On the equivalent flexural and shear moduli of laminated beams: Definition and determination by bending tests

Composites: Part A 175 (2023) 107802 Available online 21 September 2023 1359-835X/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). On the equivalent flexural and shear moduli of laminated beams: Definition and determination by bending tests Faustino Mujika a , * , Mireia Olave b , M. Asunci´ on Cantera a , Ugutz Garitaonaindia a , Miren Isasa a , Ainhoa Arrese a a Mechanics of Materials Group, Department of Mechanical Engineering, University of the Basque Country (UPV/EHU), Spain b Ikerlan Technology Research Centre, Basque Research and Technology Alliance (BRTA), P◦J.M. Arizmendiarrieta 2, 20500 Arrasate-Mondrag´ on, Spain ARTICLE INFO Keywords: A. Hybrid composites B. Sandwich material C. Equivalent modulus D Laminated beam ABSTRACT Equivalent flexural and shear moduli of laminated beams of rectangular cross-section made of orthotropic layers are defined by an analytical approach. Equivalent flexural and shear modulus are defined as those that would correspond to a homogenous beam. In order to check the suitability of the moduli defined, three-point bending tests varying the span are carried out in a sandwich material and in hybrid laminates. In the case of the sandwich material, it is shown that a three-point bending test could be considered a shear test from the stiffness point of view. In the case of hybrid materials, virtual tests are carried out by the Finite Element Method. The agreement between equivalent moduli obtained from the analytical approach and the bending tests has been checked, being quite good in all cases. Finally, the effect of dimension uncertainty in three-point bending tests is analyzed by a Monte Carlo simulation. 1. Introduction Shear effects in the stiffness of beams of rectangular cross section depend on the ratio between flexural modulus and shear modulus. In isotropic materials, this ratio is less than 3 and shear contribution in displacements is negligible. Nevertheless, shear effects can influence stiffness properties of some orthotropic materials as wood, used for wing-beam materials in the early stages of aviation [1]. Usually, Firstorder Shear Deformation Theory (FSDT) is applied, named also Timoshenko’s beam theory [2], assuming that shear strains are uniform in the thickness of the beam. Shear correction factors are applied to take into account the actual variation of shear stresses and strains in the thickness. Bert [3], determined the shear factor of a non-homogeneous cross-section using a simple mechanics-of-materials approach. Adams and Miller [4], applied Classical Laminated Plate Theory (CLPT) to the three-point beam bending problem to determine the flexural modulus and the energy absorbed for both impact and static loadings of hybrid laminates. Bert and Gordaninejad [5], analyzed the transverse shear effects in bimodular materials, which have different elastic moduli in tension and compression, based on equivalent shear strain energy. Raman and Davalos [6], derived a general expression for the shear correction factor of laminated rectangular beams with an arbitrary lay-up configuration. Results were compared with existing results for composite beams and plates. He and Zhang [7], analyzed the bending of rectangular, simply supported, antisymmetric angle-ply laminated plates, using a refined shear deformation theory to obtain closed-form solutions. Pai and Schulz [8], carried out a new derivation of shear correction factors in the case of isotropic materials, being energy-consistent and showing its physical meaning. Altenbach [9], proposed a method to determine shear stiffness for sandwich and laminated plates and compared it with results from other authors. Ghugal and Shimpi [10], presented a review of displacement and stress based refined theories for isotropic and anisotropic laminated beams, underlining some critical issues. Gibson [11], proposed simplified mechanics of materials equations for predicting both flexural and shear components of transverse deflections in composite sandwich beams, concluding that shear deflections were greater than flexural deflections. He compared proposed equations with experimental measurements and numerical results. Navarro et al. [12], developed an analytical model for the static indentation of sandwich beams with a foam core. Good correlation was obtained from the comparison of the model with experimental and finite element method (FEM) results. Gao et al. [13], investigated analytically and experimentally the shear effects on the deflection of a building floor panel made of Carbon Fibre Reinforced Polymer (CFRP), including analytical * Corresponding author. E-mail