Geometric stiffness

Basic elastic and geometric stiffness properties of the individual supporting columns are synthesized into a stiffness matrix compatible with an axisymmetrical shell element by a series of transformations. These are to be used in conjunction with a finite element representation of the cooling tower, where the displacements are decomposed into Fourier... 

The current stiffness matrix is modified by considering the last yielded hinge identified at Stage (2) and it is returned to Stage (1) for the next pushover step. Note that second-order (P-Delta) effects can be readily considered in IRSA by adding the so-called geometric stiffness matrix to the first order stiffness matrix at each pushover step. 

K = elasto-plastic stiffness matrix as a function of the current state of plastic deformation Ka = initial stress geometric stiffness matrix. 

ISil = stress stiffness eontribution or geometric stiffness... 

The constitutive properties, geometric stiffness influences, and deformation-associated inertia effects are summarized in the matrix Psit), while the applied loads, piezoelectric coupling implications of the electric parameters, as well as initial state inertia effects, are joined in the vector p it)-... 

Inversion of the overall stiffness matrix P (f2) is necessary to find the wanted steady-state displacements, rotations, and electric potentials encapsulated in the vector Ug. Prerequisite is the determination of the initial internal loads contained in the geometric stiffness matrix G(i ) in accordance with Remark 9.2 and the considerations of Section 8.3. 

This latter technique is the one which is used in this paper, modified to make this method compatible with the 0(n) formulation. As given in Ref. [14], the element geometric stiffness is... 

Finite element codes, such as NASTRAN can compute the geometric stiffness associated with a prescribed distributed load by first calculating the associated element stresses, these element stresses then being used in the determination of the element geometric stiffness matrices as per Eq.(6). Finally, the element geometric stiffness matrices are assembled into the body geometric stiffness matrix. ... 

The overall geometric stiffness matrix for y is constructed from... 

The nonlinear force-based fiber element is also used to model the columns of the DBF. The beams and braces of the DBF are modeled using linear elastic truss elements. The gravity frames are idealized using the concept of a lean-on column, where an elastic beam-column element with geometric stiffness is used to model the lean-on column. The section properties of the lean-on column are obtained by taking the sum of the section properties of each gravity column within the tributary area (i.e., one quarter of the floor plan) of the MRF and the DBF. The MR dampers are modeled using the MNS MR damper model implemented into OpenSees by Chae et al. (2010). The MR damper is assumed to be located between the top of the brace and the adjacent beam-colunm joint, as shown in Fig. 8. The results reported in this paper are for MR dampers that are passive cOTitroUed with a constant current input of 2.5 A. Studies with the MR dampers in semi-active control mode are presented in Chae (2011). 

The effect of axial force acting through the relative transverse displacement of the member ends known as P-5 effect can be taken into account in the modeling by using the geometric stiffness matrix, [Kg], as ... 

In contrast, because of the spatially variable (inhomogeneous) nature of material in a composite stiffener, the bending stiffness cannot be separated into a material factor times a geometric term as in Equation (7.6). Instead, the composite stiffener bending stiffness is... 

In a mechanical system, the degrees of freedom indicate how many numbers are required to express its geometrical position at any instant. In machine-trains, the relationship of mass, stiffness, and damping is not the same in all directions. As a result, the rotating or dynamic elements within the machine move more in one direction than in another. A clear understanding of the degrees of freedom is important in that it has a direct impact on the vibration amplitudes generated by a machine or process system. 

There are different techniques that have been used for over a century to increase the modulus of elasticity of plastics. Orientation or the use of fillers and/or reinforcements such as RPs can modify the plastic. There is also the popular and extensively used approach of using geometrical design shapes that makes the best use of materials to improve stiffness even though it has a low modulus. Structural shapes that are applicable to all materials include shells, sandwich structures, and folded plate structures (Fig. 3-8). These widely used shapes employed include other shapes such as dimple sheet surfaces. They improve the flexural stiffness in one or more directions. 

From a mechanical point of view the polymer may be regarded as a composite consisting of an alternative stiff (crystalline) and soft-compliant (disordered) elements. Given the geometrical arrangement of these two alternating phases and the hardness value of both of them, the arising question is to predict the hardness value of the material. On the other hand, it is known that density is a crystallinity parameter... 

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