An investigation of material characterization of

Mechanics of Advanced Materials and Structures

ISSN: 1537-6494 (Print) 1537-6532 (Online) Journal homepage: http://www.tandfonline.com/loi/umcm20

An investigation of material characterization of pultruded FRP H- and I-beams S. B. Singh & Himanshu Chawla To cite this article: S. B. Singh & Himanshu Chawla (2018) An investigation of material characterization of pultruded FRP H- and I-beams, Mechanics of Advanced Materials and Structures, 25:2, 124-142, DOI: 10.1080/15376494.2016.1250021 To link to this article: https://doi.org/10.1080/15376494.2016.1250021

Accepted author version posted online: 27 Oct 2016. Published online: 21 Feb 2017. Submit your article to this journal

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Date: 14 December 2017, At: 00:57

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES , VOL. , NO. , – https://doi.org/./..

ORIGINAL ARTICLE

An investigation of material characterization of pultruded FRP H- and I-beams S. B. Singh and Himanshu Chawla

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Civil Engineering Department, Birla Institute of Technology and Science Pilani, Pilani, India

ABSTRACT

ARTICLE HISTORY

This study presents the material characterization of pultruded fiber-reinforced polymer beams of different sizes using experimental and analytical methods. Various tests were performed to determine the material properties such as Young’s and shear moduli using tensile and bending tests of coupons and beams. The stiffness measured from the four-point bending test of both beams is in close agreement with analytical methods. It is shown that the numerically obtained flexural response of the beam using ABAQUS software with material properties obtained from the four-point beam bending test is close to the experimental response.

Received  August  Accepted  October 

1. Introduction The study of material characterization is very important for the application of fiber-reinforced polymer (FRP) in construction. Pultruded FRP beams have been widely attracted by marine and aerospace industries, while construction industries interest in the use of FRP beams is increasing. Due to inherent high specific stiffness and strength, and corrosion resistance, FRP beams are seen by construction industry as the sense of replacement of steel beams. For the design prospect, it is very important to determine the physical and mechanical properties of FRP beams. Composition of fiber and stiffness varies from manufacturer to manufacturer; therefore, it is necessary to predict the material characterization of FRP beams using suitable test methods. Bank [1] has performed extensive research in the field of material characterization of FRP pultruded beams and has suggested a method to determine Young’s and shear moduli of pultruded beams by the consideration of Timoshenko beam theory in conjunction with the three-point bending test of beams. Based on this method, other experimental programs were conducted by Netoand Rovere [2], Brooks and Turvey [3], and Robert and Ubaidi [4] for measurement of Young’s and shear moduli. They have compared the results of stiffness measured from this bending test with stiffness from tensile testing of coupons as well as analytical equations. From the study of Neto and Rovere [2], it is concluded that Young’s modulus obtained from three-point bending test of coupons is different from those obtained from uniaxial tension test and the three-point bending test of full length of pultruded beams. Brooks and Turvey [3] have also shown the significant difference between Young’s modulus obtained from the uniaxial tension testing and

KEYWORDS

Analytical method; compressive characteristics; Young’s modulus; experimental tests; finite element analysis; inter-laminar shear strength; shear modulus; tensile characteristics; pultruded beams

three-point bending test of coupons. Robert and Ubaidi [4] have performed the uniform and nonuniform torsion test for measurement of shear modulus and rigidity. They have observed the good correlation of Young’s modulus obtained from bending and torsional test. It is shown [4] that warping torsional modulus determined from nonuniform torsion is closer to the modulus obtained from minor axis bending; however, the Saint-Venant torsional shear modulus obtained from uniform torsion testing is slightly higher than the transverse shear modulus obtained from the major axis bending. In another study, Howard [5] has performed the three-point bending and torsional tests for measurement of beam stiffness. Results are close to those obtained from approximate classical lamination beam theory (CLBT). Minghini et al. [6–7] have proposed the different positions of loading in four-point bending test of beams, to measure flexural and shear rigidity. From their experimental investigation, they concluded that their proposed loading gives very less coefficient of variation for shear modulus as compared to modulus obtained from three and four-point bending tests. In other investigations, Correia [8] and Corriea et al. [9] have determined Young’s modulus of beams by tensile and flexural testing of coupons. The value of elastic modulus is closer to the modulus obtained from three-point bending test of coupons. Maji et al. [10] experimentally investigated the static and dynamic modulus and noted that dynamic modulus is 5–10% higher than static modulus. Few researchers have determined elastic properties of pultruded beams by a micromechanics approach. They have measured the stiffness of all plies of flanges and web; then, combined the stiffness to obtain the full section properties. Nagaraj and GangaRao [11] have presented an approximate classical lamination theory. They have predicted the anisotropic modulus in

CONTACT S. B. Singh [email protected]; [email protected] Civil Engineering Department, Birla Institute of Technology and Science Pilani, Vidya Vihar Campus, Pilani-, India. Color versions of one or more figures in this article are available online at www.tandfonline.com/umcm. ©  Taylor & Francis Group, LLC

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MECHANICS OF ADVANCED MATERIALS AND STRUCTURES

the structural coordinate system that takes into account the lamina properties in the transverse directions. Davalos et al. [12] have proposed a laminated beam theory to calculate the stiffness of beam from stiffness of plies. In their theory, coupling of the beam due to un-symmetrical layup is also considered that is missing in Nagaraj and Ganga Rao [11] theory. Bank [13] has determined the shear modulus and strength from the Losipescu shear test method (ASTM D5379). Feraboli and Kedward [14] have performed the four-point bending test of laminates to examine the inter-laminar shear strength of cross-ply and quasi-isotropic laminates. They validated their results with ANSYS and have shown good correlation between the experimental and numerical results. Luciano and Barbero [15] have proposed a method to determine the shear stiffness and Young’s modulus of a lamina based on periodic microstructure. This method is well suited to predict the shear stiffness of continuous strands mat (CSM). Hashin and Rosen [16] have determined the shear stiffness of a lamina by the composite cylinder model. These two methods, i.e., periodic microstructure and composite models were followed by Davalos et al. [12] and the shear stiffness calculated by these two models is higher than that obtained by the rule of mixture was shown. In an another study, Ning [17] has used the Manera method, which is nothing but the modified rule of mixture. This method is most suitable to determine the stiffness of CSM layers. After an extensive literature review, it is observed that little work has been done on the validation of measured stiffness from experimental tests with those of analytical and numerical methods. There are lot of tests to determine Young’s modulus of FRP components, but its detailed comparison with analytical and numerical studies is still missing. Along with Young’s modulus, shear modulus also plays a major role in the deformation of FRP I-beams. Therefore, to reproduce the behavior of FRP beams using theoretical or numerical analyses, it is important to assess the accuracy of Young’s and shear moduli obtained from tests. The main objective of this paper is to present the detailed investigation on the material characterization of pultruded FRP I and H-beams. Detailed investigation includes determination of resin content, Poisson’s ratio, tensile strength, compressive strength, shear strength, and flexural characteristics of FRP

125

elements used in I-beam. The stiffness obtained from various tests is verified with analytical methods. FRP beam was also modeled using commercial software ABAQUS with properties obtained from experimental tests, and results of flexural deflection of beam are compared with experimental results of the three-point and four-point bending tests. Based on the variation of results, a method is suggested to determine the flexural and shear stiffnesses of the beam. Moreover, an analytical approach for prediction of the stiffness of the FRP beams is presented in Appendix.

2. Experimental program This research is carried out on glass-fiber-reinforced polymer I and H-beams, manufactured by two different companies using pultrusion process. The sizes of H and I-beams used were 100 × 100 × 8 mm and 150 × 75 × 6.5 mm (height × flange width × thickness). In this study, these beams are denoted by P100 and P150, respectively, where “P” represents the pultruded beam and the numerical term represents the depth of the section. As per supplier, polyester resin has been used in both beams. In order to measure the stiffness from analytical methods, layup of the beam was predicted by burning of coupons and pultruded beams of length 100 mm. Description of the burnt out test (see Figure 1) is given in the following sections. The layup of both beams observed after burnt out test is shown in Figure 2. It is noted that the flange of beam P100 has un-symmetrical layup, while the web has symmetrical layup. Therefore, coupling will be produced in beam due to unsymmetrical layup. The beam P100 has three types of layers such as CSM, woven fabrics and roving, while the beam P150 has CSM and roving only. In order to measure Young’s properties of beams, coupons were cut on a “Power hack saw” machine in the workshop at Birla Institute of Technology and Science, Pilani (BITS-Pilani), followed by grinding the edges of coupons, to remove the extra fibers and maintaining the required shape of coupons. Different sizes of coupons were cut as per the recommendation of codes such as ASTM D2584, ASTM D3039, ASTM D3410, and ASTM D790. Samples were prepared for tensile, compressive, flexural, and shear testing. The coupons were cut from interior area of flanges and web of pultruded

Figure . Ignition of samples in a muffle furnace for removal of resin. (a) Before burning. (b) After burning.