address: [email protected] (F. Mujika). Contents lists available at ScienceDirect Composites Part A journal homepage: www.elsevier.com/locate/compositesa https://doi.org/10.1016/j.compositesa.2023.107802 Received 3 May 2023; Received in revised form 8 September 2023; Accepted 17 September 2023 Composites Part A 175 (2023) 107802 2 and experimental results. Sayyad and Ghugal [14], carried out a critical review of the literature concerning bending, buckling and free vibration of laminated composite and sandwich beams and proposed future research directions. Mujika et al. [15], determined out-of-plane elastic properties in honeycomb sandwich panels by experimental, numerical and analytic methods. The results obtained by three-point bending at different spans in that article are analyzed with a novel perspective in the current study. Lim and Kim [16], dealt with the thermo-elastic effects on shear correction factors for functionally graded beams, assuming temperature dependence of material properties. They concluded that thermal effects have influence on the shear correction factors. Bisheh and Wu [17], investigated the effects of the transverse shear and rotary inertia in wave propagation in piezoelectric coupled laminated composite cylindrical shells, considering different stacking sequences and fibre orientations. Cao and Niu [18], developed a new model for the buckling analysis of composite sandwich panels that took into account interlayer shear effects. By comparing to FEM, the solution proposed led a better accuracy than previous analytical solutions. Giordano et al. [19], carried out three-point bending tests in a composite sandwich structure, with carbon fibre woven face sheets and a relatively compliant foam core. Displacements and strains were obtained by Digital Image Correlation (DIC). Comparing experimental values with previous models, the best agreement was obtained with the First Order Shear model [20]. Garg et al. [21] carried out a review of the analysis of sandwich structures including theories and analysis methods of problems related to statics, vibration and buckling. Zhou et al. [22] investigated the bending behaviour of novel magnesium alloy facesheet and 3D-printed PLA lattice core sandwich panels, by comparing experimental results obtained by three-point bending with numerical analysis. Belarbi et al. [23] developed new finite element models for the analysis of symmetric and assymmetric sandwich panels, based on a layerwise approach and extended the formulation [24] to the analysis of the free vibration of multilayer sandwich plates. The bending behaviour of functionally graded single-layered, symmetric and non-symmetric sandwich beams by the finite element analysis, using a novel parabolic shear deformation theory was also analyzed by them [25]. Moreover, they developed also a two-node beam element [26] to investigate the case of curved beams of functionally graded sandwich beams. Garg et al [27–29] developed a C 0 finite element-based higher-order zigzag formulation to solve the bending, free vibration and buckling problems of sandwich plates and beams. In [30] their model, incorporated transverse normal and shear stress condition at interfaces. Mei et al. [30] studied the bending behavior of foam filled composite X-core sandwich panels by three-point bending tests, by analytical and numerical models. Vinh et al. [31] devoloped a novel, enhanced first-order mixed plate element, for static bending and free vibration analysis of functionally graded sandwich plates. Sayyad et al. investigated the effect of concentraded loading in laminated sandwich arches [32] and in laminated composites shells [33], using various equivalent single layer shell theories. Sayyad and Ghugal [34] presented a trigonometric shear deformation theory taking into account transverse shear deformation as well as transverse normal strain effect in their formulation, for dealing with the static flexure of symmetric and anti-symmetric cross-ply laminated beams. In their conclusions, they state that the results of transverse shear stresses obtained by the integration of equilibrium equations are better than those obtained by constitutive relations. The analytic approach of the current study is based on the main hypothesis of a linear strain field, which corresponds to Euler-Bernouilli hypothesys, and a plane stress state, without the assumption of any displacement field. Then, the complementary strain energy or coenergy corresponding to the beam is determined and the displacement in threepoint bending is obtained by the Engesser-Castigliano theorem. By comparison with the equation that correspond to a homogeneus beam, equivalent flexural and shear moduli are