Truss Stiffness Matrix Calculator
The detailed drawings and fabrication drawings are prepared and and the material-take-off is worked out. building stiffness matrix, generate load. finite element analysis of stresses in beam structures. (5-6) This equation together with Eqs (5-4) and (5-5) yields: p = BkBtv. (7 marks) b) Displacements of nodes 2 and 3. 2d truss solution via matlab rice university. The system matrix A in Eq. Let us now solve for the displacements in a small truss using the above approach (Fig. Now you have very similar model to the previous example. Please note that only the stiffness matrix is modified in this step. 0 f~fi+,_~00} T (1) where M is the mass matrix, K is the stiffness matrix, f is the vector of external dynamic loads, d is the vector of dynamic displacements, and 1 is the number of nonzero components of the dynamic load vector. derive its stiffness matrix and load vector in local AND global coordinate system) Step 3: Describe the behavior of the entire truss by putting. Higher Degrees of Indeterminacy • Consider the beam that is statically indeterminate to degree two, as shown below. the stiffness and load matrix. McCormac and James K. In solid mechanics [k] is called stiffness matrix and ffg is called load vector. E = 200 GN/m2. For comparison purposes, the following are the results of the same beam using a matrix method. For example consider a planer truss shown in Fig. EXAM 6: MONDAY, APRIL 30TH, ROOM 223B (COMPUTER LAB) See the website for a sample exam. 2 Equations of equilibrium for spatial beams An initially straight beam is considered. Two methods (Multiplication & Summation) were used to determine the global stiffness matrix. The structural stiffness matrix is formed with the stiffness matrix of each element. Obviously: r = ku. 1) Developed a code to generate stiffness matrix of each truss element for varying E. (20 points) The truss above is pin-connected and composed of two members with constant area A and second moment of area I, and made of material with Poisson's ratio v and Young's modulus E. The forces and displacements are related through the element stiffness matrix which depends on the geometry and properties of the element. The solution of the system of algebraic equations gives an approximation of the solution to the PDE. For the truss system shown, E-200000MPa. Now you have very similar model to the previous example. The same thing is true for the bridge of the truss. After loads and load cases are specified the displacements and forces are calculated by solving a bandwidth minimized stiffness matrix using an. • K11*v1 + K12*v2 = S1 • K21*v1 + K22*v2 = S2 S2 v1 v2 = S1 Gurley refers to the axial displacement as “v” and the internal force as “S” in the local coordinate system. A-Truss is a 2D Truss analysis using the stiffness matrix method. Note that in developing the new dynamic frequency analysis solver, it was discovered that the old solver missed some higher order frequencies in some models. The assembled global stiffness matrix of the truss is of the order 8×8. 6 Analysis of Multi-storied Structures by Cantilever Method Although the results using the Portal Method are reasonable in most cases, the method suffers due to the lack of consideration given to the variation of structural response due to the difference between sectional. The cross sectional area of the truss is constant and equal to A0. A block diagonal matrix containing these element stiffness matrices is known as the unassembled stiffness matrix of the structure, denoted by k. - 20 truss types template - support any unit system - calculation details: - member forces - nodes translations - supports reaction - PDF reports. This is the stiffness matrix of a one-dimensional truss element. Aerospace Engineering Department, Cairo Uiniversity % Egypt (17th Oct. A structure that fails at large loads is expected to be dense and stiff. Nodal Coordinates. 6 Analysis of Multi-storied Structures by Cantilever Method Although the results using the Portal Method are reasonable in most cases, the method suffers due to the lack of consideration given to the variation of structural response due to the difference between sectional. Global stiffness matrix matlab. The three last words refer to the fact that the displacements are expressed in the local axis of the bar. The text in that box of that figure is supposed to be placed on a Notebook cell. midas Gen provides SPC, which calculates stiffness data for any shape or form. Problem 7-3. Obviously: r = ku. 5 shows a module that evaluates and returns the 4×4 stiffness matrix of a plane truss member (two-node bar) in global coordinates. Matrix Stiffness Method or Direct Stiffness Method or Matrix Displacement Method or Finite Element Method is a structural analysis method that is used as a b. Given the flight case conditions, Nastran calculates the rigid Aerodynamic Influence Coefficient (AIC) matrix using a vortex lattice panel method; This AIC matrix can then be transformed into an incremental stiffness matrix due to aerodynamic loads (K_AIC) Nastran then solves the following equation: Where q is the free-stream dynamic pressure. The shape and properties of the generated section can be exported to midas Gen. Stiffness matrix method is used for all types of structure. However, the earliest mathematical papers on Finite Element Analysis can be found in the works of Schellbach [1851] and Courant [1943]. Alex Tomanovich: 2018 07: FRAME_(metric) Analysis of portal and gable rigid frames using the stiffness matrix. But in reality there are many members in a truss. What is in the detailed solution? The app is able to determine whether the structure type is a truss, beam or frame. This is the stiffness matrix of a one-dimensional truss element. 