# The Rolling Shear Influence on the Out-of-Plane Behavior of CLT Panels: A Comparative Analysis

^{*}

## Abstract

**:**

## 1. Introduction

_{R}) involves significant shear deformations in the layers that are orthogonally oriented with respect to the direction of the tangential stresses (here named transverse layers) (Figure 1). As a result, out-of-plane deflections and vibrations increase in CLT floors under service conditions, producing discomfort for the users. Low values of perpendicular-to-grain shear strength have been measured by experimental tests; then, the rolling shear failure mode in the transverse layers has been frequently observed [5,6]. In Eurocode 5 [7], a characteristic value of rolling shear strength equal to 1.0 MPa is a reliable value suggested for wood, independently from its species and class strength.

## 2. Problem Statement

#### 2.1. Rolling Shear Phenomenon

_{R}—significant shear deformations arise in wooden elements [6,8,9,10,11,12]. There are few structural cases of interest in which this phenomenon could occur, probably the case of the CLT panels working as plates loaded out-of-plane is the most frequent in the structural field. Under this load condition, the transverse layers of boards (i.e., whose grains direction is perpendicular with respect to the plane of bending) are subjected to perpendicular to grain tangential stresses. Shear deformation which arises in these layers could be not negligible and its magnitude growth the maximum deflections (and vibrations), modifying the internal stresses of CLT floors.

_{R}) cannot be considered an intrinsic property of material: its value not only depends on wood density and annual rings width, but also on the cutting pattern of the boards (i.e., location of pith), size, and geometry of boards cross-section (width-to-thickness of the lamination ratio), as thoroughly highlighted in [12]. In more detail, for CLT panels, it has been demonstrated that, for decreasing width-to-thickness of the lamination ratios (below to 1.33), the rolling shear decreases disproportionally; consequently, its effect is greater for higher levels of rolling shear moduli [20].

_{R}=65 MPa and a five-percentile equal to 54 MPa are suggested to be used by the Italian Technical Document CNR DT 206-R1/2018 [16] for designing glue-laminated timber elements, which also corresponds to 1/10 of the elastic tangential modulus in the longitudinal direction.

#### 2.2. Sectional Behavior of CLT Panels

_{R}). In Figure 3b, the normal and tangential stress distributions obtained in this intermediate case are also schematically represented.

## 3. Methodology

#### 3.1. IR Case

_{90}) with respect to those in parallel direction E

_{0}(E

_{90}⋍ E

_{0}/30), their contribution has been neglected in the calculations herein performed. Instead, a is the distance between the center of gravity of the i-th layer with respect to the center of gravity of the whole cross-section and A the cross-sectional area of the single layer of the board (Figure 3).

#### 3.2. Modified γ-Method

_{0}has been used for the layers stressed in parallel to the grains direction, while E

_{90}for the ones stressed in the orthogonal to grains direction. The coefficient γ is a weighted coefficient depending on both geometrical and mechanical properties of the connection system. It can be evaluated as follows:

_{R}·α, while the relative displacement between two consecutive layers is δ = h·α; the angle α, and the height h are represented in Figure 4. Then, by substituting τ and δ in Equation (4), the shear stiffness to used in Equation (3) becomes:

#### 3.3. Shear Analogy Method

_{A}and (EI)

_{B}being the flexural stiffness of beams A and B, respectively, written as follows:

#### 3.4. Deflection

_{eff}is evaluted as follows:

_{R}has been considered for the layer stresses in the orthogonal to grains direction, while the shear modulus G for those stressed parallel to the grains direction.

_{inst}) and the creep (f

_{creep}) contribution:

_{def}, which takes into account the service class of the structural element [7,16].

## 4. Case Studies and Finite Element Modelling

_{k}= 1.76 kN/m

^{2}and live loads Q

_{k}= 2.00 kN/m

^{2}.

_{0}and G) have been assigned to shell elements of the first, third, and fifth layer, while the orthogonal-to-grains elastic moduli (E

_{90}and G

_{R}) to the second and fourth layers. The multi layered modeling allows any number of layers to be defined in the thickness direction, each with an independent location, thickness, behavior, and material elastic properties. The multi-layered shell model is based on the elastic plate theory, where, for bending a Mindlin–Reissner formulation, is used, which always includes transverse shear deformation; thus, the rolling shear transverse deformation is taken into account [10].

## 5. Discussion of the Results

_{inst}) evaluated by applying the three considered analytical procedures, for each considered L/H slenderness ratio, is represented. It should be noted that: (a) deflections calculated accounting for the rolling shear effect (γ-method, SAM, and 2D shell- models) are greater than the ones provided by the IR case, especially in the case of low H/L ratios (stocky panels), thus confirming the crucial role played by the transverse layers on the total deflection; (b) both modified γ-method and SAM restitute similar values of deflections, comparable with the ones reinstituted by 2D-shell numerical models.

_{90}) with respect to the Young’s modulus of the longitudinal layers (E

_{0}). Instead, for the case IR, the normal stresses are exactly equal to zero because of the contribution of the transverse layers being completely neglected.

- (a)
- a good matching between modified γ-method and SAM in terms of both normal and tangential stress distributions has been obtained, independently from L/H ratio;
- (b)
- a substantial equality correspondence of the normal stresses obtained with analytical and numerical methods can be noted, already for H/L greater than 20;
- (c)
- a greater discrepancy of the tangential stress distributions between analytical and numerical approaches results in the case of low H/L ratios (L/H < 20), probably due to the fact that the beam theory—on which the analytical methods are essentially based—tends to fall in the case of shear-dominated elements (stocky elements). Instead, a good agreement among the tangential stress values obtained from the analytical methods, also for low H/L ratios, results.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Normal (σ) and tangential (τ) stress distribution in CLT panels: (

**a**) infinitely rigid transverse layers, (

**b**) intermediate deformable transverse layers, and (

**c**) infinitely deformable transverse layers.

**Figure 9.**Normal and tangential stresses diagrams for different H/L ratios. (

**a**) L/H=10; (

**b**) L/H = 20; (

**c**) L/H = 30; (

**d**) L/H = 40.

Properties | [MPa] |
---|---|

Young’s modulus par. to grain E_{0} | 11600 |

Young’s modulus orth. to grain E_{90} | 390 |

Shear modulus G | 720 |

Rolling shear modulus G_{R} | 72 |

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**MDPI and ACS Style**

Sandoli, A.; Calderoni, B.
The Rolling Shear Influence on the Out-of-Plane Behavior of CLT Panels: A Comparative Analysis. *Buildings* **2020**, *10*, 42.
https://doi.org/10.3390/buildings10030042

**AMA Style**

Sandoli A, Calderoni B.
The Rolling Shear Influence on the Out-of-Plane Behavior of CLT Panels: A Comparative Analysis. *Buildings*. 2020; 10(3):42.
https://doi.org/10.3390/buildings10030042

**Chicago/Turabian Style**

Sandoli, Antonio, and Bruno Calderoni.
2020. "The Rolling Shear Influence on the Out-of-Plane Behavior of CLT Panels: A Comparative Analysis" *Buildings* 10, no. 3: 42.
https://doi.org/10.3390/buildings10030042