defined. Then, the suitability of those moduli is checked by previous experimental results [15] of sandwich specimens, where shear effects are dominant, and by numerical analysis of symmetric and asymmetric hybrid laminates, where bending is dominant. The results of bending tests at different spans in the same specimen are used to determine the equivalent moduli in both cases, using a previously defined procedure [35], and proposing new regression schemes. Finally, an error analysis is included, taking into account the effect of the uncertainty of different dimensions of the specimen and the test. The main novelties of the current study are: 1. The analytical approach is based on the Euler-Bernouilli hypothesis concerning the linearity of axial normal strains through the thickness, without the assumption of any displacement field. Moreover, it is not necessary to define any shear correction factor. 2. Equivalent flexural and shear moduli that would correspond to an equivalent homogeneous beam are defined in the general case of a laminated beam constituted for orthotropic layers of rectangular cross-section. 3. The methodology based on the span variation testing the same specimen is used to determine the equivalent moduli previously defined. It is applied to sandwich and hybrid laminates. 4. It is shown that in the case of sandwich laminates, for small spans, bending test could become a shear test from the stiffness point of view. Then, the equivalent shear modulus can be determined by a single test at low span. It constitutes the counterpart to determine the equivalent flexural modulus for large spans, when bending is dominant. 5. The importance of the accurate determination of the span length is shown, using a Monte Carlo simulation. 2. Analytical approach to define equivalent flexural and shear moduli 2.1. Normal stresses The reference system is located in the mid-plane of the beam, according to Fig. 1. Assuming that the laminate has n laminae, the naming is made from top to bottom. The analysis developed is valid for a laminated beam of rectangular cross section, the beam axis being one of the principal directions of orthotropy of each lamina. The main assumptions of the analytic approach are: 1. Normal strains are linearly distributed through the thickness. 2. Plane stress state in the zx plane and σ z =0. 3. The stress–strain behaviour of each ply is linear elastic and orthotropic. 4. Residual thermal stresses are not considered. According to the first assumption: Fig. 1. Laminated beam constituted by n laminae. F. Mujika et al. Composites Part A 175 (2023) 107802 3 ε x= ε 0+κz (1) Eq. (1) is the strain field that results form the Euler-Bernouilli Beam Theory, where it is assumed that sections of the beam remain plane and perpendicular with respect to the beam axis. ε 0 is the strain of the middle surface of the beam and κ is the bending curvature of the beam. According to the second assumption σ y= σ z=0. Then, normal stresses in the k lamina, are: σ k x=Ek x ε x⇒ σ k=Ek ε =Ek( ε 0+κz)(2) Where E k is the elastic modulus of the lamina k. The normal force and the bending moment in a section are the resultant and the resultant moment of normal stresses in the section, respectively: N=∑ n k=1∫zk zk−1 Ek( ε 0+κz)wdz =A ε 0+Bκ M=∑ n k=1∫zk zk−1 Ek( ε 0+κz)zwdz =B ε 0+Dκ (3) where w is the specimen’s width. In matrix form Eq. (3) is: {N M}=[A B B D ]{ ε 0 κ}(4) Stiffness coefficients in Eq. (4) are: A=w∑ n k=1 Ek(zk−zk−1)B=w 2∑ n k=1 Ek(z2 k−z2 k−1)D =w 3∑ n k=1 Ek(z3 k−z3 k−1)(5) A and D are the axial and bending stiffness coefficients, respectively, and B is the normal-bending coupling stiffness coefficient. The inverse form of Eq.(4) is: { ε 0 κ}=[a b b d ]{N M}(6) Where the normal, coupling and bending compliance coefficients of the beam are, respectively: a=D AD −B2b=−B AD −B2d=A AD −B2(7) Replacing Eq. (6) in Eq. (2), normal stresses are: σ k=Ek( ε 0+zκ) = Ek[N(a+bz) + M(b+zd) ] (8) 2.2. Interlaminar shear stresses According to the assumptions carried out, the following equilibrium equations must be satisfied in each lamina: σ k x,x+ τ k zx,z=0 τ k zx,x+ σ k z,z=0(9) Differentiating Eq. (8) and taking into account that M,x=V, being V the shear force, the derivative of the normal stress is: σ k x,x=VEk(b+zd)(10) Replacing Eq. (10) in Eq. (9) 1 and after integrating, shear stresses are: τ k zx(z) = V[−Ek(bz +dz2 2)+ck](11) Boundary conditions of shear stresses in each lamina are: τ 1 zx(z0) = τ n zx(zn) = 0 τ k−1 zx (zk−1) = τ k zx(zk−1)(12) Imposing the first condition of Eq. (12) in Eq. (11), c 1 is obtained: c1=E1(bz0+dz2 0 2)(13) Applying recursively the other boundary conditions, the integration constants c k are given by: ck=ck−1+ (Ek−Ek−1)[bzk−1+dz2 