Shear Moduli of Structural Composites from Torsion Tests

Shear Moduli of StructuralComposites from Torsion Tests

JULIO F. DAVALOS

Department of Civil and Environmental Engineering

West Virginia University

Morgantown, WV 26506-6103, USA

PIZHONG QIAO* AND JIALAI WANG

Department of Civil Engineering

The University of Akron

Akron, OH 44325-3905, USA

HANI A. SALIM

Department of Civil Engineering

University of Missouri

Columbia, MO 65211-2200, USA

JEREMY SCHLUSSEL

Department of Civil and Environmental Engineering

West Virginia University

Morgantown, WV 26506-6103, USA

(Received August 10, 2000)(Revised November 9, 2000)

ABSTRACT: A combined experimental/analytical approach for effective evaluationof in-plane and out-of-plane shear moduli of structural composite laminates fromtorsion is presented. The test samples are produced by pultrusion and consist ofE-glass fiber systems and vinylester resin. Three types of rectangular samples areused: unidirectional samples cut from plates; angle-ply and angle/cross ply samplescut from wide-flange beams. The shear moduli are obtained from the experimentaltorsional stiffnesses (T/�) and data reduction techniques based on torsion solutionsby Lekhnitskii and Whitney. The classical approach of using paired samples ofdifferent widths but same material orientation can grossly underestimate out-of-plane shear moduli. Thus, to overcome this problem, samples with materialorientations normal to each other are used, and to obtain samples of largerdimensions than the available thicknesses of the material, the pultruded laminatesare bonded flat-wise, from which transverse strips of different widths are cut.Consistent results are obtained by the two data reduction techniques, and theexperimental values are compared to predictions by micro/macro-mechanics models.

*Author to whom correspondence should be addressed. E-mail: [email protected]

Journal of COMPOSITE MATERIALS, Vol. 36, No. 10/2002 1151

0021-9983/02/10 1151–23 $10.00/0 DOI: 10.1106/002199802023486� 2002 Sage Publications

http:\\www.sagepublications.com

In relation to more elaborate methods, the proposed torsional approach can be easilyimplemented in the laboratory, and by testing a representative volume of thematerial with this method, it is expected to obtain reliable shear moduli data forstructural composites.

KEY WORDS: fiber-reinforced plastic composites, laminates, in-plane and out-of-plane shear moduli, torsion, micro/macro-mechanics, data reduction techniques.

INTRODUCTION

FIBER-REINFORCED POLYMER (FRP) composite beams, columns, and cellular panels areincreasingly used in structural applications. Structural sections, such as wide-flange,

box, and other shapes, consist typically of arrangements of flat walls. The most commonprocessing method for FRP structural members is pultrusion, followed by vacuum-assisted resin transfer molding (VARTM). The constituents are typically E-glass fibersystems and vinylester or polyester resins, and the ply materials may include unidirectionalfiber bundles or roving, continuous or chopped randomly-oriented strand mats, andstitched fabrics with �� orientations. In the pultrusion process, the sections are notproduced by lamination lay-up, but the arrangement of the constituent materials can besimulated as a layered system, and the stiffness properties of each panel can be predictedby lamination theory in terms of the ply stiffnesses computed through micromechanics [1].Since it is nearly impossible to test the individual ‘‘plies’’ of a pultruded section, it isrequired to determine the material properties at the laminate level through reliable test andanalysis methods.

Shear deformations for pultruded FRP composites can be significant [1] due to therelatively low shear moduli of the constituents used, and therefore, the modeling of FRPstructural components should account for shear effects. In general, the evaluation of shearstiffness and strength is a difficult task, and while a number of test methods are available,as discussed for example by Chiao et al. [2], there can be a considerable discrepancy in thereported results.

For fiber reinforced composite materials, it is difficult to design a test in which only pureshear is applied. Generally, there are five acceptable test methods for evaluation ofcomposite shear properties [3], namely, the Iosipescu test, � 45� coupon tension test, theoff-axis tension test, the rail shear test, and the torsion test. The torsion test can induce apure shear state of stress, provide average shear properties over a large volume of thespecimen, and minimize localized material and stress concentration effects. For arectangular bar under torsion, the cross-section is a plane of elastic symmetry [4], and thecenter of twist is assumed to remain constant along the length of the bar. In this state, onlytwo of the six components of stress are not zero; i.e., �xy and �xz. Therefore, the shearmoduli Gxy and Gxz can be evaluated from torsion tests, which allow for larger samples tobe tested when compared to other test methods available.

Since most structural composites consist of thick or moderately-thick laminates (0.2500–0.500 or thicker), the number of test methods for determining shear moduli and strengthsis limited. In particular, torsion tests and side-notched (Iosipescu) tests are commonlyused. The torsion tests make use of either cylindrical rods [2,5] or rectangular samples [6],while the Iosipescu shear test [3] uses a flat specimen with side notches, similar to thesample proposed by Arcan et al. [7], to develop a state of uniform stress in the gaugesection. The computational procedure of shear moduli from torsion tests and linear elastic

1152 JULIO F. DAVALOS ET AL.

torsion theory for anisotropic bars [8] is well established [9]. Similarly, the Iosipescu testmethod has been extensively discussed by various researchers [6,10,11].

The Iosipescu test is the most common test method to compute the in-plane shearstiffness for composite materials, including pultruded FRP composites. The majordisadvantages of this method are the small effective sections tested, which may not berepresentative of the material [11], and the stress concentration at the notched section ofthe specimen [12,13]. For pultruded FRP composites [11], it has been observed that there issignificant variability in the results obtained with the Iosipescu test, which can beattributed to the presence of material defects (resin-rich and fiber-rich zones) over thesmall area tested by this method. Other drawbacks of this method are the labor-intensesample preparation, the large number of required repetitions, and the required fabricationof special testing fixtures. Moreover, the Iosipescu test is generally used to obtain onlyin-plane shear properties.

