Use of FRP pultruded members for electricity transmission towers

Composite Structures 105 (2013) 408–421

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Use of FRP pultruded members for electricity transmission towers Ahmed Godat a,⇑, Frédéric Légeron b, Vincent Gagné b, Benjamin Marmion b a b

Department of Construction Engineering, École de Technologie Supérieure, Université de Québec, Montreal, Québec H3C 1K3, Canada Department of Civil Engineering, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada

a r t i c l e

i n f o

Article history: Available online 31 May 2013 Keywords: FRP pultruded members Electricity transmission towers Experimental results Prediction models Design equations Cost estimate

a b s t r a c t This study investigates the replacement of traditional materials (steel, wood and concrete) in electricity transmission lines by fiber glass pultruded members. The first part of the study summarizes a comparison between different design approaches to experimental data for glass fiber pultruded sections. For this purpose, a total of fifteen specimens made of E-glass and either polyester or vinylester matrix are tested: (i) angle-section, square-section and rectangular-section specimens are subjected to axial compression; (ii) I-section and W-section specimens are tested under bending. The experimental results are summarized in terms of the failure mode, critical buckling load and load–displacement relationships. Design equations available in FRP design manuals and analytical methods proposed in the literature are used to predict the critical buckling load and compared to the experimental results. Design of various FRP pultruded sections and cost estimate are conducted for 69 kV electricity transmission portal frame and a total distance of 10 km. The significance of the present findings with regard to economic solutions is discussed. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Fiber reinforced polymers (FRP) represent a new class of construction materials. Advances in pultrusion technology lead to produce larger pultruded parts capable of serving as structural members [1,2]. The characteristics of low conductivity, light weight and ease of assembly present the opportunity of FRP to replace the steel in electricity transmission towers. In addition, FRP is very durable as it is characterized by its extensive use in the marine industry over a long period. This makes the FRP an interesting material to be evaluated structurally. However, most fabricators of FRP transmission lines produce FRP pole products similar to wood transmission lines by providing the capacity based on the pole height. Even if this approach has proven its effectiveness for some projects, there is no study in the literature, up to the authors’ knowledge, to evaluate the possibility of direct designing FRP pultruded members for electricity transmission lines. This approach would improve the optimization of structures at the line system level according to line configurations and loadings. Using such optimization, engineers can design electricity transmission lines with the most appropriate material (wood, steel, concrete or FRP). In addition, structures can be designed with different configurations such as wood portal frames, steel tower or hybrid structure using different materials.

⇑ Corresponding author. E-mail addresses: [email protected] (A. Godat), [email protected] (F. Légeron), [email protected] (V. Gagné), [email protected] (B. Marmion). 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.05.025

At this moment there is no standard for the design of FRP pultruded members. However, in the past two decades, numerous studies described the structural behavior of FRP pultruded members. A considerable amount of experimental tests were carried out to investigate the behavior of I-section FRP pultruded members pertaining axial compression [3,4,5,6,7,8,9,10,11,12]. Relatively, few researches exist on the buckling behavior of angle-section members [13,14], T-section members [15] channel-section members [16,17,18,15,19] as well as box-section members [20,21,14]. Some studies were performed to investigate the local buckling and lateral-torsional buckling of beams under transverse load [22,23,24,25,26,27,28,29,17,30,31,32,33,34,35,36]. A study examined the shear deformability of a beam with a U-section under transverse load is found [37]. Valuable studies investigated a frame made of FRP pultruded members and subjected to vibration are carried out by Boscato and Russo [38] and Russo [39]. Buckling is the dominant failure mode that occurs in FRP pultruded members due to its relatively low stiffness associated with small thickness. Whether it is in compression or bending, buckling generally appears at low stress and prevents FRP pultruded members from attaining their full capacity. Even though substantial research studies on the behavior of FRP pultruded members have been conducted, reliable design criteria is still absent. In addition, there is no detailed study to evaluate design manuals proposed by different FRP manufacturers to predict the critical buckling capacity against reliable experimental data. Therefore, it remains difficult for structural engineers to decide the appropriate design procedure when designing structures made of FRP pultruded members.

A. Godat et al. / Composite Structures 105 (2013) 408–421

This study collates experimental data from full-scale laboratory tests to investigate the structural behavior of different FRP pultruded members made of E-glass and either polyester or vinylester matrix. It compares the results of the tests to analytical calculations. The accuracy of the available design manuals to predict the strength of FRP pultruded members is examined. The paper is organized in three major sections: (1) experimental tests; (2) validation of theoretical predictions; and (3) design example and cost estimate of FRP electricity transmission portal frame. 2. Experimental program 2.1. Specimens design The study included fifteen off-the-shelf FRP pultruded specimens manufactured and supplied by Strongwell with a serial numbers 500/525/625. The specimens were made of laminated plates of E-glass fibers impregnated with polyester or vinylester matrix. The E-glass fiber type was selected in this study because it is sparingly used in construction, while other fiber types (carbon and aramid) are more costly. The cross sections considered were six equalleg angle-section specimens, three square-section specimens, three rectangular-section specimens, two I-section specimens and a Wsection specimen. The dimension of the specimens were selected close to that might be used in electricity transmission towers and summarized in Table 1. In the table, L is the length, A is the cross-section area and I is the second moment of inertia. No imperfection measurements were carried out in this study. Results provided by the manufacturer indicated that maximum deviation due to out-of-straightness for the various specimens considered is well within ASTM D3917-08 [40] tolerance. 2.2. Material mechanical properties In this study, coupon tests in tension and compression were carried out a priori to determine relevant orthotropic tensile material constants as well as the ultimate strength. Four coupons with appropriate dimensions were tested for each specimen in accorTable 1 Geometrical dimensions of the tested specimens (dimensions in mm).

