Global stiffness matrix size = Number of nodes X degree of freedom per node . The shape functions in order at node 1 of a 1D quadratic element will be Option A: 1,0,0 Option B: 0,1,0 Option C: 1,0,0,0 Option D: 0,1,0,0 Q12. An indeterminate truss is supported and loaded as shown above, using the direct stiffness method, obtain the displacements, support reactions, and internal forces that are induced in the members due to the externally applied loads, (EA = Constant, dimensions in mm). The Potential-Energy Approach. of nodes c) Degrees of freedom per node d) No. Explanation: In assembly of global stiffness matrix, the solution for finite element equations can take advantage of symmetry and sparsity of global stiffness matrix. Explanation: Global stiffness matrix method makes use of the members stiffness relations for computing member forces and displacements in structures. For example, element 3, which is connected to nodes 2 and four will contribute stiffness terms to elements 22, 24, 42, and 44 of the global stiffness matrix. Fig. For the spring assemblage shown in Figure P1.5, obtain (a) the global stiffness matrix, (b) the displacements of nodes 2 , 3 and 4 , (c) the global nodal forces, and (d) the local external forces. Stiffness Method for Frame Structures For frame problems (with possibly inclined beam elements), the stiffness method can be used to solve the problem by transforming element stiffness matrices from the LOCAL to GLOBAL coordinates. PDF Chapter 2 - Introduction to the Stiffness (Displacement ... 7. . The free body diagram method 2. Furthermore, the matrix is symmetric. This corresponds to the fact that the element exhibits two nodal displacements (or degrees of freedom) and that the two displacements are not independent (that is, the body is continuous and elastic). 10. A stiffness matrix is a positive definite. The stiffness matrix is an inherent property of the structure. It is a square matrix and . This . Third step: Assemble all the elemental matrices to form a global matrix. Introduction . 30.1 Introduction Stiffness matrix method is used for structures such as simply supported, fixed beams and portal frames. 19) Define displacement vector. Assemble member stiffness matrices to obtain the global stiffness matrix. Finite Element Method MCQ - Two Dimensional Isoparametric Elements - Four Node Quadrilateral . 3. For four noded quadrilateral element, the global load vector can be . The next step would be . summarize of stiffness method : to select a suitable displacement as unknown quantity, and the unknown forces in terms of these displacement, use the equation of equilibrium to solve the displacement Example 2-5 a solid circular steel cylinder S is encased in a hollow circular copper C subjected to a compressive force P for steel : E s, A s for . Matrix Algebra. In the penalty approach, the magnitude of the stiffness constant should be at least _____ times more than the maximum value in the global stiffness matrix. Discussion. 17. MCQ on CAD/CAM/CAE. 16. Write the global load-displacement relation for the plane frame. Impose boundary conditions on the load-displacement relation. It's clear that it is just an eigenvalue problem. Ans: Main steps in FEA are: 1. 9 - Question. of nodes c) Degrees of freedom per node d) No. Stiffness matrix [k] = ∫ [B] T [D] [B] dv [B] - Strain displacement matrix [row matrix] [D] - Stress, Strain relationship matrix [Row matrix] 42) Write down the expression of stiffness matrix for one . development of stiffness equations that only take into account bending deformations, i.e., ignore axial member, a.k.a. The size of the global stiffness matrix (GSM) = No: of nodes x Degrees of freedom per node. The points in the entire structure are defined using the coordinates system is known as. Starting point: consistent mass matrix 2. In particular, for basis functions that are only supported locally, the stiffness matrix is sparse. • To define the stiffness matrix • To derive the stiffness matrix for a spring element • To demonstrate how to Page 3/7. 2. Structures such as simply supported, fixed beams and portal frames can be solved using stiffness matrix method. a) One b) Three c) Two d) Four Answer: c Clarification: In assembly of global stiffness matrix, the solution for finite element equations can take advantage of symmetry and sparsity of global stiffness matrix. A. 4. manner as stiffness matrix relating elastic forces to displacements. This procedural 1. 