The torsion test, on the other hand, produces pure shear state of stress in the specimen,and the analysis of the experimental results can be easily interpreted. The torsion test alsoprovides average shear properties over a larger volume of the sample minimizing thelocalized effect problems encountered in the Iosipescu test. In the literature, however, thetorsion test method has been used mainly for unidirectional laminates, and the procedureconsists of testing laminates with the same material orientation but varying widths [14].In this paper, we show that this technique grossly underestimates the out-of-planeshear modulus. A better approach is to test ‘‘paired’’ samples with two distinct materialorientations.

OBJECTIVES

The objective of this paper is to propose a comprehensive combined experimental andanalytical method to effectively evaluate shear moduli for FRP structural laminates fromtorsion. By simulating the pultruded panel as a layered system, the panel shear stiffnessproperties are predicted by classical macromechanics and modified transverse sheardeformation theory in terms of the ply stiffnesses and compliances computed throughmicromechanics. The experimental program includes testing three distinct pultruded FRPlaminates (unidirectional, angle-ply, and angle/cross-ply) under torsion, and the shearmoduli are evaluated through a new approach using paired samples with materialorientations normal to each other, as opposed to classical methods that can grosslyunderestimate the out-of-plane moduli. Torsion solutions by Lekhnitskii [8] and Whitney[13] are implemented as data-reduction methods to obtain the shear moduli. The testmethods and data-reduction techniques presented in this paper provide practicalguidelines for evaluation of shear moduli by investigators, material producers anddesign engineers.

SHEAR STIFFNESSES BY MICRO/MACROMECHANICS

Overview of the Theory

Micromechanics and Macromechanics models are considered as fundamental tools forthe evaluation of composite materials. In this study, combined micro/macromechanics are

Shear Moduli of Structural Composites from Torsion Tests 1153

used to predict the laminate elastic properties. Pultruded FRP shapes are not laminatedstructures in a rigorous sense; however, they are produced with material architectures thatcan be simulated as laminated configurations [1]. A typical pultruded section may includethe following four types of layers (see examples in Figure 1): (1) a thin layer of randomly-oriented chopped fibers (Veil) placed on the surface of the composite. This is a resin-richlayer primarily used as a protective coating, and its contribution to the laminate responsecan be neglected; (2) continuous or Chopped Strand Mats (CSM) of different weights,consisting of continuous or chopped randomly-oriented fibers; (3) Stitched Fabrics (SF)with different angle orientations, and (4) roving layers that contain continuousunidirectional fiber bundles, which contribute the most to the stiffness and strength of asection. Each layer is modeled as a homogeneous, linearly elastic, and generallyorthotropic material. The modeling of FRP structural composites as laminates andprediction of their stiffness properties require accurate ply stiffnesses, which can be

(c)

(b)

(a)

Figure 1. Material architecture of test samples: (a) unidirectional (Sample U); (b) angle-ply (Sample A) and(c) angle- and cross-ply (Sample AC).


evaluated through micromechanics by estimating the individual ply thickness and thefiber volume fraction (Vf) of the constituents. Further information on modeling ofpultruded FRP sections and computation of Vf for simulated layer systems can be foundin Davalos et al. [1].

Several formulas of micromechanics of composites have been developed and used overtime [15], and the degree of correlation between experimental data and theoreticalpredictions depends on the accuracy of the model used. In this study, we compute the plyshear stiffnesses for the roving and stitched fabric layers using primarily a micromechanicsmodel for composites with periodic microstructure [16]; as an illustration, the expressionfor the computation of the in-plane shear modulus G12 is

G12 ¼ Gm � Vf �S3

Gmþ

1

Gm � Gf

� �ð1Þ

where Gm and Gf are the shear moduli of the matrix and the fibers, respectively, and S3 isgiven as

S3 ¼ 0:49247� 0:47603Vf � 0:02748V2f

In addition, the composite cylinder model developed by Hasin and Rosen [17], which isbased on the self-consistent theory, can provide reasonably accurate stiffness predictionsfor roving and SF layers. The formula for G12 given by the composite cylinder model is

G12 ¼ Gmð1þ Vf ÞGf þ ð1� Vf ÞGm

ð1� Vf ÞGf þ ð1þ Vf ÞGmð2Þ

For comparative purposes and because of its popularity, we also compute the elasticproperties of roving and SF layers by the inverse rule of mixtures approach [18]. Forexample, the expression for G12 is given by

G12 ¼GmGf

ð1� Vf ÞGf þ VfGmð3Þ

Based on the assumption that the material is isotropic in the plane [19], the in-planeshear stiffness of the CSM layers are estimated from the approximate expressions given byHull [20] for randomly oriented composites:

G12 ¼1

8E1 þ

1

4E2 ð4Þ

where, E1 and E2 are computed from any of the micromechanics models described above.Using macromechanics, the in-plane shear stiffness of a laminate can be predicted from

the shear compliance (�66), which is computed from ply material properties and the inverseof the extensional stiffness matrix [A] as

Gxy ¼1

�66hð5Þ


where, h is the thickness of the whole laminate. The in-plane shear modulus computed byEquation (5) is valid for the case of the laminate under in-plane loading. For a laminateunder torsion, the in-plane shear stiffness of the laminate is computed from the twistcompliance coefficient (�66), which is dependent on the stacking sequence and is evaluatedfrom the inverse of the bending stiffness matrix [D] as