k−1 2](14) Eq. (14) can be expressed as: ck=E1(bz0+dz2 0 2)+∑ k i=2 (Ek−Ek−1)[bzi−1+dz2 i−1 2](15) According to Eq. (11), if V is uniform along the length of the beam, as it is in three-point bending, it results that τ k zx,x=0. Then, considering Eq. (9) 2 , it results that σ k z,z=0, being σ z uniform through the thickness of each lamina. As σ z=0 at the top and bottom surfaces of the beam, they are null in the first and last laminae and consequently, in the whole section. Therefore, in the case of three-point bending, σ z=0, is a consequence of equilibrium equations. Nevertheless, it has been assumed as initial hypothesis to be able to relate directly σ x with ε x in Eq. (2). The zones of load application and reactions, with particular stress states that include transverse normal stresses, are not considered in the current study. 2.3. Engesser-Castigliano theorem The displacement of the load application point is determined by the Engesser-Castigliano theorem, where the complementary strain energy or strain coenergy is used. The name coenergy has its origin in magnetic forces [36], but it has been used also in the field of mechanics [37]. The derivation of the coenergy of a laminated beam is developed in Appendix A. According to Engesser-Castigliano’s theorem, the generalized displacement δ k of the application point of the generalized force F k is: δk=C,Fk=C ′ (16) In Eq. (16) prime means derivative with respect to F k . When F k is a force, δ k is the displacement in the same direction. When F k is a moment, δ k is the rotation in the same direction. Then, differentiating Eq. (A.16), it results: δk=C,Fk=C ′ =a∫Lx NN ′ dx +d∫Lx MM ′ dx +s∫Lx VV ′ dx +b(∫Lx N ′ Mdx +∫Lx NM ′ dx)(17) 2.4. Displacement in three-point bending Fig. 2 shows a three-point bending test configuration of a laminated beam. In the case of a three-point bending test where the span is L, with an Fig. 2. Three-point bending configuration of a laminated beam. F. Mujika et al. Composites Part A 175 (2023) 107802 4 applied force F in C, section forces and moments and their derivatives with respect ot F, indicated by primes, are: N=0N ′ =0 V=1 2F V ′ =1 2 M=1 2Fx M ′ =1 2x (18) It is worth pointing out that in a three-point bending test, normal forces and their derivatives are null, even in the case of normal-bending coupling, where the coupling compliance coefficient is b∕= 0. It is due to the freedom of the specimen to axial deformation, as it is supported on two rollers. Nevertheless, in a tensile test with b∕= 0, bending moments at the grips would arise, preventing bending curvatures. The displacement of the load application point is: δC=2⎛ ⎜ ⎝d∫1 2L 0 1 4Fx2dx +s∫1 2L 0 1 4Fdx⎞ ⎟ ⎠=FdL3 48 +FsL 4(19) Eq. (19) can be expressed in terms of an equivalent flexural modulus E eq and an equivalent shear moduli G eq , which are defined as: d=(EeqI)−1where I=1 12wh3 s=(5 6Geqwh)−1(20) Where h is the total thickness; w is the width; and I is the moment of inertia. If the displacement is determined by the testing machine, the displacement due to the flexibility of the testing system and the deformation of the specimen through the thickness, has to be included. Those contributions are included in the term defined as system-stiffness and it is assumed to be linear with respect to the applied force [36]. Being K s the coefficient of the system-stiffness, it depends on the deformability of the specimen through the thickness, and on the stiffness of the testing frame, the fixtures and the load cell. Replacing the equivalent moduli of Eq. (20), the displacement δ Cs including the system-stiffness is written as: δCs =FL3 4Eeqwh3+FL 4(5 6Geq)wh +F Ks (21) In Eq. (21), it is assumed that the effect of the span variation due to the change of contact between the specimen and the support rollers is negligible [38]. Eq. (21) is similar to that resulting in the case of homogeneous orthotropic materials. In that case, being E the flexural modulus in the beam direction and G the out of plane shear modulus, from Eqs. (5) and (7) with n =1 and from Eq. (20), it results that Eeq = E. Moreover, from Eq. (A.14) with n =1 and from Eq. (20), Geq =G is obtained. Then, E eq and G eq are the moduli that correspond to an equivalent homogeneous material. Therefore, knowing the elastic properties and the thicknesses of the laminae, the equivalent moduli can be determined. The explicit form of the equivalent flexural modulus, taking into account Eqs. (5), (7) and (20), is: Eeq =12 wh3 AD −B2 A(22) On the other hand, the explicit form of the equivalent shear modulus, taking into account Eqs. (15), (A.11), (A.14) and (20) is: Geq =(5 6swh)−1 (23) 3. Determination of E eq and G eq by three-point bending tests Eq. (21) can be expressed in different forms, taking into account the contribution of bending, shear and system-stiffness: δCs =FL3 4Eeqwh3⎡ ⎢ ⎢ ⎣ 1+Eeq (5 6Geq)(h L)2 +4wEeq Ks(h L)3⎤ ⎥ ⎥ ⎦ δCs =FL 4(5 6Geq)wh ⎡ ⎢ ⎢ ⎣ 1+(5 6Geq) Eeq (L h)2 +4w(5 6Geq) Ks(h L)⎤ ⎥ ⎥ ⎦ δCs =F Ks ⎡ ⎢ ⎢ ⎣ 1+Ks 4wEeq (L h)3 +1 4w Ks (5 6Geq)(L h)⎤ ⎥ ⎥ ⎦ (24) The sub-index s indicates that the effect of the system-stiffness is included. Being ms=F δCs the experimental slope of the load–displacement curve, the apparent values of the flexural modulus E 3ps , shear modulus G 3ps and system stiffness K 3ps obtained experimentally are defined as: E3ps =msL3 4wh3G3ps =msL 4wh K3ps =ms(25) Replacing Eq. (25) in Eq.