409

dance with the ASTM D638–03 [41]. The coupons were cut from each structural element in a given cross section, i.e., the flanges and web of the I and W-sections and each wall of the box and angle-sections. The mechanical properties as reported by the manufacturer as well as those measured are given in Table 2. In the table, the subscript x is used to denote for properties parallel to the fiber orientation, while y is used for those perpendicular to fiber orientation. From Table 2, it is seen that properties reported by the manufacturer are conservative. Manufacturers attempt to provide safe value that would cover variation in property data inherent to the manufacturing process. Note that the angles, boxsections and W-section specimens were series 500 (polyester resin), while the I1-section beam was series 525 (polyester resin). The I2-section beam was series 625 (vinylester resin). 2.3. Test setup and instrumentation In this study, two series of tests were performed: (i) axial compression, and (ii) bending. The axial compression test setup for the square-section and rectangular-section specimens was different from the one carried out for the angle-section specimens. The bending test was performed for the I-section and W-section specimens. On days of testing, the room temperature was between 20 and 25 °C in accordance with the ASTM procedure. 2.3.1. Angle-section test In this study, six angle-section specimens were tested with a testing frame that is designed to perform tension or compression tests. Note that this number of the angle members was chosen in order to evaluate the variability of the member properties. As seen in Fig. 1, the specimens were connected to a gusset plate part of a stiff testing frame. The connections were similar to what could be found in the practice. The angle-section specimen is inclined at 45°, while the load is applied at the top horizontal member of the loading frame via a displacement-controlled at a rate of 0.6 mm/min. The bottom horizontal member of the frame was kept fixed. Outof-plane displacement of the frame was prevented by a cable system. The angle members were connected to the loading frame with

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Table 2 Material mechanical properties of the tested specimens. Reported Longitudinal tensile strength rt,x (MPa) Transverse tensile strength rt,y (MPa) Longitudinal compressive strength rc,x (MPa) Transverse compressive strength rc,y (MPa) Shear strength rs (MPa) Longitudinal bending strength rb,x (MPa) Transverse bending strength rb,y (MPa) Longitudinal elastic modulus Ex (MPa) Transverse elastic modulus Ey (MPa) Shear modulus G (MPa) Longitudinal poisson ratio mx Transverse poisson ratio my 

*

207 48 207 103 31 240 100 17250 5500 2930 0.33 0.11

Measured 500/525 series

Measured 625 series

301(47) 61(21) – –– –– – –– 19550(3090)** 9720(2580) – – –

246(27) – – – – – – 20870(1920) – – – –

These values are valid for temperature between 20 and 60 °C according to Strongwell. Average (standard deviation).

**

three steel 15 mm bolts and a 20 mm thickness gusset plate at each end. Three linear variable differential transducers LVDTs were installed along the specimen length at each end and the centre to monitor the axial shortening. Data including the applied load and axial displacement were collected during testing by a data acquisition system. The sampling frequency was one second. Loading was continued up to failure characterized by a large drop in strength. 2.3.2. Box-sections test Six box-sections, three rectangulars and three squares were tested with a compression machine having a capacity of 5500 kN. It was assumed that the number of specimens considered is sufficient to evaluate the variability of the results. The test configuration was designed to cause local buckling in the specimens. As shown in Fig. 2, a steel plate was placed on the top of the section to ensure equal distribution of load to the top surface. Transverse displacements due to buckling were measured at one face by LVDTs fixed along the length of the specimen as shown in Fig. 2. The base of the specimens was seated on steel plates. Therefore, specimens were simply supported along the reaction surfaces but effectively restrained against radial movement by friction. The load was applied by a displacement control method with fast rate up to 50% of the estimated failure load then the speed of the loading rate was lowered as the expected failure load was approached. The specimens were visually inspected during the loading. 2.3.3. Bending test Three simply supported beams with a 2.2 m span were tested under monotonic static loading up to failure. The simple supports consisted of steel rollers between thick bearing plates (12.7 mm) at each end to prevent crushing failure. The load was applied by a bearing centered under the actuator to distribute the load evenly with different configurations, as shown in Fig. 3. For the W and I1

Fig. 2. Compression test setup for square and rectangular-section specimens.

specimens, the force was applied by a thick steel plate. I2-section beam was subjected to the four-point bending test. Its setup involves two steel rollers beneath the actuator with 270 mm distance between the load points. The three types of beams were selected in order to provide different failure modes: (i) the top flange of the W-section was expected to buckle locally; (ii) the I1-beam was presumed to fail in tension and compression without instability, while (iii) I2-beam was supposed to fail in lateral torsional buckling. At each support, the presence of bracing at the top flange provided the restraint against twisting, thus providing the assumed boundary conditions for the theoretical analyses. Vertical displacement was measured at the center of the span using LVDTs, while the lateral displacement of the top flange was measured by strain gauge transducer. To minimize the effect of the inherent geometric imperfections, the web needed to be vertical. This was achieved by

Fig. 1. Test setup for angle-section specimens.

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A. Godat et al. / Composite Structures 105 (2013) 408–421

plumb bobs at the free ends of the beam to ensure that the web was indeed vertical.

30

The experimental results presented in the following sections are in terms of critical buckling load, load–displacement relationships and failure modes. The critical buckling load reported is the ultimate load that the specimens can support at failure. In this study, the average critical buckling loads are reported due to small discrepancy between the test results. The small discrepancy can be attributed to the manufacturing process, which might induce geometrical imperfections and material variation.

Load kN

3. Experimental results

6,35

40

76,2

20

10

0 0

10

3.1. Load–displacement relationships

20

30

40

50

Lateral Displacement mm

3.1.1. Angle-section specimens Fig. 4 presents the average load–displacement relationship for the angle-section specimens. Overall results were very much reproductible on the six angles. The average critical buckling load was 32.6 kN with a standard deviation of 0.7 kN. The variation in the critical buckling load between the six specimens was 3%. From the figure it was observed that the load–displacement behavior is linear when the angle-section is only affected with the axial compression. When the bending due to eccentricity of load application initiated to influence the behavior, the relationship changed to nonlinear. This behavior continued up to failure of the specimen. The change in the slope of the load–displacement in low load value marked the initiation of global buckling, which indicated the end of the linear elastic behavior. The application of the load in one leg of the specimen provided a slight flexure, while the free leg worked as a stiffener. 3.1.2. Box-section specimens The average load–displacement relationships for the squaresection and rectangular section specimens are presented in Figs. 5 and 6, respectively. For the square-section specimens, it was observed that the load–displacement relationship is linear elastic until failure. The load–displacement plot of the rectangular-section (Fig. 6) showed a linear response up to 80% of the critical buckling load. With the load increase, a significant reduction in the slope of the load–displacement was observed. The slope stayed fairly constant up to failure. The change in the slope was explained by the onset of local buckling on the longer side of the specimens result-

