Study of asymmetric glass reinforced plastic beams in off-axis four-point bending

Abstract

This study deals with the computational study of asymmetric glass reinforced plastic beams in off-axis four-point bending and the comparison of the induced results with experimental and analytical results. The measurement of the interlaminar shear strength of composite beams, an important design variable in many applications, may be successfully performed by the asymmetric bending test. A three-dimensional finite element analysis is adopted throughout the composite beams in order to, on the one hand, correlate with the experimental results and, on the other hand, to obtain the stress distributions at the supports and at the loading points where usually there is an abrupt variation due to the indentation existing because of the noses. From the finite element analysis and the experimental investigation possible crack initiation positions are determined.

Keywords

Asymmetric glass reinforced plastic beams off-axis four-point bending computational analysis stress analysis composite materials interlaminar shear strength

Introduction

The glass reinforced plastic (GRP), pipes, and components are widely used in the process plant and chemical industries in applications requiring corrosion resistance. Also the need to reduce vehicle weight and consideration for mass production techniques, among other things, has led the automotive industry to consider randomly oriented chopped-fiber reinforced plastics as near-term substitutes for steel in structural panels.

In order to perform stress and stiffness analyses of chopped-fiber/resin composites, it is essential that the properties be known since in general these materials appear as anisotropic and heterogeneous. But if the fibers are randomly distributed with respect to orientation and position then samples of material which contain statistically significant numbers of fibers will appear to be isotropic. If several such samples are compared, the composite will appear to be homogeneous. If the fibers have a preferred orientation, then the composite will appear to be anisotropic.

Since the composite panel properties depend on the elastic properties, and volume fractions of the constituent materials are affected by the fabrication techniques and most of these composites are stiffness or strength designed it is useful to investigate this subject.

In reinforced composite structures the interlaminar shear strength, which characterizes the interface between fiber and matrix¹ has a huge importance. Because of low interlaminar shear strength, failure of the composite often occurs in the interlaminar region. Possible interlaminar shear failures have been observed due to high transverse shear forces arising as a result of large changes in bending moment along or around a beam or a shell under combined loading. Such conditions can occur in a pipe where the thickness changes, are abrupt and excessive, and can be produced with internal pressure² (or other types of loading³), or in smooth pipe bends under flexure.^4,5 Consequently the interlaminar shear strength is an important design parameter in many applications. Since there is a need for reliable data, a testing procedure for determining interlaminar shear properties has already been developed.⁶ The interlaminar fracture toughness in end notched flexure specimens⁷ and the interlaminar shear fracture of interleaved graphite/epoxy composites⁸ have been examined. The fracture of sandwich beams in three point bending has also been considered.^9,10

The very interesting problem of the bending behavior of off-axis composites has already been investigated by several researchers. In Grediac¹¹ the longitudinal off-axis bending compliances of composites using four-point bending tests was examined whereas in Mujika et al.^12,13 analytical, numerical, and experimental results have been developed in order to study the displacements field for any fiber orientation. On the other hand, the nonlinear analysis of off-axis pultruded composite beams has been performed in Killic and Rami¹⁴ and the behavior of woven fabric composites in off-axis end-loaded bending using the theory of plastica in Majumdar et al.¹⁵ Finally in Manolo¹⁶ the shear behavior of sandwich structures was evaluated under short beam and asymmetrical beam shear tests.

The main aim of this paper is to continue to shed some more light into the performances of thick composite beams under asymmetric off-axis four-point bending (Figure 1) and to study the interlaminar shear failure by investigating the asymmetric GRP laminates. For this reason composite specimens having resin-rich layers (RRL) thus forming asymmetric constructions have been considered (laminate T3) as a continuation of a previous work carried out on symmetric laminates¹⁷ made only from chopped strand mat material (laminate T1) (Figure 2). The materials properties used in the analysis are given in Table 1.

Table 1.

Material properties of the constituent materials.

Material	E_x (GPa)	E_y (GPa)	E_z (GPa)	$ν_{xy}$	$ν_{xz}$	$ν_{yz}$	G_xy (GPa)	G_xz (GPa)	G_yz (GPa)
R.R.L.	4.52	4.52	4.52	0.354	0.354	0.354	1.669	1.669	1.669
C.S.M.	9.14	9.14	5.48	0.328	0.197	0.328	3.44	1.371	1.371

According to a series of experiments in asymmetric four-point bending, which had been performed on these laminates it has been observed that the failure form, i.e. shear failure at the mid thickness or bending failure at a surface of the specimen depends on the parameters introduced in the analysis of the off-axis bending problem (refer to the following section).