(24) it results: (E3ps)−1=[(Eeq)−1+(5 6Geq)−1 x2+4w(Ks)−1x3] (G3ps)−1=[(Eeq)−1x−2+(5 6Geq)−1 +4w(Ks)−1x] 4w(K3ps)−1=[(Eeq)−1x−3+(5 6Geq)−1 x−1+4w(Ks)−1] (26) Where x=h L.Depending on the relative influence of bending, shear and system stiffness, if one of those terms is dominant, the apparent values become the actual values of moduli and stiffness: E3ps =Eeq G3ps =5 6Geq K3ps =Ks(27) In the current study, in the case of composite laminates, when great spans are used, shear and system terms could be negligible and E eq can be determined. In the case of sandwich beams, shear term could be dominant and in that case, G eq can be obtained directly. In the case of an indentation test, the span can be considered near zero and then K s could be determined directly. Nevertheless, it is necessary to take into account that the slope of the load–displacement curve has to be determined in the linear zone, once the contact area between the load roller and the specimen remains constant [36]. Eq. (26) can be written as: yE(x) = A+Bx2+Cx3 yG(x) = Ax−2+B+Cx yK(x) = Ax−3+Bx−1+C (28) where: yE=(E3ps)−1yG=(G3ps)−1yK=4w(K3ps)−1 A=(Eeq)−1B=(5 6Geq)−1 C=4w(Ks)−1(29) Doing N bending tests at different spans in the same specimen, A, B and C can be determined by regression of the experimental data in any of Eqs. (28). The sum of squares of residuals is defined as [39]: F. Mujika et al. Composites Part A 175 (2023) 107802 5 fI(A,B,C) = ∑ N i=1 [yI(xi) − yIi ]2(30) where I is E, G or K; yI(xi)correspond to Eq. (28) and yIi correspond to experimental values. Minimizing fI(A,B,C)in Eq. (30) with respect to A, B and C, the following system of equations is obtained, where the coefficient matrix is symmetric: ⎡ ⎣ a11 a12 a13 a12 a22 a23 a13 a23 a33 ⎤ ⎦I ⎧ ⎨ ⎩ A B C⎫ ⎬ ⎭ =⎧ ⎨ ⎩ b1 b2 b3⎫ ⎬ ⎭I (31) Using any of Eq. (28) for obtaining A, B and C, the other two equations can be used to analyse the trend of the other variables. For instance, if y E is used for obtaining A, B and C, replacing those values in y G and y K we can see the variation of G 3ps and K 3ps , respectively, and the location of the experimental data with respect to the global curve. The coefficient values for each regression curve are reported in Appendix B. The coefficient of determination R 2 is defined as [41]: R2=1−∑N i=1[yIi −yI(xi) ]2 ∑N i=1[yIi −yI]2(32) where yI=1 N∑N i=1yIi is the mean value of the experimental data. The coefficient gives a measure of the fit of experimental points to the regression curve. On the other hand, if the system-stiffness has been determined by an indentation test, the actual displacement due to bending and shear δ C can be determined as: δC=δCs −F Ks (33) In this case, Eq. (26) becomes: (E3p)−1=[(Eeq)−1+(5 6Geq)−1 x2] (G3p)−1=[(Eeq)−1x−2+(5 6Geq)−1](34) Where E 3p and G 3p are the apparent flexural and shear moduli disregarding system-stiffness. In Eqs. (34), the experimental parameters that depend on the slope m=F δC are: E3p=mL3 4wh3G3p=mL 4wh (35) The relation between the moduli affected by the system stiffness given in Eq. (25) and those not affected by it given in Eq.(35), depend on the relation between the slopes m and m s , being: ms m=F δCs δC F=δCs −F Ks δCs =1−F δCsKs =1−ms Ks (36) According to Eqs. (25) and (36), the relation between moduli affected and not affected by the system-stiffness are: E3ps =E3p(1−ms Ks)G3ps =G3p(1−ms Ks)(37) Therefore, E 3p and G 3p can be experimentally determined by two methods: •Defining a channel for δ C in the software of the testing machine, according to Eq. (33), and determining the moduli based on the slope m. •Determining E 3p and G 3p from Eq. (37), after having determined values of E 3ps and G 3ps based on experimental values of m s . Eq. (34) can be written as: zE=A+Bt zG=At−1+B(38) where: zE=(E3p)−1zG=(G3p)−1t=x2=(h L)2 In the case of Eq. (38), A and B can be obtained by linear regression. 