6) in solving problem using finite element method 3. stiffness matrix for a bar arbitrarily oriented in space. joint truss under loads, where in this way the moments and internal forces in the members can be obtained "in one go". Use the method of virtual work to calculate all deflections at node N. The connections in trusses are always assumed to be nominally pinned. (20 points) The truss above is pin-connected and composed of two members with constant area A and second moment of area I, and made of material with Poisson's ratio v and Young's modulus E. Depending on one’s perspective, FEA can be said to have its origin in the work of Euler, as early as the 16th century. 3) is true along the centroidal axis of the truss member. you would be probably talking about a spring characteristic when saying “spring constant” during a physics class. Compare this to the fink truss, which has a few less webs and hence the computations are less 21 x 21 matrix (441 values). What are the types of. becomes sparse, with nonzero terms only for the matrix components that correspond to overlapping ij:s. Indicate the degrees-of freedom in all the stiffness matrices. Apply 1,2 20 For the truss given below, a horizontal load of P = 4000 lb is applied in. • This is useful when the transformation matrix is not square - i. The structure stiffness matrix will be a 30 x 30 matrix (900 values), its no wonder they didn't do this sort of thing prior to our modern computers, imagine trying to calculate this by hand. This is an ultimate collection of all my structural programs to solve beams, frames updated porticos and trusses using the Stiffness Matrix Method or Displacements method, giving you the results step by step, drawing the beam, frame or truss, cerchqs also an animation of the structure. M: Direct stiffness method – introduction. k 3 = A 3 E/L 3 = 7. Application of the stiffness method requires subdividing the structure into a series of discrete finite elements and identifying their end points as nodes. For example consider a planer truss shown in Fig. STC/STCT/DTC Roof Truss Clips. Other types of elements have different types of stiffness matrices. T is a row vector of evenly spaced times. Static cone to compute static settlement over sand. Let us now solve for the displacements in a small truss using the above approach (Fig. Analysis involves assembling the stiffness matrix and fixed-end-forces then solving P=KU through brute force by inverting the stiffness matrix. Strain influence factor I z is interpolated for intermediate values of ratio l/b (1 < l/b < 10). Spring elements, direct stiffness method, truss elements, coordinate transformations, stress in bar elements, bar elements in 3D space Analysis of trusses including: Initial set-up, truss geometry, section properties, study properties and settings, boundary conditions, loads, meshing, solving, post processing Beam equations, distributed loading,. For 2D problems only one angle is required to describe the member direction. The solution of the system of algebraic equations gives an approximation of the solution to the PDE. 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. Step 1: Divide the truss into bar/truss elements connected to each other through special points (“ nodes”) Step 2: Describe the behavior of each bar element (i. 3 Fundamentals of the Stiffness Method. - Solutions include node displacements, member axial forces and stress. Given the flight case conditions, Nastran calculates the rigid Aerodynamic Influence Coefficient (AIC) matrix using a vortex lattice panel method; This AIC matrix can then be transformed into an incremental stiffness matrix due to aerodynamic loads (K_AIC) Nastran then solves the following equation: Where q is the free-stream dynamic pressure. Convert into distributed masses. CE6020 Finite Element Techniques Syllabus Notes Question Papers 2 Marks with Answers Question Bank with answers Anna University CE6020 FET Notes Syllabus 2 Marks with answers Part A Part B Problems Anna University Civil 8th Semester – Regulation 2013 8th Semester Syllabus Notes Anna University CE6020 Finite Element Techniques Notes Syllabus 2 marks with answers Part A […]. 002 m 3 x 53. The load is always acting on the joints. Finite element analysis (FEA) in computer-aided design; finite element formulation; first-order and second-order elements; stiffness matrix; integration points and stress recovery; convergence and mesh refinement; FEA of plane stress and plane strain problems; FEA of axisymmetric problems; FEA of nonlinear materials; FEA of contact problems; FEA of large deformation problems; FEA of dynamic. 3-7 Element Stiffness Matrix We will use the potential energy approach to derive the element stiffness matrix [k] for the 1-D element. 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. Take A = 450 Mm?, E = 200 GPa. org It’s also worth noting that at no point did the fact that the truss us statically indeterminate feature in our discussion. 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. finite element analysis of stresses in beam structures. What is the relation between flexibility and stiffness matrix? 34. , the element stiffness matrix coefficients aligned with the global coordinate system and All global coordinate stiffness equations are expressed by (17. what will be the grade of steel for the truss ? yield or tensile strength. zip - the download file includes complete open source code. 1) Developed a code to generate stiffness matrix of each truss element for varying E. Fit a curve of the form N = abt and estimate N, when t = 7. If a structure is stable it is called as statically determinate. The actual truss geometry cannot be altered within SketchUp it is hard coded into the plugin when it is created by the calculator. Depending on one’s perspective, FEA can be said to have its origin in the work of Euler, as early as the 16th century. 