In the sequel, a three-dimensional (3D) linear finite element analysis (FEA) is carried out in the undamaged composite beams in order to, on the one hand, correlate with experimental results and, on the other hand, to obtain the stress distribution versus the vertical axis of the beam at the supports and at the loading points where usually there is an abrupt variation due to the indentation existing because of the noses. Finally from the FEA, possible crack initiation positions have been identified and compared with those from the experimental work.

Analysis of the off-axis bending problem

Let us consider the asymmetric four-point loading shown in Figure 1(a) and (b) where the applied load F is divided into unequal parts P and Q, which are equal to the supports forces at points A and C, respectively.

From equilibrium and geometrical considerations, the following is easily obtained

P = \frac{{FS}_{2}}{1 + λ}, Q = \frac{{FS}_{1}}{L},

(1)

\frac{P}{Q} = \frac{S_{2}}{S_{1}} = λ, Q = \frac{F}{1 + λ}, a = \frac{λ - 1}{λ + 1} L

(2)

In the above relationships L is the distance between supports (span length), a is the distance between the support A (or C) and the loading point at B (or D) (Figure 1(b)), and λ is the loading factor (Appendix 1).

The stresses at any point in the beam can be calculated to a first approximation by using the mechanics of materials theory.⁶ Thus the normal stress $σ_{xx}$ has its maximum value at points B and C (Figure 1(b)) as

σ_{xx} = \frac{6 F (λ - 1) L}{(1 + λ) 2 bt 2}

(3)

where b and t designate the width and thickness of the beam respectively as seen in Figure 1(a).

Similarly the maximum shear stress at the part (BC) of the beam is

σ_{xz} = \frac{3 (1 - λ) F}{2 (1 + λ) bt}

(4)

Dividing the last two equations, the following can be obtained

\frac{σ_{xx}}{σ_{xz}} = \frac{4 L}{(1 + λ) t} = C

(5)

where C denotes the ratio of maximum normal stress to maximum shear stress at failure.

Thus it can be observed that in the off-axis four-point test method in addition to span length/thickness $(L / t)$ ratio, which is the main parameter in the three-point bending and symmetric four-point bending tests, the loading factor λ (= $P / Q$ ) defining the ratio of the forces at the supports becomes an important parameter, which characterizes the type of failure, i.e. shear failure at the neutral axis near the middle plane or tensile (or compressive) flexural failure at the upper or lower part of the laminate beam.

According to the Bernoulli–Euler theory, the elastic deflection arising from a flexural stress at point B (Figure 1(b)) in a rectangular beam is⁶

δ_{F} = \frac{8 λ (λ - 1) 2 FL 3}{E (λ + 1) 5 bt 3}

(6)

The elastic shear deflection at the same point is

δ_{S} = \frac{12 λ (λ - 1) FL}{5 G (λ + 1) 3 bt}

(7)

The ratio of shear modulus to elastic modulus $(\frac{G}{E})$ is much lower for composites than for the metals and hence shear deformation is more significant in composites.

The total deflection $δ_{t}$ is the sum of the bending and shear components, that is

δ_{t} = δ_{F} + δ_{S} = \frac{4 λ (λ - 1) FL}{(λ - 1) 3 bt} {\frac{2 (λ - 1)}{E (λ + 1) 2} (\frac{L}{t}) 2 + \frac{3}{5 G}}

(8)

and the ratio of the two deflections is

\frac{δ_{S}}{δ_{F}} = \frac{3}{10} (\frac{E}{G}) (\frac{t}{L}) 2 \frac{(λ + 1) 2}{λ - 1}

(9)

It can be seen that $\frac{δ_{S}}{δ_{F}}$ depends not only on the ratio of the two modules and the span length-thickness ratio, which characterize symmetric three-point and four-point bending tests, but also on the loading factor λ.