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2.1.1. Fiber content The composition of the fiber in pultruded beams was measured by ignition of samples as per the method suggested by ASTM D2584 [19]. FRP specimens of size of 25 × 25 × t mm were extruded from the web and flanges of beams. Thickness “t” varies from specimen to specimen. Samples were kept in muffle furnace at 550°C for half an hour as shown in Figure 1. Ignition loss of resin content was determined by weighing the specimen before and after igniting. The resin content was determined by Eq. (1) [19] as given below: Resin content = Percentage of weight loss

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=

(W1 − W2 ) × 100, W1

(1)

where W1 and W2 represent the weight of specimen and residue, respectively. As can be seen from the above method, the percentage of resin content in flanges and web of beam P100 is 27% and 33%, respectively, while the corresponding values for beam P150 is 42%. The layer of each fabric was separated out from the burnt coupons of both beams. From the physical observation, it seen that fiber is not uniform especially at edges of flanges. Therefore, coupon should not be extruded from the edges of flange. It is also noted that beam P100 has seven layers in web and six layers in flanges. The beam P150 has an equal number of layers as well as the same composition of fabrics in flanges and web. The layerwise description of each fabric in flanges and web of P100 and P150 beams are shown in Figure 2.

Figure . Schematics and layup configuration representations of type of fiber in flange and web. (a) P. (b) P (all dimensions are in mm).

I-beam. The junction of flange and web was avoided for coupons due to nonuniformity of fiber. Moreover, edges of flanges were also avoided because of broken edges and irregularities in thickness. For each test, five coupons were cut and name was given to each specimen based on the type of test and the origin of specimen as presented in Table 1. The first letter of each specimen name denotes the name of test, like “T” stands for tension testing, “C” stands for compression testing, “S” stands for shear testing, “F” stands for flexural testing of coupons, while the numeric term 100 or 150 represents the specimen taken from pultruded beam P100 or P150, respectively. The letter “F” or “W” after numeric terms 100 or 150 represents the origin of the specimen, i.e., specimen taken from flange or web of the beam and the last numeric term denotes the test number. The composition of the fiber in pultruded beams can be measured by ignition of samples as per ASTM D2584 [19]. Young’s modulus was obtained through tension test as described in ASTM D3039 [20]. Compressive strength of specimens was measured as per specifications recommended by ASTM D3410 [21], while ASTM D790 [22] specifies the procedure for calculation of flexural modulus of FRP beams and coupons by three-point bending test. 2.1. Physical and mechanical properties of specimens The physical and mechanical properties of specimens obtained using different tests are described in the following sections.

2.1.2. Tensile characteristics To predict the mechanical properties such as Young’s modulus, tensile strength, and Poisson’s ratio, tensile tests were performed as per ASTM D3039 [20]. Five rectangular coupons of approximate size 250 × 25 × t mm were extruded from P100 and P150 beams, where thickness “t” varied from specimen to specimen. The tension tests were conducted on a UTM of 100kN capacity. Rate of displacement of crosshead was 2 mm/min. The coupon was inserted in between the pressure wedge grips at both ends and pressure of 6.89 MPa was set in grips. With this pressure, there was no slippage and breakage of the specimen inside the grips. While placing the specimen inside the grips, it was ensured that coupons are perfectly straight. For the measurement of Poisson’s ratio, two electrical resistance strain gauges of gauge length 10 mm were bonded along the longitudinal as well as the transverse axes (see Figure 3) of the coupon. The load, deflection, and strains were recorded via a data acquisition system, and a report was generated on a computer. The failure of the samples was sudden and the sound produced was like a gunshot, and the failure was in the form of delamination of continuous strand mat and cracking of rovings in longitudinal as well as transverse direction. The stress–strain curves obtained from tensile test of beams P100 and P150 are shown in Figures 4 and 5, respectively. From Figure 4, it is noted that all tensile specimens of beam P100 show bi-linear stress– strain behavior. Young’s modulus was calculated from the first linear portion of stress–strain curve. In the stress–strain curve, for the particular strain, coupons of web (W-series) have slightly higher stress than that for coupons of flange (F-series). It is due to different composition of fiber in web and flanges. Web of P100

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Table . Description of specimens for different tests. Specimen Id

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TF- TF- TF- TW- TW- TF- TF- TF- TW- TW- CF- CF- CF- CW- CW- CF- CF- CF- CW- CW-

Type of test Tensile test Tensile test Tensile test Tensile test Tensile test Tensile test Tensile test Tensile test Tensile test Tensile test Compression test Compression test Compression test Compression test Compression test Compression test Compression test Compression test Compression test Compression test

♦ Inter-laminar shear test,

Origin of the specimen

Specimen Id

Flange of P Flange of P Flange of P Web of P Web of P Flange of P Flange of P Flange of P Web of P Web of P Flange of P Flange of P Flange of P Web of P Web of P Flange of P Flange of P Flange of P Web of P Web of P

SF- SF- SF- SW- SW- SF- SF- SF- SW- SW- FF- FF- FF- FW- FW- FF- FF- FF- FW- FW-

Type of test Short beam test♦ Short beam test Short beam test Short beam test Short beam test Short beam test Short beam test Short beam test Short beam test Short beam test Flexural test♣ Flexural test Flexural test Flexural test Flexural test Flexural test Flexural test Flexural test Flexural test Flexural test

Origin of the specimen Flange of P Flange of P Flange of P Web of P Web of P Flange of P Flange of P Flange of P Web of P Web of P Flange of P Flange of P Flange of P Web of P Web of P Flange of P Flange of P Flange of P Web of P Web of P

♣ Three-point bending test of coupons.

Figure . Stress vs. strain curves obtained from tensile test of coupons from P beam.

Figure . Tensile test of coupon with strain gauges mounted along longitudinal as well as transverse directions.

has one additional layer of woven fabric in comparison to that of flanges, i.e., more fiber volume fraction, so it has more stiffness in longitudinal direction. Hence, the web has little higher tensile strength and Young’s modulus than flanges. However, average energy absorption capacity is the same. The stress–strain curves of P150 obtained from tensile test are linear as shown in Figure 5. Young’s modulus of coupons of the web and flanges of P150 is much closer due to the same fiber volume fraction in flanges and web. The calculated value of Young’s modulus and energy ratio for each specimen is presented in Table 2. The total energy absorbed is measured by area under the load-deflection curves, while the elastic energy is calculated by the area under the elastic slope. The elastic slope is calculated from the equation below [18]: S=

Figure . Stress vs. strain curves obtained from tensile test of coupons from P beam.

P1 S1 + (P2 − P1 ) S2 , P2

(2)

where S is the slope of elastic region, S1 is the slope of the first linear portion, S2 is the slope of the second linear portion, P1 is

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Table . Tensile characteristics of pultruded FRP beams P and P. Specimen Id TF- TF- TF- TW- TW- TF- TF- TF- TW- TW-

Young’s modulus, GPa

Tensile strength, MPa

Elastic energy, kN m

Total energy absorbed, kN m

% inelastic energy ratio

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

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the maximum load of the first linear portion, and P2 is the maximum load of second linear portion. Poisson’s ratios calculated from tensile testing of coupons of P100 and P150 are 0.26 and 0.32, respectively. 2.1.3. Compressive characteristics The compressive strength of the specimen of pultruded beams was measured as per the guidelines given in ASTM D3410 [21]. Five specimens each of size 125 × 25 × t extruded from each beam were tested, where t is the thickness of the specimen. The test was performed under a displacement control mode with a constant rate of movement of crosshead as 1.5 mm/min. In this way, incremental compressive load was applied until the specimen failed. Figure 6 shows the mode of failure of specimen in compression. After the failure of the specimen, it was noted that all specimens failed by delamination (see Figure 6) of fabric layers, which is consistent with the failure mechanism reported by

Table . Compressive strength of the specimens. Specimen Id

Compressive strength, MPa

CF- CF- CF- CW- CW- CF- CF- CF- CW- CW-

         

Correia [8]. Due to high stress concentration near the grip, an external layer got folded as well as delamination occurred in the between the layers. In Table 3, compressive strengths of coupons from flanges and web of P100 and P150 beams are presented. Variation of compressive strength of each coupon is due to small precrack inside the coupons as well as nonhomogeneity of fibers in the coupons. 2.1.4. Flexural modulus from three-point bending test of FRP coupon The ASTM D790 [22] specifications were used to measure the stiffness of FRP coupons. The coupons of size 360 × 15 mm (length × width) were extruded from the FRP beam, with L/d ratio of 60, for minimizing the effect of shear. A special fixture was self-fabricated and attached to the bottom head of actuator with loading nose of 12-mm diameter, and sample was placed on the supports of 6-mm diameter as shown in Figure 7. The test was continued until the complete specimen failure, i.e., visible delamination and cracks in ply (or flaw) occurred under the loading. Tests were conducted under a displacement control mode, with the rate of displacement (R) 0.5 mm/s. The displacement rate was calculated using Eq. (3) [22]: R=

Figure . Compression testing of specimen.

