and is ignored [2]. This presliding results in a hysteretic damping from elastic deformations, but is different from that due to bending deformations. The presliding hysteretic damping is purely due to shear strains and accounts for even a smaller percentage of the total damping as compared to that from the bending hysteresis. With the presence of macroslip friction, in debonded layered composites, the system behavies bilinearly [3,4]. This can be characterized by a single point discontinuous softening load-deflection curve comprised of an initial modulus and a secant modulus as shown in Figure 2. Fig. 2. Layer behaviour for (a) stiction and sliding bilinear representation and (b) frictional and shear stresses To define the moduli of the bilinear behaviour, the system needs to be separated into two categories: stiction and sliding. When stiction occurs at the interfaces, the layers are locked together and act as one. The layered system can then be treated as a single layer following Equivalent Single Layer (ESL) theories. When sliding occurs, the layers act as separate individual layers and motion for each layer must be resolved with respect to the adjacent layer. To calculate the force-deflections for the bilinear curve, the stiction portion will first be calculated. To initiate this, the interfacial forces along the beam will need to be known. If no compression is present on the beams, then small load steps will need to be made to determine if the nodal forces are great enough to overcome the stiction or not. When a cantilever beam has a concentrated tip force, the moment along the length of the beam is linear. The maximum moment is at the fixed-end and the moment approaches zero where the force is applied at the free-end. When a cantilever beam has a uniformly distributed load, or a triangularly distributed load, the moment along the length of the beam is nonlinear from the fixed-end to the free-end. This nonlinear moment would indicate that the tip force could be replaced with a parabolic distributed load to represent the same behaviour as the concentrated tip force [4]. To find this parabolic function, the parabolic load can be discretized into a multilinear curve comprised of multiple triangular loads. If the first triangular load is considered, the force along the length of the beam can be deduced by evaluating the constant distributed force coefficient for a given length of the beam through the use of proportionality since there is a constant slope. The nodal forces can then be expressed as To calculate the distributed force coefficient β, each sub-coefficient must firstly be considered for each node. These subcoefficients are then summed and the product of their inverse, and the tip deflection from Equation 3, are used to arrive at Once the vertical nodal forces are calculated, they must be resolved into vector components. Each node within the beam will have a different temporal x and y coordinate that will behave nonlinearly. Ascertaining the force vectors is accomplished through kinematically tracking each of the nodes to estimate their final position. Assuming that the interface between each layer remains frictionless, for the time being, the elastic curve, for the system can be tracked 2 2 x L Fv β = ( ) ff eff n n E Ie x L x 6 3 4 − = γ 1 ( ) 2 − Σ = n y L L β δ γ (3) (4) (5) F δy stiction sliding Ffr, τmax δy stiction sliding (a) (b) 291
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