Descripción: equivalent spring constants for beams
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Stiffness of soil
2014 - In ETABS, shell or area element has two types of stiffnesses i.e. inplane stiffness refers as f11, f22 and f12 and out-of-plane stiffness refers ass modifiersFull description
Descripción: una reflexión sobre la película Matrix
Introduction to Finite Elements Lecture 2 Matrix Structural Analysis of Framed Structures
Introduction In this chapter, we shall derive the element stiffness matrix [k] of various one dimensional elements. Only after this important step is well understood, we could expand the theory and introduce the structure stiffness matrix [K] in its global coordinate system.
Introduction As will be seen later, there are two fundamentally different approaches to derive the stiffness matrix of one dimensional element. The first one, which will be used in this chapter, is based on classical methods of structural analysis (such as moment area or virtual force method). Thus, in deriving the element stiffness matrix, we will be reviewing concepts earlier seen.
Introduction The other approach, based on energy consideration through the use of assumed shape functions, will be examined later. This second approach, exclusively used in the finite element method, will also be extended to two dimensional continuum elements.
Influence Coefficients • In structural analysis an influence coefficient Cij can be defined as the effect on d.o.f. i due to a unit action at d.o.f. j for an individual element or a hole structure. Examples of Influence Coefficients are shown in Table 2.1. • It should be recalled that influence lines are associated with the analysis of structures subjected to moving loads (such as bridges), and that the flexibility and stiffness coefficients are components of matrices used in structural analysis.
Influence Coefficients
Flexibility Matrices
Remember the Virtual Force Method!
Flexibility Method
Stiffness Coefficients
Force Displacement: Axial Def.
Force Displacement: Flexural Def.
Force Displacement: Flexural Def.
Force Displacement: Torsional Def.
Force Displacement: Torsional Def.
Force Displacement: Shear Def.
Force Displacement: Shear Def.
Force Displacement: Shear Def.
Effect of Translation on Shear Def.
Effect of Rotation on Shear Def.
Putting it All Together • Using basic structural analysis methods we have derived various force displacement relations for axial, flexural, torsional and shear imposed displacements. At this point, and keeping in mind the definition of degrees of freedom, we seek to assemble the individual element stiffness matrices [k]. We shall start with the simplest one, the truss element, then consider the beam, 2D frame, grid, and finally the 3D frame element. • In each case, a table will cross-reference the force displacement relations, and then the element stiffness matrix will be accordingly defined.
Truss Element
Beam Elements There are two major beam theories: • Euler-Bernoulli which is the classical formulation for beams. • Timoshenko which accounts for transverse shear deformation effects.