3 x 10 5000 x 4-3EI FL y 6 3 The deflection is 2 mm downwards. K' = member stiffness matrix which is of the same form as each member of the truss. This function works for 2D and 3D trusses (type "help TrussStiffness" for details). - 20 truss types template - support any unit system - calculation details: - member forces - nodes translations - supports reaction - PDF reports. A bar is considered uniform if its cross-sectional area and elastic modulus are constant along the length of the bar. We've gone through beam curvature, we looked at singularity functions for doing beam deflections, we did beam deflections in the last prac using superposition techniques, and now we're going to look at statically indeterminate beams. Calculate element results. Generate the load vector; Apply boundary conditions. To assemble the Global Stiffness matrix, element-by-element addition is undertaken. "This comprehensive textbook combines classical and matrix-based methods of structural analysis and develops them concurrently. The displacement, u, is given by the solution to the differential equa-tion. Calculate element results. 5 a) Analyse the beam by stiffness matrix method. Jan 25,2021 - Structural Analysis - 1 | 15 Questions MCQ Test has questions of GATE preparation. the reduced theoretical stiffness matrix (K ) with the Identified stiffness matrix (K ). Extension of bars. Module 2: Calculation Process (Analysis) Procedures in module 2 are associated with the theory of the structure and the analytical methods used, e. Solve global system of equations (=> displacements) 6. 6 B Analyze the beam shown in Fig. Alex Tomanovich: 2018 07: FRAME_(metric) Analysis of portal and gable rigid frames using the stiffness matrix. The detailed drawings and fabrication drawings are prepared and and the material-take-off is worked out. Now to calculate the overall stiffness of a part made out of two different materials you would treat the problem the same way you would to calculate the overall resistance of an electrical circuit. CEE 379 ELEMENT STIFFNESS MATRIX FOR PLANAR TRUSS σ= Eε θ final ~ θ initial Element End Forces Element End Displacements q) Q If you decide a priori that the element forces are linearly related to the displacements then and Q ⎭ i. This app can give you the solution in seconds and you can quickly check if you have reasonable results. From the above results and the analysis of the stiff-ness matrix of the bicycle frame elements, the boundary conditions and load are set in order to obtain the nor-mal stress on the tubes and the local and global displa-cements. FEM-Design is an advanced modeling software for finite element analysis and design of load-bearing concrete, steel, timber and foundation structures according to Eurocode with NA. Element Stiffness Matrix The elemental properties like young’s modulus (E),cross-sectional area (A), the length of the element (L), direction cosines (,) are used for assembling stiffness matrix of each element denoted by K. • Convert the indeterminate structure to a. Serving the engineering simulation community, Ansys & Rocky DEM users - news, workshops, seminars, training, webinars, tips & tricks, & more. W: Comparison to flexibility and slope deflection methods. The Element Stiffness Module As our first FEM code segment, the top box of Figure 4. For truss analysis, the finite elements are represented by each of the members that compose the truss, and the nodes represent the joints. The extension to a three dimensional space truss will be. Learn more about Hooke's law and how to calculate the spring constant including the formula, insight on a spring's impact on force, and an example problem!. Continuing from: Frame Analysis with Excel - 3, Continuous beam or frame Download Frame1. An icon used to represent a menu that can be toggled by interacting with this icon. We propose a fast stiffness matrix calculation technique for nonlinear finite element method (FEM). k 1 = A 1 E/L 1 = (86. 7 (b) Prove that the stiffness matrix and flexibility matrix are inverse of each other for the beam shown in Figure 6. In case of elements carrying distributed loading, the reaction forces that the distributed load would cause on a single beam element, are added to the load vectors of the. Strain influence factor I z is interpolated for intermediate values of ratio l/b (1 < l/b < 10). In the considered simple case for two finite elements of length L stiffness matrices and the load vectors can be easily calculated: [k1] = [k2] = a L " 1 ¡1 ¡1 1 # ff1g = bL 2 (1 1); ff2g = bL 2 (1 1) + (0 R) (1. The analysis of truss can be done by maintly two methods, that is method of joints and method of sections. FEM-Design is the most user friendly FEM software for building analysis on the market. For the truss system shown, E-200000MPa. Calculator assessment of timber structures to AS1720; Calculator for assessment of cold formed steel structures to AS4600; Calculator for assessment of steel structures to AS4100; Calculation of Plane Truss; Cold Formed Steel Sheds Australia Height Span Limits of C-Sections; Composite Column; Concrete Beam Design (CSA A23. Previous question Next question Transcribed Image Text from this Question. This operation uses the code vectors of the truss members. And of course all special calculation options are also available, including elimination of tension / pressure rods, GNL and FNL analyzes, non-linearly resilient rod connections and imposition and stability analysis. the reduced theoretical stiffness matrix (K ) with the Identified stiffness matrix (K ). 