DIRECT STIFFNESS METHOD
As one of the methods of structural analysis, the direct stiffness method (DSM), also known as the displacement method or matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a matrix method that makes use of the members' stiffness relations for computing member forces and displacements in structures. The direct stiffness method is the most common implementation of the finite element method (FEM). In applying the method, the system must be modeled as a set of simpler, idealized elements interconnected at the nodes. The material stiffness properties of these elements are then, through matrix mathematics, compiled into a single matrix equation which governs the behaviour of the entire idealized structure. The structure’s unknown displacements and forces can then be determined by solving this equation. The direct stiffness method forms the basis for most commercial and free source finite element software.
The direct stiffness method originated in the field of aerospace. Researchers looked at various approaches for analysis of complex airplane frames. These included elasticity theory, energy principles in structural mechanics, flexibility method and matrix stiffness method. It was through analysis of these methods that the direct stiffness method emerged as an efficient method ideally suited for computer implementation.
Assembly
Once the individual element stiffness relations have been developed they must be assembled into the original structure. The first step in this process is to convert the stiffness relations for the individual elements into a global system for the entire structure. In the case of a truss element, the global form of the stiffness method depends on the angle of the element with respect to the global coordinate system (This system is usually the traditional Cartesian coordinate system).
(for a truss element at angle β)
After developing the element stiffness matrix in the global coordinate system, they must be merged into a single “master” or “global” stiffness matrix. When merging these matrices together there are two rules that must be followed: compatibility of displacements and force equilibrium at each node. These rules are upheld by relating the element nodal displacements to the global nodal displacements.
The global displacement and force vectors each contain one entry for each degree of freedom in the structure. The element stiffness matrices are merged together by augmenting or expanding each matrix in conformation to the global displacement and load vectors.
(for element (1) of the above structure)
Finally, the global stiffness matrix is constructed by adding the individual expanded element matrices together.
Solution
Once the global stiffness matrix, displacement vector and force vector have been constructed, the system can be expressed as a single matrix equation.
For each degree of freedom in the structure, either the displacement or the force is known.
After inserting the known value for each degree of freedom, the master stiffness equation is complete and ready to be evaluated. There are several different methods available for evaluating a matrix equation including but not limited to Cholesky decomposition and the brute force evaluation of systems of equations. If a structure isn’t properly restrained, the application of a force will cause it to move rigidly and additional support conditions must be added.
APPLICATIONS- DIRECT STIFFNESS METHOD
The direct stiffness method was developed specifically to effectively and easily implement into computer software to evaluate complicated structures that contain a large number of elements. Today, nearly every finite element solver available is based on the direct stiffness method. While each program utilizes the same process, many have been streamlined to reduce computation time and reduce the required memory. In order to achieve this, shortcuts have been developed.
One of the largest areas to utilize the direct stiffness method is the field of structural analysis where this method has been incorporated into modeling software. The software allows users to model a structure and, after the user defines the material properties of the elements, the program automatically generates element and global stiffness relationships. When various loading conditions are applied the software evaluates the structure and generates the deflections for the user.
The DSM steps, major and minor, are summarized in Figure 2.5 for the convenience of
the reader. The two major processing steps are Breakdown, followed by Assembly &
Solution. Apostprocessing substep may follow, although this is not part of the DSM proper.
The first 3DSMsubsteps are: (1) disconnection, (2) localization, and (3) computation of member
stiffness equations. Collectively these form the breakdown. The first two are flagged as
conceptual in Figure 2.5 because they are not actually programmed as such: they are implicitly
carried out through the user-provided problem definition. Processing actually begins at the member-stiffness-equation forming substep
Stiffness method
Stiffness method is an efficient way to solve complex determinant or indeterminant structures (Fig. 1). It is also called finite element method, which is a powerful engineering method and has been applied in numerous engineering fields such as solid mechanics and fluid mechanics. The idea of stiffness method is as following:
· Subdividing the structures into a series of discrete elements
· Formulating the stiffness matrix for each of the elements
· Assembling the global matrix
· Applying the boundary conditions to obtain the reduced matrix
· Inverting the reduced matrix
· Multiplying the inverted reduced matrix with the forces to get the displacements of the nodes
· Post-processing to obtain the stresses and strains of elements
Assembling the Stiffness Matrices and Force MatricesThe global stiffness matrix can be obtained by summing the stiffness matrix for each element, the formulation is
Where
K = Global stiffness matrix
k = Local element matrix
N = Total number of element
e = Index
Similarly, the force can be assembled as following:
Where
F = Global force matrix
f = Local force matrix
N = Total number of element
e = Index
What is meant by thermal stresses?
Thermal stresses are stresses developed in a structure/member due to change in
temperature. Normally, determine structures do not develop thermal stresses. They can absorb
changes in lengths and consequent displacements without developing stresses.
What is meant by lack of fit in a truss?
One or more members in a pin jointed statically indeterminate frame may be a little
shorter or longer than what is required. Such members will have to be forced in place during the
assembling. These are called members having Lack of fit. Internal forces can develop in a redundant frame (without external loads) due to lack of fit.
What is the effect of temperature on the members of a statically determinate plane truss.
In determinate structures temperature changes do not create any internal stresses. The
changes in lengths of members may result in displacement of joints. But these would not result in internal stresses or changes in external reactions.
Differentiate the statically determinate structures and statically indeterminate structures?
statically determinate structures statically indeterminate structures
1. Conditions of equilibrium are sufficientto analyze the structure------Conditions of equilibrium are insufficient to analyze the structure
2. Bending moment and shear force is independent of material and cross
sectional area.------Bending moment and shear force is dependent
of material and independent of cross sectional area.
3. No stresses are caused due to temperature change and lack of fit.------Stresses are caused due to temperature change and lack of fit.
DIFFERENCE BETWEEN DETERMINATE AND INDETERMINATE STRUCTURES
S. No. / Determinate Structures / Indeterminate Structures1 / Equilibrium conditions are fully adequate to analyse the structure. / Conditions of equilibrium are not adequate to fully analyse the structure.
2 / Bending moment or shear force at any section is independent of the material property of the structure. / Bending moment or shear force at any section depends upon the material property.
3 / The bending moment or shear force at any section is independent of the cross-section or moment of inertia. / The bending moment or shear force at any section depends upon the cross-section or moment of inertia.
4 / Temperature variations do not cause stresses. / Temperature variations cause stresses.
5 / No stresses are caused due to lack of fit. / Stresses are caused due to lack of fit.
6 / Extra conditions like compatibility of displacements are not required to analyse the structure. / Extra conditions like compatibility of displacements are required to analyse the structure along with the equilibrium equations.