4. Comparison between defined and experimental values The equivalent moduli defined in Eqs. (22) and (23) are determined based on the elastic properties of the orthotropic layers of the laminate. Those moduli can be also obtained by three-point bending tests, varying the span in the same specimen. To check the suitability of the defined equivalent moduli, the values obtained from Eqs. (22) and (23)are compared with values obtained by regression of bending test results. That comparison is carried out in the case of a sandwich specimen, and two hybrid laminates, one symmetric and one asymmetric. 4.1. Sandwich specimen 4.1.1. Material properties In the case of a sandwich specimen, bending stresses are supported by the skins and shear stresses are supported by the core. The data of a previous study carried out by some of the authors are used in order to obtain new conclusions [15]. Sandwich specimens made of Polypropylene Honeycomb Core - OpenThermHex® and skins of aluminium Mangealtok 30-H111 were used. The total thickness of the sandwich was 16.9 mm and the skins were 1 mm thick. Equations of sandwich standard ASTM C393 [40] were used in order to obtain the equivalent flexural modulus and test at different spans were carried out in order to obtain G. In reference [15], G eq was the value that has been called 5 6Geq in the present study. This nomenclature was due to the fact that shear stresses in the core of a sandwich material have a uniform distribution and consequently, the shear factor 5/6 associated to the parabolic distribution in a homogeneous rectangular section becomes 1. In order to avoid confusion, the sandwich shear modulus G san is defined as: Gsan =5 6Geq (39) The elastic properties used for calculations are given in Table 1. The elastic modulus of the core has been obtained from a formula that takes into account the geometry and properties of the honeycomb [41]. The Table 1 Properties of the skins and core of the sandwich. h (mm) E G Skins 1 70 GPa 26.9 GPa Core 15 0.344 MPa 20.7 MPa Table 2 Equivalent moduli obtained from different theoretical approaches. E eq (MPa) G san (MPa) ASTM C365-03 21,913 20.8 Current approach 21,914 20.8 F. Mujika et al. Composites Part A 175 (2023) 107802 6 shear modulus of the honeycomb core is the value obtained by FEM [15]. 4.1.2. Results obtained from the analytical approach Table 2 shows that the equivalent moduli obtained from the current approach, replacing the values of Table 1 in Eqs. (22) and (23), and the simplified equations given in the standard ASTM C365-03 [42] agree. Fig. 3 shows normal and shear stresses normalized with respect to the maximum values, obtained from the current analytical model, showing that normal stresses act on the faces and shear stresses act on the core. 4.1.3. Experimental results The experimental data corresponding to a specimen of reference [15] are included in Table 3. f G and f K are defined as the relative influence of shear and system-stiffness with respect to bending, in the displacement of the load application point. From Eq. (24) 1 they are: fG=Eeq Gsan (h L)2 fK=4wEeq Ks(h L)3 (40) Shear effects are much greater than bending effects in all cases, even for the largest span, where the span-to-thickness ratio is 20. The effect of the system-stiffness decreases as span increases, being less than the shear effect. Table 4 shows the values of E eq , G san and K s obtained using y E and y G as regression functions. The values of E eq and G san obtained by the regression of y E agree better with those of Table 2. Moreover, the determination coefficient is greater in the case of y E . The differences of E eq and K s values in Table 4 are much greater than those obtained for G san . Regression curves are analyzed in order to explain the source of these differences. Fig. 4 shows the regression curves and experimental data that correspond to E 3ps and G 3ps , with values of A, B and C obtained from the regression of y E . The curves obtained with the values of A, B and C coming from the regression of y G and y K are qualitatively similar. According to Fig. 4(a), spans needed to obtain the equivalent flexural modulus are too great to test specimens in three-point bending. Moreover, as experimental values are far from the asymptotic value, small variations in those values can lead to different E eq values. This could be the reason of the difference between values of E eq obtained from different regression functions in Table 4. In the case of testing with spans Fig. 3. Stress distribution of a sandwich in bending: (a) Normal stresses; (b) Shear stresses. Table 3 Experimental values of bending modulus and influence factors of shear and system-stiffness. L(mm) E 3ps (MPa) f G (%) f K (%) 120 837 2180 332 160 1516 1226 140 200 2227 785 72 240 3192 545 41 300 4672 349 21 340 5769 272 15 Table 4 Equivalent values obtained by regression of y E and y G functions. Regression