0 is a 3D Finite Element program that uses truss and beam elements to model a wide range of Mechanical and Civil structures. , 4th Edition, John Wiley. 7 by finite element method and determine the end reactions. Therefore, it is very easy to use and for many years it has been the best choice for structural engineering. Element Stiffness Matrix in Local Coordinates • Remember Kij = the amount of force required at i to cause a unit displacement at j, with displacements at all other DOF = zero • For a truss element (which has 2 DOF). equation to develop a stiffness matrix. Review of matrix analysis for pin-jointed trusses. Solving, 3 m + r = 16 and 3 j + e c = 14, Again, this structure is found to be 2 degrees internally statically indeterminate. For example consider a planer truss shown in Fig. assemble global stiffness matrix. Du Nodal deflections matrix Dk Boundary condition of nodes matrix E Modulus of Elasticity (MPa) fy Steel yield strength fu Steel tensile strength FEA Finite element analysis G Dead load h height of geodesic dome k’ Member stiffness matrix kc,i Internal combination factor kc,e External combination factor ka Area reduction factor. 7 A Discuss the generalized three dimensional stiffness matrix of a beam Element. I've also ignored that the axial tension in AB will cause position A to move, dragging member AC and hence C with it, plus the axial compression in AC will move point C even further. With the option SEPR, the stiffness matrix or load matrix can be printed to the output window if desired. You can replace each bar with a spring as shown in the right side of the following illustration. Space-Truss Analysis. Displacement (Stiffness) Method Express local (member) force -displacement relationships in terms of unknown member displacements. The first set is. All the joints in the truss are frictionless. After finding the determinant of the two stiffness matrices, the first set of boundary conditions are not valid for a static problem because the determinant of [K] is zero, and therefore the matrix cannot be inverted. Stiffness Influence Coefficients 176 6. Consider a Two-Member (Two-Dimensional) Truss:. stiffness matrix, assembly process, solution techniques; truss elements, beam elements; triangular and quadrilateral elements; iso-parametric formulation; plane stress and plane strain applications; penalty and Lagrangian methods; software applications. Truss Member 11 The transformation matrix given in (11) is valid for all space truss member orientations with the exception of a vertical truss memberas shown in Fig. , the element stiffness matrix coefficients aligned with the global coordinate system and All global coordinate stiffness equations are expressed by (17. To assemble the Global Stiffness matrix, element-by-element addition is undertaken. 4 Solid Mechanics Part I Kelly196 Figure 7. Nodal Coordinates. 21) we can construct that stiffness matrix for element 1 defined in the table above. •Partition the structure stiffness matrix for easier calculations •Determine the unknown joint displacement D x, the support reactions Q x. In a linear static analysis the model’s stiffness matrix is constant, and the solving process is relatively short compared to a nonlinear analysis on the same model. Why is it Convenient? Using hand calculations, the stiffness method can take hours and it is difficult to know if you are on the right track. Beam Stiffness Matrix. System stiffness equations. Take A = 450 Mm?, E = 200 GPa. >>When you're done reading this section, check your understanding with the interactive quiz at the bottom of the page. Element stiffness matrices b. The form factor is based on the shape of the cross section and for a rectangular area STAAD uses a factor of 2/3 or 0. In structural analysis, the stiffness method is an easy way to estimate the mechanical behaviour of a structure composed of slender elements (or beam elements), assuming that linear elasticity apply. 1 Introduction 4. (5-7) Therefore, K = BkBt (5-8) is obtained. Problem 1: Indicate by circling the correct answer whether each matrix below could be a stiffness matrix for a truss. h) The stiffness matrix from a static analysis no longer has to be saved. s arranged as shown in Fig. Truss Member 11 The transformation matrix given in (11) is valid for all space truss member orientations with the exception of a vertical truss memberas shown in Fig. Figure 7-3(a) Solution: In the case of unit load method the deflection at a point of beam is given as. Dropping these files into the SketchUp folder sets up the menu item. Merge individual member stiffness matrices into stiffness matrix, K, for entire frame 9. Application of the stiffness method requires subdividing the structure into a series of discrete finite elements and identifying their end points as nodes. It the number of unknowns is equal or less than the number of equlibrium equations then it is statically determinate. A-Truss 3D is an application for structural engineers or engineering students who want to analyze the space truss. Other types of elements have different types of stiffness matrices. To this base triangle, two more members are added to locate a new joint, and the process continued to form the complete truss. • Example 1: The figure shows a planar truss. The element stiffness matrix for this problem is shown above. , when the matrix inverse is not defined. 