Test methods for determination of shear properties of sandwich panels

Abstract

This paper presents analysis and comparison of test methods for determining transverse shear strength and shear modulus of steel-faced sandwich panels commonly used in construction. The test methods are taken from the governing European standard EN 14509:2013. Two-point loading and four-point loading test methods as well as a full-scale test method are examined. Based on extensive experimental work on sandwich panels with varying core thickness, comprising mineral wool (MW) and polyisocyanurate (PIR) and encompassing both roof and wall panels, this study provides details of the test setup for the four-point loading and vacuum box methods with which a pure shear failure is obtained. Such details are missing from EN 14509. This paper highlights that the two-point loading method fails to consistently produce shear failure, especially in thicker panels, indicating it does not accurately measure transverse shear strength. The results of the experiments conducted in this study indicate that the four-point loading and full-scale test methods provide consistent shear failure for thicker panels while yielding greater transverse shear strength than the two-point loading test in general.

Introduction

Sandwich panels are widely recognized as an effective cladding solution with adequate load-bearing capacity. When sandwich panels are utilized for external walls or roofing, they are exposed to transverse loads from wind and snow, resulting in bending and shear stresses. While the faces of sandwich panels resist bending moments, the core contributes significantly to carrying shear forces. ¹ The shear properties of sandwich panels are typically evaluated through testing. However, with the increasing thickness of sandwich panels in the market, the appropriateness of conventional test methods must be re-evaluated, and more suitable test procedures should be introduced when necessary. Sandwich panels used in construction typically consist of thin steel faces and a solid core made of materials such as Mineral wool (MW), Glass wool (GW), Polyurethane (PU), Polyisocyanurate (PIR) or Polyvinyl Chloride (PVC). The physical properties of these MW core materials are highly dependent on their density and Fiber orientation, which varies in the degree of order of interfibrillar contacts. ² Studies have also shown that the strength characteristics and deformability of mineral wool depend on specimen thickness.^3,4 Previous studies ⁵ on polymer foam core sandwich panels have demonstrated that visible shear failures occur during shear tests, despite the presence of stress concentrations and local indentations in the core material, which significantly contribute to initiating failure. The shear properties of foam cores significantly depend on variations in foam cell sizes and densities throughout the core thickness^6–8 implying that sandwich panel beams can resist lower shear stress than the shear strength of the foam core itself. ⁹ Due to the complexity of describing and modelling the core materials, the mechanical properties are reliably obtained through testing.

In Europe, sandwich panel production and design are governed by the standard EN 14509:2013, ¹⁰ which is currently undergoing revision. Additionally, a new section of EN 1993 ¹¹ is being developed specifically for the design of steel-faced sandwich panels. Standard EN 14509:2013 provides guidelines for determining the transverse shear strength and shear modulus of sandwich panels used for self-supporting and structural applications in roofs, external and internal walls, and ceilings through testing. The test involves subjecting a full-sized panel to a uniform load or a small beam cut from the panel to two-point loads (referred to as a four-point test in general literature), in a span small enough to ensure shear failure (typically around 1000 mm). However, the two-point loading test method is only suitable for panels that are not too thick and have a PU/PIR core; for thicker panels, other failure modes may occur, making it challenging to determine the actual transverse shear strength. To overcome these issues, the EN 14509:2013 standard suggests employing four-point loads (referred to as a six-point test in general literature) in the “shear beam” test, but no details on the test setup are given.

The ASTM C393/C393M-20 ¹² and ASTM D7250/D7250M-20 ¹³ standards also recommend a short beam test method to determine the shear properties of a sandwich panel. This involves subjecting the sandwich beam to various load configurations. The span length is determined based on the criteria provided in ASTM C393/C393M-20. For thicker panels, this criterion results in a shorter span length, which could lead to failure modes other than shear failure. These can include indentation failure due to load concentration beneath the loading plates and face sheet wrinkling failure, as demonstrated in a previous study ⁷ The ASTM D7250/D7250M-20 standard sets a minimum width for the beam sample, a requirement that the MW core cannot satisfy. A sandwich panel with an MW core consists of multiple lamella slabs, which have negligible shear resistance. Therefore, to prepare the beam sample for testing, it is essential to exclude the joints within the width of beam.

The aim of the present study is to quantify the differences of the test methods of EN 14509:2013 and to provide insight on the failure modes appearing in the tests. Another goal is to provide recommendations for the four-point loading test procedure, primarily effective for thicker MW panels, to complement the information missing from EN 14509:2013. The research is based on a comprehensive testing campaign comprising wall panels (flat or slightly profiled faces) and roof panels (one strongly profiled face), both having MW and PIR cores. The test setups were adjusted such that visible core shear failure would be obtained, if possible. The two-point loading tests and full-scale tests were carried out according to EN 14509:2013. The two-load point tests acted as a reference, and their purpose was to illustrate the need for the four-point loading test and to quantify the difference in transverse shear strength between the test methods. The results of the implemented test procedures are compared in terms of failure mode and the value of transverse shear strength and shear modulus obtained.

The paper is organized as follows. Section 2 outlines the details of the test setups employed in this study. Section 3 provides the calculation process and analytical solution. The results of the tests are presented in Section 4, and the paper ends with a conclusion given in Section 5.

