# Evaluating Timoshenko Method for Analyzing CLT under Out-of-Plane Loading

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Models for Deflection Calculation in CLT Panels

#### 2.1. Timoshenko Method

_{i}= bh

_{i}.

_{i}and G

_{i}are the elastic modulus and the shear modulus in MPa of the i-th layer. b

_{i}, h

_{i}, z

_{i}and a are width of the panel, height of the i-th layer, distance of the neutral axis of the i-th layer from the neutral axis of the panel in mm, and the distance between the neutral axis of the outer layers, respectively. Shear form factor, k is determined by using Equation (3) [21].

^{3}).

#### 2.2. Shear Form Factor

^{2}was proposed by Mindlin [24] and shear form factor was proposed to be a function of Poisson’s ratio by Cowper [25].

## 3. Finite Element Modelling

#### 3.1. Material Model

_{R}and E

_{T}) and Poisson’s ratio (${\nu}_{RT}$ and ${\nu}_{TR}$) normal to each other in the radial–tangential plane becomes equal. Further, the shear moduli in the other two perpendicular planes become equal. The additional constraint can be expressed as

_{RT}), commonly referred to as the rolling shear modulus, is dependent on the Poisson’s ratio and the modulus of elasticity of that plane. It is worth noting, that determining rolling shear modulus using physical tests is easier than determining Poisson’s ratio.

#### 3.2. Geometry and Boundary Conditions

#### 3.3. Material Properties

_{L}) used in the study is slightly higher than that of C24 grade, which is representative of other species including hardwoods. CLT was modelled assuming timber being transversely isotropic, and the corresponding material properties are shown in Table 2. The rolling shear modulus (G

_{RT}) of the cross-layers in each model was varied between 50 MPa and 250 MPa with an increment of 50 MPa. The range of rolling shear modulus was chosen based on the values reported for Norway spruce (a softwood species) [32] and those for hardwood species [33]. The radial and tangential elastic moduli in the cross-layers of models were varied according to their relationship with rolling shear modulus and Poisson’s ratio as shown in Table 1. The Poisson’s ratio values were obtained from experimental research conducted on Norway spruce [34].

#### 3.4. Verification of the Numerical Modelling Technique

#### 3.5. Numerical Modelling of Glue Laminated Timber (GLT)

## 4. Reassessing the Suitability of Timoshenko Method for CLT Panels

#### 4.1. Comparison between Timoshenko Method and CLT Panels

#### 4.2. Parameters Affecting the Performance of Timoshenko Method

#### 4.3. Effect of Rolling Shear Modulus

#### 4.4. Effect of Length and Depth

## 5. Modified Shear Form Factor

_{eff}the corresponding bending stiffness calculated using Equation (1) along with respective length and uniformly distributed load (q) is used in Equation (4). Using the estimated shear stiffness (GA)

_{eff}in Equation (2), the form factor k

_{m}for the model is estimated

#### 5.1. Modification Co-Efficient for Shear Form Factor

_{m}) to the shear correction factor obtained from the Timoshenko method (k), as shown below.

#### 5.2. Formulating Equation for Modification Co-Efficient

#### 5.3. Relationship between Modification Factor, Length and Rolling Shear Modulus

_{RT}is the rolling shear modulus, and L is the span length.

_{mod}, is the modified equation for mid-span deflection of three-layered CLT panel; ${w}_{s}=\frac{5}{384}\frac{q{L}^{4}}{{\left(EI\right)}_{eff}}$, is the shear free mid-span deflection; c, is the modification co-efficient factor evaluated by Equation (10); L, is the span length of the panel; (EI)

_{eff}, is the effective bending stiffness evaluated by Equation (1); and (GA)

_{eff}, is the effective shear stiffness evaluated by Equation (2).

