Numerical and Theoretical Analyses of Laminated Veneer Lumber Beams Strengthened with Fiber-Reinforced Polymer Sheets

Abstract

This study outlines a method of utilizing the finite element method and a simple mathematical model to predict the behavior of laminated veneer lumber (LVL) beams strengthened with composite sheets. The numerical models were created using the Abaqus 2017 software. The LVL was considered as a linearly elastic or elastic–plastic material, factoring in Hill’s yield criterion. The composites were simulated as linearly elastic–ideally plastic materials. The mathematical models were predicated on the methodology of transformed cross-section. The theoretical and numerical outcomes were juxtaposed with previous empirical investigations. The comparison encompassed load-bearing capacity, stiffness, and deformation under peak force. Furthermore, presentations of normal stress maps in the LVL and composite have been illustrated. The derived maps were juxtaposed with the delineations of failure modes. An adequate correlation was identified between the theoretical, numerical, and empirical values in the case of beams reinforced with aramid, glass, and carbon sheets. The relative deviation varied from several to multiple percentages. This technique is not applicable for evaluating load-bearing capacity and deformation when only dealing with sheets with low elongation of rupture. This is a consequence of their premature failure. The proposed models may be utilized by researchers and engineers in the design of reinforcements for timber structures.

1. Introduction

In scientific endeavors, significant emphasis has been placed on ecological and economic matters, consequently necessitating the rational management of limited resources. Consequently, the quantity of necessary experimental investigations has been reduced to the bare minimum and the bare minimum was based on the premise of achieving repeatable results. The repeatability of outcomes has typically been understood through the attainment of pertinent statistical indices regarding the measures of dispersion of the results. Greater emphasis has instead been placed on the ability to anticipate the behavior of actual entities. These predictions are conducted via theoretical and numerical investigations. This methodology appears to be justified and has been utilized to examine non-strengthened and strengthened timber components.

A common method for evaluating the bending behavior of complex sections composed of constituents with different mechanical characteristics is the employment of the transformed cross-section method. The main assumptions of this method were described in the study carried out in [1] regarding the analysis of solid beams reinforced with steel sections. Timbolmas et al. [2] utilized this approach to assess the mechanical performance of glued laminated timber beams made from varied wood species amalgamations and a Carbon Fiber-Reinforced Polymer (CFRP). The maximal mean value of resulting coefficient of variation was 15%. Alam et al. [3] used it to analyze the possibility of repairing solid timber beams using reinforcements inserted into near surface grooves. They deduced, among other things, that the most effective reinforcement is a steel profile or CFRP and that the most optimal configuration proposes reinforcing both the compressed and tensile zones. Mosallam [4] characterized the transformed cross-section method as a simplified methodology for the design of reinforced timber beams employing composite laminates and honeycomb-structured sandwich panels. The study comprised both experimental outcomes and a case study regarding employing an analytical method for predicting the load-bearing capacity of the reinforced element. According to the author’s assessment, the depicted reinforcement technique will lead to a 300% enhancement in load-bearing capacity. Soriano et al. [5] examined the mechanical characteristics of glued laminated timber beams reinforced with steel rods. This research used the transformed cross-section methodology to assess the bending stiffness of a composite cross-section. Upon conducting a parametric analysis, the significance of the proportionality ratio between the elasticity moduli of both wood and reinforcement was emphasized. Specifically, the higher the ratio, the higher the reinforcement efficiency. Nadir et al. [6] employed a transformed cross-section method to estimate the bending stiffness of glued laminated timber beams reinforced with glass and carbon fiber sheets. The experimental data validated the theoretical findings. Mirski et al. [7] studied the potential of reinforcing glued laminated timber beams using smooth and ribbed steel and basalt rods. Similar to other authors, they derived comparable theoretical stiffness values using a transformed section and an experimental one. Fossetti et al. [8] introduced an analytical model employing transformed section attributes to evaluate the performance of glued laminated timber beams reinforced with cords and composite rods. They assumed a linear stress distribution over the depth of the cross-section, and considered the potential for lamellae with various mechanical characteristics within the cross-section of the timber. They achieved high accuracy between experimental and theoretical outcomes. In their research, Borri et al. [9] conducted a comparative analysis between a linear and a non-linear model relative to the performance of beams strengthened with CFRP laminates. Brol and Markowska [10] introduced fundamental information concerning the methodology for estimating the load-bearing capacity of timber beams strengthened with FRP materials according to the American standard PCF-5100 and PCF-6046 while considering selected directives from European standards.