ZL2 , 6d

(3)

where L is the span length (mm), Z is the strain rate (0.01 was taken as per recommendations of ASTM D790 [22]), and d is the depth (mm) of coupon. Responses of coupons of beams P100 and P150 under flexural loading are shown in Figures 8 and 9, respectively. Table 4 represents the flexural modulus and strength of three-point bending test of coupon. Flexural modulus was determined by measuring the slope of an initial linear portion of the curve, and flexural strength was determined from

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Table . Flexural modulus and strength obtained from three-point bending test of coupons. Specimen Id

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FF- FF- FF- FW- FW- FF- FF- FF- FW- FW-

Flexural modulus, GPa

Flexural strength, MPa

. . . . . . . . . .

. . . . . . . . . .

where L (mm) is the span length, m (kN/mm) is the slope of the initial linear portion, b (mm) is the width of specimen, and d (mm) is the depth of the specimen. Modulus of elasticity of P100 calculated from three-point bending test is nearly similar to results of tensile testing, while the modulus of elasticity of P150 is significantly different from that obtained from uniaxial tensile test.

Figure . Three-point bending test of coupon.

2.1.5. Interlaminar shear strength To determine the inter-laminar shear strength of the specimen, short beam tests were performed as per ASTM D2344 [23]. Short beams specimens were cut from beams P100 and P150 in sizes of 50 mm (length) × 15 mm (width) × 8 mm (thickness) and 40 × 15 × 6.5 mm, respectively. The span length-to-depth ratio was maintained 4 during testing. Overhang on each support was kept equal to thickness of specimens. The test setup of short beam shear test is shown in Figure 10a. The beam was loaded

Figure . Load vs. deflection curve of three-point bending test of coupons from P.

Figure . Load vs. deflection curve obtained from three-point bending test of coupons from P.

the maximum bending moment and section modulus. The flexural modulus (E, GPa) of a specimen measured from three-point bending test [22] is given by E=

L3 m , 4bd 3

(4)

Figure . Response of short span coupon under flexural loading. (a) Short beam shear test of coupon. (b) Failed coupon due to inter-laminar shear stress.

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S. B. SINGH AND H. CHAWLA

Table . Inter-laminar shear strength of coupons. Specimen Id

Shear strength, MPa

SF- SF- SF- SW- SW- SF- SF- SF- S W- S W-

. . . . . . . . . .

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until the inter-laminar crack appears in the thickness of coupon as shown in Figure 10b. This figure shows the failed specimen under flexural loading with inter-laminar shear crack through the thickness of specimen. The shear strength of the specimen was determined by Eq. (5) [23]: σS =

0.75P , bh

(5)

where P is the maximum load, b is the width, and h is the thickness of the specimen. The inter-laminar shear strength of coupons is presented in Table 5. It is observed that the coupons of both beams P100 and P150 have almost the same shear strength, for flanges and web. There is a little variation in strength of coupons of P100, due to the misalignment of fiber in flanges and web.

provided at the supports to resist the rotation as well as translation of the beam. A special fixture was fabricated in the workshop at BITS-Pilani to provide the translation as well as torsional restraints for different geometry of beams. Hence, in order to test the beams of different sizes, restraints kept movable on the fixture. They are connected to the fixture with bolts and nuts, so as to adjust the distance between them as per the size of beams. Beams were allowed to bend in the plane of loading only; therefore, out of plane displacement of the beam was restrained by providing the torsional restraint near to the application of the load as shown in Figure 11. Friction between the surface of restraints and beam was reduced through a thin film of grease. A linear variable differential transducer (LVDT) was installed laterally under the loading to measure the lateral deflection of beams, and another LVDT was used to measure the vertical displacement of the beam. Load was applied through the bearing plates in order to reduce the stress concentration at the application of load, similarly bearing plates were provided upon supports to reduce the stress concentration on the joint of flange and web. The load and deflections were recorded with frequency of 30 Hz and saved in the personal computer. 2.1.6.2. Three-point bending test of profiles. The maximum deflection of a beam due to flexure and shear under three-point bending can be measured using the following equation [25]: w=

2.1.6. Mechanical properties using bending test of beams 2.1.6.1. Experimental setup. In order to test the beams with simply supported boundary conditions, torsional restraints were

PL PL3 , + 48EIz 4GKY A

(6)

where w is the deflection at midspan, P is the applied load, L is the distance between the supports, E is the flexural modulus,

Figure . Flexural testing of beam. (a) Three-point bending test. (b) Four-point bending test.

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES

131

Iz is the moment of inertia of beam about z-axis, G is the shear modulus, A is the area of section, and Ky is the shear coefficient, which is equal to tw (H − tf )/A [4], where H is the depth of section, tf and tw is the thickness of flange and web, respectively. Equation (6) can be re-written as   1 L 2 4Aw 1 = + . (7) PL 12E r GKy The above equation represents the equation of a line. Flexural modulus can be calculated from three-point bending test of Ibeams with different values of ( Lr )2 . The slope of graph between 4Aw and ( Lr )2 can be used to measure the flexural modulus, E of PL I-beam. Hence, flexural modulus of a beam is given in Eq. (8): E=

1 . 12 × Slope

(8)

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On the other hand, the shear modulus is given in Eq. (9): G=

1 . Ky × inercept

(9)

(i) Test program. In this test program, three-point bending test was conducted on the full length of the pultruded beams. The ( Lr )2 ratio as measure of slenderness was taken as 100, 200, 300, and 400. Based on these slenderness ratios, H and I-beams were cut into the lengths of 0.40, 0.65, 0.80, and 0.95 m, and 0.75, 1.0, 1.15, and 1.35 m, respectively. Overhang on each support was kept equal to half of the depth of section. Load was applied at the midspan of simply supported beam and was statically increased until deflection reaches to maximum deflection of L/350. Long beams were tested with lateral-torsional restraints at supports and near the midspan (see Figure 11) for preventing deflection in lateral direction, i.e. lateral-torsional buckling. A graph is and ( Lr )2 for the load of 1500 N plotted between 4Aw PL (see Figure 12). Young’s modulus is calculated by Eq. (8) by measuring the slope of this graph. Similarly, shear modulus is calculated using Eq. (9)after evaluating the intercept of the graph. Young’s modulus of P100 and P150 is observed to be 20.83 and 15.59 GPa, respectively. The shear modulus of P100 and P150 is 2.57 and 1.05 GPa, respectively. The intercept values as shown in

Figure . Regression analysis of beams for measurement of Young’s and shear modulus.

Figure . Response of beam under four-point loading. (a) P. (b) P.

Figure 12, for beams P100 and P150 are 0.52 and 1.14, respectively. (ii) Apparent modulus of the beam. The apparent modulus of elasticity is determined from the true modulus of elasticity and slenderness ratio of the beam. The apparent modulus of both beams is calculated from Eq. (10) as given below:   Eb /Gb Ky 1 1 1 + 12 . (10) = Ea Eb (L/r)2 Figure 14 illustrates the variation of ratio of apparent to flexural modulus with the slenderness ratio of both beams. It is

Figure . Variation of apparent to flexural bending modulus ratio with different L/r ratios.

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S. B. SINGH AND H. CHAWLA

Table . Ply properties of beams. MM♠

Rule of mixture (ROM) Beam P

P

Lamina

PM♣

CM♦

Thickness (mm)

vf

E (GPa)

E (GPa)

G (GPa)

E(GPa)

G (GPa)

G (GPa)

G (GPa)

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. — — — . —

. — — — . —

. . . . . .

. . . . . .

CSM Woven Roving-Ϯ Roving-♥ CSM Roving

ϮRoving of Flange, ♥Roving of the web, ♠Manera method, ♣Periodic microstructure, ♦Composite cylinder model.