The apparent flexural modulus $E_{α}$ can be found by assuming that the total deflection $δ_{t}$ is due to flexure only, and is given by

E_{α} = \frac{8 λ (λ - 1) 2 FL 3}{(λ + 1) 5 bt 3 δ_{F} (1 + \frac{δ_{S}}{δ_{F}})}

(10a)

For the apparent flexural modulus to be as nearly equal to the true flexural modulus as possible, the shear deflection must be compared to the flexural deflection that is

\frac{δ_{S}}{δ_{F}} << 1

(10b)

Effects of surface resin-rich layer

The existence of nonuniformity in a laminate generally influences the stresses and the moduli. In particular, a RRL on either or both of the surfaces can significantly affect the flexural modulus.

Since the elastic modulus of the fiber is much larger than that of the resin, the direct contribution of the latter to the ability of a fiber reinforced laminate to resist flexural deformation is usually small, and can be neglected as a first approximation. Therefore, resin layers on the outer surface of a laminate have little influence on the deflection of a laterally loaded beam. However, surface RRLs contribute to the laminate thickness, which is used in computing moduli and can, therefore, have a significant effect on the computed flexural modulus.

If a laminate composite of total thickness t has RRLs on either or both surfaces of combined thickness, $t_{RRL}$ , then the thickness of the part of the composite containing reinforcing fibers is

t_{0} = t - t_{RRL}

(11)

At first, it is easily observed that the existence of surface RRL can affect the determination of the fiber content, $ε_{f}$ , of a composite as it is obtained from burn-off tests. Allowance can be made for this using the following expression (Appendix 2)

ε_{fc} = \frac{1}{1 - \frac{t_{RRL}}{t}} ε_{f}

(12)

From equation (3) by taking into account the RRL it can be obtained that

σ_{xx} = \frac{6 F (λ - 1) L}{(λ + 1) 2 {bt}_{0}^{2}}, σ'_{xx} = \frac{6 F (λ - 1) L}{(λ + 1) 2 bt 2},

where the bending normal stress of the laminate of thickness t is

σ'_{xx}

and for the laminate without RRL is

σ_{xx}

Thus, by dividing

\frac{σ'_{xx}}{σ_{xx}} = \frac{t_{0}^{2}}{t 2} = (\frac{t - t_{RRL}}{t}) 2 = (1 - \frac{t_{RRL}}{t}) 2 ≅ 1 - 2 \frac{t_{RRL}}{t}

(13)

Similarly for the shear stress from equation (4) by taking into account the RRL can be obtained that

σ_{xz} = \frac{3}{2} \frac{(1 - λ) F}{(1 + λ) {bt}_{0}}, σ'_{xz} = \frac{3}{2} \frac{(1 - λ) F}{(1 + λ) bt}

where

σ'_{xz}

and

σ_{xz}

denote the shear stress of the laminate of thickness t and the laminate without RRL respectively. Again by dividing

\frac{σ'_{xz}}{σ_{xz}} = \frac{t_{0}}{t} = \frac{t - t_{RRL}}{t} = 1 - \frac{t_{RRL}}{t}

(14)

Equations (13) and (14) show that laminate stresses are sensitive to $t_{RRL}$ which is highly variable in a hand lay-up composite and suggests that caution should be exercised in interpreting test results.

Let the apparent flexural modulus of the laminate of thickness t be $E'_{α}$ and for the laminate without RRL be $E_{α}$ . From equation (10a) if the laminate deflections under load F are equated

E'_{α} = E_{α} (\frac{t_{0}}{t}) 3

(15)

Using equation (11) this can be written in terms of total thickness and RRL thickness

E'_{α} = E_{α} (1 - \frac{t_{RRL}}{t}) 3 ≅ E_{α} (1 - \frac{3 t_{RRL}}{t})

(16)

It can be observed that the RRL has a more significant effect on the apparent flexural modulus than on the stresses when equation (16) is compared with equations (13) and (14). This effect is double and triple with respect to the other two cases for the stresses.

Experimental work and results

The materials used during the off-axis bending experiments were flat GRP laminates produced by Resinform Ltd using Atlac 382-OSA polyester resin modified by Bisphenol and reinforced with powder bound glass fiber chopped strand mat (CSM).

The lamination procedure gives usually rise to resin-rich surface layers (RRL), which in practice is provided for corrosion resistance. In order to study the effect of a RRL and also to create an asymmetric laminate, the uneven (rough) side of the laminate was machined, thus avoiding the variations in the thickness of the specimens, factor which plays important role during bending.

The fiber content fraction M_f was determined from burn-off tests according to British Standards BS 2782. The result was $M_{f} = 0.33 \pm 0.01$ . Specimens were cut from each of the two perpendicular directions using a band-saw. The edges of the specimens were machined in a milling machine to the shape of a prismatic specimen having total nominal length of 10 mm and nominal width of 11 mm according to the above BS.