Experimental program

The general cross section of the test panel is shown in Figure 1(a), indicating the area where the shear beam sample are extracted across the full width, B _tot , including the total thickness of the panel, D _c , and the arrangement of MW lamellas is shown in Figure 1(b).

Figure 1.

(a) Schematic diagram showing the types of panels used for testing, (b) Internal view showing the arrangement of lamellas in MW core sandwich panel.

Material characteristics

The density of core materials was measured following the European standard EN 1602, ¹⁴ with eight specimens tested for each panel type. Three tensile tests for each steel facing were carried out following the instructions given in the standard EN ISO 6892-1:2019 ¹⁵ using the test sample as depicted in Figure 2(a). Flat MW panels used during the tests were manufactured at two different production plants, designated by A and B. The letter ‘R’ in the panel type designates a roof panel. The value of the nominal thickness of the steel faces is given in parentheses. The steel facing of all examined panels was of the S280GD grade (nominal yield strength 280 MPa), had a thin layer of coatings, and exhibited a variable nominal thickness. The measured properties of steel facing are given in Table 1.

Figure 2.

Summary of material test methods. (a) steel face tensile test sample, (b) core compression test setup, (c) core tensile test setup, (d) typical stress-strain curve of steel facing, (e) Force-displacement curve from core compression test (f) Force-displacement curve from core tensile test.

Table 1.

Test panel with their measured core thickness (D _c ), mean density value (ρ), thickness of outer (T _F1 ) and inner (T _F2 ) faces with their nominal thickness in parentheses, and their corresponding mean yield strength (R_eHF1) and (R_eHF2).

Panel	Type	D c	ρ	TF1	TF2	ReHF1	ReHF2
		(mm)	(kg/m3)	(mm)	(mm)	(MPa)	(MPa)
B-MW 100	Flat	100	116.3 ± 3.5	0.52 (0.6)	0.42 (0.5)	465 ± 6	411 ± 3
B-MW 200	Flat	200	111.4 ± 2.6	0.53 (0.6)	0.43 (0.5)	493 ± 6	461 ± 3
A-MW 100	Flat	100	92.5 ± 3.8	0.52 (0.6)	0.52 (0.6)	343 ± 2	332 ± 1
A-MW 200	Flat	200	92.3 ± 4.2	0.52 (0.6)	0.52 (0.6)	338 ± 5	336 ± 9
PIR 100	Flat	100	36.8 ± 0.2	0.41 (0.5)	0.38 (0.4)	375 ± 7	421 ± 4
PIR 200	Flat	200	37.7 ± 0.1	0.42 (0.4)	0.42 (0.5)	484 ± 7	480 ± 2
R-MW 140/100	Profiled	100	122 ± 2.8	0.52 (0.6)	0.42 (0.5)	460 ± 8	375 ± 9
R-MW 190/150	Profiled	150	114 ± 2.3	0.53 (0.6)	0.43 (0.5)	443 ± 8	482 ± 7
R-PIR 140/100	Profiled	100	37.6 ± 1.0	0.40 (0.5)	0.36 (0.4)	442 ± 9	451 ± 2
R-PIR 210/170	Profiled	170	37.5 ± 0.2	0.41 (0.5)	0.43 (0.5)	433 ± 4	384 ± 5

Compressive and tensile strength of the core material with its corresponding modulus of elasticity was determined by following the instructions given in Annex A.2 of the standard EN 14509:2013. The schematic of the test methods is depicted in Figure 2(b) and (c). The core material properties were determined using the measured force-displacement curves as depicted in Figure 2(d) and (e) by the procedure of EN 14509:2013.

The MW test specimens were prepared by including the full width of the lamellas. Conversely, PIR test specimens were cut into a square shape, with the depth of the specimen kept equal to the core thickness. Eight specimens for each panel type were prepared and kept in the laboratory for at least 48 h before the tests as instructed in EN 12085. ¹⁶ The obtained mechanical properties of the core materials are given in Table 2. All results are close to the manufacturer’s declared values, except for the compressive modulus of PIR-200, which was lower than expected. This discrepancy may be due to changes in foam properties as thickness increases.

Table 2.

Mean values of the compressive strength (f _Cc ), compressive elastic modulus (E _Cc ), tensile strength (f _Ct ), and tensile elastic modulus (E _Ct ) of the core materials, and the corresponding standard deviations obtained from the compression and tensile tests.

	f Cc (MPa)	E Cc (MPa)	f Ct (MPa)	E Ct (MPa)
B-MW-100	0.13 ± 0.005	8.63 ± 0.71	0.15 ± 0.04	9.35 ± 1.41
B-MW-200	0.11 ± 0.010	8.93 ± 0.61	0.13 ± 0.01	7.33 ± 0.63
A-MW 100	0.08 ± 0.007	7.38 ± 0.34	0.15 ± 0.03	10.96 ± 1.54
A-MW 200	0.07 ± 0.006	6.34 ± 0.88	0.20 ± 0.02	9.04 ± 0.64
PIR 100	0.15 ± 0.004	3.27 ± 0.19	0.09 ± 0.02	4.03 ± 0.59
PIR-200	0.10 ± 0.005	2.43 ± 0.13	0.06 ± 0.01	1.40 ± 0.12
R-MW 140/100	0.12 ± 0.006	6.93 ± 0.23	0.09 ± 0.17	7.40 ± 1.66
R-MW 190/150	0.11 ± 0.006	8.94 ± 0.82	0.11 ± 0.03	8.17 ± 1.26
R-PIR 140/100	0.11 ± 0.002	2.82 ± 0.16	0.08 ± 0.02	4.18 ± 0.53
R-PIR 210/170	0.12 ± 0.002	2.91 ± 0.17	0.06 ± 0.01	1.72 ± 0.20

