# Application of Transformed Cross-Section Method for Analytical Analysis of Laminated Veneer Lumber Beams Strengthened with Composite Materials

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Transformed Cross Section Method

_{f}—modulus of elasticity of composite material in fibre (main) direction; E

_{d}—modulus of elasticity of laminated veneer lumber. The modulus of elasticity of the veneer and CFRP laminate was assumed based on experimental results of 14 GPa and 195 GPa, respectively. The modulus of elasticity of the CFRP sheet was assumed according to the data declared by the material manufacturer—265 GPa [36].

_{z}was determined separately for the reinforcement glued to the sidewalls (assuming a sheet thickness of 0.333 mm and carbon tape of 1.4 mm for the calculations) and glued to the underside (the width over which the reinforcement was applied), according to the formulas:

- for composite parts oriented in vertical fashion:

- for composite parts oriented in horizontal fashion:

_{f}—thickness of composite; b

_{f}—width of composite; n—number of composite layers.

#### 2.2. Methods

#### 2.2.1. Modulus of Rupture (MOR)

_{max}—maximum bending moment; z—distance from neutral axis to external compressed fibers; I

_{y}—moment of inertia of transformed or solid cross-section.

_{theo}carried by transformed cross-section was evaluated using reversed analysis based on Formula (4), assuming the maximum normal stress for the unreinforced LVL beam from the experimental test. For the calculations, a maximum normal stress in LVL equal to 57.8 MPa was assumed.

#### 2.2.2. Bending Stiffness EI

- A method based on the curvature of the beam during bending (conducted using experimental data);
- A method based on the characteristics of the equivalent cross-section using the elastic modulus of laminated veneer (conducted using simplified mathematical model);
- A method based on a formula that describes the deflection value of a beam loaded with two concentrated forces (conducted using experimental data).

_{1}of unreinforced and reinforced full-length beams was determined as the product of the bending moment at the center of the element span M and the radius of curvature ρ (Figure 4) in the plane of bending xy according to the formula [37]:

_{1}—bending stiffness; M—bending moment; ρ—radius of curvature. A constant value of curvature and bending moment for the analyzed part of beam was assumed.

_{l}—distance between point P1 and P2; f—displacement of point P2 from line drawn between points P1–P3.

_{2}was determined as the product of the moment of inertia of the equivalent cross-section Iy (or solid cross-section) and the modulus of elasticity of the veneer E

_{d}in bending according to the formula:

_{2}= E

_{d}· I

_{y},

_{d}—modulus of elasticity of LVL; I

_{y}—moment of inertia of transformed or solid cross-section.

_{3}was estimated using the transformed deflection formula for a beam subjected to four-point bending according to the formula:

#### 2.2.3. Position of Neutral Axis

## 3. Results

#### 3.1. Modulus of Rupture (MOR)

_{theo}and experimental maximum bending moment M for the tested beams are shown in Figure 6. The most significant values of MOR are recorded for the B series and the smallest for the D series beams. An inverted order can be seen in the case of the maximum bending moment of the cross-section—the smallest values are for the B and E series beams and the largest for the D series. There are several reasons for this: the incompatible reality assumption of linear stress distribution over the height of the cross-section, the way the composite reinforcement is distributed, and the degree of reinforcement of the cross-section. The stress distribution in the compression zone is bilinear and linear in the tension zone; the mathematical model assumed linear in both zones. The higher the degree of reinforcement, the higher the moment of inertia of the equivalent cross-section, and, thus, the lower the value of MOR—when in the calculations, the value of the bending moment between the different test series is similar. The distribution of sheets along the lateral surfaces causes a proportional thickening of the cross-section, which does not significantly translate into the value of the shift of the neutral axis of the cross-section.

#### 3.2. Bending Stiffness EI

_{1}as a function of the bending moment M from the beginning of the test until the maximum value of the bending moment is reached, determined according to method 1. The graphs describing the behavior of elements reinforced with composite materials carry the curve of the course of changes in stiffness for beam A3, representing the upper limit of values for unreinforced beams.