27.Give the formula for the size of the global stiffness matrix. (A) Flexibility matrix is a square symmetrical matrix (B) Stiffness matrix is a square symmetrical matrix (C) Both (A) and (B) (D) None of the above Answer: Option C Question No. Discretization, 2. Furthermore, the matrix is symmetric. stiffness matrix K. Mass matrices for individual elements are formed in local coordinates, trans-formed to global, and merged into the master mass matrix following exactly the same techniques used for K. In practical terms, the assemblers for K and M can be made identical. c. both a & b. What are the basic unknowns on stiffness matrix method? In the previous cases examined, the aforementioned matrix . DEPARTMENT OF MECHANICAL ENGINEERING ME6603 - FEA FINITE ELEMENT ANALYSIS FORMULA BOOK BY ASHOK KUMAR .R (AP / MECH) 11 [] Global stiffness matrix {} Temperature matrix [ ] = [ 1 −1 −1 1 ] [ 1 −1 −1 1 ] { 1 2 } = { 1 2 } A Area of the wall - m2 k Thermal conductivity of wall - W . 6. The The full stiffness matrix A is the sum of the element stiffness matrices. Interpolation of results . 5. Manolis Papadrakakis, Evangelos J. Sapountzakis, in Matrix Methods for Advanced Structural Analysis, 2018. In the penalty approach, the magnitude of the stiffness constant should be at least _____ times more than the maximum value in the global stiffness matrix. The size of the global stiffness matrix (GSM) =No: of nodes x Degrees of free dom per node. Composite Materials: Given the material properties of a unidirectional lamina, this calculator constructs the stiffness matrix [C] and the compliance matrix [S] of the lamina in the principal directions.If the angle of rotation of the lamina is given (other than 0 degree), this calculator also constructs the stiffness matrix [] and the compliance matrix [] in the rotated directions. After computation of the global stiffness matrices k ¯ i of the members, the next step is the assembly of the global stiffness matrix of the entire structure. 17. Designed to provide engineers with quick access to current and practical information on the dynamics of structure and foundation, this unique work, consisting of two separately available volumes, serves as a complete reference, especially for those Lecture 13: Trusses & Grids -Stiffness Method Washkewicz College of Engineering By an analogous procedure S 12 and S 22 are 3 3 3 3 2 2 2 2 12 cos sinJ L EA L EA S 4 4 3 2 3 3 2 2 2 2 22 sin L EA L EA L EA S J The two expressions on this page as well as the two from the previous page constitute the stiffness matrix [S]. Note that in addition to the usual bending terms, we will also have to account for axial effects . 28.List the properties of the stiffness matrix. (1) Establish the x and y global coordinate system. There are two methods to identify them. 28.List the properties of the stiffness matrix. Assemble member stiffness matrices to obtain the global stiffness matrix of the plane frame. The global stiffness matrix for the problem is assembled from that for the individual elements as discussed in §3.9.2. Multiple Choice Questions with Answer Key. The size of the global stiffness matrix for the system shown below with minimum number of elements will be Option A: 2x2 Option B: 3x3 Option C: 4x4 Option D: 6x6 Q11. Analyse plane truss by the direct stiffness matrix. What are the properties of a stiffness matrix? Q. Coordinates and Shape Functions. The determinant being zero is a necessary condition for obtaining a vector u (the mode shape f n) that is not equal to zero (i.e., a solution other than the trivial . Formula for global stiffness matrix is ____ a) No. List the properties of the stiffness matrix The properties of the stiffness matrix are: It is a symmetric matrix The sum of elements in any column must be equal to zero. Multiple Choice Questions and Answers (MCQs) on Structural Analysis 01. The unit of stiffness is Newtons per meter. Using the assembly rule and this matrix, the following global stiffness matrix [4 3 4 3 4 3 Hence Global stiffness matrix or Direct stiffness matrix or Element stiffness matrix can be called as one. For many standard choices of basis functions, i.e. 2 Gauss elimination method: In linear algebra, it is a method for solving systems of linear equations by using elementary row operations till the echelon form of matrix is obtained. 18) Write then stiffness matrix for a 2 D beam element. Gaussian Elimination. Stiffness matrix method is used for structures such as simply supported, fixed beams and portal frames. 