6) in solving problem using finite element method 3. AISC Home | American Institute of Steel Construction. This imposes compatibility of element DOF and structural DOF. Write the element stiffness matrix for a truss element. This free online structural frame calculator will generate and find the bending moment and shear force diagrams of a 2D frame structure. You will certainly be asked for output:. The matrix K is singular since the boundary conditions of the. I'm going to load up all of the section geometries I can for a drop-down, then add an arbitrary section property calculator. Spring elements, direct stiffness method, truss elements, coordinate transformations, stress in bar elements, bar elements in 3D space Analysis of trusses including: Initial set-up, truss geometry, section properties, study properties and settings, boundary conditions, loads, meshing, solving, post processing Beam equations, distributed loading,. Solve first with hand calculations and then with SAP 2000. Now we can calculate a global element stiffness matrix for each element in our structure that encodes the orientation of the member. 4 KB) by Dorian Depriester This small program computes the stiffness matrix of a truss, assuming that this truss is composed of slender elastic elements (beams). 1 To obtain a listing of the stiffness matrix and load vector from an ANSYS model, use the “Substructure” option in the Solution processor, and solve. s arranged as shown in Fig. Displacement and Force Transformation Matrices. The above relation (24. For the vertical truss member, Cx= Cz= Cxz= 0 and (11) is not numerically defined. What is in the detailed solution? The app is able to determine whether the structure type is a truss, beam or frame. Because plastic flow is nonassociated, the use of concrete damaged plasticity results in a nonsymmetric material stiffness matrix. As you already know, those “small” element stiffness matrixes correspond with certain global Degrees of Freedom. Build element geometrical stiffness matrices due to N. rb files for them. Gavin 1 Finite Deformation Analysis of a Bar in Local Coordinates Recall that in local coordinates the four deflections at the bar ends are u 1, u 2, u 3, and u 4; the four forces at the bar ends are q 1, q 2, q 3, and q 4, and the element stiffness matrix in local. • Press the “F2” key. However, during college you start using a more broad term, stiffness. Say I have a 9 x 9 matrix K representing a three-member truss. 3 shows that 0 0 1 C 0 0 0 C 0 [ ] y y vert (12). STC/STCT/DTC Roof Truss Clips. flexural stiffness is 53. a) Global stiffness matrix. The system stiffness matrix, K, is assembled from the element stiffness matrices, K’, where. The reduced theoretical stiffness (K ) is generated by condens-ing the theoretical stiffness matrix (K ) to 2-DOF stiffness matrices. Section Property Calculator An unit structures such as a frame, an arch, a truss, a plate and a shell. A structure that fails at large loads is expected to be dense and stiff. Compare flexibility method and stiffness method. Assemble the system global stiffness matrix and calculate the global displacements of the unconstrained node; b. The differential equation governing the displacement of the body (x) and time (t) is given by dx +5 − 12 x = 0 2 dt dt Find the displacement (x) at time, t = 2, given that x(0) = 4 and y ( 0) = – 5. 1) 0 Figure 6 BICE-016 4 500. (5-7) Therefore, K = BkBt (5-8) is obtained. 2d truss solution via matlab rice university. Using the equation shown in (3. L AE k= is the stiffness of the truss member and is defined as the force required for unit deformation of the structure. A truss element can only transmit forces in compression or tension. Space-Truss Analysis. Global stiffness matrix truss. The free version allows you to input frames with a maximum of 3 members with applied point loads and moments for 2D frame analysis. After loads and load cases are specified the displacements and forces are calculated by solving a bandwidth minimized stiffness matrix using an. Once you get to advanced structures, you can just put every point in the structure in a 'stiffness' matrix and solve for all displacements, stresses, etc when a force/s is applied. The new equivalent cross section is assumed to be made completely from material 1. 1 Introduction 4. Is it possible to develop the flexibility matrix for an unstable structure? 33. STC/STCT/DTC Roof Truss Clips. Static cone to compute static settlement over sand. zip - the download file includes complete open source code. View/Download from: Publisher's site. In a linear static analysis the model’s stiffness matrix is constant, and the solving process is relatively short compared to a nonlinear analysis on the same model. Two ways to make the same stiffness matrix K: (i) use the entry-by-entry formula for Kij, or (ii) assemble K from "element matrices": K = K1+ K2 + K3. Indicate the degrees-of freedom in all the stiffness matrices. Convergence Values: CNVTOL: By default, the program checks the out-of-balance load for any active DOF. Solve global system of equations (=> displacements) 6. 95 ×10 6 lb. Accordingly, two forces and one moment can be applied to each node. This function works for 2D and 3D trusses (type "help TrussStiffness" for details). As mentioned in step 1, each 2D truss member is assigned a code vector consisting of 4 numbers. At a high level, the global stiffness matrix is created by summing the local stiffness matrices: where the matrix [ k i] is the local stiffness matrix of the i th element. A simple truss. The 2D Finite Element Analysis (FEA) tool allows the analysis of any 2D structure using beam elements. It then outputs a stiffness matrix, displacements, and member forces. 