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observed that for the L/r ratio greater than 40, beam P100 has an apparent flexural modulus equal to true flexural modulus. Similarly, for P150 if the L/r ratio is greater than 50, then shear deformation has a negligible role in the deformation of the beam. This indicates that the requirement of Lr limit to consider the effect of shear deformation on apparent modulus depends on layup of the beam and its material. 2.1.6.3. Four-point bending test of beams. In this test program, supports were provided in between the loading and the stiffness of beam is predicted as per the method prescribed by Minghini et. al. [7], while the beams were loaded at ends as shown in Figure 11(b). As per availability of lengths, beams P100 and P150 were tested for the length of 1.9 and 2.1 m, respectively. Supports were provided under the beams (P100 and P150) at end distance of L/6. Wooden stiffeners were provided in between flanges, to restrain the warping and torsion of beam about longitudinal axis. Vertical deflections were measured at the ends and midspan of the beam. Responses of beams P100 and P150 are shown in Figure 13(a) and (b), respectively. The flexural and shear rigidities of beam are determined using Timoshenko’s beam theory [7]. The rigidities of the beam are given by following Eqs. (11) and (12): EI = −Pa kGA =

(L − 2a) 8v 2

2

Pa 4a(3L − 4a) and μ = , v 1 + μv 2 3(L − 2a)2

(11)

3.1. Stiffness of lamina The stiffness of laminas is determined using methods such as the rule of mixture, periodic microstructure, and composite cylinder model. Based on the study of Davalos et. al [12], it is observed that Young’s modulus of a lamina obtained from the rule of mixture, periodic microstructure, and composite cylinder model is the same, but shear modulus determined using the rule of mixture is 50% lower for the CSM layer and 34% lower for roving layers than values obtained from both models, i.e., periodic microstructure and composite cylinder model. Hence, Young’s modulus of each lamina is derived from rule of mixture, while shear modulus is derived from the periodic microstructure. The Manera method is also followed to determine the stiffness of CSM layer. An approach for analytically evaluating the stiffness of a panel is presented in Appendix and the calculated value of stiffness of each lamina is presented in Table 6. It is observed that Young’s modulus of CSM layer measured from rule of mixture is much higher than the Manera method due to which overall modulus of elasticity of the beam is also much higher than experimental result. Therefore, Young’s modulus of CSM layer is determined from the Manera method, and for other layers, it is determined from rule of mixture. 3.2. Stiffness of beam Details of approximate classical lamination and mechanics of laminated beam theories for predicting the overall stiffness of beam are given in the following sections.

(12)

where P is the applied load at each end, v 2 is the vertical deflection of the beam at midspan, and v1 is the deflection at beam ends located at distance a from supports. Deflections corresponding to the total load of 9 kN (i.e., 4.5 kN on each end) are used to determine the elastic properties of the beam. Young’s modulus of P100 and P150 beams is observed to be 21.25 and 21.13GPa, respectively. The shear modulus of beams P100 and P150 is 2.47 and 2.18GPa, respectively.

3.2.1. Approximate classical lamination theory The stiffness of the beam can be calculated by measuring the stiffness of lamina of flanges and web and then combining together to obtain the stiffness of pultruded I-beam. Nagaraj and Gangarao [11] derived the approximate isotropic modulus of flanges and web, which consider the effect of stiffness of fiber in transverse direction. Therefore, this theory is known as approximate CLPT. Anisotropic modulus of a lamina can be given by Er = Q11 −

Q212 E1 E2 , Q11 = , Q22 = , and Q22 1 − v 12 v 21 1 − v 12 v 21

3. Analytical method

Q12 = v 12 Q22 .

In this section, an approach for determining the stiffness of the beam is presented using two theories, i.e., Approximate Classical Lamination theory and Mechanics of Laminated Beams theory. Following subsections provide the approach to evaluate the stiffness of lamina and beam.

Based on this theory, extensional stiffness of flanges can be calculated by

(13)

Af = b

N  r=1

Er tr ,

(14)

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES

133

where b is the width of the flange, while Er and tr are the anisotropic modulus and thickness of the rth lamina, respectively. Similarly, bending stiffness matrix of flange and web is computed by Eqs. (15) and (16), respectively,   N  tr3 2 (15) E r t r Zr + Df = b 12 r=1 d3  Er tr , 12 r=1 N

Dw =

(16)

where d is the depth of web. Overall bending stiffness of a beam is calculated by combining the stiffness of each panel, i.e., web and flanges and is given by Eq. (17):

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D = D f + A f e2o + Dw ,

(17)

where eo is the distance between center of flange and center of beam along the cross section. Shear modulus of beam is computed by addition of shear stiffness of each layer of web. Shear rigidity of flanges is neglected. Shear rigidity web is given below: Fw = d

N 

Gr tr ,

(18)

r=1

where Gr is the shear stiffness of rth lamina of the web. 3.2.2. Mechanics of laminated beam theory This method considers the effect of coupling induced in the beam due to un-symmetrical layup of beam. The compliance matrix of each panel of beam is determined using CLBT. Stiffness of each panel can be calculated from following equations:       Ai = δ11 −1 i , Bi = β11 −1 i , Di = α11 −1 i ,  −1  2 , and  = α11 δ11 − β11 , (19) Fi = α66 i where [α] = [A]−1 , [β] = [B]−1 and [δ] = [D]−1 and Ai is extensional stiffness, Bi is bending-extension coupling stiffness, Di is bending stiffness of a laminate, and subscript “”” denotes the panel number. These [A], [B], and [D]stiffness matrix are cal    culated by CLBT, while Ai , Bi , Di , and Fi are the stiffness of a panel considering the effect of coupling. Total extensional stiffness of the FRP beam considering the effect of local stiffness of the panels is given by Ax =

N 

Ai bi ,

(20)

i=1

where bi is the width of panel and N denotes the number of panels. The reference axes of the beam are considered at the centroid of cross section, i.e., at the center of the web. The depth of neutral axis of beam is represented by yn and is calculated using Eq. (21):  N    i=1 yi Ai + cos ϕi Bi bi yn = (21) Ax where yi is the distance between the center of the panel and the centroid of the beam and “φ” is the angle between the axis of panel and Z-axis of the beam. The co-ordinate system of the

Figure . Coordinate system of FRP beam.

beam followed for stiffness calculations is shown in Figure 15. The equation to evaluate bending-extension (By ), Flexural rigidity (Dz ), and shear rigidity (Fy ) of beam are given below: By =

N 

Ai [Ai (yi − yn ) + Bi cos ϕi ]bi

i=1

Dz =

N   i=1

Ai



b2 (yi − yn ) + i sin2 ϕi 12



2

+ 2Bi (yi − yn ) cos ϕi + Di cos2 ϕ bi Fy =

N 

Fi bi sin2 ϕi .

(22)

i=1

Summary of results calculated from both theories (i.e., CLPT and MLB) are presented in Tables 7 and 8. Based on above two theories, stiffness of each panel and beams is presented in the Table 8. It is observed that Young’s and shear moduli obtained using CLPT and MLB are close. However, shear modulus obtained from periodic microstructure is in close agreement with both experimental results (three-point and four-point bending tests) for low depth of beam, i.e., for beam P100 but for high depth of beam (i.e., P150 beam), results obtained using the periodic microstructure is closer to that obtained using fourpoint bending tests. In Table 8, differences in Young’s and shear

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S. B. SINGH AND H. CHAWLA

Table . Stiffness of panels from theories. Flange Beam

Theory

P

CLPT MLB CLPT MLB

P

Web

Af Ϯ GPa mm

Bf ♦ GPa mm 

Df ♣ GPa mm

Aw GPa mm

Bw GPa mm

Dw GPa mm 

Fw ¤ GPa mm

Flexural rigidity of beam D GPa mm 

  . .

— −  

   

   

   

. . . .

   

. . . .

ϮExtensional stiffness, ♦Extensional-bending stiffness, ♣Bending stiffness, ¤Shear stiffness.

Table . Stiffness (GPa) of beams from analytical and experimental methods. CLPT Beam P

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P

MLB

Tensile testing of coupons

-point bending test of coupons

Stiffness parameter

ROM∗

PM♣

ROM

PM

Flange♦

Web

Flange

Web

E G E G

. . . .

— . — .

. . .  .

— . — .

. — . —

. — . —

. — . —

. — . —

-point -point bending test bending test of beams of beams . . . .

. . . .

% error♠ . −  

∗ Properties of each lamina of beam is obtained from rule of mixture. ♣Properties of each lamina of beam is obtained from periodic microstructure. ♦Average values of Young’s modulus. ♠Error in measurement of stiffness from -point beam bending test w.r.t. -point bending test of beam.

moduli obtained using three-point and four-point beam bending tests are also presented. It is observed that for the beam P100, results obtained by three-point and four-point bending tests are closer, while for the beam P150, the four-point bending test gives higher stiffnesses than three-point bending test. As this indicates the sensitivity of equivalent Young’s and shear stiffnesses of beams to their overall geometry, fiber architectures, and constituent material characteristics.