2. The values were statistically processed and a final value for torsional stiffness of the frame was determined. the equations in matrix form made the things sufficiently simpler. Q: Formula for global stiffness matrix is _____ Q: Galerkin technique is also called as _____ Quick Links. c. both a & b. d. None of the above. Q32 - If a line joining any two of its interior points lies not completely inside are called. Lumped mass matrix Example: 2-node truss in 1D Consistent mass matrix on element level (constant density and cross-section) Lumped mass matrix by row-sum-diagonalization Row-sum-lumping 1. Q31 - If a line joining any of its two interior points lies completely within it are called. of elements Answer: a Explanation: Generally global stiffness matrix is used to complex systems. Here, the stiffness is k, applied force is F, and deflection is δ. All diagonal elements are always positive 9.What are the main steps in FEA? : a)10 b)100 c)1000 d)10000. The length of the each element l = 0.45÷3 m and area is A = 0.002×0.03 m 2, mass density of the beam material ρ = 7850 Kg/m 3, and Young's modulus of the beam E = 2.1 × 10 11 N/m. Assembly of the Global Stiffness Matrix and Load Vector . However, it is impractical to compute the coefficient cij of damping matrix directly from the size of the structural elements. Q34 - If the visit to the vertices of the polygon in the given order produces an anticlockwise loop are called. stiffness. Formula for global stiffness matrix is _____ a) No. List the properties of the stiffness matrix. Besides, the use of equations in matrix forms facilitates the application of computer methods for their solution. It can be observed from the stiffness matrix of an individual element, that it is symmetric in nature, i.e. . B. the sum of the shape function is equal to one. 5. All diagonal elements are always positive 9.What are the main steps in FEA? of nodes*Degrees of freedom per node. Analyse plane frames by the direct stiffness matrix method. shown below , by just changing values of a and b and andmultiplying the stiffness matrix by a rotation y x y x. MAE 323: Lecture 3 Shape Functions and Meshing . 3. 17. Using the minimum potential energy principle for boundary value problems under displacement boundary condition (9), in combination with the assumption of spatial homogeneity and ergodicity of random microstructure, one arrives at the result that the greater is the material domain, the softer is the ensemble-averaged stiffness.Similarly, by the minimum complementary energy principle for the . The stiffness matrix describes the elastic behavior of the ply in plane loading = 11 12 0 12 22 0 . 15) Give the formula for the size of the Global stiffness matrix. Symmetry and sparsity of the global stiffness matrix can be approached by _____ methods. Explanation: Generally global stiffness matrix is used to complex systems. The properties of the . If in a pin-jointed plane frame (m + r) > 2j, then the frame is (A) Stable and statically determinate (B) Stable and statically indeterminate (C) Unstable (D) None of 3. Apply boundary conditions 6. k_12 = k_21.The attribute that stiffness matrix is symmetric comes from the Maxwell's Reciprocal Theorem which states that for any linear elastic body, displacement produced at any point A due to certain load applied at point B . K is a banded matrix. Two methods are introduced to drive the differential equations governing the motion of multi-degree of freedom systems. of nodesDegrees of freedom per node b) No. Add all terms at each row to diagonal Example: 3-node triangle in 2D Consistent mass matrix on element level . The size of the Global Stiffness Matrix (GSM) = number of nodes * degrees of freedom per node. There are two methods to identify them. 17) Why the stiffness matrix method also called equilibrium method or displacement. Instead, damping of a MDF system is usually specified in term of damping ratio and the corresponding damping matrix can be constructed accordingly. They are banded approach and skyline approach methods. 4. Finite Element Modeling. 8. 39 — Stiffness matrix for finite elements. Matrix Algebra and Gaussian Elimination. The size of the global stiffness matrix (GSM) = No: of nodes x Degrees of freedom per node. Summary of Procedure. 3. It is a square matrix and . Determination of Element Matrix Equation, 3. Q10. Calculate reduced stiffness matrix Q ij for each material used in the laminate (if a laminate uses only one type of composite material, there will be only 1 stiffness matrix). What are the properties of a stiffness matrix? Derive the Element Stiffness Matrix and Equations-Define the stiffness matrix for an element and then consider the derivation of the stiffness matrix for a linear-elastic spring element. Calculate the strains and stresses 7. Now we can form the global stiffness matrix based on these individual stiffness matrices for each element and the connected node locations for each. Early researchers used more or less ad hoc methods for determining the element stiffness matrices, but the finite element method was placed on a more rigorous foot- 1) Give th general expression for element stiffness matrix. If in a pin-jointed plane frame (m + r) > 2j, then the frame is (A) Stable and statically determinate (B) Stable and statically indeterminate (C) Unstable (D) None of the above Where 'm' is number of members, 'r' is reaction components and 'j' is number of joints