Two-point loading test setup

The two-point loading test was conducted following the instructions of the European standard EN 14509:2013, ¹⁰ as illustrated in Figure 3. The beam specimen was extracted from the full-sized panel by a bandsaw with fine blade to ensure that the original characteristics of the material, especially the core-face interface, were not compromised. For the profiled panel the beam specimen does not include the profiled face and the specimen was extracted from the flat part of the panel as shown in Figure 1(a). The width of the beam used for the test is outlined in Table 3. All the MW panels have the same width, which corresponds to the width of the lamella as depicted in Figure 1(b).

Figure 3.

Schematic of the two-point loading test setup.

Table 3.

Width of beam, B, specimen used in shear beam tests.

Panel type	MW lamella	PIR 100	PIR 200	R-PIR 140/100	R-PIR 210/170
B (mm)	121	140	240	140	220

In the case of the MW beam sample, it is important to avoid any lamella joints (see Figure 1(b)) over the entire length of the sandwich beam as these joints will have negligible resistance to shear which can affect the test results. The width of the loading plate (U _p ) and support plate (S _p ) are selected based on the panel thickness, with the minimum and maximum widths being 100 mm and 200 mm, respectively. Wooden plates with chamfered edges were used as a load-spreading plate, which reduces the stress concentration at the edges, minimizing the likelihood of local indentation failure.

Depending on the thickness of the sandwich beam, the force was applied by a hydraulic cylinder at the rate of 3-5 mm/min, with a load measuring accuracy class of 1. Displacement during the test was measured using Gefran LT67 displacement transducers, which have an accuracy of 0.01 mm. Two displacement sensors O ₁ and O ₂ were positioned on the edges of the beam and one sensor O ₃ was placed at the mid-span to record the displacement of the beam, which will be utilized to build a force-displacement curve.

Four-point loading test setup

The four-point loading test setup was developed based on a process of trial and error by conducting a series of trial tests. The primary objective of the trial tests was to determine appropriate width and spacing the loading plates such that the biaxial stress state underneath the plates, induced by normal compression stress and bending moments, would not create undesired failure modes such as local indentation, delamination, and wrinkling. Trial tests were conducted on sandwich beams with different spans using loading and support plates of varying widths (100 mm to 200 mm). Narrow plates caused local deformation and early failure modes like indentation and wrinkling. For MW panels, core crushing was an issue with narrow plates. Hence, wider plates were used until consistent core shear failure was achieved. The sample preparation, loading rate, and type and placement of displacement sensors followed the two-point test setup. Free distance from the outer edge of the loading plate U _p1 to the inner edge of support plate S _p was set to 1.2d _c , where d _c is the core thickness, as suggested in EN 14509:2013. ¹⁰ The final test setup for the four-point loading test is shown in Figure 4.

Figure 4.

Schematic of four-point loading test setup.

Based on the trial tests, the width of the outer loading plate, U _p1 , and the support plate, S _p , was set to 200 mm while the width of the inner loading plates, U _p2 , was set to 150 mm. The adjustable distance between the loading plates U _p1 and U _p2 ranges from 5 mm to 10 mm. In all four-point loading test cases, this was set to 5 mm. Likewise, the distance between the two inner loading plates, U _p2 , can be adjusted from 10 mm to 15 mm. This was set to 10 mm in all the four-point loading test cases. The only variable was the distance between S _p and U _p1 .

Full-scale test setup

In the full-scale test, a vacuum box was employed to conduct the shear test on a full-width sandwich panel. Before the test, the dimensions of the profiled or flat face were measured at multiple locations to obtain the value of its cross-sectional area, A _F1 , and the moment of inertia, I _F1 . Sandwich panel with appropriate span length was placed on the support plate with a width of 300 mm over the beam as shown in Figure 5. The support beam on the left side was fixed while the one on the right was set as a roller.

Figure 5.

Schematic of a full-scale test using the vacuum box.

Each beam was equipped with two calibrated HBM HLC A1 2.2 t load cells, which were fixed to prevent all translational and rotational movements. However, the beam was permitted to rotate around the Z-axis. This configuration enabled the loadcells to accurately measure the vertical force acting on the panel during the test. Additionally, the loadcells were not affected by the high pressure induced inside the box, thus enabling it to record only the force acting on the sandwich panel. The sandwich specimen was covered with thin butyl rubber and enclosed within a steel frame ensuring an airtight seal. Four displacement sensors with a precision of 0.01 mm, labelled D ₁ -D ₄ , were used to measure the deflection of the panel during the test as shown in Figure 6(a) and (b). The vacuum was created by a load-controlled electric pump, imposing uniform pressure over the surface of the sandwich panel based on the given input loading rate, until the specimen failed 5-7 min from commencing the test.