4. Numerical investigation In order to check the accuracy of experimental results of threepoint bending test, beams P100 and P150 were modeled in ABAQUS 6.10 [26]. The span lengths of P100 and P150 beams were 1500 and 1173 mm, respectively. Beams were modeled with shell element S4R and meshed with fine size 0.005 of length of member. The load was applied over the top flange, i.e., at the midspan of the beam. Similarly, boundary conditions were applied at the ends of the bottom flange of the beam. At the ends, beam was not allowed to rotate about the centroidal axis and translation along y and z-axes was also restrained. Beams were analyzed with the Riks algorithm, with consideration of geometric nonlinearity and imperfection of 0.1% of the depth of beam. The flexural response of the beam was predicted using Young’s modulus obtained from tests such as uniaxial tensile test, flexural test of coupons, and flexural beam tests (three-point and four-point bending test). The deflected shape of the beam due to flexural loading is shown in Figure 16.

shear force is zero in between the supports; therefore, deformation of beam due to shear deformation is negligible. Young’s modulus obtained from three-point bending of coupons and four-point bending test of beam is the same it is because in the flexural testing of coupons, the L/d ratio was kept 60, due to which measured Young’s modulus has negligible effect of shear. The stiffness of beam P150 obtained from tensile testing is lesser than other methods except three-point beam bending, and it is because fibers are not aligned in the longitudinal direction of the beam. Shear modulus of the both beams P100 and P150 obtained from four-point bending is closer to the analytical results obtained from the periodic microstructure. In Figure 14, it is seen that beam P100 has high shear modulus than the beam P150. Therefore, the anisotropy ratio defined by the ratio of modulus of elasticity to the shear modulus (Eb /Gb ) of beam P150 is higher than that of beam P100. For comparison of results, P100 and P150 beams were modeled in ABAQUS with Young’s modulus predicted from tensile testing of coupons, flexural testing of coupons, and beams. In modeling of beam using ABAQUS, the shear modulus was taken from that data

5. Results and discussion It is observed from Table 8 that in beam P100, Young’s and shear modulus determined from three-point and four-point bending test are same, while for beam P150 stiffness from both tests is different. This is due to difference in overall geometry and material characteristics of beam. In four-point bending test,

Figure . In-plane bending of pultruded beam under flexural loading.

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES

Figure . Load vs. deflection response of beam P.

135

Figure . Young’s and shear modulus of four-point beam bending test, calculated w.r.t. three-point bending test.

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for predicting the numerical/analytical responses: obtained from flexural test of beams, while Poisson’s ratio was taken from tensile testing. On the other hand, the same dimension of the beam was tested experimentally using three-point and four-point loading conditions. Load-deflection responses of pultruded beams P100 and P150 predicted from ABAQUS and experimental testing are shown in Figures 17 and 18, respectively. From Figure 17, it is noted that numerical responses for beam P100 obtained using ABAQUS software from the stiffnesses determined using four-point beam bending test are closer to that obtained using stiffnesses determined by other methods. However, corresponding responses (Figure 18) for the beam P150 using stiffness from three-point beam bending test are different from those obtained using stiffnesses determined by four-point bending test and other methods. Thus, it is observed that stiffness obtained from four-point beam bending test closely approximates the experimental responses of the beam. Based upon the results obtained from three-point and four-point bending tests, a relation is developed between the stiffnesses (Young’s and shear modulus) determined from both tests for prediction of error in the measured value of stiffness from three-point bending test. Figure 19 shows the apparent relationship between Young’s and shear modulus obtained from four-point beam bending test (i.e., E4 and G4 ) with those obtained from three-point beam bending tests (i.e., E3 and G3 ). Equations (23) and (24) give, respectively, Young’s and shear moduli predicted from four-point beam bending test to be used

Figure . Load vs. deflection response of beam P.

E4 = 21.58 − 2.37 × 10−4 E3 D −3

G4 = 2.74 − 6.14 × 10 G3 D,

(23) (24)

where D is overall flexural rigidity of beam (Table 8), E3 and G3 are the shear moduli obtained from three-point bending test, respectively. E4 is Young’s modulus, while G4 is shear modulus measured from four-point bending test of beam. Percentage error in measurement of stiffness from three-point bending test is given by ErrE =

(E3 − E4 ) × 100 E4

(25)

ErrG =

(G3 − G4 ) × 100, G4

(26)

where ErrE and ErrG are percentage errors in measurement of Young’s and shear modulus obtained from three-point bending test, respectively.

6. Conclusions and recommendations In this paper, material characterization of pultruded beams of H and I-sections is presented. It involves the measurement of resin content, Poisson’s ratio, tensile, compressive, flexural, and shear characteristics. Various methods were adopted for measurement for complete material characterization of FRP beams. The results obtained from experimental tests are verified with mechanics of laminated beam (MLB) theory and approximate CLBT. From this study, the following concluding remarks can be made. 1. Composition of fiber and analytical approach/theory used for determining the stiffness plays a vital role on tensile, compressive, flexural, and shear characteristics of a pultruded FRP beam. 2. Young’s modulus determined from the MLB theory and approximated laminated beam theory is in good agreement. It means coupling of laminates due to unsymmetrical layup does not have much effect on stiffness measurement. 3. Stiffness determined from four-point bending test of beams and three-point bending of coupons are in good

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S. B. SINGH AND H. CHAWLA

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agreement. It is because shear deformation has negligible effect in estimation of Young’s modulus. 4. The response obtained through ABAQUS using Young’s and shear moduli obtained from four-point bending of beams is closer to experimental response than that using stiffness obtained from three-point bending test of beam. Therefore, four-point bending test of beams is a suitable method for measurement of Young’s and shear moduli, for prediction of numerical and analytical responses. 5. Due to the high anisotropy ratio (Eb /Gb ) of P150, the L/r ratio required for minimizing the shear deformation effect on bending is also higher than P100. The higher the required anisotropy ratio, higher the L/r ratio required to diminish the effect of shear deformation.

Funding The authors are thankful to CSIR New Delhi, DST New Delhi and Aditiya Birla Group for financial assistance provided for effective execution of projects.

References [1] L.C.Bank, Flexural and shear modulli of full-section fiber reinforced plastic (FRP) pultruded beams, J. Test. Eval., vol. 17, no. 1, pp.40–45, 1989. [2] A.S.Neto and H.L.Rovere, Flexural stiffness characterization of fiber reinforced plastic (FRP) pultruded beams, Compos. Struct., vol. 81, no. 2, pp.274–282, 2007. [3] R.J.Brooks and G.J.Turvey, Lateral buckling of pultruded GRP Isection cantilevers, Compos. Struct., vol. 32, no. 1–4, pp. 203–215, 1995. [4] T.M.Roberts and H.Al-Ubaidi, Flexural and torsional properties of pultruded fiber reinforced plastic I-Profiles, J. Compos. Constr., vol. 6, no.1, pp. 28–34, 2002. [5] I.L.Howard, Practical approaches for evaluating bending and torsion of fiber-reinforced polymer components using instrumented testing, J. Appl. Sci. Eng. Tech., vol. 2, no. 1, pp. 24–30, 2008. [6] F.Minghini, N.Tullini, and F.Laudiero, Full section properties of pultruded FRP profiles using bending tests, Proceedings of the 6th Int. Conference on FRP Composites in Civil Engineering, pp.1–8, Rome, 2012. [7] F.Minghini, N.Tullini, and F.Laudiero, Identification of the shortterm full-section moduli of pultruded FRP profiles using bending tests, J. Compos. Constr., vol. 18, no. 1, pp. 1–9, 2014. [8] M.M.Correia, Structural behavior of pultruded GFRP profiles experimental study and numerical modeling, Inst. Superior Técn., vol.1, pp. 1–14, 2012. [9] J.R.Correia, F.A.Branco, N.M.F.Silva, D.Camotim, and N.Silvestre. First-order, buckling and post-buckling behavior of GFRP pultruded beams, Part 1: Experimental study, Comput. Struct., vol. 89, no. 2122, pp. 2052–2064, 2011. [10] B.A.K.Maji, R.Acree, D.Satpathi, and K.Donnelly, Evaluation of pultruded FRP composites for structural applications, J. Mat. Civil. Eng., vol. 9, no. 3 pp. 154–158, 1997. [11] V.Nagaraj and H.V.S.GangaRao, Static behavior of pultruded GFRP Beams, J. Compos. Constr., vol. 1, no. 3, pp. 120–129, 1997. [12] J.F.Davalos, H.A.Salim, P.Qiao, W.Virginia, and E.J.Barbero, Analysis and design of pultruded FRP shapes under bending, Compos. B, vol. 95, no. 3-4, pp. 295–305, 1996. [13] L.C.Bank, Shear properties of pultruded glass FRP materials, J. Mat. Civil. Eng., vol. 2, no. 2, pp.118–122, 1990.