In conclusion, numerous instances verify the applicability of the transformed cross-section method. Undeniably, it was found to be suitable by multiple scientists and civil engineering practitioners. Its main advantage is its simplicity. Computations can be executed in spreadsheets formulated in both free and commercially available software. It can be deployed for comprehensive parametric investigations when integrated with material databases (wood and composite materials). The parametric investigations themselves could potentially be augmented with economic analysis. Paradoxically, this simplicity also constitutes its most significant disadvantage. Fulfilling a range of assumptions—such as those relating to linear stress distribution or the manner of destruction—restricts its utility.

A more intricate approach employed for the prediction the behavior of both non-strengthened and strengthened timber beams involves conducting a numerical analysis based on the finite element method. The execution of these necessitates the possession of specialized software and a vast knowledge of modeled experiments. The prerequisites for input, therefore, substantially exceed the comparison of the conditions necessitated for applying the transformed cross-section approach. Undoubtedly, it was evident that we received considerably more information regarding the behavior of the beam. Several studies that have utilized simulations to describe bending behavior when incorporating wooden elements are delineated below.

Li et al. [11] examined the behavior of steel–wood composite floors. The experimental outcomes were juxtaposed with numerical simulations. Timber was regarded as an orthotropic material. The researchers utilized numerical simulations to investigate the influence of the pattern of self-drilling screws and the width of the flange on the bearing capacity of the floor. Ye et al. [12] conducted numerical simulations on bent beams with circular wood cross-sections with lap joints strengthened using CFRP sheets. The wood was formulated as a linearly elastic (orthotropic) material. The perfect connection of the wood and reinforcement was assumed, disregarding the influences of an adhesive or slippage. The interaction properties assumed no penetration between entities in the normal direction and a specific value of the friction coefficient in the tangential direction. The authors achieved a significant correspondence between the experimental and numerical equilibrium paths within the elastic limit with these premises. Navarantnam et al. [13] conducted numerical analyses on wood beams strengthened using glass fiber-reinforced polymer profiles. The models were prepared using the Abaqus software. The effects of joints, GFRP profiles, and wood grain orientations on the behavior of bent elements were analyzed. Ghanbari-Ghazijahani et al. [14] studied lightweight I-beams reinforced with composite materials. A notable enhancement in load-bearing capacity and ductility was observed in comparison to the reference beams. The numerical models were found to be effective in predicting the behavior of the beams with a high degree of accuracy. Kula et al. [15] conducted an experimental, analytical, and numerical analysis of bent I-beam timber beams with section weaknesses in the tension zone and the concept of reinforcing them with CFRP strips. An optimal elastic–plastic material model was implemented for all components, including oriented strand board (OSB), solid wood, and CFRP. A similar assumption was made regarding the perfect connection between the parts [12]. A quantitative analysis revealed a notable influence of the length of the reinforcement on the load-bearing capacity of the reinforced section.

The main goal of this article is to assess the suitability of using the transformed cross-section and finite element methods to predict the behavior of laminated veneer beams reinforced with sheets bonded to external surfaces. This goal will be achieved through a comparison of the results of experimental, numerical, and theoretical investigations. The findings of the numerical studies were presented using the commercial software Abaqus. Two distinct material models were considered for the LVL: linear elastic and linear elastic–perfectly plastic, with Hill’s plasticization criterion considered. The composite materials were modeled as elastic–perfectly plastic materials. A theoretical study was conducted utilizing the transformed cross-section method. The comparison included several basic mechanical parameters, such as load-bearing capacity, stiffness, and deflection at maximum force. Furthermore, the typical stress distributions in the veneer and composite material at the maximum force are illustrated—the issues raised in the article present scientific and engineering-relevant concerns. The main novelty of the paper is the description of conditions for which both presented methodologies could be utilized from engineering and scientific points of view.

2. Materials and Methods

This section presents only the most important information regarding the work’s subject matter. Further details of the experimental study can be found in [16].