Gxy ¼12

�66h3ð6Þ

For computation of the out-of-plane shear moduli of a laminate, the constitutiverelations for the laminate shear stress resultants are given by Vinson and Sierakowski[21] as

Qx

Qy

� �¼

A55 A45

A45 A44

� ��xz

�yz

� �ð7Þ

where

Aij ¼5

4

XNk¼1

ð �QQijÞk zk � zk�1 �4

3z3k � z3k�1

� � 1h2

� i, j ¼ 4, 5

and �zzk is the distance from the mid-surface of the kth ply to the middle surface of thelaminate. Ply shear stiffness quantities ð �QQijÞk are computed from the individual ply shearmoduli G13 and G23, which are obtained from micromechanics models. Then the out-of-plane modulus of a laminate, Gxz, is

Gxz ¼A55

hð8Þ

The out-of-plane modulus in Equation (8) is useful when looking at laminatemacroscopic responses such as bending with shear deformation, buckling loads andvibration frequencies. For a laminate in torsion, we adopt a modified shear deformationtheory [13,22] to evaluate the effective out-of-plane shear modulus. In this case, thetransverse shear constitutive relations are of the form [22]

Qx

Qy

� �¼

F55 F45

F45 F44

� ��xz

�yz

� �ð9Þ

where

Qx

Qy

( )¼

Z h=2

�h=2

�xz

�yz

( )dz

F44 ¼R55

R44R55 � R245

; F55 ¼R44

R44R55 � R245

; F45 ¼R45

R44R55 � R245

;

Rij ¼9

4h2

XNk¼1

ð �SSijÞk zk � zk�1 �8

3ðz3k � z3k�1Þ

1

h2þ 16ðz5k � z5k�1Þ

1

h4

� i, j ¼ 4, 5


and ð �SSijÞk denotes the ply transverse shear compliance. Then the out-of-plane shearmodulus of a laminate, Gxz, is

Gxz ¼F55

hð10Þ

The shear modulus in Equation (10) is based on the integral of weighted shearcompliances and on the assumption that the transverse shear stress distributions areparabolic and continuous at layer interfaces.

The predicted values from Equations (6) and (10) are representative of the laminateresponses under torsion loading, and they are used in this study as baseline values forcomparisons with experimental data.

Analytical Predictions for Test Specimens

The in-plane and out-of-plane shear moduli for three pultruded laminates used in thisstudy are predicted by the above micro/macro mechanics approach. The laminatesconsidered are: (1) a unidirectional sample, designated as U and obtained from a flat plate,consisting of two continuous strand mat (CSM) layers and one roving layer [Figure 1(a)];(2) an angle-ply (A) sample, cut from a 1200 1200 1/200 wide-flange beam, with sevenCSM layers, seven angle-ply stitched fabric (SF) layers, and six roving layers [Figure 1(b)];and an angle/cross-ply (AC) sample, obtained also from a 1200 1200 1/200 wide-flangebeam, consisting of seven CSM layers, four cross-ply and three angle-ply SF layers, and sixroving layers [Figure 1(c)]. The reinforcements are all E-glass and the resin is Vinylester,with material properties listed in Table 1.

The material properties and fiber volume fractions of the different plies are computedfollowing Davalos et al. [1] and are given in Table 2, along with the in-plane shear moduli

Table 2. Ply properties and in-plane shear moduli.

Ply Properties G12 (GPa)

SampleType Lamina

t(mm) Vf

PeriodicMicrostructure

CompositeCylinder

Inverse Ruleof Mixture

U 1 oz. CSM 0.508a 0.236 4.357 4.424 3.945U 61 yield roving 8.509b 0.410 3.536 3.652 3.907A, AC 3/4 oz. CSM 0.381a 0.236 4.205 4.276 4.237A, AC �45� SF 0.660a 0.362 3.205 3.294 3.489A, AC 62 yield roving 0.851b 0.627 5.852 6.219 7.138AC 0�/90� SF 0.660a 0.370 3.256 3.350 3.553

aUnconsolidated thickness provided by supplier.bBased on total thickness of laminate.

Table 1. Material properties of the constituents.

Material E (GPa) G (GPa) m q (g/cm3)

E-glass fiber 72.395 28.841 0.255 2.550Vinylester resina 5.061 1.634 0.30 1.136

aFrom tension and torsion tests of circular resin rods [28].


values evaluated from micromechanics formulae. Then, using Equations (5), (6), (8) and(10), the in-plane and out-of-plane shear properties of the laminates (Gxy and Gxz) areevaluated and are given in Table 3; the predicted values by Equations (6) and (10) arecorrelated with experimental results in subsequent sections.

TORSION SOLUTIONS

The first solution for torsion was presented by Coulomb in 1784. About 40 years later,Navier presented a rigorous solution for torsion of circular shafts. In 1855, St. Venantdeveloped a torsion solution for rectangular bars that included a warping function. Adetailed discussion of torsion theories is presented by Hsu [23].

Lekhnitskii [8] applied St. Venant’s formulation for torsion to orthotropic materials andobtained solutions for circular and rectangular bars with three mutually perpendicularplanes of material symmetry. Davalos et al. [9] applied this approach to orthotropic woodsamples and presented a combined analytical/experimental method to obtain principalshear moduli. Lekhnitskii’s solution can be successfully used for orthotropic samples; formore complex materials, Whitney [13] presented a solution incorporating transverse sheardeformation for the torsion of rectangular laminated anisotropic composite plates. In thefollowing two sections, we briefly describe the computational algorithms of Lekhnitskii’sand Whitney’s torsion solutions for anisotropic materials.