[14] P.Feraboli and K.T.Kedward, Four-point bend interlaminar shear testing of uni- and multi-directional carbon/epoxy composite systems, Compos. A, vol. 34, no. 12, pp. 1265–1271, 2003. [15] R.Luciano and E.J.Barbero, Formula for the stiffness of composites with periodic microstructure, Int. J. Solids Struct., vol. 31, no. 21, pp.2933–2944, 1994. [16] Z.Hashin and B.W.Rosen, The elastic modeling of fiber-reinforced materials, J. Appl. Mech., vol. 31, no.1, pp. 223–232, 1964. [17] P.Ning, The elastic constants of randomly oriented fiber composites: A new approach to prediction, Sci. Eng. Compos. Mat., vol. 5, no. 2, pp.63–72, 1996. [18] N.F.Grace, A.K.Soliman, A.Sayed, and K.R.Saleh, Behavior and ductility of simple and continuous FRP reinforced beams, J. Compos. Constr., vol. 2, no. 4, pp. 186–194, 1998. [19] ASTM D 2584. Standard Test Method for Ignition Loss of Cured Reinforced Resins, American Society of Testing and Materials. Philadelphia, USA, 2008. [20] ASTM D 3039. Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials, American Society of Testing And Materials, Philadelphia, USA, 2014. [21] ASTM D 3410. Standard Test Method for Compressive Properties of Polymer Matrix Composite Materials with Unsupported Gauge Sections by Shear Loading, American Society of Testing and Materials, Philadelphia, USA, 2008. [22] ASTM D 790. Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials, American Society of Testing and Materials, Philadelphia, USA, 2002. [23] ASTM D 2344. Standard Test Method for Short-Beam Strength of Polymer Matrix Composite Materials and Their Laminates, American Society of Testing and Materials, Philadelphia, USA, 2000. [24] R.M.Jones, Mechanics of Composite Materials, McrGaw-Hill, New York, pp. 37–45, 1975. [25] J.N.Reddy, Mechanics of Laminated Composite Plates and Shells, CRC Press, New York, 2003. [26] Abaqus Standard User’s Manual, Version 6.10, Hibbit, Kalsson & Sorensen Inc., Abaqus, Pawtucket RI, vol. 2010. [27] L.P.Kollár and G.S.Springer, Mechanics of Composite Structures, Cambridge University Press, Cambridge, UK, 2003.

Appendix In this section, an analytical approach to calculate the overall stiffness of pultruded laminated I-beam is presented. Stiffness is calculated for a typical case of pultruded beam, P100 (Figure 2a) that has unsymmetrical layup in flanges and symmetrical in web. Stiffness is derived using (i) approximate classical lamination beam theory (CLPT) and (ii) mechanics of laminated beam theory (MLB) approaches that includes coupling of the beam due to unsymmetrical layup. Mechanical properties of fiber and matrix of P100 Young’s modulus of glass fiber, Ef = 74 GPa Shear modulus of glass fiber, Gf = 29 GPa Young’s modulus of matrix, Em = 2.60 GPa Shear modulus of matrix, Gm = 1.25 GPa I. Stiffness calculation of each lamina The layup of flange of P100 is represented in Figure 2a. Elastic properties of each lamina are evaluated in the following sections: a. Continuous strands mat (CSM) This lamina consists of randomly oriented glass fibers having weight of 550 GSM (gram per square meter). Thickness of lamina, t = 0. 9 mm Density of mat, ρ = 2300 kg/m3 w × 10−3 Fiber volume fraction, V f = ρt = 2300550 = 0.27 × 0.9 × 10−3 Matrix volume fraction, Vm = 1 − 0.27 = 0.73.

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES

Young’s modulus (E) and shear modulus (G) is calculated by the rule of mixture and the Manera method, in addition to these two methods, shear modulus is also calculated by periodic microstructure and composite cylinder model. The detailed calculation are as follows: i. Rule of mixture [22] Young’s modulus of a lamina based on a micromechanics approach is calculated by E = E f V f + EmVm = 74 × 0.27 + 2.60 × 0.73 = 21.88 GPa. Similarly, shear modulus is calculated by mechanics of material approach [22]

137

i. Rule of mixture Young’s modulus (E) of woven layer is E = E f V f + EmVm = 74 × 0.21 + 2.60 × 0.79 = 17.59 GPa. Similarly, shear modulus is calculated by G12 =

G f Gm G f Vm + GmV f

=

29 × 1.25 = 1.56 GPa. 29 × 0.79+1.25 × 0.21

ii. Periodic microstructure S3 = 0.49247 − 0.47603 V f − 0.02748 V f2

= 0.49247 − 0.47603 × 0.21 − 0.02748 × 0.212 = 0.3913  −1 29 × 1.25 S3 1 = 1.69 GPa. = G12 = + G12 = Gm − V f − G f Vm + GmV f 29 × 0.73 + 1.25 × 0.27 Gm Gm − G f −1  1 0.3913 ii. Manera method [16] + = 1.25 − 0.21 × − = 1.85 GPa.   1.25 1.25 − 29 16 8 E = Vf E f + 2Em + Em = 0.27 45 9 iii. Composite cylinder model   8 16 G f (1 + V f ) + Gm (1 − V f ) × 74 + 2 × 2.60 + × 2.60 = 10.82 GPa × G12 = Gm 45 9 G f (1 − V f ) + Gm (1 + V f )   2 3 1 29 × (1 + 0.21) + 1.25 × (1 − 0.21) G = Vf E f + Em + Em = 0.27 = 1.85 GPa. = 1.25 × 15 4 3 29 × (1 − 0.21) + 1.25 × (1 + 0.21)   2 3 1 c. Rovings of flange × × 74 + × 2.60 + × 2.60 = 4.06 GPa. 15 4 3 Thickness of layer of roving = 3.3 mm Number of roving per unit width (mm) = 0.280 iii. Periodic microstructure [14] m Linear density of fiber, i.e., yield of fiber, γ = 104.78 kg  −1 S3 1 Density of roving, ρ = 2670 kg/m3 + , G12 = Gm − V f − 1 Gm Gm − G f Area of rovings, Ar = γ1ρ = 104.78 × 103 × = 2670 × 10−9 2 3.57 mm . where S3 is computed by Volume fraction of rovings, 2 S3 = 0.49247 − 0.47603 V f − 0.02748 V f nr Ar 0.280 × 3.57 = 0.30 = Vf = 2 t 3.3 r = 0.49247 − 0.47603 × 0.27 − 0.02748 × 0.27 = 0.3619 −1  Matrix volume fraction, Vm = 1 − 0.30 = 0.70. 1 0.3619 + = 2.08 GPa. G12 = 1.25 − 0.27 × − Young’s modulus (E) is calculated by the rule of mixture and 1.25 (1.25 − 29) shear modulus (G) is calculated by the rule of mixture, periodic microstructure and composite cylinder model. The detailed caliv. Composite cylinder model [15] culation are as follows: G f (1 + V f ) + Gm (1 − V f ) i. Rule of mixture G12 = Gm Young’s modulus of lamina: G f (1 − V f ) + Gm (1 + V f )

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G f Gm

= 1.25 ×

29 × (1 + 0.27) + 1.25 × (1 − 0.27) 29 × (1 − 0.27) + 1.25 × (1 + 0.27)

= 2.07 GPa. b. Woven fabrics Thickness of lamina, t = 1.1 mm Density of mat, ρ = 2400 kg/m3 Weight, w = 550 gm/m2 w × 10−3 Fiber volume fraction, V f = ρt = 2400550 = 0.21 × 1.1 × 10−3 Matrix volume fraction, Vm = 1 − 0.21 = 0.79. Young’s modulus (E) is calculated by the rule of mixture and shear modulus ( G) is calculated by the rule of mixture, periodic microstructure and composite cylinder model. The detailed calculation are as follows:

E1 = E f V f + EmVm = 74 × 0.30 + 2.6 × 0.70 = 24.02 GPa E2 =

E f Em E f Vm + EmV f

=

74 × 2.6 = 3.66 GPa. 74 × 0.70 + 2.6 × 0.30

Shear modulus of lamina of rovings: G12 =

G f Gm G f Vm + GmV f

=

29 × 1.25 = 1.75 GPa. 29 × 0.7 + 1.25 × 0.3

ii. Periodic microstructure S3 = 0.49247 − 0.47603 V f − 0.02748 V f2 = 0.49247 − 0.47603 × 0.30 − 0.02748 × 0.302 = 0.3472

138

S. B. SINGH AND H. CHAWLA

G12

 −1 S3 1 = Gm − V f − + Gm Gm − G f −1  1 0.3472 + = 1.25 − 0.30 × − 1.25 (1.25 − 29)

II. Evaluation of overall stiffness using Approximate Classical Lamination Beam Theory

= 2.21 GPa. iii. Composite cylinder model G12 = Gm

G f (1 + V f ) + Gm (1 − V f ) G f (1 − V f ) + Gm (1 + V f )

= 1.25 ×

Q11 = Q22 =

29 × (1 + 0.30) + 1.25 × (1 − 0.30) 29 × (1 − 0.30) + 1.25 × (1 + 0.30)

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= 2.20 GPa. d. Rovings of web Thickness of layer of roving = 2.2 mm Number of roving per unit width (mm) = 0.28 m Linear density of fiber, i.e., yield of fiber, γ = 104.78 kg Density of roving, ρ = 2670 kg/m3 1 Area of rovings, Ar = γ1ρ = 104.78 × 103 × = 2670 × 10−9 2 3.57 mm × 3.57 Volume fraction of rovings, V f = nrtAr r = 0.2802.2 = 0.45 Volume fraction of matrix, Vm = 1 − 0.45 = 0.55. Young’s modulus (E) is calculated by the rule of mixture and shear modulus (G) is calculated by the rule of mixture, periodic microstructure and composite cylinder model. The detailed calculation are as follows: i. Rule of mixture Young’s modulus of lamina: E1 = E f V f + EmVm = 74 × 0.45 + 2.6 × 0.55 = 34.73 GPa E2 =