E eq (MPa) G san (MPa) K s (MPa) Coef. R 2 (%) y E (MPa −1 ) 22,282 20.3 6002 99.9 y G (MPa −1 ) 28,413 18.6 13,277 94.5 Fig. 4. Experimental values and regression curves including the stiffness of the system: (a) Bending modulus E 3ps ; (b) Shear modulus G 3ps . F. Mujika et al. Composites Part A 175 (2023) 107802 7 greater than 2.5 m, the bending of the specimen would be probably affected by its own weight. In the case of Fig. 4(b), G 3ps shows a maximum value and the regression curve does not reach G san , as the effect of the system-stiffness increases as span decreases. Since experimental values are near this maximum value, values of G san are less affected by small variations of experimental data, as seen in Table 4. According to Fig. 5, K 3ps values increase asymptotically as span decreases. As K s value corresponds to 0 span, this asymptotic behaviour indicates that it is very sensitive to small variations of experimental values. Therefore, it seems reasonable to obtain K s by an indentation test [36] and use Eq. (38) for the reduction of experimental data. The experimental value obtained by indentation tests was K s =4800 N/mm. As shear effects have more influence than bending effects, Table 5 shows the values of G 3p obtained from Eq. (35), where the slope m does not include the influence of the system-stiffness. f E being the relative influence of bending with respect to shear in the displacement of the load application point, from Eq. (24) 2 , considering that Ks→∞, it is: δCs =FL 4Gsanwh [1+fE]where fE=Gsan Eeq (L h)2 (41) Table 5 shows that the influence of shear (100 %) is more important than bending in all cases. Even for the largest span, where the span-to-depth ratio is L/h =20, the bending effect is the 38 % of the shear effect. Therefore, in a sandwich specimen, in spite of the fact that increasing the distance between faces increases the flexural stiffness, the global stiffness of the specimen decreases noticeably due to the great value of the ratio Gsan Eeq . In this particular case, this ratio is around 1000. In the case of an isotropic material, it is less than 3 in all cases. Table 6 shows the values of E eq and G san obtained by z E and z G of Eq. (38)as regression functions. The differences in values of equivalent modulus in Table 6 are less than in Table 4. Moreover, the value of G 3p obtained for the minimum span of 120 mm in Table 5 is very close to G san of Table 6, as the influence of bending is 5 % of the shear influence. Fig. 6 shows the curves that correspond to E 3p and G 3p , having obtained A and B from z G . Fig. 6(a) is similar to Fig. 4(a), being E3p>E3ps. In the case of Fig. 6(b), for small spans, the curve tends asymptotically to G san . Consequently, G san could be obtained directly performing a low span three-point bending test, where the influence of shear is great enough. It would be similar to obtain E eq for great spans when bending effects are dominant. Therefore, from a stiffness point of view, a threepoint bending test becomes a shear test when the shear is dominant. Nevertheless, as the indentation effect increases as span decreases, it is necessary to determine the slope of the load–deflection curve in the linear zone after indentation, when the contact area between the specimen and the load roller does not change any more [36]. According to Table 6 and Fig. 6, the determination of E eq and G san having determined previously K s , gives more accurate results than the simultaneous determination of E eq , G san and K s . Fig. 5. Experimental and regression values of the stiffness K 3ps . Table 5 G 3p experimental values disregarding the effect of system-stiffness. L(mm) G 3p (MPa) f E (%) 120 20.1 5 160 19.6 8 200 17.8 13 240 17.4 19 300 16.0 29 340 15.2 38 Table 6 Equivalent values obtained by regression of z E and z G functions. Regression E eq (MPa) G san (MPa) Coef. R 2 (%) z E (MPa −1 ) 20,001 21.2 99.7 z G (MPa −1 ) 22,301 20.8 95.4 Fig. 6. Experimental values and regression curves disregarding the stiffness of the system: (a) Bending modulus E 3p ; (b) Shear modulus G 3p . Table 7 Elastic properties of unidirectional materials. E 1 (GPa) G 13 (GPa) E-Glass/Epoxy 41 4.3 Carbon fibre/Epoxy 147 7.0 F. Mujika et al. Composites Part A 175 (2023) 107802 8 4.2. Hybrid laminates 4.2.1. Material properties Hybrid laminates made up of plies of expoxy matrix reinforced with carbon fibres and glass fibres, have been analyzed by FEM. The material properties are given in Table 7 [42]: A symmetric and an asymmetric hybrid laminate made of carbon/ epoxy and glass/epoxy layers in three point bending have been modelled in the software Abaqus. Incompatible plane-stress elements of 0.2 mm have been used. The force and