Figure 6.

Full-scale test using vacuum box. (a) Flat panel, (b) Profiled panel.

The vacuum box system used in this research was fully automated to control the loading rate based on the response of load cells and stop the test after the failure of the specimen. The data recording from load cells and displacement sensors was synchronized to produce the exact force-displacement curve of the test.

Analytical evaluation of shear properties

Transverse shear strength and shear modulus are obtained from the test results by applying the well-known theory of layered beams in bending .^1,17,18 The shear deformation of the core and the slip between the faces and the core are considered. The analysis is linear elastic. In the following, only results relevant to this study are presented. The sandwich panels considered in this study can be analysed by solving the governing differential equation of the deflection, w, analytically. For panels with flat or slightly profiled faces, this is

$w^{″}\left(x\right)=-\frac{M_s\left(x\right)}{B_s}+\frac{V_s^{\prime }\left(x\right)}{A_c{G}_{eff}}$

(1)

where M _s and V _s are the bending moment and the shear force, respectively, acting in the cross-section at location x along the longitudinal axis of the panel. B _s is the bending stiffness of the sandwich part. The primes denote differentiation with respect to x. $G_{eff}=G_{c}e/d_c$ is the effective shear modulus of the core ¹ where e is the measured depth between the centroids of the faces, and d _c is the core thickness. Finally, A _c = Be is the effective cross-sectional area of the core, where B is the width of the panel.

The total deflection can be divided into two components, w _b and w _s , called the bending component and shear component, respectively, such that w = w _b + w _s . These components can be obtained by solving equation (1) with only $\left(-M_s/B_s\right)$ or $V_s^{\prime }/\left(A_c{G}_{eff}\right)$ as the right-hand-side term.

The roof panels considered in this study have a profiled face, which complicates the analysis because the bending stiffness of the profiled face cannot be neglected. On the other hand, the bending stiffness of the flat face is negligible, and it is not included in the calculation. The differential equation of deflection for panels with profiled faces subjected to a uniform line load, q, can be written as ¹⁷ :

$w^{⁗}\left(x\right)-{\left(\frac{\lambda }{L}\right)}^2{w}^{″}\left(x\right)=\frac{1}{B_{tot}}{\left(\frac{\lambda }{L}\right)}^2\left(M\left(x\right)+\beta L^{2}q\left(x\right)\right)$

(2)

where B _tot = B _F + B _S is the total bending stiffness of the panel, and B _F = B _F1 + B _F2 = EI _F1 + EI _F2 is the bending stiffness of the faces. B _S is the bending stiffness induced by the composite action in the sandwich cross-section:

$B_s=\frac{E_{F1}{A}_{F1}{E}_{F2}{A}_{F2}}{E_{F1}{A}_{F1}+E_{F2}{A}_{F2}}{e}^2$

(3)

where E _F1 and E _F2 are Young’s modulus, and A _F1 and A _F2 are the cross-sectional area of the upper and lower steel faces, respectively. The dimensionless parameters $\alpha$ , $\beta$ and $\lambda$ are calculated as:

$\alpha =\frac{B_F}{B_S},\beta =\frac{B_S{d}_C}{B{e}^2{G}_C{L}^2},\lambda =\sqrt{\frac{1+\alpha }{\alpha \beta }}$

(4)

Shear properties from two-point loading tests

The solution to the two-point loading tests is based on the design equations presented in EN 14509:2013. ¹⁰ The transverse shear strength, f _Cv , of the panel can be calculated as:

$f_{Cv}=F_u/2Be$

(5)

Where F _u is the ultimate load obtained from the test and B is the width of the beam specimen.

Equation (5) is valid also for panels with profiled faces as the test specimen in the two-point load test is cut such that it contains only the flat parts of the faces. The shear modulus, G _C , can be calculated using the slope obtained from the linear part of the load-displacement curve as ¹⁰ :

$G_C=\frac{∆F\hspace{0.17em}L}{6\hspace{0.17em}{A}_c∆w_s}$

(6)

Where ΔF is the difference in force corresponding to two points in the linear part of the load-deflection curve, L is the span length, and Δw _s is the change in shear deflection calculated as:

$∆w_s=∆w-∆w_b$

(7)

where Δw is the change in the total deflection at the mid-span recorded at force interval ΔF.

The bending component of deflection, Δw _b , is calculated as ¹⁰ :

$∆w_b=\frac{∆F{L}^3}{56,34B_s}$

(8)

The compression stress below the loading plate (or support plate), f _CUp , due to the force, F_Lp, acting below (or above) the loading (or support) plate of area A _Lp is calculated as:

$f_{CUp}=\frac{F_{Lp}}{A_{Lp}}$

(9)

Shear properties from four-point loading tests

The free-body diagram of the four-point loading test is shown in Figure 7. Because the test specimen in the four-point load test is cut as in the two-point loading test, the transverse shear strength, f _Cv , can be calculated by equation (5). To evaluate the shear modulus, the shear deflection needs to be determined. It can be obtained by equation (7) once the bending component of the deflection is known.

Figure 7.

Free body diagram of four-point load test.

