Direct Stiffness Method - Article Example

Introduction Direct Stiffness Method better known as DSM is the simplest computation of structural analysis and offers better understanding of Finite Element Analysis (FEA). In fact, all the command codes involved in DSM are found in FEA (Carlos, n.d). FEA is a structural analysis mostly used for complex structures involving several degrees of freedom. It is usually applied in computerized structural analysis where special computer software is used to analyze structures. FEA uses the basic principles of DSM and this gives the best relationship between the two. While FEA is used for complex structures, DSM is usually applied for simple structures such as a two-node structure, simple frames, or truss that are rigidly supported. The rigid support is to ensure that, displacements and forces acts only at the joints. Direct Stiffness Method criteria. Breakdown. The individual elements making up the structure are indentified and separated. This involves disconnecting the member elements at the nodes where the structure is connected. The forces and displacement relations are related through the element stiffness matrix that is dependent on the properties and geometry of the element. (Colorado education, n.d). Therefore, each member has to be analyzed separately for the member stiffness equation to be computed. There are two degrees of freedom at each node, which are the horizontal and vertical displacement and this when put as a matrix gives a two by two matrix. A frame element can withstand bending moments apart from compression and tension. This results into three degrees of freedom (Colorado education, n.d). This gives rise to the 6 by 6-stiffness matrix due to the horizontal, vertical displacements and the rotation outside the plane. Assembly. This is done after the first process to get them into the original structure after computing the individual stiffness matrix. The first thing in this process involves converting the individual stiffness relations into a global matrix system representative of the entire structure (Colorado education, n.d). This involves combining all the element matrices into a master stiffness matrix from the Cartesian coordinates. Displacements and force experienced at each node are the cardinal elements in this process. The two are related together by introducing the load vectors and then the matrix is expanded. The expanded individual element matrices when added together give the global stiffness matrix. Solution. After the global stiffness matrix and the displacement vector as well as the force vector have been put in place, the next step involves putting the whole system into a single matrix. One variable is usually unknown that is either the displacement or the force at the nodes. Solution is affected substituting the known valuables and then using one of several methods to compute the matrix. In DSM, analysis both simple and complex structures are treated alike Carlos, n.d). This is due to the matrix computation, which can effectively analyze both in the same way by following the above set up guidelines. Computation. In DSM calculations, t is convenient to work with a sample representing the actual structure. This process is called idealization. (Carlos, n.d). This means that a structure that represents the actual item with results of analysis representative to the actual situation. Consider the frame or truss above. The frame has three members; that is, the three sides of the frame that would be indicated by numbers 1, 2, and 3 with lengths l1, l2, and l3. The frame has three joints and as in the above explanation of FEA, the joints are referred to as nodes and would be indicated by j, n, and i. The sides of the frame are known as elements instead of members in FEA. the Elastic Modulus E is considered constant as well as the Area of the frame, which is denoted by A. the geometry of the whole structure is in line with the common Cartesian coordinates that are usually indicated as (x, y). This is well known as global or overall coordinate system (Carlos, n.d). The major assumption in the frames while analyzing using the FEA or DSM is that all forces have their reactions at the joints (Carlos, n.d). The axial forces on these joints are then indentified by a subscript x or y to differentiate between the two axial components. This is indentifying the forces according to the Cartesian coordinates in the x and y-axis in the direction of the axial force. The components at any joint are thus denoted as follows: fx and fy depending on the direction of the force. The six component forces at all the joints; that is 2-force component in the x and y direction per every joint, are then arranged into a column vector containing six components and is denoted by f. Therefore f= [fx1 fy1 fx2 fy 2 fx3 fy3 ] As noted above, another important aspect is the displacement at the joints. The displacement expression gives a more generalized Finite Element Theory (Carlos, n.d). The displacement of the whole truss is given by the displacement of the rigid frame at the joints. Displacement is represented by the letter u and like the force vector, it has a subscript x or y according to the direction of displacement on the global coordinates, and indicated as; ux and uy depending on the direction of the displacement on the Cartesian coordinates. The six displacement components indicating displacements at the three joints are then arranged into a 6-column vector component similar to the force vector as shown below. U= [ux1 uy1 ux2 uy2 ux3 uy3 ] Therefore, there are two vector components: the nodal force and the displacement vector at the joints. In direct stress method, the above vector components represent the variables or the unknown components and are known as the degrees of freedom or the state variables of the system (Carlos, n.d). The displacement boundary conditions are important in indentifying which components of f and u are unknown and which ones are known before the computation. In computer finite element analysis, the boundary conditions are defined at the end of computation. This is because, the reliability of the data arrangement is more important than the volume of data to be handled. The above conditions lead to the master stiffness equation. The equation is a relation between all the nodal forces f to all the displacements u before the supporting conditions are factored in. The assumption made in this case is that there is a direct relation between the nodal forces and the displacements. If the forces at the joints are zero, then the displacements are expected to be zero also (Carlos, n.d). This provides a case of homogeneity and linearity for the ease of computation in that, for there to be a direct relation between the forces and displacements then, there must be a linear relation of the components. The master equation is then set as below. f=k u k in this case is called the stiffness matrix , global stiffness matrix, or the overall stiffness matrix (Carlos, n.d). It represents a 6 x 6 square matrix with all the 36 stiffness coefficients. For better understanding of the coefficient k in three degrees of freedom, an example is given below where the value of ux2 is set as all the other components are zero. Therefore: u=[0 0 1 0 0 0] f= [kx1x2 ky1x2 ky2x2 ky2x2 kx3x2 ky3x2] Ky1x2 means the y force at joint 1 when an x displacement is applied at joint 2 and all the other components are zero. This is the interpretation given to all the stiffness coefficients in carrying out the structural analysis of any structure. The Finite Element Analysis is based on this principle but is more a generalized way of structural analysis. Several features characterize a direct stiffness matrix. First, the coefficient factor k is dependent on the cross sectional area denoted by A and the Length L. the modulus of elasticity E is constant since it depends on the properties of the materials. When no forces or displacements have been applied then, k=0 and this signifies that no constraints have been applied on the frame. When the displacement and the applied forces are proportional, the matrix becomes a symmetrical matrix. Therefore; kxy= kyx. These are the properties involved with structural analysis using the Direct Stiffness Method. List of References 1. Carlos F (n.d). Direct Stiffness Matrix: Encyclopedia. All Experts.com. website. . www.en.allexperts.com/e/d/di/direct_stiffness_method.html. Retrieved on 27th March 2010. 2. Colorado education. (No date). Direct stiffness analysis 1. Website. www.colorado.edu/engineering/CAS/courses.d/.../IFEM.Ch02.pdf. Retrieved on 27th March 2010. Read More