Experimental tests were carried out on non-strengthened and strengthened laminated veneer lumber beams with nominal dimensions of 45 × 100 × 1700 mm purchased from Steico (Czarnków, Poland) [17]. The beams were reinforced with composite sheets bonded to the bottom surface only over the entire length of the beam (scheme 1) and to the bottom surface and half the height of the side surfaces (scheme 2), as shown in Figure 1. Four sheet types manufactured by S&P Reinforcement Poland (Malbork, Poland) were used for reinforcement including aramid fiber-reinforced polymer (AFRP) [18], glass fiber-reinforced polymer (GFRP) [19], carbon fiber-reinforced polymer (CFRP) [20], and ultra-high modulus carbon fiber-reinforced polymer (CFRP UHM) [21] ones. The test series were marked with letters depending on the type of fiber used: aramid (A), glass (G), carbon (C), carbon ultra-high modulus (CH). The suffix U refers to the configuration of the U-type reinforcement. An epoxy resin-based adhesive (manufacturer’s reference, S&P Resin 55 HP) was used to bond the sheets [22]. Figure 1 shows a schematic of the strengthening and test bench.

The bending was carried out according to the PN-EN 408+A1:2012 [23] and PN-EN 14374:2005 [24] standards. A 4-point bending test scheme was used. The distance between the points of concentrated force was 600 mm. The distance from the point of concentrated force to the axis of the nearest support was 500 mm. The length of the beam at the axes of the supports was 1600 mm. The overall length was 1700 mm. The loading was carried out using the actuator’s displacement speed control. On the supports and at the point of load application, steel guide plates with a section size of 10 × 40 mm were placed.

3. Numerical Model

This section gives a general description of the numerical models created and the material models assumed. Due to the system’s bisymmetry, only ¼ of the actual beam was prepared. The displacements U1, U2, and U3 and the rotations R1, R2, and R3 are assigned to the x, y, and z axes of the global coordinate system, respectively.

3.1. Model Description

Laminated veneer beams were prepared as three-dimensional deformable bodies by extending a 22.5 × 100 mm profile to 850 mm distance. The composite-covered corner has been rounded over its entire length for beams reinforced in a U-shape configuration. The radius of the rounding was 6.25 mm. The sheets were modeled as three-dimensional deformable shells by extending the profiles to a length of 850 mm. For the sheets applied to the underside, the width of the profile was 22.5 mm, and for the U-scheme, the width was approximately 72.5 mm. A sketch of the LVL cross-section was used to create the sheet profiles. The steel guide plates placed on the supports and at the point of load application were modeled as discrete rigid bodies—the dimensions of the plate were 10 × 40 mm, and its length for visualization was 22.5 mm. Boundary conditions and loads were then assigned to these reference points.

Due to the symmetrical layout, the front and side surfaces were assigned corresponding symmetry conditions about the x and z axes of a global coordinate system. In the initial step, the possibility of moving and rotating (except for rotation about the x-axis) reference points placed on steel plates was blocked. In the following calculation step, the load was realized by assigning a displacement equal to U2 = −40 mm to the reference point on upper steel guide plate.

The meshing was performed on individual parts. An approximate maximum mesh size of 5 mm was assumed. The C3D8R (an 8-node linear brick with reduced integration and hourglass control) for solids, S4R (a 4-node doubly curved thin or thick shell with reduced integration and hourglass control) for composite sheets, and R3D4 (a 4-node 3-D bilinear rigid quadrilateral) steel guide plate finite element types were used. The finite element shape for orthogonal elements adopted was Hex, Structured; for curved surfaces, it was Hex, Sweep along with the algorithm for the medial axis.

The “Surface-to-Surface Contact” option was applied to account for the contact between the laminated veneer laminated surfaces and the steel guide plate was applied in initial step. For the contact properties, the ‘Hard Contact’ option assuming that no penetration of bodies is possible and a coefficient of friction of 0.3 were used. The bond between the LVL and the composite material has been modeled using the ideal bond for the ‘Tie’ constraint. This does not consider the adhesive’s effect on the bending beams’ behavior.

The calculation was performed in the static range. For this purpose, a step of type “Static, General” was created. The increment parameters were chosen to obtain a solution convergence. The main assumptions of the numerical model are shown in Figure 2.

Forces were read at the steel guide plate reference point on the top of the beam. Deflection values were read from the node located at the midspan of the outermost compressed fibers.