The total bending deflection can be calculated by superimposing the deflection caused by four-point loads as shown in Figure 7. The loads are symmetrically applied on both sides of the mid-span of the beam, therefore, using the Macauley’s convention the total deflection caused by two-point loads (left side from the mid-point) applied at x ₁ and x ₂ is calculated and multiplied by two to get the total deflection as ¹ :

$∆w_b\left(\xi \right)=\frac{∆F{L}^3}{6B_s}\left[e_1\left(1-\xi \right)\left(2\xi -{\xi }^2-{e_1}^2\right)+e_2\left(1-\xi \right)\left(2\xi -{\xi }^2-{e_2}^2\right)\right]$

(10)

where ξ = x/L is the non-dimensional longitudinal coordinate and e ₁ = x ₁ /L and e ₂ = x ₂ /L. The values of x ₁ and x ₂ can be calculated from the geometry of the test setup given in Section 2.3 (see Figure 4).

The change in bending deflection at mid-span is

${\mathrm{\Delta }w}_b={\mathrm{\Delta }w}_b\left(0.5\right)=∆F{L}^3\left[(0.5e_1\left(0.75-{e_1}^2\right)+0.5e_2\left(0.75-{e_2}^2\right)\right]/12B_s$

(11)

The change in the shear component of the deflection, Δw _s , can be calculated from equation (7), using Δw _b from equation (11). The shear deflection, w _s , can also be obtained by integrating twice equation (1) with only the shear force term on the right-hand side to get the deflection caused by a point load. ¹ The change in total shear deflection is then calculated by superimposing the deflection caused by two separate point loads as:

$\mathrm{\Delta }{w}_s\left(\xi \right)=\frac{∆FL}{G_{eff}{A}_c}\left[e_1\left(1-\xi \right)+e_2\left(1-\xi \right)\right]$

(12)

Thus, after slightly rearranging equation (12) and using the value of ξ = 0.5 the unknown value of shear modulus can be calculated as:

$G_c=\frac{∆FL\left(e_1+e_2\right)}{4\mathrm{\Delta }{w}_s{A}_c}$

(13)

Shear properties from full-scale test

The sandwich panel in the full-scale (vacuum box) test is a simply supported beam under uniform load q as shown in Figure 5. The analysis of panels with flat and profiled faces is carried out separately, because in the former case, the bending stiffness of the faces can be neglected which simplifies the analysis ¹

Panels with flat faces

For a simply supported sandwich panel, the maximum bending displacement occurs at mid-span, and it can be calculated as:

$w_b=\frac{{5F}_u{L}^3}{384B_s}$

(14)

The shear displacement of the panel can be calculated using equation (7). During the full-scale test performed using a vacuum box, the total length, L _tot , of the panel is covered (see Figure 5). Since the total length of the panel is uniformly loaded, a reduction of the force acting on the support area should be considered. In this case, EN 14509:2013 Annex 4.5.2 ¹⁰ suggests reducing the measured total load, F _u , and consequently the transverse shear strength by the ratio $\left(L_{tot}-L\right)/\left(L_{tot}+L\right)$ . Therefore, the transverse shear strength obtained from full-scale tests is calculated by modified equation (5) as:

$f_{cv}=\frac{F_u}{2Be}\left(\frac{L_{tot}-L}{L_{tot}+L}\right)$

(15)

The shear modulus of the core can be calculated using the force-displacement curve as ¹⁰ :

$G_c=\frac{\left(\frac{∆F}{{∆w}_s}\right)B_{s}L{d_c}^2}{8{\hspace{0.17em}A}_c\hspace{0.17em}{e}^2\hspace{0.17em}\left(B_s+B_F\right)\left(1+\alpha \right)}$

(16)

Panels with a profiled face

The roof panels considered in this study have a profiled face, which implies that their deflection is obtained by solving equation (2). For a simply supported profiled sandwich panel, the six boundary conditions are w (0) = 0, w’’ (0) = 0, γ′ (0) = 0, γ′(L) = 0, w’’(L) = 0 and w’’(L) = 0. Consequently, the change in total deflection at mid-span is

$\mathrm{\Delta }w=\frac{∆F{L}^3}{\left(B_F+B_s\right)}\left[\frac{5}{384}-\frac{\cosh \left(\frac{\lambda }{2}\right)}{\alpha {\lambda }^4\hspace{0.17em}\cosh \left(\frac{\lambda }{2}\right)}+\frac{1}{8\alpha {\lambda }^2}\right]$

(17)

which can be divided into two components, denoted the bending deflection and shear deflection as follows:

${\mathrm{\Delta }w}_b=\frac{∆F{L}^3}{\left(B_F+B_s\right)}\left[\frac{5}{384}-\frac{\cosh \left(\frac{\lambda }{2}\right)}{\alpha {\lambda }^4\hspace{0.17em}\cosh \left(\frac{\lambda }{2}\right)}\right]$

(18)

$\mathrm{\Delta }{w}_s=\frac{∆F{L}^3}{\left(B_F+B_s\right)}\frac{1}{8\alpha {\lambda }^2}$

(19)

The shear modulus of the core, G _C , can be solved from equation (19) (see equation (4)) as:

$G_c=\frac{\left(\frac{∆F}{{∆w}_s}\right)B_{s}L{d}_c}{8B\hspace{0.17em}{e}^2\hspace{0.17em}\left(B_s+B_F\right)\left(1+\alpha \right)}$

(20)

Note that G _C depends on ${∆w}_s$ , which again depends on G _C through λ (see equations (7) and (18)). Therefore, G _C is determined iteratively. The shear force acting on the sandwich cross-section at the support is partially carried by the core and partially by the profiled face. The component of shear force carried by the core, V _s , can be expressed as ¹ :

$V_S\left(\xi \right)=\frac{F_u}{\left(1+\alpha \right)}\left[\frac{1-2\xi }{2}+\frac{\sinh \left(\frac{\lambda \left(1-2\xi \right)}{2}\right)}{\lambda \hspace{0.17em}\cosh \left(\frac{\lambda }{2}\right)}\right]$

(21)

The transverse shear strength can be calculated using the shear force obtained from equation (21) by setting $\xi =0$ as:

$f_{cv}=\frac{F_u}{A_c\left(1+\alpha \right)}\left[\frac{1}{2}-\frac{\sinh \left(\frac{\lambda }{2}\right)}{\lambda \hspace{0.17em}\cosh \left(\frac{\lambda }{2}\right)}\right]$

(22)

Test results

The goal of the tests is to produce shear failure in the core. However, it is known that in some cases other failure modes may appear. Regardless of the failure mode, the expressions for evaluating the transverse shear strength presented in Section 3 can be employed. But if actual shear failure was not obtained in the test, the calculated value of transverse shear strength does not correspond to the full shear capacity of the panel. Therefore, as the first step in analysing the test results, the failure modes observed in the tests are examined. The failure modes observed in the two-point loading test are depicted in Figure 8. Several failure modes other than shear were obtained, especially as the thickness of the panel was increased.

Figure 8.

Failure modes observed in two-point loading tests. (a) MW Panels, (b) PIR panels.

In general, the use of load-spreading and a support plate with smaller width (100 mm – 150 mm) resulted in the specimen failing with the crushing of the core at the support plate. Increasing the width of the load/support plates (>150 mm) solved the issue of core crushing at the support plate in most cases. Nonetheless, failure modes such as debonding, local indentation, and partial shear with wrinkling as the primary failure mode were observed, mainly for thicker MW panels.

In the four-point loading test, the load is applied at four different locations, thereby reducing the compressive stress below the loading plates. This increases the chance of avoiding unwanted failure modes, mainly face wrinkling. For all panel thicknesses and core materials tested, shear failure was obtained as the primary failure mode as shown in Figure 9 with the four-point loading test method. This suggests that the four-point loading test method, with the parameters specified in this study, can be recommended for measuring the transverse shear strength of sandwich panels, especially with thicker MW core.

Figure 9.

Failure modes observed in Four-point loading tests. (a) MW core, (b) PIR core.

All sandwich panels tested using the full-scale test failed in shear failure as shown in Figure 10. In some cases, buckling of the profiled face at mid-span was also observed in conjunction with shear failure. However, it is difficult to declare the primary failure mode of the test specimen, as it was covered during the test.

Figure 10.

Failure modes observed in Full-scale test. (a) MW core, (b) PIR core.

Force-displacement curves obtained from the tests were utilized to evaluate the results based on the expressions presented in Section 3. The force-displacement graphs obtained from the full-scale test of all the panel types are depicted in Figure 11. The graphs are plotted for a single specimen from each panel type, which represents the average behavior. The PIR panel typically showed a linear force increase until shear failure, except for R-PIR 140/100, which experienced preliminary buckling at mid-span. The MW panel showed a gradual force decrease after the elastic regime until shear failure.

Figure 11.

Force-displacement graph obtained from full-scale test method.

Figure 12 displays the force-displacement graphs of the shear beam tests. This figure illustrates the behaviour of a beam with a similar cross-section under two distinct loading configurations.

Figure 12.

Force-displacement graphs obtained from two-point and four-point loading test.

It offers insights into the maximum shear force attained in each testing method, revealing that the four-point method provides a greater ultimate load compared to the two-point loading test. Most significantly, it enables the visualization and comparison of the slope of the curve derived from the two different testing methods.

Table 4 presents the transverse shear strength of the specimens obtained from all three types of tests conducted including the width of loading plate used in two-point loading test. The results of each individual test are detailed in the Supplemental Material.

Table 4.

Summary of mean transverse shear strength, f _cv , obtained from the, n, number of tests including the span length, L, of the test specimens and width of loading plate, U _p .

