Force-displacement relationship

The non-linearity of the force-displacement relationship between a tip and a sample is sketched in Fig. 13.3(a). The force is a non-linear function of the tip-sample displacement, and depends on whether approach or retraction is under way. If the tip and the sample are well separated, at the far right on the graph, there is negligible interaction. During the approach, from right to left on the graph, initially there is an attractive interaction. At closer approach the force becomes repulsive. When reversing the displacement, the tip and the surface adhere until the contact is broken at a certain pull-off distance, which... 

Damrau, E., Normand, M.D., and Peleg, M. (1997). Effect of resolution on the apparent fractal dimension of jagged force-displacement relationships and other irregular signatures. 

Figure 7.1 Sketch to show how the so-called geometric factors act as an interface between the stress-strain and force-displacement relationships.

Gruben, K.G., Halperin, H.R., Popel, A.S., and Tsithk, J.E. Canine sternal force-displacement relationship during cardiopulmonary resuscitation. IEEE Trans. Biomed. Eng., 46, 788-796, 1999. 

Let us consider the force-displacement relationship for the two atoms in Fig. 2.3 in more detail. First, note that continuous function and has a minimum value when u=0, implying that dequilibrium spacing. As < ) is continuous, one can expand this function around the equilibrium position as a Taylor series, i.e.,... 

Fig. 2.35 Test for force-displacement relationship. Measuring the force with zero displacement (a) and with a displacement s (b). Reprinted from [Lee et al. (2005)].

The developed bimorph beam model of IPMC was validated using the finite element method (FEM) and the used software was MSC/NASTRAN. As the software does not directly support the electromechanical coupling, the thermal analogy technique as described in [Lim et al. (2005) Taleghani and Campbell (1999)] was used. The simulated versus measured force-displacement relationship of an IPMC actuator is shown in Fig. 2.39. The relative errors for A = 0 between the calculated values and the measured data for 2V and 3V are 2.8% and 3.7%, respectively. The equivalent Young s moduli estimated from the equivalent beam model and the equivalent bimorph beam model are 1.01 GPa and 1.133-1.158 GPa, respectively, which are very close. However, the values from the equivalent beam model... 

The stiffness properties k( and force-displacement relationships of the uniaxial elements are defined according to constitutive stress-strain relationships implemented in the model for concrete and steel (Fig. 20.2) and the tributary area assigned to each uniaxial element. The reinforcing steel stress-strain behavior implemented in the wall model is the well-known nonlinear relationship of Menegotto and Pinto (1973) (Fig. 20.2b). The hysteretic constitutive relation developed by Chang and Mander (1994) (Fig. 20.2a) is used as the basis for the relation implemented for concrete because it is a general model that provides the... 

In FEA, a problem is nonlinear if the force—displacement relationship depends on the current state of the displacement, force, and stress—strain relations. NonUnearity in a problem can be classed as material nonlinearity, geometric nonlinearity, and bound conditions. 

Figure 3 The moment-rotation relationship for a plastic hinge, and the force-displacement relationship of a diagonal representing masonry infill. C represents the cracking of masonry infill,Y the yielding of reinforcement, M the maximum force or moment, and NC the near collapse limit state.

Viscoelastic (VE) Dampers A wide class of energy dissipation devices whose force-displacement relationship has viscoelastic mechanical properties. In recent decades, VE dampers have been widely used to reduce vibration in civil engineering structures caused by various excitations. 

The force-displacement relationship for each element is given by... 

In order to obtain the contact forces, the force-displacement relationship is indispensable. Here, the force-displacement relationship between two particles is assumed separately for normal and tangential components by the model shown in Fig. 1 using two rigid spheres, a spring, and a dashpot. In order to take account of the friction between spheres, a slider element is used for tangential force. A Coulomb-type friction law is adopted in this simulation. 

For two particles interacting cohesively, we set up four types of degrees of freedom normal, sliding, bending, and torsional displacements [46, 57]. To keep the model simple, force-displacement relationships are used for hindering motion for these degrees of freedom, with a spring constant associated with each of them. The model further assumes monodisperse hard sphere particles. These can be modeled... 

There are no exact instructions as to how to determine the parameters of the SDOF model since the mass, the damping c, and the stiffness all depend on the assumed displacement shape. In order to obtain a practical solution to Eq. 10, the analyst has to determine the force-displacement relationship of the SDOF model taking into account certain hysteretic rules, which are often assumed to be similar to those used for the structural components. However, the displacement of the SDOF model can be simply determined according to Eq. 9. Some further explanation is needed in order to be able to better understand the background for the determination of the forces of the SDOF model from the pushover curve. In general, the product of the stiffness A and m represents a force in the spring of the SDOF model ... 

Equation 10 can be solved by numerical integration which is implemented in conventional software for response history analysis. However, the solution depends, in general, on the ground motion and assumptions adopted for the determination of the parameters of the SDOF model, i.e., the mass w, the damping model, the force-displacement relationship F — m, and the hys-teretic rules. 

Idealize the pushover curve F — u, using a bilinear or multi-linear shape or the shape of any other curve which can be used to simulate theF — M force-displacement relationship of the SDOF model. 

Determination of the SDOF models. This step requires the transformadmi of top displacement and base shear to the displacement (Eq. 9) and force (Eq. 14 or Eq. 18) of the SDOE model. The force-displacement relationship of the SDOE model should be multilinear. Negative post-capping stifliiess can also be applied in order to estimate the collapse capacity. It is usually assumed that the hysteretic behavior is the same as that prescribed for the components of the stmctural models. The mass of the SDOE model should be determined according to Eq. 7a, whereas the damping can be proportional to mass and/or stif iess (Eq. 24). 

Step 7. The pushover curves were idealized by means of a trilinear force-displacement relationship, also taking into account negative post-capping stiffness. Transformation of the... 

In order to conduct nonlinear analysis, the force-displacement relationships corresponding to the equivalent strut model must be adequately defined. The modeling of hysteretic behavior increases not only the computational complexity but also the uncertainties of the problem. 

If the target displacement d, is quite different from the displacement used to determine the idealized elastic-perfectly plastic force-displacement relationship, an iterative procedure may be applied. 

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