Stiffness matrix

About: Stiffness matrix is a research topic. Over the lifetime, 8926 publications have been published within this topic receiving 173938 citations.

Crack band theory for fracture of concrete

Abstract: A fracture theory for a heterogenous aggregate material which exhibits a gradual strain-softening due to microcracking and contains aggregate pieces that are not necessarily small compared to structural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeard crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the complicance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three parameters—fracture energy, uniaxial strength limit and width of the crack band (fracture process zone), while the strain-softening modulus is a function of these parameters. A method of determining the fracture energy from measured complete stres-strain relations is also given. Triaxial stress effects on fracture can be taken into account. The theory is verified by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three material parameters involved, namely the fracture energy, the strength, and the width of crack band front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is also indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture theory as well as the strength theory. The applicability of fracture mechanics to concrete is thus solidly established.

Reduction of stiffness and mass matrices

Abstract: Just as it is often necessary to reduce the size of the stiff­ness matrix in statical structural analysis, the simulta­neous reduction of the nondiagonal mass matrix for natural mode analysis may also be required. The basis for one such reduction technique may follow the procedure used in Ref. 1 for the stiffness matrix, namely, the elimination of coordinates at which no forces are applied.

Behavior of Concrete Under Biaxial Stresses

Abstract: The results of an extensive series of tests of three types of concrete under biaxial loadings are used to develop stress-strain relations for concrete subjected to biaxial stress states. By means of a decomposition of the stresses and strains into their hydrostatic and deviatoric portions, it was possible to express the relations between octahedral normal stresses and strains, and octahedral shear stresses and strain through use of bulk and shear moduli. These moduli can be expressed as functions of the octahedral shear stress only; formulas and coefficients are given for the values of the tangent and secant, bulk and shear moduli for the three types of concrete. The deformational behavior is described as the material reaches its failure stage. The application of these nonlinear stress-strain relations to stress analysis is indicated; a material stiffness matrix for use in finite element analysis is presented, and a partial differential equation with variable coefficients for analysis of plane-stress problems is shown.

Reducing the bandwidth of sparse symmetric matrices

Abstract: The finite element displacement method of analyzing structures involves the solution of large systems of linear algebraic equations with sparse, structured, symmetric coefficient matrices. There is a direct correspondence between the structure of the coefficient matrix, called the stiffness matrix in this case, and the structure of the spatial network delineating the element layout. For the efficient solution of these systems of equations, it is desirable to have an automatic nodal numbering (or renumbering) scheme to ensure that the corresponding coefficient matrix will have a narrow bandwidth. This is the problem considered by R. Rosen1. A direct method of obtaining such a numbering scheme is presented. In addition several methods are reviewed and compared.

A Model for the Mechanics of Jointed Rock

Abstract: The representation of discontinuities in analysis of blocky rock is discussed. A linkage type element is developed for addition of rock joint stiffness to the structural stiffness matrix describing the behavior of a system of rock blocks and joints. Several basic problems of jointed rock are studied. These examples demonstrate the marked influence joints may have on the stress distribution, displacements, and failure pattern of an underground opening or other structures in jointed rock. A new classification of joints is introduced, based on the application of the joint element to finite element analysis of structures in jointed rock. Normal stiffness, tangential stiffness, and shear strength are used as parameters in the classification system. The methods discussed in this paper allow a jointed rock mass to be treated as a system of blocks and links. Just as analysis of a reinforced concrete building requires detailed knowledge of the behavior of concrete alone and steel alone, the joint stiffness approach calls for and uses detailed description of the behavior of rock blocks and rock joints independently.