E f Em E f Vm + EmV f

=

Steps for applying the approximate CLBT are presented below: a. Isotropic stiffness of continuous strand mat, Ec Young’s modulus of the CSM is calculated based on the Manera method,

74 × 2.6 = 4.60 GPa 74 × 0.55 + 2.6 × 0.45

E 10.82 = 11.54 GPa = 1 − v 12 v 21 1 − 0.25 × 0.25

Q12 = v 12 Q22 = 0.25 × 11.54 = 2.89 GPa Ec = Q11 −

Q212 2.892 = 10.82 GPa. = 11.54 − Q22 11.54

b. Isotropic stiffness of Woven fabrics Q11 = Q22 =

= 18.76 GPa Q12 = v 12 Q22 = 0.25 × 18.76 = 4.69 GPa Ew = Q11 −

G f Gm G f Vm + GmV f

=

29 × 1.25 = 2.20 GPa. 29 × 0.55 + 1.25 × 0.45

ii. Periodic microstructure S3 = 0.49247 − 0.47603 V f −

G12

0.02748 V f2

= 0.49247

− 0.47603 × 0.45 − 0.02748 × 0.452 = 0.2727  −1 S3 1 = Gm − V f − + Gm Gm − G f −1  1 0.2727 + = 1.25 − 0.45 × − 1.25 1.25 − 29 = 3.02 GPa.

iii. Composite cylinder model G12 = Gm

G f (1 + V f ) + Gm (1 − V f ) G f (1 − V f ) + Gm (1 + V f )

29 × (1 + 0.45) + 1.25 × (1 − 0.45) = 1.25 × 29 × (1 − 0.45) + 1.25 × (1 + 0.45) = 2.20 GPa.

Q212 4.692 = 17.59 GPa. = 18.76 − Q22 18.76

c. Isotropic stiffness of roving of flange E1 E2 3.66 × 0.25 = 0.038 = , v 21 = v 12 v 21 24.02 E1 24.02 Q11 = = = 24.25 GPa 1 − v 12 v 21 1 − 0.25 × 0.038 E2 3.66 = 3.70 GPa = Q22 = 1 − v 12 v 21 1 − 0.25 × 0.038 Q12 = 0.25 × 3.70 = 0.93 GPa

Shear modulus of lamina of rovings: G12 =

E 17.59 = 1 − v 12 v 21 1 − 0.25 × 0.25

Er f = Q11 −

Q212 0.932 = 24.02 GPa. = 24.25 − Q22 3.70

d. Isotropic stiffness of roving of web E1 E2 4.60 × 0.25 = 0.033 = , v 21 = v 12 v 21 34.73 E1 34.73 Q11 = = 35.02 GPa = 1 − v 12 v 21 1 − 0.25 × 0.033 E2 4.60 Q22 = = 4.64 GPa = 1 − v 12 v 21 1 − 0.25 × 0.033 Q12 = v 12 E2 = 0.25 × 4.64 = 1.16 GPa Erw = Q11 −

Q212 1.162 = 34.73 GPa. = 35.02 − Q22 4.64

Extensional stiffness of flange A f = b1

N 

Er tr = 100 × (4 × 10.82 × 0.9

r=1

+ 17.59 × 1.1 + 24.02 × 3.3) = 13756.7 GPa mm2 .

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES

139

× 10 = 17.40 GPa. Young’s modulus, E = DIzz = 65.94 3.79 × 106 To calculate the overall shear modulus of beam material, shear stiffness of flanges is neglected and shear stiffness of lamina is calculated based upon the periodic microstructure 6

F∼ = Fw = b2

N 

Gr tr = 84 × (4 × 2.08 × 0.9 + 2

r=1

× 1.85 × 1.1 + 3.02 × 2.2) = 1528.97 GPa mm2 Figure A. Distance of lamina of flange from the reference plane in CLPT.

Extensional stiffness of web Aw = b2

N 

Er tr = 84 × (4 × 10.82 × 0.9

r=1

+ 2 × 17.59 × 1.1 + 34.73 × 2.2)

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= 12940.7 GPa mm2 . Bending stiffness of the flange D f = b1

  t3 Er tr Zr2 + r , 12 r=1

N 

where Zr is the distance from the reference plane of the laminate to the middle of the lamina as shown in Figure A1    0.93 + 10.82 = 100 × 10.82 × 0.9 × 3.552 + 12     0.93 2.23 × 0.9 × 2.652 + + 24.02 × 2.2 × 1.12 + 12 12  3 1.1 + 24.02 × 1.1 × (−0.55)2 + 12   1.13 + 17.59 × 1.1 × (−1.65)2 + 12   0.93 2 + 10.82 × 0.9 × (−2.65) + 12   0.93 2 + 10.82 × 0.9 × (−3.55) + 12 = 100 × [123.38 + 69.042 + 85.25 + 10.66 + 54.63 + 69.04 + 123.28] = 53538 .20 GPa mm4 ⇔ 53538 GPa mm4 . Bending stiffness of the web Dw =

N b2 3  843 × (4 × 10.82 × 0.9 + 2 Er tr = 12 r=1 12

× 17.59 × 1.1 + 34.73 × 2.2) = 7.61 × 106 GPa mm4 . Combined flexural stiffness of I-beam   Dz = 2 × D f + A f e2o + Dw Dz = 2 × (53538 + 13756.7 × 462 ) + 7.61 × 106 = 65.94 × 106 GPa mm4 I = 3.79 × 106 mm4 .

Aw = d × t = 84 × 8 = 672 mm2 . = 2.28 GPa. Shear modulus of the beam, G = AFww = 1528.97 672 III. Mechanics of laminated beam theory Stiffness matrix of a lamina ⎤ ⎡ v 12 E2 E1 0 ⎥ D ⎢ D ⎥ E2 v 12 E2 Q=⎢ ⎣ 0 ⎦, D D 0 0 G12 where D = 1 −

E2 2 v = 1 − v 12 v 21 . E1 12

i. Stiffness matrix of continuous strand mat (CSM) E1 = E2 = 10.82 GPa,v 12 = v 21 = 0.25 and D = 1 − 0.252 = 0.94 ⎡ ⎤ 11.54 2.89 0 ⎢ ⎥ 0 ⎦ GPa. Qc = ⎣ 2.89 11.54 0 0 2.08 ii. Stiffness matrix of woven layer E1 = E2 = 17.59 GPa,v 12 = v 21 = 0.25 and D = 1 − 0.252 = 0.94 ⎡ 18.76 4.69 ⎢ Qw = ⎣ 4.69 18.76 0

0

⎤ 0 ⎥ 0 ⎦ GPa. 1.85

iii. Stiffness matrix of roving of flanges E1 = 24.02 GPa,E2 = 3.66 GPa,v 12 = 0.25, 3.66 × 0.252 = 0.99 v 21 = 0.036 and D = 1 − 24.02 ⎡ ⎤ 24.25 0.92 0 ⎢ ⎥ 0 ⎦ GPa. Qr f = ⎣ 0.92 3.70 0 0 2.21 iv. Stiffness matrix of roving of web E1 = 34.73 GPa, E2 = 4.60 GPa, v 12 = 0.25, 4.60 × 0.252 = 0.99 v 21 = 0.033 and D = 1 − 34.73 ⎡ ⎤ 35.02 1.16 0 ⎢ ⎥ 0 ⎦ GPa. Qrw = ⎣ 1.16 4.64 0

0

Evaluation of laminate stiffness

3.02

140

S. B. SINGH AND H. CHAWLA

+ Qw × ((−1.1)3 − (−2.2)3 ) + Qr f × (0−(−1.1)3 ) + Qr f × (2.23 − 0) + Qc × (3.13 − 2.23 ) + Qc × (43 − 3.13 )]

Figure A. Distance of lamina of flange from the reference plane in MLB.