#### 5.4. Validation of the Proposed Factor

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Symbols and Abbreviations

LVL | Laminated veneer lumber |

GLT | Glued laminated timber |

CLT | Cross-laminated timber |

LSL | Laminated strand veneer |

k | Shear form factor |

UDL | Uniformly distributed loading |

E_{L} | Longitudinal elastic modulus |

E_{R} | Radial elastic modulus |

E_{T} | Tangential elastic modulus |

${\nu}_{LT}$ | Poisson’s ratio in longitudinal–tangential plane |

${\nu}_{LR}$ | Poisson’s ratio in longitudinal–radial plane |

${\nu}_{RT}$ | Poisson’s ratio in radial–tangential plane |

G_{LT} | Shear modulus in longitudinal–tangential plane |

G_{LR} | Shear modulus in longitudinal–radial plane |

G_{RT} | Shear modulus in radial–tangential plane |

L/D ratio | Span-to-depth ratio |

${\left(EI\right)}_{eff}$ | Effective bending stiffness of the panel |

${\left(GA\right)}_{eff}$ | Effective shear stiffness of the panel |

q | Uniformly distributed load |

L | Span of the panel |

w | Mid-span deflection of the panel |

${k}_{m}$ | Shear form factor estimated from FE models |

c | Modification co-efficient |

w_{s} | Mid-span shear free deflection |

w_{m} | Mid-span deflection from modified Timoshenko method |

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**Figure 2.**(

**a**) Stresses relevant to an orthotropic material, (

**b**) longitudinal direction for timber, and (

**c**) cross-section showing radial and tangential directions in radial–tangential plane (cross-section).

**Figure 4.**(

**a**) The experimental test setup and (

**b**) illustration of geometric dimensions of the test specimen (all measurements in mm).

**Figure 5.**The (

**a**) FE model with rolling shear stress (shear stress radial–tangential plane) profile and (

**b**) load-displacement results from the experimental tests and FE model.

**Figure 7.**Comparison between FE predictions for CLT deflections vs. analytical predictions using Timoshenko method: (

**a**) 90 mm thick CLT panels, (

**b**) 120 mm thick CLT panels, (

**c**) 150 mm thick CLT panels and (

**d**) 180 mm thick CLT panels

**Figure 8.**Deviations of analytical vs. FE shear stiffness for the considered CLT panels: (

**a**) 90 mm thick CLT panels, (

**b**) 120 mm thick CLT panels, (

**c**) 150 mm thick CLT panels and (

**d**) 180 mm thick CLT panels.

**Figure 9.**Variation of (

**a**) deflection and (

**b**) shear stiffness of the section of panel with length to depth ratio of 11.11.

**Figure 10.**Deviation of (

**a**) deflection and (

**b**) shear stiffness against changing span-to-depth ratio of Timoshenko method from FE models.

**Figure 11.**Variation of shear stiffness with changing length for panels of thickness 90 mm and 180 mm for cross-layer characterized by 250 MPa rolling shear modulus.

**Figure 12.**Deviation of shear stiffness of Timoshenko theory to that estimated from Numerical model for panel thickness of 90 mm and 180 mm where cross-layers are characterized by 250 MPa rolling shear modulus.

**Figure 14.**Variation of the proposed modification co-efficient with (

**a**) Rolling shear modulus, (

**b**) Length and (

**c**) Depth.

**Figure 15.**Modification co-efficient against length. (

**a**) 90 mm thick CLT panels, (

**b**) 120 mm thick CLT panels, (

**c**) 150 mm thick CLT panels and (

**d**) 180 mm thick CLT panels.

**Figure 16.**Illustration of four-point bending set up [36].

Elastic Property | Direction | Transversely Isotropic Material |
---|---|---|

Elastic Modulus | Longitudinal direction | E_{L} |

Radial direction | E_{R} = E_{T} | |

Tangential direction | ||

Poisson’s ratio | Characterizing tangential normal strain to longitudinal normal strain | ${\nu}_{LT}$ = ${\nu}_{LR}$ |

Characterizing radial normal strain to longitudinal normal strain | ||

Characterizing normal tangential strain to radial strain | ${\nu}_{RT}$ = ${\nu}_{TR}$ | |

Shear modulus | Longitudinal–tangential plane | G_{LT} = G_{LR} |

Longitudinal–radial plane | ||

Radial–tangential plane | ${G}_{RT}=\frac{{E}_{R}}{2\left(1+{\nu}_{RT}\right)}$ |

Characteristic Mechanical Property | Elastic Modulus (MPa) | Poisson’s Ratio | Shear Modulus (MPa) | |||||
---|---|---|---|---|---|---|---|---|

E_{L} | E_{R}/E_{T} | ${\mathit{\nu}}_{\mathit{L}\mathit{R}}$ | ${\mathit{\nu}}_{\mathit{L}\mathit{T}}$ | ${\mathit{\nu}}_{\mathit{R}\mathit{T}}$ | G_{LR} | G_{LT} | G_{RT} | |

Longitudinal layer | 11,600 | 370 | 0.014 | 0.014 | 0.21 | 690 | 690 | 50 |

Cross-layer | 11,600 | Constrained * | 0.014 | 0.014 | 0.21 | 690 | 690 | 50–250 |

_{RT}and ${\nu}_{RT}$ as shown in Table 1.