Fig. 4. Average load–displacement relationship for angle-section specimens.

ing in a longitudinal crack. The average critical buckling loads recorded for the square-section and rectangular-section specimens were 716 kN and 399 kN with standard deviations of 14.0 kN and 2.5 kN, respectively. The test results confirmed that although the cross-sectional area of the square-section is lower compared to the rectangular section, the square-section gives higher loading capacity. This explains the influence of local buckling which is related to the inverse of the square of the aspect ratio (b/t), which is higher in the rectangular-section specimen. 3.1.3. W and I-section specimens Plots of the load–displacement relationships for the W-section and I2-section specimens are shown in Figs. 7 and 8, respectively. As can be seen in the plots, a linear load–displacement relationship was observed for the W-section, while an early small change in the slope of the load–displacement curve was observed in the I2-section. From the curves, it can be noted that the various configurations of load application do not change the load–displacement plot and both beams behaved in similar manner. The test result of the I2-section member showed significant increase in the maximum buckling load (67.2 kN) over the I1-section member (18.6 kN). The maximum buckling load of the I2-section beam was almost 3.6 that obtained for the I1-section specimen, while the difference in the section modulus (Sx–x) was 2.6.

Fig. 3. Different loading configurations for bending test.

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80

9,53

800

Load (kN)

40

9 ,5 3

12,7

Load (kN)

152,4

400

457

60

600

20

200

0

0 0

1

2

3

4

5

6

114

0

Fig. 5. Average load–displacement relationship for square-section specimens.

1 5 2 ,4

Load (kN)

229

400

300 9 ,5 3

200

100

0

0

1

2

8

12

Displacement (mm)

Displacement (mm)

500

4

3

4

Displacement (mm) Fig. 6. Average load–displacement relationship for rectangular-section specimens.

Fig. 8. Load–displacement relationship for I2-section specimen.

affecting the local buckling capacity of sections. In this study, the plate width-to-thickness ratio is 11.5, 15.0 and 27.8 for the angle-section, square-section and rectangular-section, respectively. For the rectangular-section, the ratio is calculated for the longer side. The failure mode (discussed in the subsequent section) reported for the angle-section was global buckling, whereas local buckling was observed for the square-section and rectangular-section. From the comparison between the two slenderness ratios it can be observed that for the angle-section, the possibility of occurrence of global buckling is higher than the local buckling due to the great difference between the two ratios. The two ratios are very similar for square-section; hence, the two failure modes are applicable. For the rectangular-section, the local slenderness ratio is greater than the global one. This explains the local buckling failure observed in this specimen. From this comparison it can be observed that the values of slenderness ratios follow the failure modes reported experimentally.

3.2. Effect of slenderness ratio

3.3. Failure modes

The slenderness ratio is undoubtedly one of the important parameters affecting the buckling capacity of FRP pultruded members under axial compression. In this study, the slenderness ratios of the specimens tested under axialpcompression are compared. ffiffiffiffiffiffiffi The global slenderness ratio (k ¼ L= I=A) of the angle-section is 101.85, while it is 14.5 and 12.7 for the square-section and rectangular-section, respectively. It is well known that the local slenderness ratio (plate width-to-thickness ratio) is an important factor

3.3.1. Angle-section specimens A photograph of the failure of the angle-section specimen is presented in Fig. 9. The failure was characterized with a global buckling accompanied with change in the cross section. The global buckling was a lateral displacement of the specimen in a halfwavelength associated with change in the cross-section. As seen in the figure, the rupture of the specimen occurred suddenly at the areas of highest compressive stress (bolted edges) and extended towards the middle of the specimen. The location of the crack was consistent with the location of low fiber content. This finding agrees with other researchers [4]. It is of interest to mention that specimens returned to their original shape without apparent permanent deformations, with the exception of rupture at the leg junction, after release of load.

254

40

9,53

20

9,53

Load (kN)

30

254

10

0

0

2

4

6

8

Displacement (mm) Fig. 7. Load–displacement relationship for W-section specimen.

10

3.3.2. Box-section specimens The failure modes of rectangular-section and square-section specimens are presented in Fig. 10. For both sections, the failure mode observed was the local plate buckling. The failure of the square-section specimens involved all sides at the critical cross section and occurred in a localized laminate where the compression side laminate buckled-in as the two adjacent laminates buckled-out (Fig. 10). The failure initiated with a crack occurs at corners of the section. For the rectangular-section specimens, a three-wave local buckling was observed in the wider side of the cross-section. It was noted that the smaller side of the cross section stiffens for the wider side. The location of the crack depended on the local

A. Godat et al. / Composite Structures 105 (2013) 408–421

413

Fig. 9. Buckling failure for angle-section specimen.

buckling waves. In addition, a crack along the specimen length occurred in the shorter side of the cross-section, where stresses were higher than the wider side. No audible acoustic emission was heard during the test prior to the failure. 3.3.3. W-section specimen For the W-section specimen, the failure occured due to local buckling of the loaded flange at the area under the load associated with rupture at the junction between the web and the compression flange in the buckled region, as seen in Fig. 11. The later failure was related to the low stiffness in the junction region due to fiber architecture and resin-richness. By increasing the strength and stiffness of the filet region such failure mode is prevented to cause the beam fail in a desired mode [30]. The onset of local buckling in the flanges was clearly seen from the sudden change in flange displacements. In addition, the failure was accompanied with a horizontal crack near the mid-depth of the web along the length of the loading plate. The failure mode obtained in this study was also observed by other researchers [27]. 3.3.4. I-section specimens The I-section specimens failed due to lateral-torsional buckling. In this type of failure, the flanges displaced laterally (or sideways) relative to transverse load direction and the web twisted, causing the entire beam to move out of its vertical plane. During the test, the twist of the flanges was clearly visible with strong snapping sound. Similar to that observed for the W-section, failure occurred due to a crack at the intersection between the loaded flange and the web. This is attributed to the small fibers content at these locations. The specimens were loaded and measurements were taken

Fig. 11. Local buckling failure for W-section specimen.

until no increase in load was recorded. It is necessary to mention that elastic behavior is the dominant throughout the test. At removal of the load, there are no permanent deformations. Note that no bearing failure at the support is observed. 4. Comparison of test results with prediction models In this study, buckling was the governing failure mode and the critical buckling load was directly related to the carrying capacity of the members. As such, there is a need for various design procedures to be assessed critically against reliable and relevant experimental evidence. Following the previous discussion on the behavior and buckling load capacity of the FRP pultruded members, it is of interest to see how the critical buckling loads compares with the predictions from available design manuals. In addition, the study considers analytical formulations that predict the critical buckling load of the FRP pultruded members. Design manuals considered in this study [42], Creative Pultrusions [43,44,45] presents the most comprehensive source of design procedures for FRP pultruded members. Available design equations to compute the critical buckling load is presented based on the type of failure. Stresses and critical buckling loads are calculated using mechanical properties reported by the manufacturer (Table 2) when the measured ones are not available. In addition, safety factors proposed by the design manuals are not applied. 4.1. Specimens subjected to axial compression

Fig. 10. Local buckling failure for square-section and rectangular-section specimens.