Panel type	Two-point loading				Four-point loading			Full scale test
	n	L (mm)	U p (mm)	f Cv (MPa)	n	L (mm)	f Cv (MPa)	n	L (mm)	f Cv (MPa)
B-MW 100	5	660	100	0.069±0.004	5	1160	0.068±0.003	5	2200	0.083±0.007
B-MW 200	2	1020	100	0.053±0.004	5	1400	0.056±0.004	5	2400	0.060±0.004
	1	1170	150
	2	1315	200
A-MW 100	5	660	100	0.068±0.003	5	1160	0.067±0.003	5	2200	0.086±0.004
A-MW 200	2	1020	100	0.057±0.010	5	1390	0.069±0.009	3	2700	0.071±0.008
	1	1170	150					2	2500
	2	1315	200
PIR 100	5	660	100	0.13±0.002	5	1160	0.151±0.004	3	1700	0.136±0.009
								3	1400
PIR 200	5	1170	150	0.087±0.005	6	1400	0.108±0.007	5	2190	0.113±0.009
R-MW 140/100	5	660	100	0.064±0.006	5	1160	0.067±0.003	2	2000	0.069±0.003
								2	1800
								1	1700
R-MW 190/150	5	840	100	0.051±0.005	4	1280	0.055±0.001	5	2000	0.056±0.002
R-PIR 140/100	5	660	100	0.126±0.006	5	1160	0.141±0.007	3	1700	0.115±0.015
								2	1400
R-PIR 210/170	5	1062	150	0.093±0.003	5	1330	0.105±0.006	3	1700	0.103±0.007

Table 5 presents the contribution of shear deformation to the total beam deflection for each test method, accompanied by their respective shear modulus values. The given proportion of shear deflection is derived from the linear segment of the force-displacement curve, which is where the shear modulus values are calculated. Since the shear modulus is dependent on the shear deflection value, this table offers insights into the calculated value of shear deflection, while also illustrating the proportion contributed by the shear deflection.

Table 5.

Summary of mean proportion of shear deformation and mean shear modulus obtained from the implemented test methods with their corresponding standard deviation.

Panel type	Two-point loading		Four-point loading		Full-scale
	% Of ∆w s	G C (MPa)	% Of ∆w s	G C (MPa)	% Of ∆w s	G C (MPa)
B-MW 100	97 ± 1.6	4.16 ± 0.52	90 ± 0.4	3.99 ± 0.14	75 ± 0.3	3.99 ± 0.04
B-MW 200	96 ± 1.3	2.97 ± 0.25	94 ± 0.8	3.09 ± 0.08	84 ± 0.6	3.14 ± 0.07
A-MW 100	96 ± 0.2	3.79 ± 0.22	90 ± 0.2	4.10 ± 0.11	70 ± 1.4	4.02 ± 0.28
A-MW 200	94 ± 1.6	3.93 ± 0.16	94 ± 0.3	3.54 ± 0.16	83 ± 1.4	3.36 ± 0.26
PIR 100	97 ± 0.1	2.96 ± 0.05	90 ± 0.7	3.33 ± 0.23	79 ± 3.3	4.14 ± 0.17
PIR 200	96 ± 0.2	2.56 ± 0.14	95 ± 0.3	2.55 ± 0.13	85 ± 0.8	3.11 ± 0.18
R-MW 140/100	97 ± 0.3	3.82 ± 0.36	90 ± 0.4	3.98 ± 0.20	85 ± 4.5	3.88 ± 0.41
R-MW 190/150	97 ± 0.2	2.95 ± 0.16	94 ± 0.2	2.94 ± 0.10	95 ± 0.8	2.39 ± 0.20
R-PIR 140/100	97 ± 0.1	3.03 ± 0.11	90 ± 0.7	3.24 ± 0.22	91 ± 5.1	3.62 ± 0.34
R-PIR 210/170	96 ± 0.1	2.70 ± 0.04	94 ± 0.3	2.77 ± 0.13	95 ± 0.3	3.01 ± 0.22

Comparison of test methods

The difference between the test methods is illustrated by calculating the shear per unit width of the specimen, using the measured ultimate load, as depicted in Figure 13. In general, the four-point loading test yields greater values for the shear force per unit width than the conventional two-point loading test method. Since the width of the test specimen used in both methods is identical, withstanding higher force per unit width means higher transverse shear strength. The difference is more pronounced for the PIR panels and the thick MW panel with flat faces (e.g. A-MW 200). Interestingly, there is a clear difference between nominally identical panels B-MW 200 and A-MW 200, especially in the results of the four-point loading test. This can be attributed to the fact that different MW materials were used in panels A and B, with different core densities (see Table 1) and compressive strengths (Table 2).

Figure 13.

Shear force acting across the width of the beam specimen.

Additionally, the distinctive manufacturing process employed at each production plant affects the test results. The transverse shear strength obtained from each test method is depicted in Figure 14. The two-point loading test is used as a reference to which the four-point and full.

Figure 14.

Comparison of transverse shear strength and their percentage deviation relative to the two-point test method as the reference.

The full-scale test method provides the greatest transverse shear strength for all MW core panels and the thick PIR wall panel (PIR 200). However, the shear strength of R-PIR 140/100 obtained from the full-scale test is the lowest among the test methods, which can be due to the span length of the tested panel. It is to be noted that during the full-scale test, any initial non-shear failure modes that might be detrimental to the results are not detected as the panels are covered. However, for the R-PIR 140/100 panel type, the ultimate failure mode was shear. As for the small-scale tests, the four-point test method produces greater transverse shear strengths than the two-point test in general. For the thinner MW panels (B-MW 100 and A-MW 100), both shear beam test methods yielded an identical level of shear strength.

For further insight on the shear beam test methods, the compressive stresses in the core at the support and loading plates, f _CSp and f _CUp , respectively, corresponding to the ultimate load reached in the tests are examined. Table 6 shows these stresses, calculated by equation (9), and their ratio to the mean compression strength of the core material, obtained from the material tests

Table 6.

