330:155g Finite Element Analysis

Transcription

1 :55g inite Element Analysis Nageswara Rao Posinasetti Stiffness Matrices Review Matrix Algebra given in App A. Direct stiffness method is sed which simple to nderstand This can be sed for spring, bar and beam elements.. Spring element (-dim) Parts are D Some times D yields reslts that can be applied to D nder certain circmstances Use one dimensional spring element Obeys Hooe s law Deflection is linearly proportional to the force within the spring divided by the spring rate f =

2 Nodal point = Displacement -orce. A single spring element Stiffness matrix for the spring element Spring rate, 5 6

3 7 [K] Stiffness matrix 8 9

4 Example. 5 - = -f i -5 + = -f j f i = 5 f j = -5 Let s consider an example with two springs and

5 . Assembling Total Strctre s stiffness matrix Sm of the internal forces shold be eqal to the external forces applied at each node = = - + = + = Or more compactly as [ K ]{} = { } Nmber of rows = nmber of degrees of freedom 5 5

6 6 6 = =

7 9 Interchange rows as well as colmns in the same seqence as the node element seqence Bandwidth Sch that the stiffness vales (non-zero elements) concentrated closer to the diagonal Bandwidth refers to the nmber of terms we mst move away from the main diagonal before we enconter all zeroes. 7

8 . Bondary conditions These are the restrained movements of the nodal points Homogeneos type ixed Non-homogeneos type Specified displacement 8

9 9 5 6 = + + 7

10 8 9

11 What is a DO? The nnowns in a finite element problem are referred to as degrees of freedom (DO). Degrees of freedom vary by element and analysis type. DO Type Displacement Temperatre Action orce Heat low Rate Application Strctral Thermal What is a DO? Uy Rot y Node Rot x Ux Uz Rot z

12 Node A node is a coordinate location in space where the DO are defined. The DO of this point represent the possible response at this point de to the loading of the strctre. Element An element is a mathematical relation that defines how the DO of a node relate to the next. These elements can be lines (beams), areas (-D or -D plates) or solids (brics and tetrahedrals). 5 Nodes and Elements A node has a given set of DO, which characterize the response. or strctral analyses, these DO inclde translations and rotations in the three global directions. The type of element being sed will also characterize which type of DO a node will have. Some analysis types have only one DO at a node. Examples of these analysis types are temperatre in a heat transfer analysis and velocity in a flid flow analysis. 6

13 Element Connectivity Elements can only transfer loads to one another via common nodes. No Commnication Between the Elements Commnication Between the Elements 7 Stress and Strain Review The basic stress and strain eqations: σ = A ε = σ E δ = L ε dx δ = L ΑE 8 Stress Basic eqations do not reqire the se of a compter to solve. Compter-based analysis is needed when complexity is added as follows: Geometric complexity maes the elasticity eqation difficlt or impossible to solve. Variations in material properties exist throghot the part. Mltiple load cases and complex or combined loading exists. Dynamics are of interest. 9

14 General Case The DO components of each element combine to form a matrix eqation: [K] {d} = {A} [K] = element stiffness components {d} = DO reslts (nnown) {A} = action vale (e.g., force, temperatre) Strctral EA Eqation To determine the displacement of a simple linear spring nder load, the relevant eqation is: {f} = [K] {d} Known where {f} = force vector [K] = stiffness matrix {d} = displacement vector Unnown EA Eqation Soltion This can be solved with matrix algebra by rearranging the eqation as follows: - {d} = [K] {f}

15 Calclation of σ and ε Strains are compted based on the classical differential eqations previosly discssed. Stress can then be obtained from the strains sing Hooe s law ( = x). Dynamic Eqation or a more complex analysis, more terms are needed. This is tre in dynamic analysis, which is defined by the following eqation: {f} = [K] {d} + [c] {v} + [m] {a} where {f} = force vector [K] = stiffness matrix {d} = displacement vector [c] = damping matrix {v} = velocity vector [m] = mass matrix {a} = acceleration vector Other Applications EA can be applied to a wide variety of applications sch as: Dynamics Nonlinear Materials Heat Transfer lid low Electrostatics Piping Design and Analysis 5 5

16 Qestions/ Comments? 6