Before testing, the width and thickness of each specimen were measured with a micrometer at three points inside the specimen length. From these measurements, mean values of thickness and width were calculated for each specimen.

The experimental test apparatus consisted of an adjustable anvil which was mounted on a circular base that fits onto a compressive load cell on an Instron Universal testing machine of 100 kN capacity. The radius of a loading nose and support nose is 5 mm according to BS 2782. Before the tests the load cell should be calibrated. By employing strain gages and linear variable displacement transducers (LVDT), the stress–strain and load–deflection curves can be plotted to determine material properties. A Peckel automatic data logger with an Anadix printer was used to record the strain and deflections. The cross-head was 0.5 mm/min. The specimen shape and dimensions (Figure 1) are as follows: $L = 56 mm$ , $a = 18.7 mm$ , $l' = 10 mm$ , and a loading factor $λ (= P / Q = 2)$ . The CSM layers had a thickness of $t_{0} = 15.30$ mm whereas the RRL had a thickness of $t_{RRL} = 0.68$ mm, thus obtaining a total thickness of $t = 15.98$ mm.

Four rectangular specimens were used for each laminate with a nominal thickness of 16 mm and a width of 20 mm after machining. During the tests, photos of some specimens had been taken of to record the crack formation process and the type of failure (flexural, shear, or mix) (Figure 3). From the experimental investigation the beams usually fail in shear at the (BC) region of the beam (Figure 1(a) and (b)), namely between the load P and the right support C where cracking appeared at the middle plane depending on L and λ values.

Consequently by varying L and λ (with constant laminate thickness) one can obtain the two types of failure. For shear failures the ratios of $max σ_{xx}$ to $max σ_{xz}$ at failure should be grouped below C (equation (5)) and for flexural failures the ratios of $max σ_{xx}$ at failure to $max σ_{xz}$ should be grouped above C. However, in some cases, there is a transitional (mix) case where the mode of failure changes from specimen to specimen.

The following observations had been made from the experiments.⁶ An increase in normal stress $σ_{xx}$ with increasing $L / t$ or decreasing λ is apparent. In addition, it had been observed that the shear strength depends upon $L / t$ and λ even for shear failures. This effect, which can be related to the discrepancies observed during the calculations, could be due to local stress concentrations, especially at the loading noses or the supports which are ignored in the simple beam theory. Alternatively, it could be a combined stress effect. If the shear strength is enhanced by the presence of a transverse compressive stress, then an increase in shear strength at smaller $L / t$ would be expected.

Also, the effects of indentation near the supports or loading noses where the stresses change in an abrupt manner can be the reason of the shift and the different variation mainly in the shear stress obtained also by the FEA near and far off the supports, as it is exposed later.

Finally as to the influence of the existence of a RRL in the material, the fiber weight fraction M_f was determined from burn-off tests according to BS 2782 as $M_{f} = 0.330 \pm 0.01$ . The corresponding volume fraction from equation (17) is $ε_{f} = 0.190 \pm 0.01$ (see Appendix 2). From equation (12) by excluding the RRL this value becomes $ε_{fc} = 0.197$ , whereas the experimental value from burn-off tests for the same material without RRL is $ε_{fc} = 0.200 \pm 0.01$ .

Also from equations (13) to (15) and from Figure 2 excluding the RRL of thickness $t_{RRL} = 0.68$ mm we obtain for the laminate of thickness $t ≅ 16$ mm, $σ'_{xx} = 0.915 σ_{xx}, σ'_{xz} = 0.958 σ_{xz}$ and $E'_{a} = 0.875 E_{a}$ , respectively.

The results show the influence of RRL on these properties of the laminate.

Three-dimensional finite element analyses

In this section of the paper the case of off-axis four-point bending of asymmetric glass reinforced plastic beams through the FEA is studied. A computational procedure has been used to find out what occurs near the loading points and near the supports during a bending test since there is an abrupt change in the bending moment and also due to effect of indentation at these points.

The fact that the test method used is an asymmetric off-axis bending, in our opinion, makes the investigation more difficult and complicated when compared to a symmetric four-point or three-point bending test. Therefore, a precious numerical method such as FE can contribute for understanding many issues and points and thus can act as a complementary useful tool to detect possible irregularities and obtain more accurate conclusions. Consequently, we consider that both methods experiments and FE constitute a totality and complete each other.