3.2. Material Properties

For the composite materials, an elastic–perfectly plastic material model has been adopted—

Table 1. Selected mechanical and physical properties of composite sheets [18,19,20,21].

Parameter	S&P A-Sheet 120	S&P G-Sheet E90/10	S&P C-Sheet 240	S&P C-Sheet 640
Tensile strength ft [N/mm2]	2900	3400	5100	2600
Mouduls of elasticity EFRP [N/mm2]	120,000	73,000	265,000	640,000
Thickness tFRP [mm]	0.200	0.289	0.333	0.189
Poisson’s Ratio [-]	0.3	0.3	0.3	0.3

shows selected sheet material properties taken from the manufacturer’s data. The yield strength was assumed to be the tensile strength of the composite. The sections were created by taking into account the thickness of the material. A Poisson ratio of 0.3 was assumed for each composite.

In the case of laminated veneer lumber, two types of material models were verified: a linear elastic one and a linearly elastic–perfectly plastic one considering Hill’s yield criterion.

Table 2. Mechanical properties of laminated veneer lumber [16,17].

Parameter	Value
Compressive strength, parallel to the grain fc,0 [MPa]	58.5
Compressive strength perpendicular to the grain fc,90,edge [MPa]	9.5
Bending strength (in edgewise condition) fm,0,edge [MPa]	65
Shear strength parallel to the grain in edgewise condition fv,0,edge [MPa]	4.6
Shear strength parallel to the grain in flatwise condition fv,0,flat [MPa]	2.6

and

Table 3. Orthotropic properties of laminated veneer lumber [16,17,25].

Modulus of Elasticity [MPa]			Shear Modulus [MPa]			Poisson’s Ratio [-]
E1	E2	E3	G12	G13	G23	ν12	ν13	ν23
12,000	430	430	600	600	96	0.48	0.48	0.48

show the mechanical and orthotropic properties of the material. These were determined based on our experimental studies [16], the manufacturers’ data [17], and the findings of other researchers [25]. The yield stress was used as the bending strength value determined experimentally on unreinforced members.

The “Engineering constant” option was used to enter the orthotropic properties of the veneer. The Hill function was entered via the “Potential” sub-option and calculated using Formula (1):

$$f\left(\sigma \right)=\sqrt{F{\left({\sigma }_{22}-{\sigma }_{33}\right)}^2+G{\left({\sigma }_{33}-{\sigma }_{11}\right)}^2+H{\left({\sigma }_{11}-{\sigma }_{22}\right)}^2+2N{\tau }_{12}^2+2M{\tau }_{13}^2+2L{\tau }_{23}^2}$$

(1)

In which σ_ij and τ_ij correspond to the stress tensors σ and the six constants estimated according to Formulae (2) and (3):

$$F={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{R_{22}^2}}+{\displaystyle \frac{1}{R_{33}^2}}-{\displaystyle \frac{1}{R_{11}^2}}\right),G={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{R_{33}^2}}+{\displaystyle \frac{1}{R_{11}^2}}-{\displaystyle \frac{1}{R_{22}^2}}\right),H={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{R_{11}^2}}+{\displaystyle \frac{1}{R_{22}^2}}-{\displaystyle \frac{1}{R_{33}^2}}\right),$$

(2)

$$N={\displaystyle \frac{3}{{2R}_{12}^2}},M={\displaystyle \frac{3}{{2R}_{13}^2}},L={\displaystyle \frac{3}{{2R}_{23}^2}}$$

(3)

In which R_ij constants were estimated from [26] according to Equation (4):

$$R_{11}={\displaystyle \frac{{\overline{\sigma }}_{11}}{{\sigma }^0}},R_{22}=R_{33}={\displaystyle \frac{{\overline{\sigma }}_{22}}{{\sigma }^0}}={\displaystyle \frac{{\overline{\sigma }}_{33}}{{\sigma }^0}},R_{12}=R_{13}={\displaystyle \frac{{\overline{\sigma }}_{12}}{{\tau }^0}}={\displaystyle \frac{{\overline{\sigma }}_{13}}{{\tau }^0}},R_{23}={\displaystyle \frac{{\overline{\sigma }}_{23}}{{\tau }^0}}$$

(4)

in which τ⁰ = σ⁰/√3. Estimated constants for the analysis are shown in

Table 4. Constants for numerical analysis.