Lekhnitskii’s Solution for Orthotropic Materials

A solution for the torsion of rectangular orthotropic bars in their principal materialdirections was presented by Lekhnitskii [8]. In Figure 2, the sample is subjected to atorque, T, applied about the longitudinal axis x and causing an angle of twist �. Thedimensions of the sample are: a¼width, b¼ thickness, and L¼ length. The torque versusangle of twist (T/�) can be expressed as

T

�¼ Gxy

ab3

L ð11aÞ

where, Gxy¼ in-plane shear modulus of the laminate and is given by

¼32a2Gxz

4b2Gxy

X1i¼1, 3, 5, ...

1

i41�

2a

ib

ffiffiffiffiffiffiffiffiGxz

Gxy

stanh

ib

2a

ffiffiffiffiffiffiffiffiGxy

Gxz

r !" #ð11bÞ

Table 3. Laminate material properties of experimental samples.

Gxy (GPa) Gxz (GPa)

SampleType

Ex

(GPa)Ey

(GPa)Equation

(5)Equation

(6)Equation

(8)Equation

(10) mxy

U 30.192 10.446 3.627 3.772 2.958 2.958 0.327A 26.366 13.259 6.412 6.247 3.813 4.199 0.396AC 28.310 14.858 5.343 4.475 3.813 4.199 0.300


In Equation (11), the torsional stiffness is a function of the two principal shear moduli.When Equation (11) is reduced to an isotropic case (Gxy¼Gxz¼G) and evaluated for alarge number of terms, is denoted by k and expressed as

k ¼1

31� 0:63

b

a

�

By substituting k for in Equation (11), we obtain St. Venant’s torsion solution for arectangular isotropic bar:

T

�¼ Gxy

ab3

L

1

31� 0:63

b

a

� ð12Þ

The most commonly used approach to solve for the shear moduli in Equation (11) is totest two samples with different cross-sectional dimensions but same material orientation[4]. A graphical method proposed by Tarnopol’skii and Kincis [24] is typically used tosolve for the shear moduli. The problem with this approach is that the out-of-plane shearmodulus, Gxz, is grossly underestimated. For example, Sumsion and Rajapakse [14]reported Gxz values as much as 90% lower than Gxy for unidirectional transverselyisotropic composites, for which one should expect Gxz�Gxy.

A better approach is to pair samples with either the same or different cross-sectionaldimensions but with material orientations normal to each other. Davalos et al. [9] used thisapproach for paired orthotropic wood samples manufactured with the same cross-sectional dimensions, but with principal material orientations perpendicular to oneanother. The corresponding two distinct equations were then solved by a method ofsuccessive substitutions. In this study, we modify the approach proposed by Davalos et al.[9] to obtain a more efficient solution by the bisection method [25,26]. Pultruded FRPstructural laminates are considered, and to obtain samples of variable widths in thexz-plane, the pultruded laminates [Figure 3(a)] were bonded flat-wise, and vertical stripswere cut from the bonded assemblies as shown in Figure 3(b).

Figure 2. Coordinates and dimensions of rectangular bar for torsion solution.


Whitney’s Solution for Anisotropic Materials

Whitney [13] provided solutions for both pure and free torsion of plates, and the basicassumptions of the theory are similar to those often used in macromechanics ofcomposites. For an anisotropic laminated plate subjected to torsion, a modified sheardeformation theory for displacements was used to derive the torsional stiffness as

T

�¼

4a

D�66

� 1� 2=�ð Þ tanh �=2ð Þ½ �

1� 2=�ð Þ 1� �DD66=D�66

� �tanh �=2ð Þ

� � ð13Þ

where �DD66 is given by

�DD66 ¼ D�66 �

B�16ðD

�11B

�16 �D�

16B�11Þ þD�

16ðD�16A

�11 � B�

11B�16Þ

ðD�11A

�11 þ B�2

11Þ

� �ð14Þ

A�ij ,B

�ij, andD

�ij are the coefficients of the inverse of the ABD matrix, and � is defined as

� ¼ a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�DD66ðA44A55 � A2

45Þ

A44

s24

35 ð15Þ

where a¼width of sample (Figure 3). For a symmetric orthotropic sample, we getA45 ¼ B�

16 ¼ D�16 ¼ 0, and Equation (13) can be simplified as

T

�¼

4ab3Gxy

12

� 1�

2

�

� tanh

�

2

� � �ð16Þ

where

� ¼ a

ffiffiffiffiffiffiffiffiffiffiffiffi12Gxz

b2Gxy

s" #

(b)(a)

Figure 3. Sample orientations for: (a) in-plane (xy) and (b) out-of-plane (xz) shear moduli.


The Lekhnitskii’s and Whitney’s torsion solutions summarized above are used inconjunction with torsion tests to evaluate the shear moduli for various laminates. Theapplication of torsion solutions as data reduction techniques for torsion tests is furtherdiscussed in the Results section.

TEST SAMPLES AND PROCEDURE

The three sample types shown in Figure 1 were tested using a manually operated torsionmachine that permitted accurate measurements of applied torque and angle of twist. Inthis section we describe the test specimens and methods used.

Test Samples

The unidirectional samples (U) were cut from a pultruded flat plate of thickness¼9.53mm (3/800) [Figure 1(a)], and both the angle-ply samples (A) and angle/cross-plysamples (AC) were obtained from pultruded wide-flange beams 304.8 304.8 12.7mm(1200 1200 1/200); thus the thickness of these samples was 12.7mm(1/200), and their fiberarchitectures are shown respectively in Figures 1(b) and 1(c). For each of these threesample-types, specimens of 0.432m (1700) in length were cut in four widths: 25.4mm (1.000),38.1mm (1.500), 50.8mm (2.000), and 63.5mm (2.500) [e.g., for the U samples, U10, U15, U20and U25 represent the unidirectional samples with widths of 25.4mm (1.000), 38.1mm(1.500), 50.8mm (2.000) and 63.5mm (2.500), respectively], with four replications for eachwidth. The orientation of these samples is described in Figure 3(a), with the widthdimension along the y-axis and the longitudinal dimension along the x-axis.