4.1.1. Design concept Open-section members subjected to axial compression buckle either in global buckling, local buckling or lateral torsional buckling. For box sections, the torsional rigidity is high and torsional buckling is seldom a limiting state; hence not considered for such specimens. In this study, the various possible limit states (global buckling, local buckling and lateral torsional buckling) are considered for angle-section members, while the first two limits are investigated for the box-sections. The global buckling failure mode governed the behavior of the angle-section specimens. The effect of the bending is neglected because there is no design expression to date express the global buckling-bending interaction. Three design equations are considered by the design manuals to compute the global buckling load. The Euler equation is adopted by European [42] and Pultex [43]. The Fiberline critical buckling stress (rcr) equation for global buckling is:

414

rcr ¼

A. Godat et al. / Composite Structures 105 (2013) 408–421

Nel

ð1Þ

1 þ NFel d

where Fd is the compression strength and given by Fd = A rc.x; A is the area of the cross-section; Nel is the Euler critical buckling load. Strongwell design manual (2007) developed an empirical equation based on their experimental tests as follows:

rcr ¼

EL kL0:55 56 r

ð2Þ

where EL, k, L and r are the elastic modulus parallel to fiber orientation, buckling factor, length of the member and radius of gyration, respectively. The critical buckling load (Fcr) is calculated by (Fcr = rcrA). For the global buckling, two values of k are considered: 1.0 and 0.75. The value of k = 0.75 is used to take into account the stiffness of the connections. For FRP pultruded beams or columns, local buckling can be predicted as an assemblage of orthotropic plates with consideration of the flexibility of web-flange junctions. The critical buckling load is then determined either: (i) exactly, by assuming all plates that constitute the member buckles simultaneously and the continuity conditions at web-flange junctions are respected [39] or; (ii) approximately, where plates that constitute the member buckles individually and web-flange junctions are flexible to a certain degree [46]. Recently, numerical procedures are proposed for plates under uniform compression, when the edges are considered constrained into two sides. First, two edges of the plate are constrained against rotation (box-section members and webs). Second, one edge is free and the other is constrained against rotation (flanges). In both cases, the compression stresses are applied in the other free edges. In addition to the design equations provided in the design manuals, the design equations to predict the local buckling load of Pecce and Cosenza [9] are considered. A summary of these equations is presented in Table 3. It is necessary to note that details of each design equation and their limits of application are available in the corresponding references. The definition of each symbol is presented in the appendix. The Strongwell design manual proposed empirical equations for the local buckling of angle-section and box-section members, respectively, as follows:

rcr ¼

EL  0:95 27 bt

ð11Þ

rcr ¼

EL b0:85 16 t

ð12Þ

where b and t are the plate width and thickness, respectively.

In comparison to local and global buckling, only Pultex [44] design manuals provides a design equation for the case of lateral-torsional buckling for angle-sections. The equation is given as:

rcr ¼ /

 2 E t 2ð1 þ mÞ b

ð13Þ

the orthotropic coefficient / is taken equal to 0.8. Zureick and Steffen [47] presented an equation to predict the lateral-torsional buckling of angle-section members as follows:

Gxy F cr ¼ 0:9  2 b

ð14Þ

t

where Gxy is the shear modulus. The results of the both equations are compared with the experimental loading capacity. 4.1.2. Comparison with design equations Table 4 presents the comparison between the theoretical values of the critical buckling load and the experimental ones for the specimens subjected to axial compression. The various design equations used are given with their corresponding numbers. For each section, the theoretical values are compared with the average experimental critical buckling load. The global buckling is calculated for two values of k (1.0 and 0.75). For the angle-section specimens, the average critical buckling load obtained experimentally was 32.6 kN. According to the findings shown in the table, it can be observed that the results of global buckling with k = 0.75 gives the best results compared to the average experimental buckling load. The good correlation between the experimental and predicted results confirms the type of fixity of the specimen assumed, which is intermediate between pin-connected and perfectly fixed. The results of the global buckling with k = 1.0 deviate somewhat from the experimental value. The design equations for the local buckling, except Pecce and Cosenza [9] equation, predict the critical buckling load with high discrepancy. Pultex lateral-torsional buckling equation shows acceptable agreement with the experimental load, while Zureick and Steffen equation predict the critical buckling load with high discrepancy. The average critical buckling loads obtained experimentally for the square-section and rectangular-section are 716.0 kN and 398.7 kN, respectively. For the box-section specimens, the design equations for the global buckling overestimate the critical buckling load with high discrepancy regardless of the type of fixity, while the design equations for local buckling predict close results to the experimental values. This result follows failure mode obtained experimentally, which was local buckling. For the square-section, European [43] design equation gives the best results compared to the experimental value. The local buckling equation of Strongwell

Table 3 Design equations of critical buckling stress for local buckling failure.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  D11 D22 þ D12 þ 2D66

Eurocomp (1996)

2p2 tb2

Pecce and Cosenza [39]

–

Pultex [17]

p2 EL t /k 12ð1 m2L Þ bs

(6)

Davalos et al. [20]

p2

(7)

–

(8)

pffiffiffiffiffiffiffiffiffiffiffiffi

(3)

a 2

2 pffiffiffi pffiffiffiffiffiffiffiffiffiffi  t qð2 EL ET Þ þ pðmL ET þ 2GÞ bs hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 p 1 þ 4:139nð2 D11 D22 Þ þ ð2 þ 0:62n2 ÞðD12 þ 2D66 Þ 2

12

Kollàr [44]

tb

ET EL

(4)