Ratio of mean compression stress below the loading plate (f _CUp ) and mean compression stress above the support plate (f _CSp ) to the mean compression strength of the core material (f _Cc ).

Panel type	Two-point loading		Four-point loading
	f CUp (MPa)	f CUp /f Cc	f CUp (MPa)	f CSp (MPa)	f CUp /f Cc	f CSp /f Cc
B-MW 100	0.07	0.54	0.02	0.03	0.13	0.26
B-MW 200	0.08	0.59	0.03	0.57	0.25	0.5
A-MW 100	0.07	0.81	0.02	0.03	0.2	0.4
A-MW 200	0.08	0.49	0.03	0.07	0.43	0.87
PIR 100	0.13	0.85	0.04	0.09	0.24	0.59
PIR 200	0.12	1.14	0.05	0.11	0.52	1.04
R-MW 140/100	0.06	0.53	0.02	0.03	0.14	0.28
R-MW 190/150	0.08	0.7	0.02	0.03	0.18	0.37
R-PIR 140/100	0.13	1.14	0.04	0.07	0.32	0.64
R-PIR 210/170	0.10	0.87	0.04	0.09	0.36	0.72

Note that, for the two-point loading test, the compressive stress below the support plate is equal to the compressive stress in the loading plate, because the plates have identical geometry and the force acting on each plate is F/2. The ratio of compression stress below the loading plate to the compression strength of the core material is lower in the four-point loading test than in the two-point loading test. This was to be expected because, in the four-point loading test, the total load applied on the specimen is divided among four loading plates instead of two. Moreover, the width of the loading plates is larger in most cases for the four-point loading tests. Therefore, even if a larger ultimate load was achieved for many panels by the four-point loading test, the compressive stress below the loading plate in the core remains smaller than in the two-point loading test.

For the ratio of compression stress above the support plate to the compression strength of the core material, the four-point loading test provides lower values for thinner wall panels and most of the roof panels, whereas for the thicker wall panels, the two testing methods give similar values. For A-MW 200, the four-point loading test clearly gives a higher ratio than the two-point loading test (0.87 vs 0.49). It can be concluded that crushing of the core is avoided for both test methods in general. For the PIR 200 panel (and R-PIR 140/100), the ratio is greater than 1,0 at the support plate (also at the loading plate for the two-point loading). Nevertheless, no visible crushing of the core was observed in the four-point loading tests. For the two-point loading tests, the crushing of the core was the primary failure mode for these panels. Combining the observations from the two-point and four-point loading tests with the calculated crushing of the core strength, it can be deduced that the core was close to failing by crushing in the four-point loading tests on PIR 200 panels.

Conclusion

This study suggests that the four-point loading test is more suitable for evaluating the shear properties of thicker sandwich panels than the two-point loading test. In the four-point loading test, the failure modes of both the thicker and thinner panels were primarily shear. Conversely, in the two-point loading test, both the thicker MW and PIR panels mainly failed due to core crushing, wrinkling, indentation, and debonding. For thin (100 mm) MW panels, the difference between two-point and four-point shear beam test methods is insignificant. However, in the case of thicker MW panels, the four-point test method provides up to about 20% higher transverse shear strength than the two-point test. This disparity becomes prominent as the thickness of the panel increases for both MW and PIR panels. The four-point loading method also proved to be more consistent, as it produced a smaller deviation among the test results than the two-point test method.

The full-scale test resulted in all the specimens failing in shear and it provided the greatest transverse shear strength for MW panels. For PIR panels, the full-scale test produced better results than the two-point test, but there was some discrepancy with the four-point loading test, which yielded greater transverse shear strength for the PIR panel and both roof panel types. Further investigations are needed with the vacuum box to determine the appropriate span length of PIR panels.

The detailed guidelines on the developed four-point loading test procedure presented in this study provide a good basis for testing thick sandwich panels. Slight modifications to the details may be required, particularly if panels thicker than 200 mm are to be tested. It should be noted that the profiled section of the roof panels is not included either in the two-point or the four-point loading tests. However, the profiling of the panel face contributes to the shear resistance of the sandwich panel, and this effect should preferably be considered. A more detailed study regarding the effect of the profiled section on the overall shear performance of the sandwich panel is needed. Detailed study on the distribution of shear stresses can be conducted utilizing method such as Digital Image Correlation (DIC) or Finite Element Analysis (FEA).

Finally, it should be noted that the “semi-analytical” approach of EN 14509 for determining the shear modulus of the core can be sensitive to numerical errors. These errors can arise from various factors, including the dimensional variability of the specimen and the material properties of the faces, especially Young’s modulus. More research is needed quantify the potential magnitude of these errors. As an alternative, the approach presented in the ASTM D7250/D7250 M with appropriate modifications for more general sandwich panels (with profiled or unequal faces) can be considered for determining the shear modulus.

Supplemental Material

Supplemental Material - Test methods for determination of shear properties of sandwich panels

Supplemental Material for Test methods for determination of shear properties of sandwich panels by Shekhar Silwal, Kristo Mela, and Zhongcheng Ma in Journal of Sandwich Structures & Materials