Characteristic Mechanical Property | Elastic Modulus (MPa) | Poisson’s Ratio | Shear Modulus (MPa) | |||||
---|---|---|---|---|---|---|---|---|

E_{L} | E_{R}/E_{T} | ${\mathit{\nu}}_{\mathit{L}\mathit{R}}$ | ${\mathit{\nu}}_{\mathit{L}\mathit{T}}$ | ${\mathit{\nu}}_{\mathit{R}\mathit{T}}$ | G_{LR} | G_{LT} | G_{RT} | |

Longitudinal layer | 11,000 | 370 | 0.014 | 0.014 | 0.21 | 690 | 690 | 50 |

Engineered Wood Product | Elastic Modulus (MPa) | Shear Modulus (MPa) | ||||
---|---|---|---|---|---|---|

Layer 1 E _{1} | Layer 2 E _{2} | Layer 3 E _{3} | Layer 1 G _{1} | Layer 2 G _{2} | Layer 3 G _{3} | |

CLT | 11,600 | 370 | 11,600 | 690 | 50 | 690 |

GLT | 11,600 | 11,600 | 11,600 | 690 | 690 | 690 |

Dimension (L × W × D) (mm × mm × mm) | Deflection (Timoshenko) (mm) | Deflection (Model) (mm) | Error (%) |
---|---|---|---|

1026 × 200 × 60 | 0.57 | 0.55 | 3.5 |

1800 × 200 × 120 | 0.68 | 0.68 | 0 |

1800 × 200 × 90 | 1.57 | 1.54 | 1.9 |

2500 × 200 × 120 | 2.45 | 2.40 | 2 |

Species | E_{L} (MPa) | E_{R}/E_{T} (MPa) | G_{LT}/G_{LR} (MPa) | G_{RT} (MPa) | Reference |
---|---|---|---|---|---|

Irish sitka (Softwood) | 9900 | 772.2 | 633.6 | 61.88 ^{3} | [29] |

White pine (Softwood) | 8900 | 694.2 | 462.8 | 44.5 ^{3} | [29] |

Eucalyptus urophylla (Hardwood) | 13,391.7 | 640.70 | 897.24 ^{1} | 267.83 ^{2} | [37] |

Red Maple (Hardwood) | 11,300 | 1582 | 1502 | 203.4 ^{2} | [29] |

White Ash (Hardwood) | 12,000 | 1536 | 1308 | 216 ^{2} | [29] |

Species | Geometry (l × w × h) (mm) | a (mm) | EI_{global} (×10^{10} N/mm^{2}) | Experimental vs. Analytical Predictions | |
---|---|---|---|---|---|

δ_{TM}/δ_{Test} * | δ_{mTM}/δ_{Test} | ||||

Irish Sitka [40] | 1440 × 270 × 60 | 540 | 3.69 | 0.939 | 0.963 |

1728 × 288 × 72 | 648 | 6.64 | 0.917 | 0.941 | |

2880 × 584 × 120 | 1080 | 63.61 | 0.936 | 0.962 | |

1440 × 584 × 120 | 360 | 39.44 | 0.806 | 0.922 | |

Eucalyptus urophylla [40] | 1620 × 305 × 54 | 540 | 4.59 | 0.786 | 0.938 |

White pine [38] | 2992.5 × 300 × 105 | 997.5 | 17.5 | 0.812 | 0.824 |

Red Maple [38] | 2992.5 × 300 × 105 | 997.5 | 28.6 | 0.944 | 0.960 |

White Ash [38] | 2992.5 × 300 × 105 | 997.5 | 30.6 | 0.952 | 0.970 |

_{Test}, Deflection obtained from experiment; δ

_{TM}, Deflection obtained from Timoshenko method; δ

_{mTM}, Deflection obtained from modified Timoshenko method.

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## Share and Cite

**MDPI and ACS Style**

Rahman, M.T.; Ashraf, M.; Ghabraie, K.; Subhani, M.
Evaluating Timoshenko Method for Analyzing CLT under Out-of-Plane Loading. *Buildings* **2020**, *10*, 184.
https://doi.org/10.3390/buildings10100184

**AMA Style**

Rahman MT, Ashraf M, Ghabraie K, Subhani M.
Evaluating Timoshenko Method for Analyzing CLT under Out-of-Plane Loading. *Buildings*. 2020; 10(10):184.
https://doi.org/10.3390/buildings10100184

**Chicago/Turabian Style**

Rahman, MD Tanvir, Mahmud Ashraf, Kazem Ghabraie, and Mahbube Subhani.
2020. "Evaluating Timoshenko Method for Analyzing CLT under Out-of-Plane Loading" *Buildings* 10, no. 10: 184.
https://doi.org/10.3390/buildings10100184