2

þ p tLD211 0:85

12D66 2 tb

2

p EL 12ð1mL mT Þ



(5)

tf bs

–

pffiffiffiffiffiffiffiffiffiffiffiffi D11 D22 ½15:1g 1  m þ 6ðK  gÞð1 þ mÞ for K > 1.0 2 tb pffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffi D11 D22 1K ffi Kð15:1g 1  m þ 6ðK  gÞð1 þ mÞÞ þ 7 pffiffiffiffiffiffiffiffiffiffiffiffiffi for K 6 1.0 2 tb

1þ4:12n

(9) (10)

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A. Godat et al. / Composite Structures 105 (2013) 408–421 Table 4 Comparison between experimental buckling load and design equations for specimens subjected to axial compression. Type of section

Limit state

References

Predicted load Pcr (kN)

Ppred/Pexp

Angle-section

Global buckling (k = 1.0)

Eurocomp (1996) Pultex [42] Fiberline [47] Strongwell [40] Eurocomp (1996) Pultex [42] Fiberline [47] Strongwell [40] Eurocomp [2] Pecce and Cosenza [22] Strongwell [4] Pultex [8] Zureick and Steffen [46] Eurocomp (1996) Pultex [42] Fiberline [47] Strongwell [40] Eurocomp (1996) Pultex [42] Fiberline [47] Strongwell [40] Eurocomp [39] Pultex [24] Strongwell [5] Eurocomp (1996) Pultex [42] Fiberline [47] Strongwell [40] Eurocomp (1996) Pultex [42] Fiberline [47] Strongwell [40] Eurocomp [39] Pultex [25] Strongwell [5]

17.0 17.0 16.3 24.9 30.3 30.3 27.3 29.2 18.9 36.6 62.6 37.4 16.8 4820.8 4820.8 1192.7 4130.0 8570.2 8570.2 1337.5 6003.3 758.5 148.7 609.7 6726.1 6726.1 1357.8 5244.7 11957.6 11957.6 1489.4 7623.3 250.3 49.1 396.5

0.52 0.52 0.50 0.76 0.93 0.93 0.84 0.90 0.58 1.12 1.92 1.15 0.52 6.70 6.70 1.67 5.77 11.97 11.97 1.87 8.38 1.06 0.21 0.85 16.87 16.87 3.41 13.15 29.99 29.99 3.74 19.12 0.63 0.12 0.99

Global buckling (k = 0.75)

Local buckling

Lateral-torsional buckling Square-section

Global buckling (k = 1.0)

Global buckling (k = 0.75)

Local buckling

Rectangular-section

Global buckling (k = 1.0)

Global buckling (k = 0.75)

Local buckling

[46] estimates the critical buckling load for rectangular-section specimen with a discrepancy of 1%. Pultex [44] design equation underestimates the local buckling load for square-section and rectangular-section specimens with various discrepancies (Table 4).

sional buckling capacity of FRP pultruded beams. Extensive research data demonstrated that the steel beam lateral-torsion equation developed by Mitchell [51] was modified for FRP pultruded beams and adopted in Pultex [44] and Strongwell [46] design manuals. It is given by:

4.2. Specimens subjected to bending 4.2.1. Design concept FRP pultruded members are especially subjected to local buckling under transverse loads due to the low in-plane moduli and the slenderness (width-to-thickness ratio) of the plate elements that make up the section. As mentioned earlier, the critical local buckling load (or stress) of a section is a function of the boundary conditions on the longitudinal edges of the section. In this study, to account for the critical local buckling load, the previous equations used for the specimens under axial compression are used. As well, the Kollar [48] equation is implemented to predict the capacity because it is widely accepted by many researchers [49] for FRP pultruded members under bending. It is necessary to mention that Pultex local buckling equation (Eq. (6)) is for members under axial compression. Pultex uses the local buckling equations developed by Davalos et al. [50] (Eq. (7)) for members under bending. Strongwell equation for local buckling of I-sections under bending is proposed as follow:

0:5E

rcr ¼ 1:5L b

ð15Þ

f

tf

On the other side, the doubly symmetric pultruded profiles are subjected to lateral-torsional buckling when loaded by transverse loads. Few design equations are found to estimate the lateral-tor-

Mzcr

C1p ¼ K y Lu

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   pE 2 C w Iy þ EIy Gxy J K y Lu

ð16Þ

where J is the modulus of torsion; Lu is the clear span; C1 and Cw are coefficients to adjust moment gradient and buckling, respectively; Ky is the coefficient of effective length. Values for C1 and Ky depend on the type of loading and end support restraint conditions and taken as 1.0 in this study. The factor Cw is given by: 2

Cw ¼

Iy d 4

ð17Þ

4.2.2. Comparison with design equations The comparison between the theoretical and experimental results is presented in Table 5. The various design equations calculates the critical buckling load obtained for the two ultimate states (local buckling and lateral-torsional buckling). The critical buckling load measured experimentally was 32.1 kN for the Wsection specimen, 18.6 kN for the I1-section specimen and 67.2 kN for the I2-section specimen. For the W-section specimen, the design equations of local buckling predict the load capacity with various discrepancies compared to the experimental value. The equation of lateral-torsional buckling overestimates the buckling capacity of W-section with high difference. The design equation of Eurocomp provides very reasonable accuracy compared to

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A. Godat et al. / Composite Structures 105 (2013) 408–421

the experimental value. The lateral-torsional buckling failure governed the behavior of I-section specimens. For the I1-section, all the equations overestimate the loading capacity with various ranges. The equation of lateral-torsional buckling gives the less discrepancy compared to the others. For the I2-section specimen, the equation of lateral-torsional buckling shows an excellent agreement with the experimental value. The discrepancy obtained between the lateral-torsional buckling equation and the experimental value is 1%. The local buckling design equations show various discrepancies compared to the experimental buckling load. The result obtained by this equation confirms the hypothesis that the design manuals are, sometimes, accurate to predict the capacity of pultruded members. For the three specimens, the Strongwell [46] local buckling equation overestimates the value of the critical buckling load with high discrepancy. 5. Example of structural design of electricity transmission towers