Answer: Option B 02. In the stiffness method of analysis, we write equilibrium equationsin terms of unknown joint (node) Introduction to Stiffness Analysis 2 displacements. The basic principles involved in the analysis of beams, trusses were discussed. Obtain global stiffness matrix. Structural Analysis IV Chapter 4 - Matrix Stiffness Method 12 Dr. C. Caprani 4.2.2 Assemblies of Elements Real structures are made up of assemblies of elements, thus we must determine how to connect the stiffness matrices of individual elements to form an overall (or global) stiffness matrix for the structure. Principle of superposition . Stiffness matrix is a square and symmetric matrix. List the properties of the stiffness matrix The properties of the stiffness matrix are: It is a symmetric matrix The sum of elements in any column must be equal to zero. The length of a bar with cross-sectional area and tensile force applied is shown below in Figure (1). We could know the eigenvalues and eigenvector after integrating the approximate solution and forming the mass matrix [M] and the stiffness matrix [K]. Sectional stiffness is important, because if one imagine the frame as a system of springs in series, 7.6.5 Assembly of the Global Stiffness Matrix of the Entire Frame. — soil (foundation) stiffness — spring stiffness spring P pit beam width — tributary length Units (L) (L) Units (F/L3) (F/L2) Solution sensitivit to the soil stiffness value The Winkler model requires in its formulation a single soil parameter, the soil stiffness, k. Soil stiffness is obtained in one of ways: 17. k the Lagrange formula, . Assemble the Element Equations to Obtain the Global or Total Equations and Introduce Boundary Conditions-We then show how the total stiffness matrix Ans: 1. 5. . This corresponds to the fact that the element exhibits two nodal displacements (or degrees of freedom) and that the two displacements are not independent (that is, the body is continuous and elastic). of elements Answer: a Clarification: Generally global stiffness matrix is used to complex systems. the stiffness matrix represents the system of linear equations that must be solved in order to as certain an approximate solution to the differential equation. n values into Eq. Determination of Element Matrix Equation, 3. Using the assembly rule and this matrix, the following global stiffness matrix [4 3 4 3 4 3 This . Introduction. 24.1 Introduction An introduction to the stiffness method was given in the previous chapter. It can be seen that the stiffness matrix of a frame element will be of order 12 × 12. of nodes*Degrees of freedom per node b) No. Size of stiffness matrix is defined as: Size of global stiffness matrix=No. Since there are total N+1 degrees of freedom, the global structural stiffness matrix K would be of size N+1. piecewise linear basis functions on triangles, there are simple formulas for the element stiffness matrices. Stiffness matrix method is used for structures such as simply supported, fixed beams and portal frames. The sectional stiffness of the frame was determined in a similar manner (using displacements at points 4 to 6). List the properties of the stiffness matrix The properties of the stiffness matrix are: It is a symmetric matrix The sum of elements in any column must be equal to zero. Several important comments will now be made regarding the global stiffness matrix for the linear one-dimensional problem discussed earlier: 1. 3. 8. 6 Analysis of a single element stiffness matrix. 35. Figure 29: Global stiffness matrix and partition ... 33 Figure 30: Calculation of nodal deflection ... 34 Figure 31: Calculation of strut axial force ... 35 Figure 32: Effective length factors for members for idealized conditions of end restraint . 20) Write a note on global stiffness matrix. Explanation: Stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. Give the formula for the size of the Global stiffness matrix. 1 CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES 2 INTRODUCTION • We learned Direct Stiffness Method in Chapter 2 - Limited to simple elements such as 1D bars • we will learn Energy Methodto build beam finite element - Structure is in equilibrium when the potential energy is minimum 32 W L x x (A) 0.211 L (B) 0.25 L (C) 0.234 L (D) 0.5 L Answer: Option A Question No. Give the formula for the size of the Global stiffness matrix. K is symmetric. This follows from the fact that each node has only one degree of freedom. Stiffness matrix is a square and symmetric matrix. Stiffness matrix of element 1 d1x d2x d1x d2x Stiffness matrix of element 2 ⎥ ⎦ ⎡ = 2 2) 2 2 k-kˆ d2x 3x 2x d3x Global stiffness matrix ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = + 2 2 1 1 2 2 1 1 0 -k k-k k k -k k -k 0 K d 2x d3x d2x d3x d1x d1x Examples: Problems 2.1 and 2.3 of Logan Example 2.1 Compute the global stiffness matrix of . Figure 29: Global stiffness matrix and partition ... 