R11	R22	R33	R12	R13	R23
1.0	0.162	0.162	0.136	0.136	0.077

.

As mentioned above, the analyses neglected the effect of the adhesive on the static behavior of the beams. In the following section, a distinction has been made between the numerical models according to the material model used for the laminated veneer. The letters EL indicate elastic models, and the letters EL-PL indicate the elastic–plastic model.

4. Transformed Cross-Section Method

The transformed cross-section method was used for the theoretical analysis. This model assumes a proportional increase in the timber cross-section to replace the reinforcement elements. This creates sections made from the same material. The work of [1,2,27], among others, describes the model’s assumptions in detail. Its application to the assessment of the performance of full-size laminate and CFRP sheet-reinforced laminated veneer lumber beams was described in [27]; the behavior of LVL slabs reinforced with FRP sheets was described in [28]; and CFRP-LVL sandwich structures were described in [29].

Based on the proportionality coefficient η obtained from Formula (5), the increase in area was estimated [27]:

$$\eta ={\displaystyle \frac{E_{FRP}}{E_{LVL}}},$$

(5)

where E_LVL is the modulus of elasticity of laminated veneer lumber (Table 3), and E_FRP is the modulus of elasticity of composite material (Table 1). This factor is then multiplied by the horizontal dimension of the composite. In the case of the reinforcement bonded to the bottom, this is its width. For the composite placed on the sides it is the sheet thickness.

Equation (6) determines the position of the center of gravity of the section:

$$z_c={\displaystyle \frac{S_y}{\sum _{i=1}^n{A}_i}},$$

(6)

where S_y is a static moment of the surface area concerning the assumed reference system; A_i is the surface area of the elementary figure. The initial position of the datum was such that the cross-section fell entirely within the first quadrant of the coordinate system, and its outline was adjacent to the assumed axes.

Equation (7) was used to determine the moment of inertia of the section:

$$I_y=\sum _{i=1}^n\left(I_{yi}+A_i\times {\left(z_c-z_i\right)}^2\right),$$

(7)

where I_yi is the moment of inertia of the elementary field, and z_i is the distance from the center of gravity of the elementary field to the assumed datum system.

Equation (8) was used to estimate the maximum value of the loading force:

$$f_{m,0,edge}={\displaystyle \frac{M_{max}\times z}{I_y}}\to M_{max}={\displaystyle \frac{f_{m,0,edge}\times I_y}{z}}$$

(8)

where f_m,0,edge—the bending strength of the non-strengthened laminated veneer lumber determined from experimental tests; z—distance from the center of gravity to the outermost compression fibers.

The bending stiffness coefficient k was estimated as the quotient of the loading force and the deflection according to Formula (9) [27]:

$$k=F/u$$

(9)

where u is the deflection value at the center of the beam span; F is the value of the force loading.

Figure 3 shows schematics of the transformed cross-sections and their designations.

5. Results and Discussion

Basic information on about beams’ behavior, including their load-carrying capacity, stiffness, and stress distributions obtained in the composite reinforcement and laminated veneer lumber, is presented in this section.

5.1. Load-Bearing Capacity

Figure 4 plots the relationship between loading force and deflection measured at the center of the horizontal span of the extreme tensile fibers. The red color with markers indicates the curves for the linearly elastic model. In contrast, the blue color with markers represents the elastic–plastic model of laminated veneer behavior obtained from the numerical analyses. The green color with markers indicates the results obtained from the theoretical model. The black curves represent the experimental results obtained from [16]. The green and red curves exhibit linear behavior throughout the test range. The models labeled FEM-EL were plotted significantly beyond the scope of the experimental studies. The models designated as THEO were plotted from the beginning of the test until the theoretical maximum loading force was reached. The FEM-EL-PL model curves show a linear relationship from the beginning of the test until the proportionality limit is reached. After that curve course becomes non-linear. Compared to the other models prepared, they are the most representative of the experiment.

Table 5. Comparison of experimental [16], theoretical, and numerical values of maximum load.

