Proceedings e report - Firenze University Press

3. Methods and models
WOOD SCIENCE FOR CONSERVATION OF CULTURAL HERITAGE
For the comprehensive analysis of the construction of historic pianofortes, the determination of the
strength properties of the component, the loading due to stringing as well as the distribution of forces
in the structure are necessary regarding climate and humidity fluctuations plus known damages. Due
to the geometrically complex structure and the nonlinear material behaviour, numerical methods of
structural analysis are required.
One common approach of numerical simulation is the Finite-Element-Method (FEM) used in many
fields in engineering and material science. The FEM is the main simulation-tool of modern structural
analysis. Using the FEM, profound knowledge on the mechanical properties and predictions of the
structural behaviour can be obtained without destroying the construction by experiments.
The basic idea of the FEM is the division of a complex not analytically calculable structure into
discrete finite elements. The load-deformation-dependency of element-types like solids, plates and
bars can be described easily basing on analytical relations, energy principles and suitable material
models. By assembling the finite elements, the numerical calculation of stresses, strains and
deformations of the whole structure are possible, considering boundary conditions (forces and
constraints) and continuity conditions on the element border. In the case of a load-dependent forcedisplacement-relationship
due to e.g. nonlinear material behaviour of creeping and cracking, an
iterative-incremental solution algorithm (e.g. Newton-Raphson scheme) is carried out. All
mathematical functions of the FEM are formulated consistently in vector and matrix notation.
With respect to the constitutive description of wooden material, models are introduced, which
represent the moisture and temperature-depending anisotropic elastic material behaviour, ductile
failure due to compression loading and brittle failure and cracking due to tension and shear loading.
These models are already used for applications like simulation of the load-bearing behaviour of wood
connections and components. An extension of the material formulations will be obtained by finding
and modifying the sets of engineering input parameters for the wood species in construction of historic
pianofortes. The material models are classified by the elastic, the ductile and the brittle formulation. A
tensor notation is used for the following mathematical formulas to support more ready understanding
in combination with a short syntax.
3.1. Elasticity
The cylindrical anisotropic elastic behaviour of wood is based on the strain-energy-density function:
1
= E : C : E
2
Ψ (1)
In this formulation, the elasticity tensor C includes the Young’s moduli E r , E t , E l , the shear moduli
G rt , G tl , G rl and the Poisson’s ratios v rt , v tl and v rl , where the indices r, t and l stand for the
cylindrical material directions of wood (see Fig. 4). With the relation E i /E j = v ji /v ij , the stress tensor:
is determined.
∂Ψ
S = 2 = C : E
∂E
Fig. 4. Material directions of wood
205
(2)