlinearity. The effect of material nonlinearity on the structural stability can be assessed by comparing the variation of load factors. Newton-Raphson method was used to solve the nonlinear equations in the finite element analysis. In order to show the gap and differences between geometrical nonlinear analysis (GNA) and geometrical and material nonlinear analysis (GMNA), load-displacement curves of node 252 (see Fig. 7) in the S-T shell are shown in Fig. 8. The reason of selecting node 252 is that the maximal displacement occurs at this point by GNA. Table 3 shows the load factors of GNA and GMNA. As indicated in Fig. 8 and Table 3, there is a significant difference in load factors for the cases with and without considering material nonlinearity. Without considering material nonlinearity, the load factors (K1) of single-layer
Fig. 4. Explanation for grid size of the cooling tower.
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Table 1 Structural schemes of steel cooling towers. Structural systems
Type of reticulated shell
Form of girds
Side length of grids (m)
Thickness of grids (m)
Structural weight (t)
Stress ratio of members
Range of section sizes
S-T
Single-layer
Triangle
6
4
3462.3
<0.70
S-R
Single-layer
Rectangle
6
4
3437.3
<0.70
D-T
Double-layer
Triangle
4
4
3461.6
<0.90
D-R
Double-layer
Rectangle
4
4
3468.7
<0.92
D-S
Double-layer
Square pyramid
4
4
3487.7
<0.70
Φ159 � 6 – Φ900 � 18 Φ159 � 6 – Φ1000 � 12 Φ159 � 6 – Φ450 � 12 Φ159 � 6 – Φ400 � 12 Φ159 � 6 – Φ350 � 10
Note: S-T, S-R, D-T, D-R, and D-S refer to Fig. 3.
Fig. 5. Average wind pressure distribution coefficient of the cooling tower.
Table 2 Values for αk. α0 0.34387
α1
α2
α3
α4
0.40025
0.51139
0.41500
0.13856
α5 0.06904
α6
α7
α8
0.07317
0.01357
0.03466
Fig. 7. Position of node 252 in the S-T shell. Fig. 6. Area where loads (except self-weight of elements) are applied. 6
α9 0.00851
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Fig. 9. Load-displacement curves of the perfect and imperfect S-T shells by GNA and GMNA.
Fig. 8. Load-displacement curves of the perfect S-T shell. Table 3 Load factors by GNA and GMNA. Structural system
S-T
S-R
D-T
D-R
D-Q
Load factor: GNA (K1) Load factor: GMNA (K2) Variation: (K1–K2)/ K1 � 100%
13.166 4.556 65.4%
12.879 4.301 66.6%
63.360 4.079 93.6%
46.578 3.552 92.4%
67.874 3.722 94.5%
3.2. Effects of imperfection shapes In order to investigate the randomness of imperfection distributions, the first 20 eigenmode shapes were individually assigned as imperfec tions to the structure with imperfection amplitude H/300 (H refers to the height of the cooling towers, i.e., 150 m). The amplitude of imper fection of span/300 is recommended for spatial truss structures by Chinese code “JGJ7-2010” [38], which is widely used in China. The load factors of the five shells by GMNA with imperfection amplitude H/300 are listed in Table 4. The load-displacement curves of node 252 in the S-T shell by GNA and GMNA are shown in Fig. 9. To demonstrate the gap and differences between perfect and imperfection models, Fig. 10 shows the load-displacement curves of node 252 in the ST shell by only GMNA and the last points at the end of the curves correspond to the divergence of the calculation. The points corre sponding to the occurrence of the first yielding are also shown on the curves, which indicates that the occurrence time of the first yielding of imperfect shell is earlier than that of perfect shell. In Figs. 9 and 10, the 15th eigenmode shape with imperfection amplitude H/300 is assigned as imperfection to the imperfect structure. As shown in Fig. 10, the load-displacement curves are basically
Note: variation refers to the change of load factors before and after considering material nonlinearity.
nonlinearity, the load factors of S-T and D-T shells reduced by 65.4% and 93.6%, respectively. It is indicated that the influence of material nonlinearity on the stability of the double-layer reticulated shells is much higher than that of the single-layer ones. The above observations indicate that when considering geometric nonlinearity only, it may lead to incorrect conclusions of load factors associated with single-layer reticulated shells and double-layer ones. Therefore, both material and geometric nonlinearity were considered in the following analysis of imperfection sensitivity for hyperbolic steel cooling towers.