6 and you want, stresses that the state department of the dimensions of structure. Follow 153 views (last 30 days) abdelrahman alhammadi on 12 Oct 2018. The total strain energy of the truss may be computed by adding together the strain energy of each element: It is more convenient to express W in terms of the global displacement vector, u. 3D Vector Calculator nbsp 11 Dec 2001 To reduce the 3D stress matrix to the 2D plane maximum of the shear stress transformation equation and solving for. 1) 2 0 2 1 u k xdx kQ Q = ∫ = (3. 3) is true along the centroidal axis of the truss member. If you'd like to validate this calculator use our closed-form beam calculator to compare results: Beam Calculator. matrix that will convert the displacements within the element into strains. Determine the forces in the truss shown in Fig. graphs desktop-application finite-element-analysis finite-element-methods truss-solver python-gui truss-analysis stiffness-matrix truss-software force-matrix. Vibrations in a long floor span and a lightweight construction may be an issue if the strength and stability of the structure and human sensitivity is compromised. –[K] is square, symmetric, positive definite, non-singular, invertible, and. 3 Three-Dimensional'Ihlsses 114 4. Retrieve the global matrices; Perform static analysis. 002 m 3 x 53. This means that the first 3 rows and columns correspond to the first node, the second set of three rows and columns with the second node, and the third with the. To start, what if you had to different materials that are connected in parallel? To find the stiffness of the overall part refer to the image below. The program part named "FemBase" is massively parallelized: It calculates the result combinations and returns all results to the graphics, tables, and sections. 2) Displacements are generated for N (=10^5) simulations and a histogram is plotted to determine the. A structure that fails at large loads is expected to be dense and stiff. 5 September : Systematics in truss analysis. Question 1: For the plane (2D) truss shown below, evaluate the transformation matrix [T] and the stiffness matrix in the local axis system [KL] of all elements. (a) Differentiate between stiffness and flexibility method. Let us now solve for the displacements in a small truss using the above approach (Fig. Member Global Stiffness Matrix. Basically, this stiffness shear area is a form factor times the axial area. 6 and you want, stresses that the state department of the dimensions of structure. External Loads. The use of natural coordinates in expressing approximate functions is advantageous because special integration formulas can often be applied to evaluate the integrals in the element matrix. Figure 2 shows a two-member plane truss supported by a linearly elastic spring. (18 marks) b) Calculate all the nodal displacements and all the member forces for the truss. 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. A bar is considered uniform if its cross-sectional area and elastic modulus are constant along the length of the bar. The solution of the system of algebraic equations gives an approximation of the solution to the PDE. Show numbers where known (stiffness matrix terms, support displacements, applied loads). stiffness matrix for a bar arbitrarily oriented in space. With clever use of an X-Y scatter plot and cell entries, the bridge is drawn to scale when unique span, radius, and point loads are entered. Derivation Of Global Stiff Matrix For A Truss (Part 1) FREE: 8:42: 3. the design program provides an economic and efficient design of the main frames and allows the user to utilize the program in different modes to. % 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 stiffness matrix for each beam is generated as before. 1) Developed a code to generate stiffness matrix of each truss element for varying E. Note: The two unknowns can also be solved for using only matrix manipulations by starting with the initial conditions and re-writing: Now it is a simple task to find γ 1 and γ 2. This operation uses the code vectors of the truss members. • Press the “F2” key. That is the basis for modern computational analysis techniques, finite element method. Next the internal and external virtual work can be formed and equated to develop the stiffness matrix. Use the method of virtual work to calculate all deflections at node N. 3) is true along the centroidal axis of the truss member. fea code in matlab for a truss structurea ppt mathworks. The roller boundary condition of Truss 1 allows rotation of the entire truss as a rigid-body. Two ways to make the same stiffness matrix K: (i) use the entry-by-entry formula for Kij, or (ii) assemble K from "element matrices": K = K1+ K2 + K3. The extension to a three dimensional space truss will be. Write the element stiffness matrix for a beam element. Lecture 19 : (Section 3. k = stiffness (N/m, lb/in) F = applied force (N, lb) δ = extension, deflection (m, in). For instance, the stiffness matrix of element A corresponds with deformations and loads in nodes (1) and (2). Why is it Convenient? Using hand calculations, the stiffness method can take hours and it is difficult to know if you are on the right track. Once you get to advanced structures, you can just put every point in the structure in a 'stiffness' matrix and solve for all displacements, stresses, etc when a force/s is applied. The analysis of truss can be done by maintly two methods, that is method of joints and method of sections. Having set forth the foundation on which the direct stiffness method is based, we will now derive the stiffness matrix for a linear-elastic bar (or truss) element using the general steps outlined in Chapter 2. In which, K is stiffness matrix of truss; P is nodal load vector of truss Equilibrium constraints satisfies the equation (2. The immediate objective is the development of a load-dependent element tangent stiffness matrix that can be