The uncracked laminates T3 are studied numerically in asymmetric off-axis four-point bending by using the general purpose finite element program ANSYS.¹⁸ In the 3D FEA, the entire beam is modeled and the domain is filled with 8-node solid brick elements (SOLID 45). In order to verify our numerical results three finite elements meshes have been used with different mesh refinement with 18, 36, and 54 elements in the thickness direction at the part (BC) (Figure 1) of the beam where the maximum shear stresses are appeared. The results for the 36 and 40 elements in the thickness direction are almost similar.

Analysis of the T3 specimen

In the FEA the specimen shape and dimensions considered in the experimental verification (section “Experimental work and results”) have been taken into account. The width of the beam is discretized with 20 elements, the thickness of the beam with 37 elements, and the length of the beam at the parts (AB) (= (CD)) and (BC) (Figure 1) with 30 and 40 elements respectively. In the thickness direction 6 elements are considered in the upper RRL and 31 elements in the lower CSM. Finally the mesh considered shown in Figure 4(a), has 84360 3D brick elements (SOLID 45) (13680 RRL, 70680 CSM). The loads at failure are taken P = 14,800 N, and Q = 7400 N from the experimental investigation.⁶

The deflection obtained by the finite element method (FEM) at point B (Figure 1(b)), are taken at the same surface points where the deflection are measured in the experiments and an average value of them

δ [= (δ L_{1} + δ L_{2}) / 2]

is given in Table 2. The maximum deflection appeared in the middle section at the end of the beam (x = 94.7 mm, y = 10 mm, z = 15.98 mm).

Table 2.

Deflections from the finite element analysis for laminate T3.

Deflection	max (x = 94.7 mm)	B (x = 28.7 mm)
δ	1.545 mm	1.179 mm

From the stress analysis of the simple supported composite beam of laminate T3 in asymmetric bending, the maximum normal stresses appeared (Figure 4(b)) at the point B where the load P is applied and at the support C (Figure 1(b)), where the moments and the shear forces take their maximum values. The maximum shear stresses appeared at the parts (BC), (AB), and (CD) of the beam where the shear forces take their maximum values.

From the 3D FEA the maximum reasonable normal and shear stresses are also given (Table 3). The maximum

σ_{xx}

and

σ_{xz}

stresses are determined at a middle longitudinal cross section of the beam. The max

σ_{xx}

and

σ_{xz}

are found at a distance of about 2.5 mm to the left of the point B (Figure 1) where the load P is applied inside the CSM material and close to the interface (

z ≅ 1.1

mm) between the CSM and RRL materials of the T3 specimen. High

σ_{xx}

and

σ_{xz}

stresses are also observed near the right support C, and high

σ_{xz}

near the middle of the part (BC) of the beam.

Table 3.

The $max σ_{xx}$ and the $max σ_{xz}$ stresses from the finite element analysis for laminate T3.

Maximum stresses	Close to B ( $x ≅ 26.12$ mm)	Close to C ( $x ≅ 76$ mm)	Between B and C ( $x ≅ 47.4$ mm)
$max σ_{xx}$	175.51 MPa	140.695 MPa	2.087
$max σ_{xz}$	43.475 MPa	36.358 MPa	33.301

In Figure 5 the variation of normal $σ_{xx}$ stress distributions obtained from FEM calculations are given near the point B (x = 28.7 mm) and at different distances from it, where the maximum reasonable $σ_{xx}$ is obtained. It is observed that at the point B the $σ_{xx}$ distribution tend to very high values as it was expected from linear theory of elasticity. The $σ_{xx}$ distributions as we move away from the support approach the linear variation. In addition the $σ_{xz}$ distributions close to the point B and at different distances from it are given in Figure 6. It is observed that the $σ_{xz}$ distributions as we move away from the singular point B tend to become parabolic as it was expected from laminate theory. However, by taking into consideration equation (5), the ratios of the maximum normal to the maximum shear stresses from the FEA (Table 3) at the parts of the beam where the maximum stresses appeared, were always lower than the values of C (= 4.673) for laminate T3. Hence it is also verified from the FEA that the laminate T3 fails in shear. Furthermore from the FEA and the maximum $σ_{xx}$ , $σ_{xz}$ stresses (Table 3) developed in the beam it is verified that the RRL material has already failed in flexure. This happens because the max $σ_{xx}$ stresses overpass the stress failure in flexure of the RRL material about 47.5 MPa determined experimentally.⁶ In Figures 7 and 8 the normal $σ_{zz}$ and $σ_{yy}$ stresses in the direction of the width and the thickness of the cross section respectively are given near the point B and at different distances from it. It is observed the high stresses are appeared only near the singular point.