	LVL	A	AU	G	GU	C	CU	CH	CHU
Experimental	18.87	21.78	21.93	22.63	23.12	24.49	25.56	18.30	19.22
Theoretical	19.50	20.98	22.24	20.84	22.02	25.09	29.37	27.28	32.96
Numerical	20.86	21.46	21.96	21.43	21.90	22.72	24.59	22.91	24.61
Theo./Exp.	1.03	0.96	1.01	0.92	0.95	1.02	1.15	1.49	1.71
Num./Exp.	1.11	0.99	1.00	0.95	0.95	0.93	0.96	1.25	1.28

and

Table 6. Comparison of experimental [16], theoretical, and numerical values of deflection corresponding to maximum load.

	LVL	A	AU	G	GU	C	CU	CH	CHU
Experimental	32.5	35.6	41.6	38.1	40.5	39.8	38.8	27.5	24.8
Theoretical	30.2	30.6	31.3	30.5	31.12	32.1	34.2	33.1	35.8
Numerical	34.6	33.9	34.1	33.7	33.92	32.5	33.6	31.1	31.6
Theo./Exp.	0.93	0.86	0.75	0.80	0.77	0.81	0.88	1.20	1.44
Num./Exp.	1.06	0.95	0.82	0.88	0.84	0.82	0.87	1.13	1.28

present the experimental [16], theoretical, and numerical results. The analysis is presented in terms of the maximum loading force and corresponding midspan deflection. The mean value of all measurements was calculated to obtain experimental results. The numerical results for the elastic–plastic model were used. Furthermore, the tables present quotients of the theoretical and numerical results with the experimental results.

The theoretical model obtained a slightly higher accuracy of the maximum loading force for non-strengthened elements, with a quotient of 1.03 in comparison to the numerical model. In the case of strengthened beams, a numerical model is typically more accurate than a theoretical model. The relative error in estimating the maximum force is approximately 5%, and the deflection is approximately 5% to 10%. The differences are significant only for beams strengthened with ultra-high modulus sheets. This phenomenon can be attributed to the low elongation value at the point of material failure. Consequently, the material is subjected to preliminary failure, as elucidated in [16]. It is crucial to acknowledge that the theoretical and numerical models apply only when failure is initiated in the wood by a fracture due to tension or crushing in the compression zone. Similarly, as previously stated in [28], these models are not applicable in the event of shear failure. The analytical model demonstrated comparable accuracy in estimating the maximum bending moment for full-size non-strengthened and strengthened laminated veneer lumber beams with CFRP composites, as reported in [27]. The transformed cross-section method provided considerably less accuracy when estimating the so-called rupture modulus [30].

In most instances, the estimated deflection values are less than the experimental values (Table 6).

5.2. Stiffness

The bending stiffness was determined by a factor calculated using Equation (9). Due to the stiffness degradation of the real and numerical elements for the elastic–plastic model, the coefficient was estimated for a load range of 0.1 to 0.4 F_max. The estimated stiffness comparison is presented in

Table 7. Comparison of experimental [16], theoretical and numerical values of stiffness coefficient.

	LVL	A	AU	G	GU	C	CU	CH	CHU
Experimental	0.655	0.722	0.710	0.738	0.734	0.815	0.834	0.851	0.868
Theoretical	0.647	0.685	0.711	0.683	0.707	0.780	0.858	0.825	0.921
Numerical (EL)	0.681	0.745	0.768	0.753	0.776	0.865	0.945	0.895	0.991
Numerical (EL-PL)	0.636	0.692	0.719	0.702	0.727	0.800	0.886	0.825	0.928
Theo./Exp.	0.99	0.95	1.00	0.93	0.96	0.96	1.03	0.97	1.06
Num. (EL)/Exp.	1.04	1.03	1.08	1.02	1.06	1.06	1.13	1.11	1.14
Num. (EL-PL)/Exp.	0.97	0.96	1.01	0.95	0.99	0.98	1.06	0.97	1.07

. As demonstrated by the findings of [5,7,27,28,29], a high degree of accuracy was observed between the experimental and theoretical stiffness values, with maximum differences of 6%. A comparable degree of accuracy was also achieved in numerical models in which the LVL is represented as an elastic–plastic material. As in the work [28], the most significant discrepancies were observed in the case of CFRP UHM sheet reinforcement. In the case of numerical models with elastic modeling, the results obtained in each case analyzed indicated that the gain was more significant than the experimental value. At most, the difference between the two values was several per cent.