To obtain samples with larger dimensions than the available material thickness in thexz-plane [see Figure 3(b)], flat strips were cut from the plate and beam components andbonded flat-wise with epoxy. The bonded layered specimens were subsequently cut intovertical strips to obtain samples with the same dimensions as those obtained directlyfrom the pultruded components; i.e., samples with thickness¼ 9.53mm (3/800) or 12.7mm(1/200) and width¼ 25.4mm (1.000), 38.1mm (1.500), 50.8mm (2.000) and 63.5mm (2.500). Thisapproach was used to produce paired samples with same dimensions, but materialorientations normal to each other (see Figure 3). To denote the bonded samples for eachtype of laminate, the letter B is added to the corresponding designation; e.g., UB identifiesthe unidirectional bonded samples.

Test Procedure

The samples were tested in a manually-operated torsion machine. The torque wasapplied through a large wheel at one end and a counterbalance pendulum at the otherend. To increase the accuracy of angle of twist measurement, a circular gear was fittedconcentrically through the specimen at each gage point, and the rotation of the gearwas transferred by a plastic chain to an outside small pulley in order to magnify thedisplacement of a transducer (LVDT) suspended from the pulley by a plastic thread.The angle of twist between two sections sufficiently away from the grips (ASTM D198)was computed from the displacements of the transducers and the geometry of thegear-and-pulley mechanism. In this study, the gauge length of 0.229m (9.0 in) is used tomeasure the angle of twist between two sections.


The torque was applied at a rate of about three degrees per minute, for a total angle oftwist within the material elastic limit of about 10�. Each sample was tested four times, andthe torque versus angle of twist (T/�) was obtained through linear regression of the data.For further details of the experimental procedure, see Schussel [27]. The data reductiontechniques used for various torsion solutions are discussed next.

RESULTS, DATA REDUCTION METHODS AND DISCUSSION

The torsional stiffnesses (T/�) for three types of laminated samples were obtainedexperimentally to evaluate their shear moduli from torsion solutions. The experimentalprogram is organized in two parts: (1) unidirectional (U) samples and (2) angle-ply (A)and angle/cross-ply (AC) samples.

Unidirectional Samples

This phase of the study is concerned with the response of the unidirectional samplesshown in Figure 1(a). Two approaches are used to evaluate the shear moduli Gxy and Gxz

from paired samples. The first approach consists of applying Lekhnitskii’s orthotropictorsion solution to test results of samples of different sizes but same material orientations;in the second approach, the solutions by Lekhnitskii and Whitney are used in conjunctionwith test results for samples with mutually orthogonal material orientations; i.e.,combining samples U (cut from the pultruded plate) and UB (produced by bonding) asshown in Figure 3. Further, as an approximation and to show the validity of transverseisotropy for these samples, the shear moduli is also obtained from St. Venant’s isotropicsolution (Equation 12). The results for each method used are discussed next.

UNIDIRECTIONAL SAMPLES WITH SAME MATERIAL ORIENTATIONSA commonly used method to evaluate shear moduli for specially orthotropic materials

(when the geometric and material axes coincide) is by torsion tests of samples withdifferent cross-sectional dimensions but same material orientation. However, as shown inthis study, this method can grossly underestimate the out-of-plane shear modulus Gxz.Samples with four different widths were tested in torsion, and the average dimensions andtorsional stiffnesses of the samples, including the bonded samples to be used later, aregiven in Table 4.

Following the graphical method by Tarnopol’skii and Kincis [24], the results for theunidirectional samples U with same material orientations are shown in Table 5. Based on

Table 4. Average dimensions and response of U and UB samples.

Sample Width (mm) Thickness (mm) T//(N-m)

U10 24.638 9.296 64.095 (COV = 2.623%)U15 37.694 9.373 121.517 (COV = 2.163%)U20 50.444 9.398 179.085 (COV = 1.688%)U25 63.221 9.373 239.128 (COV = 1.047%)UB10 25.781 9.296 63.482 (COV = 3.227%)UB15 37.694 9.373 107.040 (COV = 4.524%)UB20 50.419 9.550 181.686 (COV = 5.002%)UB25 65.075 9.042 213.536 (COV = 3.115%)


transverse isotropy, it is expected that the two shear moduli be approximately equal toeach other, but in this case, the out-of-plane modulus Gxz obtained is only about 27% ofthe in-plane modulus Gxy. A similar discrepancy can be observed in the results reported forunidirectional samples by Sumsion and Rajapakse [14], who provided no explanation forthe out-of-plane modulus being as low as one-tenth of the in-plane modulus. A betterapproach is to use paired samples with material orientations normal to each other, asshown next.

UNIDIRECTIONAL SAMPLES WITH DISTINCTMATERIAL ORIENTATIONS

To obtain shear moduli from torsion solutions by Lekhnitskii [Equation (11)] andWhitney [Equation (16)], an effective approach is to determine the (T/�) values for at leasttwo samples with distinct material orientations [9]. In this study, samples with the samecross-sectional dimensions but orthogonal material orientations were paired (Figure 3),and using Equations (11) and (16), the in-plane and out-of-plane shear moduli values wereobtained. Each size group consisted of four samples in the xy direction and four samples inthe xz direction to obtain 16 sets of values for Gxy and Gxz. Furthermore, by cross-pairingeach size-group of xy samples with the four size-groups of xz samples, a total of 64 sets ofGxy and Gxz values were obtained, with a final grand total of 256 sets of values.