used are reported on Table 6. The loading on the line is chosen typical to that in Canada and is made of four combinations as presented in Table 7: (i) Maximum wind corresponding to a winter storm with no ice; (ii) light ice design load 1 on a conductor combined with high wind; (iii) light ice design load 2 on a conductor combined with lower wind; (iv) heavy ice with moderate wind. These four cases are taken from real data of a project in Canada. According to the IEC, the wind speed (VRB) is converted to wind pressure (q0) with the following equation:

q0 ¼ 0:5slK R V RB

ð18Þ

where s is the air density correction factor calculated according to the IEC and reported in Table 7; l is the air mass per unit volume at temperature of 15° and taken as 1.225 kg/m3; KR is the terrain roughness factor which is usually taken as 1.0 for areas with few obstacles. The weight of the ice, g (in N/m), is calculated based on the following equation:

g ¼ 9:82  106 dptðd þ tÞ

ð19Þ 3

This section presents a complete description of the configuration and dimension of a transmission tower, calculation of loads, design of FRP pultruded members and cost estimate. An electricity transmission tower with 69 kV for a distance of 10 km is considered. Various spans between the towers range between 100 m and 400 m in increment of 50 m. For these spans, the design is carried out with one of the following FRP pultruded members: (i) circular-section; (ii) rectangular-section; and (iii) I-section. For each of these members, cost estimate is provided in order to determine the most economical solution. The loads are calculated using the international standard IEC 60826 [52]. Many parts of this code are used worldwide. It is necessary to mention that sections and mechanical properties used in this study are identical to that of Strongwell E-glass fiber. It is chosen because of its availability in North America. The mechanical properties provided by the manufacturer and presented in Table 2 are used in the design.

where d is the ice density taken as 900 kg/m ; t the ice thickness in mm; d the diameter of the iced cable, respectively. The overall height of the portal frame and the height of the cross arms is determined based on a minimum distance between the ground and the conductor as illustrated in Fig. 13. These values depend on the tension of the electrical line. Based on the Canadian standard (CAN/CSA-C-22.3 No. 1 [53]), a value of 5.5 m is selected in this research for a 69 kV line. This value is similar in most countries and it should be respected for all loads. The two loads that can produce the maximum conductor sag are the maximum operating temperature and the maximum ice load. Under these loads, the conductor sag can be calculated from the initial condition at erection. To calculate the value of initial sag, the elastic catenary equation from Irvine [54] is used with the initial tension reported in Table 6. To calculate the maximum sag under the maximum ice and the maximum temperature, the following equation is applied:

5.1. Line configuration and loading conditions

 2  2 n2 mg L3 n mg L3 H2  H1  1 ¼ L0 þ L0 aðh2  h1 Þ 24 24 EA0 H2 H1

The configuration of the portal frame considered in this study is shown in Fig. 12. It is a portal frame with two columns and crossbracing. The portal frame supports 3 conductors on a horizontal cross arm and 2 ground wires at the top of the vertical members. The phase-to-phase tension voltage of the transmission tower is 69 kV, while the phase-to-ground voltage is 42 kV. Dimensions and mechanical properties of the conductors and ground wires

where n1mg and n2mg are the total weight of the cable (including ice load when necessary), L is the span length, L0 the initial unstressed length of the conductor in the span; a is the coefficient of thermal expansion of the cable as provided in Table 6; h1 and h2 are the temperatures in the initial and loading cases, respectively. For the determination of the maximum sag, in addition to the maximum load provided at Table 7, the loading condition at maximal operat-

ð20Þ

Table 5 Comparison between experimental critical buckling load and design equations for specimens subjected to bending. Type of section

Limit state

Reference

Predicted load Pcr (kN)

Ppred/Pexp

W-section

Local buckling

I1-section

Lateral-torsional buckling Local buckling

I2-section

Lateral-torsional buckling Local buckling

Eurocomp [2] Pecce and Cosenza [22] Pultex [25] Kollar [23] Strongwell [37] Pultex and Strongwell [45] Eurocomp [2] Pecce and Cosenza [22] Pultex [25] Kollar [23] Strongwell [37] Pultex and Strongwell [45] Eurocomp [39] Pecce and Cosenza [22] Pultex [25] Kollar [23] Strongwell [37] Pultex and Strongwell [45]

24.4 46.1 60.5 46.0 103.0 298.6 49.2 65.4 113.0 94.9 148.9 36.7 41.8 117.1 18.6 94.7 751.1 68.0

0.76 1.44 1.88 1.43 3.21 9.30 2.67 3.50 6.08 5.10 8.01 1.97 0.62 1.74 0.28 1.41 11.18 1.01

Lateral-torsional buckling

A. Godat et al. / Composite Structures 105 (2013) 408–421

Ac ¼ q0 C xc Gc GL Ld

417

ð21Þ

where Cxc, Gc and GL are the cable drag coefficient (taken as 1.0), wind combined factor and factor of range, respectively. The factors Gc and GL depend on the height of the frame and on the span between the frames, respectively. Their values are taken directly from the IEC. It is necessary to note that the portal frame is subjected to horizontal forces at the locations of the attachment of the five cables. 5.3. Design of portal frame

Fig. 12. Considered electricity wooden portal frame.

Table 6 Cable dimension. Description

Symbol

Conductors

Ground wire

Diameter (mm) Cross-section area (mm2) Modulus of elasticity (MPa) Linear load (N/m) Thermal coefficient of expansion (/°C) Ultimate resistance (kN) Tension percentage of RTS (%) Tension in RTS (kN)

d A E

27.8 455 68300 14.93 1.93  105 127 25 31.75

9.1 65 172400 3.94 1.21  105 53 40 21.36

x a Tu H1

ing temperature h2 in a sunny summer day without wind is considered. Table 8 shows the maximum deflection of conductors for the various spans. The vertical height of the isolator is taken as 840 mm, which is typical to a tower with 69 kV. The distance between conductors are calculated to respect the minimum distance of the structure with the maximum swing obtained under the lateral wind load and the minimum distance between conductors. Therefore, the geometry of the portal frame is calculated according to the loads and electrical safe distances for each span length. 5.2. Calculation of load In order to design the portal frame, the traditional approach is to calculate the effect of load on the structure by: (i) calculating the loads at conductors and ground wire attachments; and (ii) applying those loads to the point of attachment in a model considering only one structure. In this traditional design, weight and wind of spans are assumed from the expected variation of spans on the line. However, in this research we have a constant span. The loads at points of attachments are calculated according to the IEC. In addition to the environmental loading, the IEC also recommends the use of safety load and unbalanced iced loads, which provide high longitudinal loads. It is understood that these loads present the same order of magnitude as transverse load but do not benefit from portal and bracing actions. The structure is modeled and analyzed with the software Advanced Design America [55], which is a powerful finite element analysis package that is able to model the structural behavior of pultruded FRP structures. The wind load (Ac) per unit length on the cable is calculated as follows:

The axial force, shear force and bending moments are obtained in the portal frame for each configuration (circular tubes, rectangular tubes and I-beam) and for each span. The dead load of the portal frame is automatically calculated by the ADA using the value of the density of the FRP pultruded members. Once the analysis is completed, the design of portal frame is made following the Eurocomp manual (1996). This manual is considered because it is comprehensive, easy to use and has shown good predictions of the experimental values as proven earlier by our comparison. Design is made for the various limit states that may influence the behavior of FRP pultruded sections mainly local and global buckling. Note that the sections selected are appropriate in terms of economy (with minimum cross section) and respecting all limit states. All members in one frame are of the same section type. In the corresponding tables, the relative size is determined based on the largest cross-section area for each solution. The relative size is given in terms of percentage from the largest size. It is realized that in real projects, engineers can use different sections to obtain the best performance (for example round vertical members with I-section cross arms and rectangular tubes for bracings). The selection of the circular-section columns (Table 9) is governed by the critical buckling moment resistance and the ultimate bending moment resistance in the main axis. The rectangular-section (Table 10) is controlled by the ultimate bending strength and the critical buckling resistance. For the horizontal members, the critical local buckling resistance in the weak axis plays an important role in determining the dimension of the circular-section as well as the rectangular-section. The dimension of the cross-bracing members in the three sections is mainly selected based on the compressive resistance. For the I-section option (Table 11), with the exception of 100 m span, the dimension of the columns is selected based on the bending moment resistance, followed by the critical local buckling moment resistance. The columns in the portal frame of 100 m span are governed by the web shear stress. For the horizontal member, some spans are controlled by the ultimate bending moment resistance and others are governed by the critical local buckling resistance. The dimensions of the tested specimens are compared to the sections of the tower for each solution. For the tested sections, the dimension of the square-section and rectangular section are 6% of the largest section in the tower in the corresponding solution, while it ranges between 3% and 4% for the I-section specimens. From the relative size, it appears that the dimensions of the tested specimens can serve as horizontal bracing and cross-bracing only. 5.4. Cost estimates Based on the design of the portal frame with different sections, the cost is carried out for the various spans. The cost of FRP pultruded members is estimated at 8 $/kg. This price may vary widely according to cost quantities of FRP pultruded members. The cost of an FRP structure is significantly expensive than similar wood or steel structures, but advantages of FRP pultruded members make it worth the difference in price. The cost estimate is only for the

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Table 7 Climatic load. Cases of combinations Climate data

Cables load

Temp. T Ice equivalent thickness

Heavy combined Light combined # 1 Light combined # 2 Maximum wind

Reference wind speed VRB Correction factor s Cables pressure q0 Ice load

(°C)

Conductor Ground Wire (mm) (mm) (km/h)

5 5 5 10

38 25 19 0

38 25 19 0

40 50 60 80

1.08 1.08 1.08 1.10

(kPa)

Conductors gcond Guard cable gcdg (N/m) (N/m)

0.082 0.128 0.184 0.333

69.43 36.65 24.69 0.00

49.70 23.67 14.83 0.00

Fig. 13. Definition of dimensions of the portal frame.

Table 8 Dimensions configuration of portal frame for various spans. Span between towers

Max. calculated deflection

Horizontal dimensions

(m)

Conductor (m)

Guard cable (m)

X1 (m)

X2 (m)

X3 (m)

X4 (m)

h1 (m)

Vertical dimensions h2 (m)

h3 (m)

h4 (m)

100 150 200 250 300 350 400

1.65 3.18 5.00 7.09 9.44 12.05 14.90

1.91 3.64 5.64 7.88 10.32 12.93 15.71

1.2 1.3 1.4 1.6 1.7 1.9 2.0

1.2 1.3 1.4 1.6 1.7 1.9 2.0

1.2 1.3 1.4 1.6 1.7 1.9 2.0

1.2 1.3 1.4 1.6 1.7 1.9 2.0

4.5 5.7 7.2 8.7 10.8 12.8 15.3

2.4 2.6 2.8 3.2 3.4 3.8 4.0

1.2 1.3 1.4 1.6 1.7 1.9 2.0

0.5 1.5 2.5 3.5 4.5 5.5 6.5

Table 9 Pultruded circular-section dimension and cost estimates for various spans. Span

a

Frame total height

Section dimension

(m)

(m)

100 150 200 250 300 350 400

8.6 10.1 11.9 14.0 16.4 19.0 21.8

Column

Cost estimate Cross arm (mm)

Rela-tive size (%)

Crossbracing (mm)

Rela-tive size (%)

Weight of one frame (kg)

Cost of one frame ($)

Number of frames in 10 km

Total cost

(mm)

Rela-tive size (%)

710  12a 815  12 850  13 900  14 965  14 1040  16 1120  17

45 51 58 66 71 87 100

315  5 375  5 340  6 380  6 375  7 415  7 440  7

8 10 11 12 14 15 16

100  2 120  2 150  2 185  2 215  2 250  2 245  3

1 1 2 2 2 3 4

933.6 1257.4 1661.9 2224.1 2978.9 3955.7 5170.9

7470 10,060 13,295 17,795 23,830 31,650 41,367

100 66.7 50.0 40.0 33.3 28.6 25.0

746,880.0 670,615 664,760 711,710 794,375 904,160 1,034180

($)

Diameter  thickness.

material, which represent a small part of the overall cost. Weight is an advantage of FRP, but connections in wooden portal frame is easier. There is no difference in the foundations and it would be similar. Overall, it is believed that the difference in cost for an FRP structure compared to a wood or steel structure is small, and

the real economic advantage can be obtained in the maintenance and durability when life cycle cost is accounted for. Tables 9–11 show the cost for each structure for a 10 km length of transmission line for different section types and spans. For better indication of the results, Fig. 14 provides the cost of each sec-