33 Figure 30: Calculation of nodal deflection ... 34 Figure 31: Calculation of strut axial force ... 35 Figure 32: Effective length factors for members for idealized conditions of end restraint . % Compute the element stiffness matrix for a 2D truss bar in global coordinates % % INPUT DATA: % X1,Y1 is the location of joint 1 of the truss bar % X2,Y2 is the location of joint 2 of the truss bar % EA is the product of the elastic modulus and the section area % % OUTPUT DATA: % K is the 4x4 truss bar element stiffness matrix in global . The size of the Global Stiffness Matrix (GSM) = number of nodes * degrees of freedom per node. With the selected global and local node numberings local-to-global node mapping matrix can be written as follows [] where the entry of the last row does not exist since the third element has only three nodes. Fig. 27.Give the formula for the size of the global stiffness matrix. 21) Define load . Give the formula for the size of the Global stiffness matrix. slope-deflection method. Add all terms at each row to diagonal Example: 3-node triangle in 2D Consistent mass matrix on element level . •Note that the local-to-global mapping has the same form as the displacement solution: Discretization, 2. Structures such as simply supported, fixed beams and portal frames can be solved using stiffness matrix method. The equation shows that the element stiffness matrix for the linear spring element is a 2 ×2 matrix. . The number of unknowns in the stiffness method of which can be as the ones shown in Figure 3.4. which can be as the ones shown in Figure 3.4. With the selected global and local node numberings local-to-global node mapping matrix can be written as follows [] where the entry of the last row does not exist since the third element has only three nodes. Assembling to form Global matrix equation, 4. 100 Give the formula for the size of the Global stiffness matrix. Assembling to form Global matrix equation, 4. Ans: 1. Lumped mass matrix Example: 2-node truss in 1D Consistent mass matrix on element level (constant density and cross-section) Lumped mass matrix by row-sum-diagonalization Row-sum-lumping 1. Multiple Choice Questions / Mechanical Engineering MCQs / Finite Element Method Mcqs / Question; . Conjugate Gradient Method for Equation Solving. Structures such as simply supported, fixed beams and portal frames can be solved using stiffness matrix method. Starting point: consistent mass matrix 2. The size of the global stiffness matrix (GSM) = No: of nodes x Degrees of freedom per node. Ans: Main steps in FEA are: 1. 2. a. the shape function has a unit value at one nodal point and zero value at the other nodes. As with a normal modes (free-vibration) analysis, to carry out a flutter analysis we require a global mass matrix (inertial forces) and a global stiffness matrix (elastic forces), which can be seen in the normal modes equations of motion: One-Dimensional Problems. Chapter 4 - Matrix Stiffness Method - colincaprani.com Multiple Choice Questions and Answers (MCQs) on Structural Analysis 01. The equation shows that the element stiffness matrix for the linear spring element is a 2 ×2 matrix. Strain approximationin terms of strain-displacement matrix Stress approximation Summary: For each element Element stiffness matrix Element nodal load vector u =N d σ=DB d ε=B d =∫ Ve k BT DBdV V S e T b e f S S T f f =∫ T X dV +∫N T dS Constant Strain Triangle (CST) : Simplest 2D finite element • 3 nodes per element If the local axes (xyz system) are chosen to coincide with the principal axes of the cross-section, it is possible to construct the 12 × 12 stiffness matrix from 2 × 2 and 4 × 4 submatrices. Substituting the shape functions in the above mass matrix equations give: 2 2 0 11 1 L xxx LLL mA dx xx x LL L 21 612 AL m Evaluating the above integral gives: Structural Dynamics Direct Derivation of the Bar Element Step 5 -Assemble the Element Equations and Introduce Boundary Conditions The global stiffness matrix and the global force vector are 1, the determinant of the matrix (k-w n 2m) automatically becomes = 0, since this w n is a root of the determinant equation (i.e., the matrix becomes singular). The Galerkin Approach. 16) List the properties of the rotation matrix. 10 B. This global matrix can be assembled from individual element stiffness matrices and using element connectivity information from topology matrix. 2. The formula for axial stiffness is expressed as, The dimension of the global stiffness K is (N x N), where N is the number of nodes. assemble the ABD matrix, follow these steps: 1. 33 Study the following statements. Stiffness is applied to tension or compression. 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