Data Reduction Based on Lekhnitskii’s SolutionTo use Lekhnitskii’s solution, Equation (11) was rewritten for each sample orientation as

Gxy ¼K1L1

a1b311

; Gxz ¼K2L2

a2b322

ð17aÞ

where

Ki ¼Ti

�i, i ¼ 1, 2 ð17bÞ

and 1 and 2 are expressed as

1 ¼32a21Gxz

4b21Gxy

X1i¼1, 3, 5, ...

1

i41�

2a1ib1


Gxy

stanh

ib12a1


Gxz

r !" #ð17cÞ

Table 5. Shear moduli results from U samples with one material orientation.

Sample Pairs Gxy (GPa) Gxz (GPa)

U10 : U15 3.647 1.269U10 : U20 3.813 1.151U10 : U25 3.806 1.151U15 : U20 4.020 0.848U15 : U25 3.944 0.910U20 : U25 3.992 0.841

Average (COV) 3.868 (3.65%) 1.027 (17.90%)


2 ¼32a22Gxy

4b22Gxz

X1i¼1, 3, 5, ...

1

i41�

2a2ib2


Gxz

rtanh

ib22a2


Gxy

s !" #ð17dÞ

Both Equations (17a) need to be solved simultaneously, and to simplify the computationaleffort we divide Equation (17c) by Equation (17d), resulting in

a2a1

� 3b2b1

K1L1

K2L2¼ �

X1i¼1, 3, 5

1=i4 1� ð2a1=ib1Þðffiffiffi�

pÞ tanhððib1=2a1Þð

ffiffiffiffiffiffiffiffi1=�

pÞÞ

h iX1

i¼1, 3, 5

1

i41� ð2a2=ib2Þð


pÞ tanh ððib2=2a2Þð

ffiffiffi�

pÞÞ

h i ð18Þ

where

� ¼Gxz

Gxy

Now, there is only one unknown, �, in Equation (18), and by solving for � by the bisectionmethod [25,26], we can obtain the in-plane (Gxy) and out-of-plane (Gxz) shear moduli as:

Gxz ¼K1L1

a31b1X1

i¼1, 3, 5

1=i4 1� ð2a1=ib1Þffiffiffi�

ptanh ððib1=2a1Þð


pÞÞ

h i( )4

32ð19aÞ

Gxy ¼Gxz

�ð19bÞ

Table 6 presents the results for paired samples of the same size and cross-paired samplesof different sizes. The results of this method support the assumption that an orthotropiccomposite material with unidirectional fibers can be modeled as transversely isotropic.Also, the results indicate that as the width of the sample increases, the corresponding shearmodulus on that plane increases (Table 6). To observe this effect in Table 6, see any rowfor Gxy and any column for Gxz.

Data Reduction Based on Whitney’s Solution

Whitney’s simplified torsion solution [13] is given in Equation (16). By mathematicallyrotating the coordinate system (Figure 4), the following system of equations are obtainedfor the xy and xz planes:

Table 6. Shear moduli results using Lekhnitskii’s solution for U samples.

Width(mm) xzPlane

Gxy (GPa) Gxz (GPa)

xy Plane xy Plane

24.638 37.694 50.444 63.221 24.638 37.694 50.444 63.221

25.781 2.959 3.250 3.384 3.535 2.683 2.644 2.627 2.61137.694 2.936 3.233 3.371 3.524 2.793 2.767 2.755 2.74350.419 2.858 3.180 3.332 3.491 3.250 3.225 3.214 3.20365.075 2.841 3.169 3.323 3.484 3.366 3.344 3.338 3.327


xy-plane : K1 ¼T1

�1¼

1

L1

4a1b31Gxy

12

� 1�

b1ffiffiffi3

pa1

ffiffiffi1

�

s !tanh

ffiffiffi3

pa1

b1

ffiffiffi�

p� " #

ð20aÞ

xz-plane : K2 ¼T2

�2¼

1

L2

4a2b32Gxy

12

� 1�

b2ffiffiffi3

pa2

ffiffiffi�

p�

tanh

ffiffiffi3

pa2

b2

ffiffiffi1

�

s !" #ð20bÞ

where

� ¼Gxz

Gxy

By dividing Equation (20a) by Equation (20b), we can obtain

K1

K2¼

L2

L1

� a1b

31

a2b32

� 1

�

1� b1=ffiffiffi3

pa1

� � ffiffiffiffiffiffiffiffi1=�

p� �tanh

ffiffiffi3

pa1=b1

� � ffiffiffi�

p� �� 1� b2=

ffiffiffi3

pa2


p� �tanh

ffiffiffi3

pa2=b2


p� �� ð21Þ

Again, by using the bisection method [26] and solving for the only unknown, �, we canobtain the in-plane (Gxy) and out-of-plane (Gxz) shear moduli as:

Gxy ¼3K1

a1b31 1� b1=

ffiffiffi3

pa1


p� �tanh

ffiffiffi3

pa1=b1


p� ��

� ð22aÞ

Gxz ¼ Gxy� ð22bÞ

Table 7 presents the results for paired samples of the same size and cross-paired samplesof different sizes. Similar to the results based on Lekhnitskii’s solution, the shear modulusincreases as the corresponding width of the sample increases.

(b)(a)

Figure 4. Orientation for: (a) in-plane shear and (b) out-of-plane shear moduli.