displacement of the load application point have been determined, assuming small displacements. In this case, the coefficient K s is related to the local deformation of the mesh near the load application and reaction points. Fig. 7 shows the cross section of both laminates. Three-point bending models have been run at different spans. The values of E eq and G eq obtained by virtual testing have been compared with those obtained by the analytical model. The thickness in both laminates is 3 mm and spans used in the simulation are (mm): 34, 40, 48, 60, 80. Fig. 7. Cross sections of the hybrid laminates: (a) Symmetric; (b) Asymmetric. Table 8 Equivalent moduli of the current approach of symmetric and asymmetric laminates. E eq (MPa) G eq (MPa) Symmetric 115,593 4796 Asymmetric 67,213 4513 Fig. 8. Stress distribution of the symmetric laminate: (a) Normal stresses; (b) Shear stresses. Fig. 9. Stress distribution of the asymmetric laminate: (a) Normal stresses; (b) Shear stresses. F. Mujika et al. Composites Part A 175 (2023) 107802 9 4.2.2. Results obtained from the analytical approach Table 8 shows the equivalent moduli obtained by the current model, replacing the values of Table 7 and the dimensions of Fig. 7 in Eqs. (22) and (23). Fig. 8 and Fig. 9 show normal and shear stresses normalized with respect to the maximum values of the symmetric and asymmetric laminates, respectively, obtained with the model of the present work. 4.2.3. Numerical results Table 9 and Table 10 show numerical results of the apparent modulus E 3ps and influence factors of shear and system-stiffness defined in Eq. (40), for the symmetric and asymmetric laminates, respectively. In spite of in both cases bending is dominant, shear effects are greater in the symmetric laminate and the system-stiffness influence is similar in both cases. Table 11 shows the values of E eq , G eq and K s obtained using y E as regression function. In this case, the results obtained by y G are the same, as there are not experimental errors. Taking as reference the values of Table 8 obtained from the analytic approach of this study, the difference in E eq is less than 1 % for values in Table 11. In the case of G eq , the difference is 2.3 % in the case of the symmetric laminate and 6.2 % in the case of the asymmetric laminate. Those differences are related to the dominant role of bending. Fig. 10 shows the regression curves and experimental data that correspond to E 3ps and G 3ps in the case of the asymmetric beam. The values of the symmetric laminate are qualitatively similar, so they are not included. As bending is dominant, experimental results of E 3ps are near the horizontal line that correspond to E eq . On the other hand, experimental values of G 3ps and the maximum of the regression curve are far from the value of G eq . According to Fig. 11, as in the case of Fig. 5 for the sandwich specimen, K s value related to the local deformation of the mesh is very sensitive to small variations of data. As in this case experimental errors do not exist, the data of K s obtained by regression have been introduced to obtain results of equivalent moduli, disregarding the effect of the local mesh deformation. Indentation simulations have been done to verify that K s obtained by regression and by indentation agree. Furthermore, it has been verified that K s depends on the local deformation of the model at load application and supports points. Fig. 12 shows the curves that correspond to E 3p and G 3p , having obtained A and B from z E . Fig. 12(a) is similar to Fig. 10(a). In the case of Fig. 12(b), the curve tends to G eq when the span goes to zero. Nevertheless, experimental values are far for from this value even in the case of the smallest span. Therefore, small variations in experimental data could affect the result of G eq . Table 9 Numerical values of bending modulus and influence factors of shear and system stiffness in the symmetric laminate. L(mm) E 3ps (MPa) f G (%) f K (%) 80 110,522 4 1 60 106,511 7 1 48 101,489 11 3 40 95,661 16 5 34 88,885 22 8 Table 10 Numerical values of bending modulus and influence factors of shear and system stiffness in the asymmetric laminate. L(mm) E 3ps (MPa) f G (%) f K (%) 80 65,396 2 0 60 63,892 4 1 48 61,935 7 2 40 59,563 9 3 34 56,665 13 6 Table 11 Equivalent values obtained by regression of y E . Specimen E eq (MPa) G eq (MPa) K s (MPa) Coef. R 2 (%) Symmetric 115,599 4906 59,260 100 Asymmetric 67,218 4810 49,772 100 Fig. 10. Numerical values and regression curves including local deformation in the asymmetric laminate: (a) Bending modulus E 3ps ; (b) Shear modulus G 3ps . Fig. 11. Asymmetric laminate: experimental and regression values of the stiffness K 3ps . F. Mujika et al.