Technology & Engineering
Stiffness Matrix
A stiffness matrix is a mathematical representation of the stiffness properties of a structure or material. It is used in engineering to analyze the behavior of structures under different loads and conditions. The stiffness matrix is a key tool in finite element analysis, allowing engineers to predict how a structure will deform and respond to external forces.
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eBook - PDFMatrix Structural Analysis
Pergamon Unified Engineering Series
- Jamal J. Azar(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
CHAPTER THREE Direct Stiffness Matrix Method 3.1 INTRODUCTION The theory and the application of the direct Stiffness Matrix method to structural analysis are presented in this chapter. The stiffness matrices of the stiffness coefficients are developed and fully generalized for each of the following structural elements: Axial rod element Linear spring element Torsional rod element Torsional spring element Rectilinear beam element Curved beam element The method is applied to various structural problems which are com-mon in Civil Engineering, Aerospace and Mechanical Engineering, and other disciplines where such problems exist. 3.2 GENERAL DISCUSSION Matrix structural analysis is predicated on the ability of the structural analyst to formulate a discrete-element mathematical model equivalent to the real structure. The necessity of structural modeling is to obtain an idealized system with a finite number of degrees of freedom upon which matrix algebra operations can be performed. In matrix structural analysis the real structure is conceived as an assembly of structural elements (rods, beams, shear panels, etc.) con-nected at specified node points. The nodes selected are based on the 47 (a) Truss Typical elements Typical nodes (b) Rigid frame Typical elements 3 (c) Fin structure 3 3 48 Matrix Structural Analysis configuration of the real structure as well as on other factors depending on the critical regions to be investigated. (See Fig. 3.1 for illustrations.*) The functional relationship between the nodal forces Q i (forces acting at the nodes) and their corresponding displacements (nodal displace-ments) forms the basis of the Stiffness Matrix approach. In its most Fig. 3.1. Structural elements. generalized form, this relationship can be written as, Q1 S S 12 S 13 Q2 S21 S22 S23 Q3 S31 S32 S33 ... ... ... ... Qi Sil Si2 Si3 ... ... ... ... Q, S31 532 J3 ' S 1 ' . ~ ~ S2i ~ ~ ~ S 2 S . S, ... ... . . . Sii . . . '' i q R i q2 q 3 ..
eBook - PDFStructural Analysis
Understanding Behavior
- Bryant G. Nielson(Author)
- 2022(Publication Date)
- Wiley(Publisher)
Indeed, the stiffness approach requires us to find element stiffness values using some 433 434 DIRECT STIFFNESS METHOD FOR BEAMS AND FRAMES FIGURE 20.1 Typical beam action. other method before a structural Stiffness Matrix can be assembled. One may quickly see that this whole process could become extremely cumbersome and time consuming if every new structure required us to first use the flexibility approach to find the stiffness terms and then the stiffness approach to solve for all structural actions. For this reason, it is useful to view a structure (whether simple or complex) as nothing more than an assemblage of a few basic structural elements whose properties are already known. DID YOU KNOW? A structure is nothing more than an assemblage of simple and basic structural elements. The type of structure you can assemble is dependent upon what elements you have available to you in your virtual toolbox. The first element you have placed in this toolbox is the axial element. This section will now add another element to this toolbox—the beam element—which increases the types of structures that may be analyzed. Q U I C K N O T E A beam element is defined by four degrees-of-freedom—a single translation and rotation at each end. This will result in a 4 × 4 Stiffness Matrix. In this section, we introduce the stiffness terms for a beam element. Referring to Figure 20.1, a beam element is an element where the load is applied in a direction perpendicular to the mem- ber or in the form of a moment. These kinds of applied forces only produce internal shears and internal moments. Further, the resulting deflected shape at any location along the beam can be fully described by a vertical displacement and a rotation. For this reason, the DOFs at each node on the beam element are a vertical translation and a rotation, as shown in Figure 20.2.