DISCUSSION FOR UNIDIRECTIONAL SAMPLESFrom Tables 6 and 7, it can be observed that the shear modulus increases as the

width of the corresponding principal plane increases. A comparative summary of averageexperimental values are given in Table 8, along with predicted values from micro/macro-mechanics formulas [see Equations (6) and (10)]. Consistent results are obtainedby all three solutions: Lekhnitskii, Whitney, and isotropic. For the in-plane shearmodulus, the predicted value is about 16.3 and 16.9% higher than the averageexperimental values of various widths from Lekhnitskii’s and Whitney’s solutions,respectively; whereas, there are correspondingly about 3.4 and 0.1% differences forthe out-of-plane shear modulus. Thus, the proposed analytical formulations canprovide reasonable shear moduli properties for unidirectional composites. A graphicalrepresentation of the results is given in Figures 5a and 5b, clearly showing the behaviorof shear moduli with increase in sample width. As discussed above, the method usingsamples of one material orientation grossly underestimates out-of-plane shear moduli(Table 5).

Angle-Ply and Angle/Cross-Ply Samples

Following the procedures described above, the angle-ply (A) and angle/cross-ply (AC)samples and their corresponding bonded pairs (AB and ACB) were tested in torsion and

Table 8. Summary of Gxy and Gxz for U samples.

ShearModulus

SamplesWidth(mm)

a/bRatio

Isotropic(GPa)

Lekhnitskii(GPa)

Whitney(GPa)

Predicted(GPa)

Gxy 24.638 2.65 2.882 2.896 2.930 3.77137.694 4.02 3.282 3.206 3.179 Equation (6)50.444 5.37 3.344 3.351 3.31663.221 6.75 3.496 3.509 3.475

Average (COV) 3.254 (8.05%) 3.241 (8.06%) 3.227 (7.16%)

Gxz 25.781 2.77 3.034 2.641 2.586 2.95837.694 4.02 3.199 3.041 2.734 Equation (10)50.419 5.28 3.241 3.227 3.18565.075 7.20 3.372 3.344 3.316

Average (COV) 3.213 (4.34%) 3.061 (10.06%) 2.958 (11.89%)

Table 7. Shear moduli results using Whitney’s solution for U samples.

Width (mm)xz plane

Gxy (GPa) Gxz (GPa)

xy Plane xy Plane

24.638 37.694 50.444 63.221 24.638 37.694 50.444 63.221

25.781 2.877 3.195 3.343 3.501 2.619 2.586 2.571 2.55737.694 2.853 3.178 3.330 3.490 2.754 2.728 2.718 2.70850.419 2.786 3.131 3.294 3.460 3.215 3.189 3.179 3.16965.075 3.199 3.199 3.285 3.453 3.337 3.314 3.310 3.303


analyzed by both Lekhnitskii’s and Whitney’s solutions. The fiber architectures are shownin Figure 1(b) (samples A) and Figure 1(c) (samples AC), and the layer and laminateproperties are given in Tables 2 and 3, respectively. The average dimensions and measuredtorsional responses (T/�) for the as-manufactured and bonded samples are summarized inTable 9.

Using the sample-pairing technique by testing samples with material orientationsnormal to each other (e.g., samples A and AB; samples AC and ACB), as discussedabove for the unidirectional samples, the shear moduli were obtained through datareduction torsion solutions. The results for Lekhnitskii’s and Whitney’s solutionsare given, respectively, in Tables 10 and 11. Each value reported in these tables is theaverage of 16 results, corresponding to four replications per sample. As observed inTables 10 and 11, consistent results are obtained for both Lekhnitskii’s and Whitney’ssolutions.

(a)

(b)

Figure 5. (a) In-plane shear modulus Gxy for U samples; (b) Out-of-plane shear modulus Gxz for U samples.


A comparative summary of average experimental values of shear moduli for bothsamples A and AC is given in Table 12, along with predicted values [Equations (6) and(10)] from micro/macro-mechanics formulas. For the A samples, the differences are 9.3and 5.9%, respectively, for in-plane and out-of-plane shear moduli between averageexperimental values based on Lekhnitskii’s solution and predicted values; whereas, thecorresponding differences are 6.0 and 7.5% for the results from Whitney’s solution andpredicted values. For the AC samples, compared to the average experimental values fromLekhnitskii’s solution, the predicted values are lower by 16.9 and 2.0%, respectively, forin-plane and out-of-plane shear moduli. Similarly, the corresponding predicted values arelower by 14.6 and 0.2% than the average experimental results based on Whitney’s

Table 10. Shear moduli results using Lekhnitskii’s solution for A and AC samples.

Width(mm)

xz Plane

Gxy (GPa) Gxz (GPa)

xy Plane xy Plane

Samples 24.994 37.719 50.470 63.754 24.994 37.719 50.470 63.754

A 25.222 7.382 6.438 6.462 7.266 3.927 4.016 4.016 3.93637.744 7.376 6.449 6.469 7.266 3.933 3.985 3.982 3.93650.394 7.446 6.479 6.489 7.280 3.862 3.901 3.900 3.87363.856 7.288 6.424 6.454 7.247 4.028 4.055 4.050 4.031

Width(mm) xz

Plane

xy Plane xy Plane

25.070 37.795 50.495 63.830 25.070 37.795 50.495 63.830

AC 25.222 5.461 5.076 5.367 5.619 4.278 4.351 4.296 4.25237.871 5.434 5.072 5.360 5.611 4.336 4.372 4.342 4.31450.571 5.490 5.099 5.379 5.618 4.215 4.241 4.220 4.25863.906 5.470 5.091 5.371 5.619 4.258 4.278 4.267 4.252

Table 9. Average dimensions and response of A, AB, AC andACB samples.