Table 4 Load factors of the five shells by GMNA with imperfection amplitude H/300. Mode order Perfect shells 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
S-T
S-R
D-T
Load factor
Variation
Load factor
Variation
Load factor
4.556 4.448 4.458 4.394 4.461 4.406 4.504 4.431 4.482 4.599 4.478 4.447 4.468 4.484 4.532 4.356 4.415 4.408 4.449 4.441 4.451
2.4% 2.2% 3.6% 2.1% 3.3% 1.1% 2.7% 1.6% 0.9% 1.7% 2.4% 1.9% 1.6% 0.5% 4.4% 3.1% 3.2% 2.3% 2.5% 2.3%
4.301 4.051 4.051 4.022 3.782 3.816 3.825 3.809 4.339 4.015 4.286 4.141 4.147 4.209 4.263 4.231 4.289 3.891 4.089 4.103 4.004
5.8% 5.8% 6.5% 12.1% 11.3% 11.1% 11.4% 0.9% 6.6% 0.3% 3.7% 3.6% 2.1% 0.9% 1.6% 0.3% 9.5% 4.9% 4.6% 6.9%
4.079 4.005 3.989 3.920 4.007 4.074 3.864 4.020 4.016 4.067 4.070 3.934 4.032 3.943 3.964 3.983 4.052 3.921 4.001 4.026 4.052
D-R Variation 1.8% 2.2% 3.9% 1.8% 0.1% 5.3% 1.4% 1.5% 0.3% 0.2% 3.6% 1.2% 3.3% 2.8% 2.4% 0.7% 3.9% 1.9% 1.3% 0.7%
D-Q
Load factor
Variation
Load factor
Variation
3.552 3.353 3.334 3.529 3.500 3.461 3.550 3.260 3.419 3.290 3.217 3.501 3.372 3.486 3.544 3.464 3.379 3.372 3.550 3.614 3.617
5.6% 6.1% 0.6% 1.5% 2.6% 0.1% 8.2% 3.7% 7.4% 9.4% 1.4% 5.1% 1.9% 0.2% 2.5% 4.9% 5.1% 0.1% 1.7% 1.8%
3.722 3.707 3.730 3.825 3.698 3.828 3.747 3.765 3.788 3.810 3.681 3.768 3.747 3.747 3.682 3.818 3.750 3.715 3.677 3.638 3.718
0.4% 0.2% 2.8% 0.6% 2.8% 0.7% 1.2% 1.8% 2.4% 1.1% 1.2% 0.7% 0.7% 1.1% 2.6% 0.8% 0.2% 1.2% 2.3% 0.1%
Note: variation refers to the change of load factors with and without considering imperfections, indicating the extent of favorable or unfavorable influence that imperfections have on the structure. It can be calculated by following equation: (K0-Ki)/Ki � 100%, where K0 is load factor for a perfect structure and Ki is load factor when taking the ith buckling mode as imperfection. The imperfection is unfavorable to the structure only if variation is negative. 7
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horizontal, thus the last points at the end of the curves can denote the load factors. Besides, in most cases the maximal displacements at the last load step are approximately 1.0 m which is not large, so the divergence of the incremental resolution of the equilibrium equation is mainly caused by excessive yielding of members. As indicated in Table 4, the load factors of the hyperbolic cooling tower shells do not increase with the increase of the eigenmode order. The correlation between load factors and modal order is not explicit. By taking different buckling modes as imperfections, some of load factors are smaller than the case when taking the first buckling mode as imperfection. The orders of buckling modes, which have larger effect than the first buckling mode, are listed in Table 5. As indicated, the first buckling mode is not necessarily the most unfavorable buckling mode for the hyperbolic steel cooling towers. For instance, with respect to the Model S-T, S-R and D-T, nearly half of the first 20 buckling modes are more much unfavorable than the first buckling mode. The above anal ysis demonstrates that CIMM is not applicable to hyperbolic steel cool ing tower. The maximum and minimum values of the load factors are shown in Fig. 11. The change of load factors before and after considering imper fections can be compared through the