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The extensional, bending-extensional and bending stiffness matrix of flange of I-beam is represented by [A], [B], and [D]. Stiffness matrix of flanges is represented by subscript “f ” and the web is represented by “w”. The extension stiffness matrix A=

N 

= 35.568Qc + 3.106Qw + 3.993Qr f ⎡ ⎤ 556.55 121.03 0 ⎢ ⎥ 3 0 ⎥ Df = ⎢ ⎣ 121.03 483.49 ⎦ GPa mm . 0 0 88.55 Stiffness parameters, α, β, and δ for the flange laminate are determined by Laminate stiffness, A, B, and D as given below:    −1 α β A B = β δ B D ⎡

Qr (zr − zr−1 ),

r=1

where zr is the distance from the reference plane to the top of rth lamina and it is shown in Figure A2 A f = Qc × ((−3.1) − (−4)) + Qc × ((−2.2) − (−3.1)) + Qw × ((−1.1) − (−2.2)) + Qr f × (0 − (−1.1)) + Qr f ×(2.2 − 0) + Qc × (3.1 − 2.2) + Qc × (4 − 3.1) A f = 3.6Qc + 1.1Qw + 3.3Qr f ⎡ ⎤ 142.21 18.60 0 ⎢ ⎥ 0 ⎥ Af = ⎢ ⎣ 18.60 74.39 ⎦ GPa mm 0 0 16.82  1  2 2 Qr Zr − Zr−1 B= 2 r=1

−0.18

0.73

⎢ [α] = ⎢ ⎣ −0.18

0

−1.59

⎢ [β] = ⎢ ⎣ 0.40 0 ⎡

5.95

0.40

0

7.66

0

0

−4.36

−0.05

0.19

⎢ [δ] = ⎢ ⎣ −0.05

0.22

0

⎤

⎥ 1 −2 0 ⎥ ⎦ × 10 GPa mm

1.42

0 ⎡

0

0

0

⎤ ⎥ 1 ⎥ × 10−4 ⎦ GPa mm2 ⎤

⎥ 1 −2 0 ⎥ ⎦ × 10 GPa mm3 1.13

2  = α11 δ11 − β11 = 0.73 × 10−2 × 0.19 × 10−2

− (−1.59 × 10−4 )2 = 1.39 × 10−5

N

+ Qc × ((−2.2)2 − (−3.1)2 ) + Qw × ((−1.1)2 − (−2.2)2 ) + Qr f × (0−(−1.1)2 ) + Qr f × (2.22 − 0) + Qc (3.12 − 2.22 ) + Qc × (42 − 3.12 )] B f = 0.5 × (3.63Qr f − 3.63Qw ) = 1.815 × (Qr f − Qw ) ⎡

9.96

⎢ Bf = ⎢ ⎣ −6.84 0

−6.84 −27.33 0

0

⎤

⎥ 2 0 ⎥ ⎦ GPa mm 0.65

 1  3 3 Qr Zr − Zr−1 3 r=1 N

D=

0.19 × 10−2 = 136.69 GPa mm 1.39 × 10−5

Af = δ11 −1 =

B f = 0.5 × [Qc × ((−3.1)2 −(−4)2 )

1 D f = ×[Qc × ((−3.1)3−(−4)3 )+Qc × ((−2.2)3 − (−3.1)3 ) 3

Bf = −(β11 −1 ) = − Df = α11 −1 = −1 Fi f = α66 =

1 GPa2 mm4

−1.59 × 10−4 = 11.44 GPa mm2 1.39 × 10−5

0.73 × 10−2 = 525.18 GPa mm3 1.39 × 10−5

1 = 16.81 GPa mm. 5.95 × 10−2

Evaluation of laminate stiffness [A], [B] and [D] matrix of web of the I-beam A=

N 

Qr (zr − zr−1 ),

r=1

where zr is the distance from the reference plane to the top of lamina and it is shown in Figure A3 Aw = Qc × ((−3.1) − (−4)) + Qc × ((−2.2) − (−3.1)) + Qw × ((−1.1) − (−2.2)) + Qrw × (0 − (−1.1)) + Qrw × (1.1 − 0) + Qw

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES

× (2.2 − 1.1) + Qc × (3.1 − 2.2) + Qc × (4 − 3.1) = 3.6Qc + 2.2Qw + 2.2Qrw ⎡ ⎤ 159.86 23.27 0 ⎢ ⎥ 0 ⎥ Aw = ⎢ ⎣ 23.27 93.02 ⎦ GPa mm. 0 0 18.20 Bw = 0 (i.e., web laminate is symmetric with respect to midsurface, i.e., reference plane) 1 3 Qr (Zr3 − Zr−1 ) 3 r=1

2  = α11 δ11 − β11 = 0.65 × 10−2 × 0.19 × 10−2 − 0

= 0.12 × 10−4 Aw = δ11 −1 =

Dw =

Dw = α11 −1 = −1 Fw = α66 =

1 × [Qc × ((−3.1)3 − (−4)3 ) + Qc × ((−2.2)3 3

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− (−3.1)3 ) + Qw × ((−1.1)3 − (−2.2)3 ) + Qr f × (0 − (−1.1) ) + Qr f × (1.1 − 0) 3

3

Dw = 35.568Qc + 6.211Qw + 0.887Qrw ⎡ ⎤ 558.05 132.95 0 ⎢ ⎥ 3 0 ⎥ =⎢ ⎣ 132.95 531.10 ⎦ GPa mm . 0 0 88.15 Laminate stiffness of the web,    −1 α β AB . = β δ BD Laminate is balanced; therefore, B = 0 [α] = [A]−1 , [β] = [B]−1 and [δ] = [D]−1 ⎡ ⎤ 0.65 −0.16 0 ⎢ ⎥ 1 −2 0 ⎥ [α] = ⎢ ⎣ −0.16 1.13 ⎦ × 10 GPa mm 0 0 5.50 ⎡ ⎤ 0.19 −0.05 0 ⎢ ⎥ 1 −2 0.2 0 ⎥ [δ] = ⎢ ⎣ −0.05 ⎦ × 10 GPa mm3 0 0 1.13

0.19 × 10−2 = 158.33 GPa mm 0.12 × 10−4

0.65 × 10−2 = 541.67 GPa mm3 0.12 × 10−4

1 = 18.18 GPa mm. 5.50 × 10−2

Combined stiffness of I-beam Extensional stiffness of the beam Ax =

N 

Ai bi = Af b1 + Aw b2 + Af b3 = 2 × 136.69

i=1

× 100 + 158.33 × 84 = 40638 GPa mm.

+ Qw × (2.23 − 1.13 ) + Qc × (3.13 − 2.23 ) + Qc × (43 − 3.13 )]

1 mm4

GPa2

Bw = 0

N

D=

141

The reference axis of the section (Figure 15) is taken at centroid of the section; therefore, yn = 0. Thus, the distance between center of web and reference axis is zero, i.e., y2 = 0. The distance from the center of the flange to reference axis is given by y1 = y3 =

t d − = 46 mm. 2 2

Orientation of panel with co-ordinate axes are as follows. φ = 0° for top flange, φ = 90° for web, and φ = 180° for bottom flange (see Figure 15) Bending-extensional stiffness of the beam By =

N 

[Ai (yi − yn ) + Bi cos ϕi ]bi

i=1

= (Af (y1 − yn ) + Bf cos 0◦ )b1 + (Aw (y2 − yn ) + Bw cos 90◦ )b2 + (Af (y3 − yn ) + Bf cos 180◦ )b3 = Af (y1 − yn )b1 + Bf b1 + Aw (y2 − yn )b2 + Af (y3 − yn )b3 − Bf b3 = 136.69 × (46 − 0) × 100 + 11.44 × 100 + 158.33 × (0 − 0)×84+136.69×(46 − 0) × 100 − 11.44 × 100 = 1.26 × 106 GPa mm3 . Bending stiffness of the beam  N    b2i 2  2 Dz = Ai (yi − yn ) + sin ϕi 12 i=1

Figure A. Distance of lamina of web from the reference plane in MLB.

 + 2Bi (yi − yn ) cos ϕi + Di cos2 ϕ bi

142

S. B. SINGH AND H. CHAWLA

   b2 = Af (y1 − yn )2 + 1 sin2 0◦ + 2Bf (y1 − yn ) cos 0◦ 12     b2 + Df cos2 0◦ b1 + Aw (y2 − yn )2 + 2 sin2 90◦ 12  + 2Bw (y2 − yn ) cos 90◦ + Dw cos2 90◦ b2    b2 + Af (y3 − yn )2 + 3 sin2 180◦ 12  + 2Bf (y3 − yn ) cos 180◦ + Df cos2 180◦ b3 A b3 = Af (y1 − yn )2 b1 + 2Bf (y1 − yn )b1 + Df b1 + w 2 12

Downloaded by [14.139.236.66] at 00:57 14 December 2017

+ Af (y3 − yn )2 b3 − 2Bf (y3 − yn )b3 + Df b3 = 136.69 × (46 − 0) × 100 + 2 × 11.44 × (46 − 0) 2

× 100 + 525.18 × 100 +

158.33 × 843 + 136.69 12

× (46 − 0)2 × 100 − 2 × 11.44 × (46 − 0) × 100 + 525.18 × 100 = 65.77 × 106 GPa mm4 . Young’s modulus, E = Fy =

N 

Dz I

=

65.77 × 106 3.79 × 106

= 17.35 GPa.

Fi bi sin2 ϕi = 16.81 × 100 × sin 0o + 18.18

i=1

×84 × sin 90o + 16.81 × 100 × sin 180o = 1527.10 GPa mm2 . Shear stiffness due to flange is zero so only area of web is taken: Aw = d × t = 84 × 8 = 672 mm2 Shear modulus, G =

Fy Aw

=

1527.10 672

= 2.27 GPa.