Xlam: evaluation of the correct stiffness matrix

Xlam or, more correctly, Cross Laminated Timber (CLT), is a material composed of a minimum of three layers of wooden boards, glued one perpendicular to the other.

Every element has different stiffness in the x or y direction of the reference axes, due to both the intrinsic orthotropy of timber and to the particular crossed-layers geometry.

To the present date, in the absence of a harmonised legislation regarding this material (the EN 16351 ‘Timber Structures – Cross Laminated Timber – Requirements’ has not yet been published in the Official Journal), every product must be covered by a specific ETA before being commercialized.

Cross-layered geometry of Xlam

The mechanical response of the material is deeply influenced by the typology of panel, so this aspect cannot be ignored during analysis.

The stiffness matrix is the element linking stresses to strains. In Finite Elements Modelling it is extremely important to define it carefully, in order to avoid mistakes leading to corrupt results.

Stiffness matrix scheme for an orthotropic material

The stiffness matrix of orthotropic materials is composed of 8 rows and 8 columns, is symmetrical and contains all the information regarding geometrical and mechanical features (E, G, J, A, and so on). Based on the loading conditions and the constraints imposed on the model, one or more of the elements composing the matrix define the panel’s strains. The matrix can be, therefore, divided in 4 parts:

• Elements defining flexural and torsional stiffness out of the panel’s plane:

D11 – flexural stiffness in x-direction;

D22 – flexural stiffness in y-direction;

D33 – torsional stiffness;

D12 – effect of transversal elongation on bending moments (depends on ν_xy and ν_yx);

D13 and D23 – assumed to be zero in CLT.

• Elements defining shear out of the panel’s plane:

D44 – shear stiffness for the v_x,z stress;

D55 – shear stiffness for the v_y,z stress;

D45 – assumed to be zero in CLT.

• Elements defining membrane stiffness in the panel’s plane:

D66 – stiffness to the elongation in x-direction;

D77 – stiffness to the elongation in y-direction;

D88 – shear stiffness for the n_xy stress;

D67 – effect of transversal elongation (depends on ν_xy and ν_yx);

D68 and D78 – assumed to be zero in CLT.

• Elements defining the panel’s eccentricity:

From D16 to D38 – they take the effects of eventual asymmetries on strains into account. In the case of CLT they are usually assumed to be zero, because the panel is built starting from single layers placed symmetrically in thickness and alignment.

Although some FEM softwares are provided with apps that automatically define stiffness matrixes, CLT is a very peculiar material. That is why we decided, in collaboration with the Engineering Departments of the Universities of Trento and Bologna, to analyse every single element of the matrix. Comparing the results obtained from softwares to the theoretical formulations and the experiments found in the Italian and international literature, we were able, after more than a year’s work, to build, test and validate an Excel file that can evaluate the stiffness matrix of every CLT panel, in the most detailed way.

Input data for the evaluation of CLT’s stiffness matrix

Given the element’s stratigraphy and the geometrical and mechanical characteristics of the single layers, the file generates all the elements of the matrix, that can be introduced in the FEM software. Such a result, besides being exact, helps saving a whole lot of time during modelling.

More information on the errors generated by a wrong or missing evaluation of the CLT’s stiffness matrix, can be found in the article ‘Xlam: Orthotropy and Modelling Failures’