eBook - PDF- Aslam Kassimali(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
Consider, for Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 9.9 Solution of Large Systems of Stiffness Equations 561 example, the analytical model of the six-degree-of-freedom continuous beam shown in Fig. 9.22(a) on the next page. The Stiffness Matrix S for this structure is also shown in the figure, in which all the nonzero elements are marked by 3s, and all the 0 elements are left blank. From this figure, we can see that, out of a total of 36 elements of S, 20 elements are 0s. Furthermore, this figure indicates that all the nonzero elements of S are located within a band centered on the main diagonal. Such a matrix, whose elements are all 0s, with the exception of those located within a band centered on the main diagonal, is referred to as a banded matrix. In general, a structure Stiffness Matrix is considered to be banded if S ij 5 0 if Z i 2 j Z . NHB (9.72) where NHB is called the half-bandwidth of S, which is defined as the number of elements in each row (or column) of the matrix, that are located within the band to the right of (or below) the diagonal element. Thus, the half-bandwidth of the Stiffness Matrix of the continuous-beam analytical model of Fig. 9.22(a) is 1 (i.e., NHB 5 1), as shown in the figure. Although the total number of nonzero elements of a structure Stiffness Matrix remains the same, their locations depend on the order in which the structure’s joints are numbered. Thus, the half-bandwidth of a structure stiff- ness matrix can be altered by renumbering the structure’s joints.
- Daryl Logan(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
For a continuous medium or structure comprising a series of elements, such as shown for the spring assemblage in Figure 2–1b, Stiffness Matrix [ K ] relates global-coordinate ( x, y, z ) nodal displacements { d } to global forces { F } of the whole medium or structure. such that { } { } 5 [ ] F K d (2.1.2) where [ K ] represents the Stiffness Matrix of the whole spring assemblage. 2.2 Derivation of the Stiffness Matrix for a Spring Element Using the direct equilibrium approach, we will now derive the Stiffness Matrix for a one- dimensional linear spring—that is, a spring that obeys Hooke’s law and resists forces only in the direction of the spring. Consider the linear spring element shown in Figure 2–2. Reference points 1 and 2 are located at the ends of the element. These reference points are called the nodes of the spring element. The local nodal forces are 1 f x and 2 f x for the spring element associated with the local axis x . The local axis acts in the direction of the spring so that we can directly measure displacements and forces along the spring. The local nodal displacements are 1 u and 2 u for the spring element. These nodal displacements are called the degrees of freedom at each node. Positive direc-tions for the forces and displacements at each node are taken in the positive x direction as shown from node 1 to node 2 in the figure. The symbol k is called the spring constant or stiffness of the spring. Figure 2–1 (a) Single spring element and (b) three-spring assemblage x x y z (b) (a) Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
eBook - PDF- S. Sivasundaram(Author)
- 2004(Publication Date)
- CRC Press(Publisher)
13 Identification of Stiffness Matrices of Structural and Mechanical Systems from Modal Data Firdaus E. Udwadia Aerospace and Mechanical Engineering, Civil Engineering, Mathematics, and Information and Operations Management, University of Southern California, Los Angeles, CA In this chapter we present a simple method for the identification of stiffness matrices of structural and mechanical systems from information about some of their natural frequencies and corresponding mode shapes of vibration. The method is computationally efficient and is shown to perform well in the presence of measurement errors in the mode shapes of vibration. The method is applied to the identification of the stiffness distribution along the height of a simple vibrating structure. An example illustrating the method’s efficacy in structural damage detection is also given. The efficiency and accuracy with which the method yields estimates of the system’s stiffness make it worthy of further exploration for damage detection. INTRODUCTION Modal testing of structures is an extensive field in civil and mechanical engineering. It is generally used to understand/predict the dynamic behavior of a structure when subjected to low-amplitude vibrations. Often modal information is also used to identify/estimate the structural parameters of a system, under the assumption that it has classical normal modes of vibration [1]. Such identification leads to improved mathematical models that can be used in either predicting and/or controlling structural response to dynamic excitations. Several different approaches to the parameter identification problem have appeared in the literature [2–10]. One approach is the so-called model updating method. Here a suit-able analytical model of a structural system is developed using the equations of motion, and its numerical representation is obtained. Validation of the numerical model through modal testing is then sought.
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