variation, which is listed in Table 6. As indicated in Fig. 11, load factors associated with all five structural systems change within a relatively small range, and hyperbolic steel cooling towers are of low imperfection sensitivity. For the researched five models, load factors of single-layer reticulated shells are higher than those of double-layer ones. Therefore, single-layer reticulated shells are more suitable for hyperbolic steel cooling towers; and among singlelayer reticulated shell models, load factors of shells with triangular grid are higher than those of shells with rectangular grid. In double-layer reticulated shell models, load factors of shells with triangular grid are generally higher than those of shells with rectangular grid and square pyramid grid. Thus, triangular grid has a better performance compared with other gird forms. According to Table 6, the absolute value of negative variation is taken as imperfection sensitivity. Stability performance of each grid form is evaluated by comparing their imperfection sensitivity accord ingly. The maximum imperfection sensitivity of shells with triangular grid is 5.3%, while for square pyramid grid that is merely 2.3%. The imperfection sensitivity of the both grid forms is at a relatively low level. Imperfection sensitivity of shells with rectangular grid reach at approximately 10% for both single-layer reticulated shell and doublelayer one. Therefore, grid forms can be sorted by imperfection sensi tivity as: rectangular grid > triangular grid > square pyramid grid. Rectangular grid is not recommended for hyperbolic steel cooling towers. Based on the above analysis, Model S-T is not sensitive to imper fections and has the highest load factor among all the five structural systems. Therefore, for hyperbolic steel cooling towers with height of approximately 150 m, the single-layer reticulated shell with triangular grid is recommended.
Fig. 10. Load-displacement curves of the perfect and imperfect S-T shells by GMNA. Table 5 Orders of buckling modes which are more unfavorable than the first buckling mode. Number
S-T
S-R
D-T
D-R
D-Q
Mode orders
3, 5, 7, 11, 15 16, 17, 19
2, 3, 4, 5, 6, 7 9, 17, 20
2, 7, 9, 10
11, 14, 19
Number of modes
8
9
2, 3, 6, 11, 13 14, 15, 17, 18 9
4
3
Fig. 11. Range of load factor for steel cooling towers with different struc tural systems. Table 6 Range of variation for load factors. Structural system
S-T
S-R
D-T
D-R
D-Q
Max of Load factor Variation Min of Load factor Variation
4.599 0.9% 4.356 4.4%
4.339 0.9% 3.782 12.1%
4.074 0.1% 3.864 5.3%
3.617 1.8% 3.217 9.4%
3.828 2.8% 3.638 2.3%
3.3. Effects of imperfection amplitude
Note: the meaning of variation refers to Table 4.
Yamada et al. [47] and Guo [48] found that the load factors of single-layer latticed shells do not necessarily decrease with the increase of imperfection value. As for the researched hyperbolic steel cooling
Table 7 Summary of the most unfavorable buckling modes of each structural system. Structural system
S-T
Modal order Load factor
3 4.394
S-R 15 4.356
4 3.782
D-T 7 3.809
3 3.920
8
D-R 6 3.864
7 3.260
D-Q 10 3.217
10 3.681
19 3.638
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Fig. 12. Buckling modes of cooling towers.
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Fig. 12. (continued).