evaluated for given element forces and is a reliable predictor of a type of behavior in the neighborhood of the loads to which those forces correspond. Build element geometrical stiffness matrices due to N. Space-Truss Analysis. The continuous beam spreadsheet presented in the previous post has been modified to deal with 2D frames where any nodes may be connected, and to deal with much bigger structures. This program is designed to apply basic matrix structural analysis to truss members. Derivation Of Local Stiff Matrix For A Truss: FREE: 9:04: 2. You will be given a non-sway frame and a truss. Assemble the system global stiffness matrix and calculate the global displacements of the unconstrained node; b. 002 m 3 x 53. org It’s also worth noting that at no point did the fact that the truss us statically indeterminate feature in our discussion. The extension to a three dimensional space truss will be. In this video I solve a simple truss/bar problem in Excel, using the matrix equations I derived in my previous vi. Then, calculate the vertical deflection of joint 1 and the force in member 2 of the truss. ? do i have to assign any support conditions ? or etab will selelct by default. Metal Truss: Light Gage Truss Design Based on AISI S100-07/SI-10 & ER-4943P: 62: Sleeve Joint Connection: Sleeve Joint Connection Design, for Steel Cell Tower / Sign, Based on AISC 360-10: 63: Moment to Column Web: Moment Connection Design for Beam to Weak Axis Column Based on AISC 360-10: 64: Beam Connection: Beam Connection Design Based on. Q&A forum for midas Civil stiffness matrix (1) story center (1) story data (1). - 20 truss types template - support any unit system - calculation details: - member forces - nodes translations - supports reaction - PDF reports. A force of 200 lbs pushes against a rectangular plate that is 1 ft. Analysis of portal and gable rigid frames using the stiffness matrix method. In case of elements carrying distributed loading, the reaction forces that the distributed load would cause on a single beam element, are added to the load vectors of the. This app can give you the solution in seconds and you can quickly check if you have reasonable results. 2D Finite Element Analysis Spreadsheet Calculator. Stiffness matrix of the member is derived in its local co-ordinate axes and then it is transformed to global co-ordinate system. What is the relation between flexibility and stiffness matrix? 14. This test is Rated positive by 86% students preparing for GATE. Remember that in the integrals of elemental stiffness matrix and elemental force vector we have shape function derivatives with respect to and that need to be converted to derivatives wrt and. To this base triangle, two more members are added to locate a new joint, and the process continued to form the complete truss. The use of natural coordinates in expressing approximate functions is advantageous because special integration formulas can often be applied to evaluate the integrals in the element matrix. Indicate the degrees-of freedom in all the stiffness matrices. stiffness matrix of the truss element subjected to tensile force N can be easily derived. Based on the matrix displacement method the original calculated result leads to the global stiffness matrix the displacement on each unknown node the moments on each unit and so on. 4 Assembly of Global Stiffness Matrix for the Banded and Skyline Solutions 116 Assembly for Banded Solution, 116 Input Data File, 119 Problems 120 Program Listing, 128 S TWO·DIMENSIONAL PROBLEMS USING CONSTANT STRAIN TRIANGLES. Calculate element results. , similarly, k 2 = A 2 E/L 2 = 2. Md + Kd - f - {0. Solve first with hand calculations and then with SAP 2000. you would be probably talking about a spring characteristic when saying “spring constant” during a physics class. Du Nodal deflections matrix Dk Boundary condition of nodes matrix E Modulus of Elasticity (MPa) fy Steel yield strength fu Steel tensile strength FEA Finite element analysis G Dead load h height of geodesic dome k’ Member stiffness matrix kc,i Internal combination factor kc,e External combination factor ka Area reduction factor. Convergence Values: CNVTOL: By default, the program checks the out-of-balance load for any active DOF. Since the resistance of braces acting in compression is ignored in the design, tension-only braced systems must be direct-acting and concentrically braced frames. This work is a prelude to the ongoing parametric study on structural behaviour of prestressed configurations under external loads. QUESTION 2 (25 MARKS) 4 m 3 m 20 KN -4 m Figure 2: Truss Construct the stiffness matrix K for the truss made form Laminated Veneer Lumber (LVL) with cross-section of 200mm x 200mm as Figure 2 above. An icon used to represent a menu that can be toggled by interacting with this icon. The theoretical stiffness matrix (K ) is obtained using the as-built drawings of the structure. Matrix equation and solution to calculate displacements. • This is useful when the transformation matrix is not square - i. Solve the unknown displacements D for using the partitioned K (K11). 