_{1}of reinforced beams is higher than that of reference beams. As the load increases, the bending stiffness decreases—the negative slope of the relationship EI

_{1}—M curves. Due to the nature of changes in the curvature of the relationship, “bending stiffness EI

_{1}—bending moment M” can be divided into three parts, representing, respectively, (Figure 8):

- Initial phase—rapid changes (decreases in stiffness) associated with stabilizing beams in the test stand. Typically, this phase lasts up to a load approximately equal to 5 kN;
- Middle phase—relative linear behavior, without significant changes in the curvature course;
- Final phase—begins at the point of change in the slope of the curve in the final stage of the test. The difference in slope is related to the nature of the failure and the accompanying change in stiffness, for example, due to crack propagation in the compression zone. This phase does not occur for beams whose failure occurs suddenly and is caused by exceeding the strength of the veneer in the tensile area, as is the case for unreinforced beams.

#### 3.3. Neutral Axis Position c

## 4. Discussion

## 5. Conclusions

- The suitability of using the equivalent cross-section method to estimate the cross-sectional capacity of laminated veneers reinforced with fiber composites has not been confirmed. This is related to the assumption of a linear distribution of stresses in the compression and tension zones. In contrast, the actual distribution is linear in the tension zone and nonlinear in the compression zone. The degree of reinforcement and the way the composite reinforcement is redistributed between the compression and tensile zones greatly influence the MOR value;
- A high correspondence was obtained between the average values of EI bending stiffness estimated according to the method of equivalent characteristics (method 2) and the values derived from experimental tests (method 1). In the case of reinforcement with CFRP sheets and CFRP tapes glued to the external surface, the difference slightly exceeds 1%, and in the case of tapes glued into the grooves, less than 5%. The most significant discrepancies are recorded for reference beams—more than 7%. Of the three methods discussed for estimating the bending stiffness EI, method 3 (based on the formula for deflection at the center of the beam) differs the most from the others;
- Changes in the shape of the curves describing changes in bending stiffness as well as the position of the neutral axis can be divided into three zones: initial (stabilization), middle (constant work), and final (decrease in stiffness). The final phase occurs only in the case of reinforced beams; the failure initiated in the compression zone—a significant reduction in bending stiffness. The equivalent cross-section most accurately describes the position of the neutral axis at failure for beams reinforced with sheets glued to the outer surfaces and CFRP laminates glued into the grooves. Beams reinforced with laminates glued to the bottom surface are the worst in this respect;
- For the reinforcement simulation, using carbon sheets with an ultra-high modulus of elasticity proves the most beneficial due to the increased bending stiffness. In contrast, the least beneficial is the use of glass sheets. As the stiffness of the reinforcement and the number of sheets used increase, the effectiveness of the reinforcement increases.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Tested series: unreinforced beam (A series); beam reinforced with one layer of CFRP sheet (B series); beam reinforced with two layers of CFRP sheet bonded in tension zone (C series); beam reinforced with two layers of CFRP sheet covering entire side surface with overlap in tension zone (D series); beam reinforced with two CFRP laminate inserts bonded into pre-drilled grooves (E series); beam strengthen with CFRP laminate bonded to bottom side (F series) [33].

**Figure 2.**Selected cross-section cut out of tested beams: (

**a**) unreinforced beam (A series); (

**b**) beam reinforced with one layer of CFRP sheet; (

**c**) beam reinforced with two layers of CFRP sheet bonded in tension zone; (

**d**) beam reinforced with two layers of CFRP sheet covering entire side surface with overlap in tension zone (D series); (

**e**) beam reinforced with two CFRP laminate inserts bonded into pre-drilled grooves (E series); (

**f**) beam strengthen with CFRP laminate bonded to bottom side (F series).

**Figure 3.**Transformed cross-section of reinforced beams: (

**a**) beam reinforced with one layer of CFRP sheet (B series); (

**b**) beam reinforced with two layers of CFRP sheet bonded one on another (C series); (

**c**) beam reinforced with two layers of CFRP sheet (D series); (

**d**) beam reinforced with two CFRP laminate inserts bonded into pre-drilled grooves (E series); (

**e**) beam reinforced with CFRP laminate bonded to bottom side (F series).

**Figure 6.**Average values of: (

**a**) modulus of rupture (MOR); (

**b**) experimental bending moment M; (

**c**) theoretical bending moment M

_{theo}.