In Figure 9 the distribution of maximum normal stress, $σ_{xx}$ versus the laminate thickness as obtained from laminate theory in equation (3) and also from FEM is illustrated. It can be observed that the stress variation according to laminate theory shows a linear variation, as expected, similar to that of a simple beam considering that the RRL layer has already failed. Nearly the same variation is also obtained from FEM calculations far off the supports (x = 26.12 mm, 25.5 mm). The stress distribution is not completely linear. It is like a small curve. It reminds the normal stress distribution in curved beams where $σ_{xx}$ stress has a hyperbolic distribution. In addition the percentage differences between the laminate theory and the FEA close to the point B where the maximum $σ_{xx}$ stresses appear, at the interface between the RRL and CSM materials and at the lower part of the laminate beam are presented in Table 4. On the other hand in Figure 10 the distribution of maximum shear stress, $σ_{xz}$ versus the laminate thickness as obtained from laminate theory in equation (4), and from FEA is illustrated. It can be observed that the stress variation according to laminate theory shows a parabolic variation as expected, similar to that of a simple beam. Nearly the same variation is also obtained from FEM calculations far off the supports ( $x = 47.4$ mm). However, the variation of shear stress obtained from FEA is very different near the support A (Figure 1(b)) i.e. at $x = 11.86 mm$ . The parabolic variation with a maximum at the neutral axis becomes a curve with a different shape the maximum of which shifts to the lower part of the laminate beam as in the case of failures due to normal stress (tensile or compressive) where the specimen fails either at its lower part or at the upper part, which it does not occur in our case where from the experiments a shear failure at the neutral axis was observed. In addition the percentage differences between the laminate theory and the FEA results for the maximum $σ_{xz}$ stresses in the middle of the part (BC) (Figure 1(b)) and close to the support A are presented in Table 5. Finally in Figures 11 and 12 $σ_{xx}$ and $σ_{xz}$ stress contours are given from the 3D FEA in the middle section through the length of the beam ( $y = 10$ mm). It is observed that because of the uniformity of the high stresses from the linear 3D FEA it is difficult to determine where failure will be started.

Table 4.

Percentage differences between the laminate theory and FE results for $max σ_{xx}$ of laminate T3 close to the point B where the force P is applied.

T3	(i) Laminate theory (Point B)	(ii) FEA x = 26.12 mm	Percentage difference (ii)--(i) (%)	(iii) FEA x = 25.5 mm	Percentage difference (iii)--(i) (%)
max $σ_{xx}$ (MPa)	151.95	175.51	13.4	158.07	3.9	Interface (z = 0.68 mm)
max $σ_{xx}$ (MPa)	158.94	175.54	9.45	169.88	6.4	Lower tensile fiber (z = 15.98 mm)

Table 5.

Percentage differences between the laminate theory and FE results for $max σ_{xz}$ stresses of laminate T3.

T3	(i) Laminate theory (Point C)	(ii) FEA x = 47.4 mm	Percentage difference (ii)--(i) (%)	(iii) FEA x = 11.86 mm	Percentage difference (iii)--(i) (%)
max σ_xz (MPa)	33.224	33.301	0.23	36.110	8.7

Conclusions

The study of asymmetric GRP beams in asymmetric off-axis four-point bending was examined experimentally and numerically.

During the experimental investigation carried specimens were tested to failure. From the experimental verification and from a wide range of tested specimens it was observed, mainly, that they fail in shear at the part of the beam between the area of application of load P and the right support of the beam (Figure 1(b)). However, from the experiments was also observed that it was not easy to find out what occurs near the loading points and near the supports during a bending test where there is an abrupt change in bending moment which influences the stresses. Thus FEA was decided to be used, which is a very useful and helpful tool in order to thoroughly investigate the state existing in the above-mentioned critical points of the beam and receive a better answer for understanding this complex situation.

A linear 3D FEA was performed since the behavior of the corresponding composite beams from the start point of the experiment up to the failure was linear.