SampleWidth(mm)

Thickness(mm) T//(N-m)

A10 24.994 12.649 311.445 (COV¼ 3.22%)A15 37.719 12.573 516.752 (COV¼ 8.75%)A20 50.470 12.573 756.929 (COV¼ 8.18%)A25 63.754 12.598 1122.146 (COV¼ 9.94%)AB10 25.222 11.786 185.585 (COV¼ 5.63%)AB15 37.744 11.862 308.939 (COV¼ 6.47%)AB20 50.394 11.811 418.072 (COV¼ 11.91%)AB25 63.856 11.735 553.605 (COV¼ 6.91%)AC10 25.070 12.675 261.299 (COV¼ 2.85%)AC15 37.795 12.675 439.600 (COV = 6.93%)AC20 50.495 12.675 662.312 (COV = 7.51%)AC25 63.830 12.675 911.575 (COV = 6.97%)ACB10 25.222 12.040 201.557 (COV = 5.78%)ACB15 37.871 12.141 351.071 (COV = 10.43%)ACB20 50.571 12.090 476.616 (COV = 5.80%)ACB25 63.906 12.065 623.885 (COV = 9.57%)


Table 12. Summary of Gxy and Gxz for A and AC samples.

SampleShear

Modulus

SampleWidth(mm)

a/bRatio

Lekhnitskii(GPa)

Whitney(GPa)

Predicted(GPa)

A Gxy 24.994 1.98 7.373 6.861 6.24737.719 3.00 6.447 6.238 Equation (6)50.47 4.01 6.469 6.331

63.754 5.06 7.265 7.141

Average (COV) 6.889(7.25%)

6.643(6.48%)

Gxz 25.222 2.14 3.974 3.885 4.19937.744 3.18 3.959 3.908 Equation (10)50.394 4.27 3.884 3.85063.856 5.44 4.041 3.979

Average (COV) 3.965(1.63%)

3.905(1.40%)

AC Gxy 25.070 1.98 5.463 5.181 4.47537.795 2.98 5.084 4.956 Equation (6)50.495 3.98 5.369 5.27563.830 5.04 5.617 5.542

Average (COV) 5.383(4.16%)

5.239(4.63%)

Gxz 25.222 2.09 4.294 4.162 4.19937.871 3.12 4.341 4.267 Equation (10)50.571 4.18 4.233 4.17163.906 5.30 4.264 4.227

Average (COV) 4.283(1.07%)

4.207(1.17%)

Table 11. Shear moduli results using Whitney’s solution for A and AC samples.

Width(mm) xz

Plane

Gxy (GPa) Gxz (GPa)

xy Plane xy Plane

Samples 24.994 37.719 50.470 63.754 24.994 37.719 50.470 63.754

A 25.222 6.864 6.243 6.334 7.149 3.865 3.920 3.911 3.84237.744 6.840 6.240 6.332 7.142 3.897 3.929 3.924 3.88250.394 6.882 6.260 6.345 7.151 3.842 3.864 3.860 3.83363.856 6.857 6.212 6.313 7.121 3.875 4.023 4.020 3.997

Width(mm) xz

Plane

xy Plane xy Plane

25.070 37.795 50.495 63.830 25.070 37.795 50.495 63.830

AC 25.222 5.197 4.960 5.281 5.549 4.169 4.207 4.156 4.11837.871 5.159 4.945 5.266 5.535 4.271 4.293 4.262 4.23950.571 5.195 4.964 5.279 5.545 4.173 4.188 4.169 4.15363.906 5.175 4.955 5.271 5.538 4.229 4.240 4.224 4.213


solution. Graphical representations of the results are given in Figures 6a, 6b, 7a and 7b forsamples A and AC, respectively.

For the U samples, we observed an asymptotic increase of shear moduli values as thesample-width increases (Figures 5a and 5b). In contrast to this behavior, there is no cleartrend on the data obtained for the A and AC samples in relation to the sample-width; ingeneral, the experimental values are randomly distributed around a narrow band, withrelatively larger discrepancies for the A samples (Figures 6a and 6b) than for the ACsamples (Figures 7a and 7b). One reason for the large variability with the A samples can beattributed to the larger difference between Gxy and Gxz (degree of anisotropy) than for theAC samples, which are nearly transversely isotropic (similar to the U samples). For both Aand AC samples, the experimental values are higher than the predictions for the in-planeshear moduli.

(a)

(b)

Figure 6. (a) In-plane shear modulus Gxy for A samples; (b) Out-of-plane shear modulus Gxz for A samples.


CONCLUSIONS

It is shown that torsion tests of pultruded FRP composites with rectangular cross-sections can be used to determine in-plane and out-of-plane shear moduli values. Thedetermination of out-of-plane shear moduli from samples with same material orientationdoes not lead to accurate results. It is necessary to test samples with two materialorientations, which can be obtained by bonding flat laminates and cutting vertical stripsfrom the assemblies. Both Lekhnitskii’s and Whitney’s solutions with paired samples arerecommended to obtain shear moduli values using the torsional stiffness from torsion testsand the data reduction methods outlined in this study. Consistent results are obtained forboth Lekhnitskii’s and Whitney’s solutions. The predicted in-plane and out-of-plane shear

(a)

(b)

Figure 7. (a) In-plane shear modulus Gxy for AC samples; (b) Out-of-plane shear modulus Gxz for AC samples.


moduli of the laminates are obtained from micro/macromechanics models for the case oftorsional loading [Equations (6) and (10)]. The combined experimental/analytical torsionsolutions presented in this paper can be efficiently used to obtain reliable shear moduli forcomposite laminates.

ACKNOWLEDGMENTS

The test samples were generously provided by Creative Pultrusions, Inc., Alum Bank,PA. Partial financial support for this study was received from NSF-CRCD program.

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Shear Moduli of Structural Composites from Torsion Tests

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