towers, there may be an imperfection, which is less than H/300 and is more disadvantageous to the structures. Therefore, the rationality of H/300 needs to be verified. Based on the data in Table 4, the two most unfavorable buckling modes and their corresponding load factors are listed in Table 7. The corresponding buckling modes are shown in Fig. 12, which were taken as imperfections in the following calculation. Imperfection amplitudes of � H/1000, � H/700, � H/500 and � H/300 were taken into consideration. For a certain buckling mode, the imperfection was applied positively or negatively. As shown in Fig. 13, when the value of imperfection amplitude is negative, the offset direction of each node is reversed, i.e., the imperfection is reversely applied. The calculation re sults are shown in Table 8 and Fig. 14. As indicated in Fig. 14, the effects of imperfection amplitudes on the structural stability can be described by “symmetrical imperfection sensitivity (SIS)” and “asymmetric imperfection sensitivity (ASIS)”. SIS means that the load-imperfection amplitude curve is symmetric, i.e., both positive and negative imperfections will cause decrease of load factors, as shown in Fig. 14(a, b, c). ASIS means that the load-
imperfection amplitude curve is asymmetric, e.g., positive imperfec tions cause decrease of load factors, but negative imperfections cause increase of load factors, as shown in the 7th mode curve in Fig. 14(d). The change pattern of stability with imperfection amplitudes is listed in Table 9. As indicated, in most cases, the buckling pattern of hyper bolic steel cooling towers is related with the SIS. Therefore, under normal circumstances, a larger value of the imperfection amplitude could result in a larger reduction in the structural stability. Hence, for hyperbolic steel cooling towers investigated in this paper, imperfection amplitude of H/300 can reasonably present the most unfavorable buckling state of the structure. 3.4. Effects of the structural weight and stress ratio The above analytical results in Section 3.1–3.3 are based on the structural schemes of cooling towers listed in Table 1, in which the five structures have the same level of weight. However, as listed in Table 1, the maximal stress ratio of members in the five researched shells has large differences under the same structural weight, e.g., the maximal 10
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Table 8 Load factors for different imperfection amplitudes. Structural system
S-T
S-R
D-T
Imperfection amplitude
3rd mode
15th mode
4th mode
7th mode
3rd mode
6th mode
7th mode
10th mode
10th mode
19th mode
-H/300 - H/500 - H/700 - H/1000 0 H/1000 H/700 H/500 H/300 The most Unfavorable amplitude
4.466 4.535 4.550 4.553 4.556 4.548 4.540 4.523 4.394 H/300
4.417 4.521 4.536 4.543 4.556 4.553 4.546 4.527 4.356 H/300
3.779 4.038 4.107 4.173 4.301 4.173 4.107 4.040 3.782 -H/300
3.809 4.080 4.161 4.221 4.301 4.221 4.153 4.080 3.809 �H/300
3.920 3.919 3.957 4.033 4.079 4.040 3.993 3.946 3.920 �H/300
3.908 3.909 3.931 3.945 4.079 3.989 3.962 3.926 3.864 H/300
3.707 3.640 3.591 3.564 3.552 3.540 3.516 3.408 3.260 H/300
3.217 3.409 3.476 3.492 3.552 3.492 3.476 3.409 3.217 �H/300
3.702 3.727 3.724 3.738 3.722 3.736 3.729 3.727 3.681 H/300
3.860 3.799 3.773 3.750 3.722 3.697 3.677 3.662 3.638 H/300
stress ratio of S-T and D-T shells is 0.7 and 0.9, respectively. The section sizes of double-layer shells are much smaller than those of single-layer shells, which may lead to the load factors of double-layer shells are smaller than those of single-layer ones. In order to evaluate the effects of structural weight and stress ratio of members on load factors, two structural systems (i.e., S-T and D-T) with three allowable stress ratio (i. e., 0.7, 0.8, and 0.9) were considered, as listed in Table 10. The corre lation between load factors and structural weight under different allowable stress ratio of members is listed in Table 11. Fig. 15 shows the load factors with structural weight of S-T and D-T shells with and without imperfections. As indicated in Table 11 and Fig. 15, single-layer reticulated (S-T) shell has higher load factors than double-layer reticu lated (D-T) shell with the similar weight, i.e., the weight of S-T shell is lighter than that of D-T shell with the same stress ratio. In addition, Fig. 16 shows the load factors with stress ratio of S-T and D-T shells with and without imperfection, and indicates that D-T shell has higher load factors than S-T shell with the same stress ratio. Furthermore, Fig. 17 shows the ratio of load factor to weight with stress ratio of members. As indicated in Table 11 and Fig. 17, S-T shell has higher load factors per weight than D-T shell, thus single-layer reticulated shell is recom mended considering costs of the structures. Note: the 1st column lists the allowable stress ratio of members, the 2nd column lists the load factor of perfect structure, the 3rd column lists the minimal load factor with imperfect amplitude of H/300, the 4th column lists the structural weight, and the 5th column lists the ratio of load factor with perfect structure to weight.