An important problem for the numerical investigation was to verify the obtained shear failure, from the experiments, at the part of the beam between the right support and the point where the load P was applied determined by the ratio $C (= 4 L / (1 + λ) t)$ (equation (5)) of maximum normal stress to maximum shear stress at failure. Thus, initial crack formation occurs in the middle plane of the T3 material and inside the CSM material.

From the proposed analysis results that the classical laminate theory in the case of asymmetric GRP beams in asymmetric off-axis four-point bending cannot accurately predict the complex situation existing at loading points and supports and also the mode of failure of these beams. On the other hand the existence of an external RRL material (T3 specimen) causes a decrease in the fiber content of this laminate with respect to laminate T1, without RRL and also influences the stresses and the elastic modulus (since $E_{CSM} = 9.14$ GPa, $E_{RRL} = 4.52$ GPa, $σ_{CSM}^{u} = 161$ MPa and $σ_{RRL}^{u} = 47.5$ MPa) but does not affect substantially the interlaminar shear strength of GRP specimens because of the low flexural strength of RRL compared with that of the CSM material, as it can be seen from the above values.

Finally another important remark is the variation of the normal $σ_{xx}$ stress and its nearly curved form, instead of a linear. This variation which looks like that of a curved beam in a symmetric bending may be due to the off-axis four-point bending.

Footnotes

Appendix 1

References

Kitching

Ashton

Sideridis

. An interlaminar shear test for fibre reinforced plastics. In: COMP 88, Symposium on phase interaction in composite materials, The University of Patras, Greece, 1988.

Clarke

. A study of differences of strength properties and geometric variations of commercially produced straight GRP piping from three manufacturers. MSc Dissertation, University of Manchester, 1980.

Kirk

. Effect of geometric design parameters on the flexural behaviour of glass reinforced plastic pipe intersections. PhD Thesis, University of Manchester, 1980.

Tan

. Strength and flexibility of glass reinforced plastic pipes and pipe bends. PhD Thesis, University of Manchester, 1983.

Myler

. Glass reinforced plastic pipe bends under a variety of loads. PhD Thesis, University of Manchester, 1985.

Sideridis

Ashton

Kitching

. Measurement of the interlaminar shear stress of glass-reinforced plastics of different construction using the off-axis four point bending test. Compos Struct 1991; 18: 139–161.

Carlsson

Gillespie

Jr Pipes

. On the analysis and design of the End Notched Flexure (ENF) specimen for mode II testing. J Compos Mater 1986; 20: 594–604.

Carlsson

Gillespie

Jr . Interlaminar shear fracture of interleaved graphite/epoxy composites. Compos Sci Technol 1992; 43: 55–69.

Theotokoglou

Hortis

Carlsson

. Numerical study of fractured sandwich composites under flexural loading. J Sandwich Struct Mater 2008; 10: 75–94.

10.

Theotokoglou

Tourlomousis

. Crack kinking in sandwich structures under three point bending. Theoret Appl Fract Mech 2010; 53: 158–164.

11.

Grediac

. Four-point bending tests on off-axis composites. Compos Struct 1993; 24: 89–98.

12.

Mujika

Mondragon

. On the displacement field for unidirectional off-axis composites in 3-point flexure – Part 1: Analytical approach. J Compos Mater 2003; 37: 1041–1066.

13.

Mujika

De Benito

Mondragon

. On the displacement field for unidirectional off-axis composites in 3-point flexure – Part II: Numerical and experimental results. J Compos Mater 2003; 37: 1191–1217.

14.

Killic

Rami

H-A

. Elastic-degrading analysis of pultruded composite structures. Compos Struct 2003; 60: 43–55.

15.

Majumdar

Fazzino

Reifsnider

. Behavior of woven fabric composites in off-axis end-loaded bending. In: International SAMPE symposium and exhibition (proceedings). Baltimore, MD, USA, 18–21 May 2009, vol. 54, Code 79037.

16.

Manolo

. Behaviour of fibre composite sandwich structures under short and asymmetrical beam shear tests. Compos Struct 2013; 99: 339–349.

17.

Theotokoglou

Sideridis

. Study of composite beams in asymmetric four-point bending. J Reinf Plast Compos 2011; 30: 1125–1137.

18.

ANSYS Engineering Analysis System. User’s manual, Swanson Analysis Systems, Inc., 2007.