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A. Godat et al. / Composite Structures 105 (2013) 408–421 Table 10 Pultruded rectangular-section dimension and cost estimates for various spans. Span

a

Frame total height

Section dimension

Cost estimate

Column

Rela-tive size

(m)

(m)

(mm)

(%)

Horizontal bracing (mm)

100 150 200 250 300 350 400

8.6 10.1 11.9 14.0 16.4 19.0 21.8

515  320  23a 545  350  25 555  390  28 610  420  30 645  460  33 685  500  36 730  545  39

39 45 53 62 73 86 100

170  170  12 180  180  13 190  190  14 200  200  15 210  210  16 225  225  17 230  230  18

Rela-tive size (%)

Cross-bracing

Weight of one frame (kg)

Cost of one frame ($)

Number of frames in 10 km

(mm)

Rela-tive size (%)

Total cost ($)

8 9 11 12 13 15 16

90  90  4 110  110  4 120  120  5 145  145  5 155  155  6 170  170  7 180  180  8

1 2 2 3 4 5 6

1254.8 1703.7 2344.4 3216.0 4404.3 5947.4 7897.6

10,040 13,630 18,755 25,728 35,235 47,580 63,185

100 66.7 50.0 40.0 33.3 28.6 25.0

1,003,840 908,640 937,760 1,029,120 1,174,480 1,359,410 1,579,520

Depth  Width  Thickness.

Table 11 Pultruded I-section dimension and cost estimates for various spans. Span

a

Frame total Section dimension height Column

Cost estimate Rela-tive size

(mm)

Rela-tive Weight of Cost of Number of Total cost size one frame one frame frames in 10 km (%) (kg) ($) ($)

85  135  6  2 95  155  7  4 110  175  8  5 130  205  10  5 130  220  12  5 145  230  14  5 155  245  16  5

1 1 2 3 3 4 5

Horizontal bracing

Rela-tive Cross-bracing size (%)

(m)

(m

(mm)

(%)

(mm)

100 150 200 250 300 350 400

8.6 10.1 11.9 14.0 16.4 19.0 21.8

875  600  39  16a 975  675  43  18 1040  770  46  19 1090  870  50  21 1190  975  54  23 1265  1090  58  25 1330  1200  62  27

33 41 49 59 72 86 100

240x240x20x6 6 245x245x22x7 7 290x260x22x8 7 325x285x23x9 9 335x300x24x9 9 355x320x25x9 10 370x335x26x10 11

2120.8 3070.7 4308.4 6140.9 8624.9 11,867.9 15,848.9

16,965 24,570 34,470 49,130 69,000 94,945 126,795

100 66.7 50.0 40.0 33.3 28.6 25.0

1,696,640 1,637,710 1,723,360 1,965,090 2,299,975 2,712,665 3,169,780

Depth  width  flange thickness  web thickness.

4000

the actual production costs and available sizes. It is better to emphasis that it is easier to assemble rectangular sections than circular section in practice. However, the I-section do not seem appropriate even for short spans.

Circular-section solution

Total cost for 10km line (x10 3)

Rectangular-section solution I-section solution 3000

6. Conclusions 2000

1000

0

0

100

200

300

400

500

Span between portal frames (m) Fig. 14. Summary of cost estimates for a line of 10 km with various spans.

tion. From the figure, it can be observed that the circular-section presents the economic solution. The circular-section shows the minimum cost when the span between portal frames is 200 m. It is the most economical solution, while a span of 150 m is the optimum for the other options. For the I-section, the cost increases very sharply with the span increase, while it is less for the circular-section and rectangular-section. Up to this end, we can conclude that the FRP pultruded circular-section with 200 m span between frames can be considered the optimum solution to replace the steel and wood in electricity transmission lines. Rectangular sections are also interesting and could be taken based on

This study presented a comprehensive program on the use of FRP pultruded members in electricity transmission towers. The investigation was conducted to obtain a better understanding of the behavior of FRP pultruded members. Critical buckling capacity was obtained for fifteen FRP specimens made of E-glass and either polyester or vinylester resins. The behavior of angle-section, square-section and rectangular-section specimens were investigated under axial load, while W-section and I-section specimens were tested under transverse load. The experimental results showed that the angle-section members failed in a combination of global buckling and bending. The local buckling is the dominant failure mode for the square-section, rectangular-section and Wsection specimens. The I-section specimens showed lateral-torsional buckling failure. In addition, the failure of FRP pultruded sections was affected with the fiber architecture, since the failure located where the content of fibers was low. All tested specimens exhibited elastic behavior during the test. The critical buckling loads measured from the experimental tests were compared to the predictions from available design manuals and analytical equations. Based on the comparison, the various design equations considered to estimate the critical buckling load predict the loading capacity with acceptable accuracy. The results obtained by design equations confirm the hypothesis that the design manuals are, sometimes, accurate to predict the capacity of pultruded members. Design equations taking into consideration

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the rotational stiffness of the plate connections are shown to be reasonably accurate. In this study, the possibility to replace the steel in electricity transmission towers was investigated by conducting structural design for towers made of FRP. The design was conducted for a transmission tower with 69 kV along a distance of 10 km. Various spans between transmission towers range between 100 m and 400 m in increment of 50 m were considered. The design was carried out with one of the following FRP pultruded members: (i) circular-section; (ii) rectangular-section; and (iii) I-section. For each of these members, cost estimate was provided in order to determine the most economical solution. It was obtained that the FRP pultruded circular-section with 200 m span between frames can be considered the optimum solution to replace the steel in electricity transmission lines. Acknowledgements The financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Hydro-Québec through operating grants is gratefully acknowledged. Frédéric Légeron (program director) is Canada Research Chair in Structural Aspects and Mechanical/Electrical Transmission Lines, and his support to the study is greatly acknowledged. The efficient collaboration of Marc Demers (research associate) and Frederic Turcot (technician) of the Université de Sherbrooke in conducting the tests is also acknowledged. Appendix A The following symbols are used in Table 3: bs bf Dij EL ET G k or K L p and q t or trs tf

g mLT mTL n

a u

half flange width flange width bending stiffness elastic modulus parallel to fiber orientation elastic modulus transverse to fiber orientation shear modulus buckling factor length of the member constants plate thickness flange thickness constant Poisson ratio parallel to fiber orientation Poisson ratio transverse to fiber orientation coefficient of restrained restrained factor coefficient taken as 0.8

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