PSI - Issue 2_B

Ershad Darvishi et al. / Procedia Structural Integrity 2 (2016) 2750–2756 Author name / Structural Integrity Procedia 00 (2016) 000–000

2751

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intended to check the performance of such structures under horizontal loading (Wanninger and Frangi 2014). The prediction of wood properties within structures and buildings are studied (Sandak et al. 2015). To design a timber structure we could refer to European standards (EN1995-1-1) for designing and the assessment of existing structures (MKS EN1995-1-1). The linear and nonlinear analysis of several materials have numerically been studied (Di Cocco et al. 2015; Namdar and Feng 2014; Namdar et al. 2016). In this paper five timber structures have been modeled and subjected to the forcing frequency in different modes in order to analyze nonlinear horizontal displacement of column located at the corner of structure, to evaluate flexibility of timber structure and to realize the effects of forcing frequency on timber structure. Totally 15 modes of frequency were applied on all models. The load displacement curve was depicted, as a main research outcome.

Nomenclature r

function of time mass matrix damping matrix

M C K R ω

static-stiffness matrix

loads

excitation frequency amplitudes of the loads amplitudes of displacements dynamic-stiffness matrix

P u S

2. Theoretical concept To perform the spatial discretization, the finite element method was applied whose equations of motion were developed by assembling the elements matrices. It is included; [M]{ r� } + [C]{ r� } + [K]{r} = {R} (1) The vector {r} is a function of time, it contains the displacements of all unconstrained degree of freedom of all nodes. The matrices [M], [C] and [K] represent the mass matrix, the damping matrix, and the static-stiffness matrix respectively which are constant for a linear system. The vector {R} denotes the prescribed loads, which are a

function of time, acting in the direction of the displacement in all nodes. For an harmonically varying load with the excitation frequency ω, {R} = {P} exp (iωt)

(2)

The response will be; {r} = {u} exp (iωt)

(3) The vectors {P} and {u} represent the (complex) amplitudes of the loads and displacements, respectively. The equations of motion (Eq. 1) are formulated as [S]{u}={P} (4) Where the dynamic-stiffness matrix [S] is specified as [S] = [K] + iω[C] – ω 2 [M] (Wolf, 1985) (5)