5.3. Stresses and Failure

The stresses presented in this section were expressed in megapascals in the figures. Normal stress maps for composites and laminated veneer lumber were juxtaposed with the typical failure modes obtained for experimental tests. The goal of that comparison was to prove the applicability of numerical simulations for predicting the location of the failure and indicating the weakest link in the composite–LVL chain.

In the case of reference beams, the maximum stresses are equal to the assumed yield strength. The maximum tensile stresses were slightly higher than the compressive stresses. The destruction of unreinforced beams was attributed to the brittle fracture of the LVL in the tension zone (Figure 5). A slight deformation in the veneer was observed at the point of application of the concentrated force [16].

For beams strengthened with relatively low modulus composite sheets bonded only on the underside (A and G series), the ratio between the maximum tensile and compressive stresses was comparable to that observed in the non-strengthened beams. In the U-scheme (AU and GU series) context, the values of the maximum tensile and compressive stresses were approximately equal when these materials were used. In the remaining cases of strengthened beams (C, CU, CH, and CHU series), a slight decrease in tensile stress relative to compression was evident (Figure 6). The mode of failure also underwent a transformation, shifting from brittle fracture to a more plastic failure due to crushing in the compressed zone (Figure 7). In each of the cases analyzed, there was a distortion of the normal stress distribution within the concentrated force—the disturbances result from the plasticization of the laminated veneer lumber. The nature of these disturbances was identical for all strengthened beams. The slight discrepancies in the dimensions of these zones were attributable to the distinctive properties of the reinforcement employed—the volume of the plasticized zone increases in direct proportion to the rise in the factors above.

Figure 8 illustrates the typical normal stress S11 distributions observed in the composite materials subjected to the maximum loading force. The following relationships can be observed: (1) as the stiffness of the sheet increases, the normal stresses also increase; (2) for the U-type configuration, slightly higher normal stresses were obtained compared to the sheet bonded only on the underside; (3) the maximum normal stresses of the sheet bonded on the underside were obtained at the center of the span, while for the sheet in the U-scheme at the axis of the loading force. The stress values obtained for aramid, glass, and carbon sheets were lower than those presented by the manufacturer [18,19,20,21]. Despite the absence of stresses over the specified limit in the simulations, composite failure was observed in the experimental tests. This resulted from the LVL undergoing excessive deformation or crack propagation (for example, a crack in the veneer locally interrupted a section of the composite sheet). In the case of a sheet with an ultra-high modulus of elasticity, slightly higher stresses were obtained than those specified by the manufacturer. This type of sheet has consistently demonstrated a tendency to fail during experimental studies. Concurrently, this destruction occurred earlier in the test and was attributable to the relatively low elongation at material rupture. It is important to note that the CFRP UHM sheet has been designed to reinforce shear zones.

6. Conclusions

This paper compares two popular methods for predicting the bending behavior of composite cross-sections. It evaluates the transformed cross-section method and numerical simulations. Previous experimental studies validated the aforementioned approach. The research involved the use of both non-strengthened and strengthened laminated veneer lumber beams. To strengthen the structure, composite sheets were bonded to the external surfaces with an epoxy resin-based adhesive.

Both approaches were appropriate for estimating load-carrying capacity, stiffness, and deflection at maximum force for most cases analyzed. The relative error exhibited a range of values, spanning from a few to several per cent. Only for the beams strengthened with CFRP UHM sheets was the presented methodology not suitable for the evaluation of load-carrying capacity and corresponding stiffness. This phenomenon can be attributed to the low elongation value at rupture of the composite material.

In most analyzed case scenarios of strengthening LVL beams with composite sheets the weakest point was the wood. For increasing the reinforcement’s effectiveness, a reinforcement should be applied to the compressive zone should be applied. Due to the type of failure mode, composite materials with elongation at rupture should be avoided for reinforcing wooden beams in the tension zone.

The authors posit that the presented transformed cross-section method, in conjunction with numerical simulations, can be employed for the design of timber structure reinforcement. It is important to note that both approaches have been validated for beams reinforced with a reinforcement placed only in the tension zone. In the case of such configurations, the statements above are accurate. Incorporating composite reinforcement within the compression zone will fundamentally alter the material’s mechanical properties.