**Figure 7.**Bending stiffness EI

_{1}versus bending moment M curves: (

**a**) A series; (

**b**) B series; (

**c**) C series; (

**d**) D series; (

**e**) E series; (

**f**) F series.

**Figure 10.**Changes in position of neutral axis for beam: (

**a**) A1; (

**b**) B1; (

**c**) C1; (

**d**) D1; (

**e**) E1; (

**f**) F1.

**Table 1.**Relative differences between the values calculated by methods 2 and 3 in respect to method 1.

Series | A | B | C | D | E | F |
---|---|---|---|---|---|---|

Method 2 | −7.57% | −1.25% | 0.24% | 1.05% | −4.97% | −1.14% |

Method 3 | −19.56% | −14.60% | −17.15% | −22.24% | −17.69% | −16.25% |

Series | Position of Neutral Axis c [cm] | |
---|---|---|

Based on Strain Distribution | Transformed Cross-Section | |

A | 0.22 | 0.00 |

B | −0.62 | −0.83 (−25%) |

C | −1.12 | −1.39 (−19%) |

D | −0.87 | −0.85 (2%) |

E | −0.44 | −0.67 (−33%) |

F | −2.04 | −1.01 (102%) |

Sheet Type | AFRP Sheet S&P A-Sheet 120 | GFRP Sheet S&P G-Sheet E 90/10B | CFRP UHM Sheet S&P C-Sheet 640 |
---|---|---|---|

Modulus of elasticity E_{f} [GPa] | 120 | 73 | 640 |

Tensile strength f_{t,f} [MPa] | 2900 | 3400 | 2600 |

Density ρ_{f} [kg/m^{3}] | 1450 | 2600 | 2120 |

Elongation at rupture ε_{f} [%] | 2.5 | 4.5 | 0.4 |

Thickness for dimensioning t_{f} [mm] | 0.200 | 0.308 | 0.189 |

**Table 4.**Predicted values of bending stiffness and position of neutral axis for reinforced LVL beams.

Sheet Type | AFRP Sheet S&P A-Sheet 120 | GFRP Sheet S&P G-Sheet E 90/10B | CFRP UHM Sheet S&P C-Sheet 640 | ||||||
---|---|---|---|---|---|---|---|---|---|

Number of Layers | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |

Bending stiffness EI_{2} [kNm^{2}] | 485 | 556 | 625 | 481 | 550 | 618 | 742 | 1051 | 1354 |

Position of neutral axis c [cm] | −0.28 | −0.54 | −0.75 | −0.26 | −0.51 | −0.72 | −1.06 | −1.55 | −1.83 |

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**MDPI and ACS Style**

Bakalarz, M.M.; Kossakowski, P.G.
Application of Transformed Cross-Section Method for Analytical Analysis of Laminated Veneer Lumber Beams Strengthened with Composite Materials. *Fibers* **2023**, *11*, 24.
https://doi.org/10.3390/fib11030024

**AMA Style**

Bakalarz MM, Kossakowski PG.
Application of Transformed Cross-Section Method for Analytical Analysis of Laminated Veneer Lumber Beams Strengthened with Composite Materials. *Fibers*. 2023; 11(3):24.
https://doi.org/10.3390/fib11030024

**Chicago/Turabian Style**

Bakalarz, Michał Marcin, and Paweł Grzegorz Kossakowski.
2023. "Application of Transformed Cross-Section Method for Analytical Analysis of Laminated Veneer Lumber Beams Strengthened with Composite Materials" *Fibers* 11, no. 3: 24.
https://doi.org/10.3390/fib11030024