D-R
D-Q
4. Conclusions The increasing demand for resources results in a need for super-large cooling towers and there is a trend of development of steel structure considering its superior performance in the large-span spatial structures. In order to ensure structural performance, the nonlinear stability and the selection of structural systems are of great importance. In this paper, the nonlinear stability analysis was conducted on a 150 m high hyperbolic steel cooling tower considering different structural systems. Some con clusions of this study can be summarized as follows. (1) Material nonlinearity should be taken into account in nonlinear stability analysis of hyperbolic steel cooling towers and could have a significant influence on the stability analysis. The effects of material nonlinearity on the stability of the double-layer reticulated shells are much higher than those of the single-layer ones. (2) For the hyperbolic steel cooling tower shells, the first-order buckling mode is not necessarily the most unfavorable mode. It is recommended that more forms of imperfection (not only the first buckling mode) should be taken into consideration in nonlinear stability analysis. (3) The correlation of the stability factors and the modal orders is not explicit. The load factors vary in a small range with different imperfection distributions. For the researched five models, hy perbolic steel cooling towers are of relatively low imperfection sensitivity, which is different from most other thin-walled shells.
Fig. 13. Imperfection applied positively (left) or negatively (right). 11
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Thin-Walled Structures 146 (2020) 106448
Fig. 14. Relationship between load factor and imperfection amplitude.
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Table 9 Stability with imperfection amplitudes. Structural system
S-T S-R D-T D-R D-S
Mode order
3 15 4 7 3 6 7 10 10 19
Change in stability after applying imperfections Imperfection applied positively
Imperfection applied reversely
Decrease Decrease Decrease Decrease Decrease Decrease Decrease Decrease Slightly decrease Decrease
Decrease Decrease Decrease Decrease Decrease Decrease Increase Decrease Slightly decrease Increase
Imperfection sensitivity pattern
SIS SIS SIS SIS SIS SIS ASIS SIS SIS ASIS
Fig. 15. Load factors with structural weight of models S-T and D-T with and without imperfection.
(4) Under the premise that the weight of each structural system is at the same level, considering the effects of imperfections, the load factors of shells with rectangular grid can be reduced by 12.1%, of which the imperfection sensitivity is relatively high. When it comes to triangular grid and square pyramid grid, the largest reductions are 5.3% and 2.4%, respectively, the imperfection sensitivity of which is relatively low. According to the load fac tors, single-layer reticulated shells are better than double-layer ones, and the triangular grid is better than the other two grid forms. The load factors of single-layer reticulated shell with triangular grid range from 4.36 to 4.60. (5) With the same level weight, the load factors of single-layer reticulated shells are higher than those of double-layer ones, while the former is lower than the latter with the same allowable stress ratio of members. Furthermore, the load factors per weight of single-layer reticulated shells are higher than those of doublelayer ones. Considering load factors, imperfection sensitivity and costs of each structural system, it is recommended that the design of steel cooling towers could give priority to the scheme of singlelayer reticulated shell with triangular grid.
Fig. 16. Load factors with stress ratio of models S-T and D-T with and without imperfection.
Table 10 Structure schemes of cooling towers under different stress ratio. Structural system
Type of reticulated shell
Form of girds
Side length of grids (m)
Thickness of grids (m)
Structural weight (t)
Stress ratio of members
Range of section size
S-T-0.7
Single-layer
Triangle
6
4
3462.3
<0.70
S-T-0.8
Single-layer
Triangle
6
4
3164.9
<0.80
S-T-0.9
Single-layer
Triangle
6
4
2937.4
<0.70
D-T-0.7
Double-layer
Triangle
4
4
3987.7
<0.70
D-T-0.8
Double-layer
Triangle
4
4
3739.1
<0.80
D-T-0.9
Double-layer
Triangle
4
4
3461.6
<0.90
Φ159 � 6 – Φ900 � 18 Φ159 � 6 – Φ800 � 16 Φ159 � 6 – Φ800 � 12 Φ159 � 6 – Φ550 � 12 Φ159 � 6 – Φ500 � 12 Φ159 � 6 – Φ450 � 12
Table 11 Correlation between load factors and structural weight under different allowable stress ratios of members. Stress ratio 0.7 0.8 0.9
S-T
D-T
Load factor (LF)
LF with imperfection
Weight (t)
LF/Weight ( � 10 3/t)
LF
LF with imperfection
Weight (t)
LF/Weight ( � 10 3/t)
4.556 4.027 3.677
4.356 3.769 3.476
3462.3 3164.9 2937.4
1.316 1.273 1.252
4.746 4.464 4.079
4.427 4.022 3.864
3987.7 3739.1 3461.6
1.190 1.194 1.178
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Fig. 17. Ratio of load factor to weight with stress ratio of members.
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