5 shows a module that evaluates and returns the 4×4 stiffness matrix of a plane truss member (two-node bar) in global coordinates. To generate the solution to the system of linear algebraic equations (denoted by the matrix equation K ⁢ u = f, where K is the global stiffness matrix, f is the load vector, and u is the desired displacement solution), a sequence of Krylov solver iterations is performed, whereby an approximate solution gets closer to the exact solution at. The length of each element in plane truss is calculated as, = −) + − ). The structural stiffness matrix is formed with the stiffness matrix of each element. Problem 1: Indicate by circling the correct answer whether each matrix below could be a stiffness matrix for a truss. FEM-Design is the most user friendly FEM software for building analysis on the market. Depending on one’s perspective, FEA can be said to have its origin in the work of Euler, as early as the 16th century. Consider element A, Fig. A truss element stiffness requires only the material elastic modulus, E, the cross‐sectional area, A, and the member length, L. After loads and load cases are specified the displacements and forces are calculated by solving a bandwidth minimized stiffness matrix using an. 7 (b) Prove that the stiffness matrix and flexibility matrix are inverse of each other for the beam shown in Figure 6. Spring stiffness and optional mass are provided via the material properties E i and ρ i. It is another good option for civil engineers to use for physical structure analysis. For the truss system shown, E-200000MPa. Having set forth the foundation on which the direct stiffness method is based, we will now derive the stiffness matrix for a linear-elastic bar (or truss) element using the general steps outlined in Chapter 2. For different truss sizes and shapes it is simply a matter of creating and storing separate. Structural Analysis IV Chapter 4 - Matrix Stiffness Method 3 Dr. Remember that in the integrals of elemental stiffness matrix and elemental force vector we have shape function derivatives with respect to and that need to be converted to derivatives wrt and. Elastic instability of frames, and the design of continuous steel beams and portal frames using plastic methods will be undertaken. This is a 32 bit Windows 95, 98 & NT Two-Dimensional frame analysis program. Literature: Schmertmann, J. It is loaded by force P at node N. , when the matrix inverse is not defined. When is a vector field the gradient of a potential?. Do truss rest on the beam ? if yes. SELF ASSESSMENT EXERCISE No. 69 which is subjected to inclined loads. inverting a matrix. Determine 6x6 stiffness matrix, k', for each of the member expressed in global coordinates 8. Programing the Finite Element Method with Matlab. As mentioned in step 1, each 2D truss member is assigned a code vector consisting of 4 numbers. Cheap paper writing service provides high-quality essays for affordable prices. 4: stresses and moments acting over a cross-section of a beam; (a) a cross-. (50 marks) a) Determine the stiffness matrix of the whole truss given in problems 14. The maximum deflection of the truss may be computed by using either strain energy method or matrix stiffness analysis programme. If we have a structural analysis problem with multiple one-dimensional truss elements, we must first define the stiffness matrices for each individual element as described in the previous section. It then outputs a stiffness matrix, displacements, and member forces. At a high level, the global stiffness matrix is created by summing the local stiffness matrices: where the matrix [ k i] is the local stiffness matrix of the i th element. T is a row vector of evenly spaced times. stiffness matrix for a bar arbitrarily oriented in space. Now we can calculate a global element stiffness matrix for each element in our structure that encodes the orientation of the member. Programing the Finite Element Method with Matlab. Preliminary Remarks. It is loaded by force P at node N. 3 Three-Dimensional'Ihlsses 114 4. FINITE ELEMENT MATRIX FORMULATION. Assemble the elemental stiffness and mass matrices to form global stiffness and global mass matrices. Let us now solve for the displacements in a small truss using the above approach (Fig. The free version allows you to input frames with a maximum of 3 members with applied point loads and moments for 2D frame analysis. Fundamental principles of the stiffness method of analysis, with automatic assembly of the stiffness matrix for rigid jointed plane frames and space structures, are presented in some detail. The differential equation governing the displacement of the body (x) and time (t) is given by dx +5 − 12 x = 0 2 dt dt Find the displacement (x) at time, t = 2, given that x(0) = 4 and y ( 0) = – 5. Truss Stiffness Matrix Calculator. RISA-3D is the next free structural analysis software for Windows in this list. (Rajan’s book page 351-353, Example 6. Structural Analysis IV Chapter 4 - Matrix Stiffness Method 3 Dr. When is a vector field the gradient of a potential?. A simple truss. However for almost all frames I enter the data, when the menu Tools/Solve is selected the software simply crashes and disappears. The shape and properties of the generated section can be exported to midas Gen. Design considerations for roof support system include cost-effectiveness, speed of construction In 2010, a precast concrete truss-girder was designed to support the roof of a coal storage facility in. Step 5: Assemblage of the global stiffness matrix for the entire system from the global stiffness matrices of the bars. Use unit load method to find the deflection at the center of the beam shown in figure 7-3(a). 3b – Engineering Problem Solving Answer Key. Dropping these files into the SketchUp folder sets up the menu item. The element attaches to two nodes and each of these nodes has two degrees of freedom. Sename (the name of the super-element matrix file) will default to the jobname. stiffness matrix for a bar arbitrarily oriented in space. fea code in matlab for a truss structurea ppt mathworks. coordinate element stiffness 19 matrix, i. The free version allows you to input frames with a maximum of 3 members with applied point loads and moments for 2D frame analysis. now we have more equations to work with PIN ROLLER 0 1 2 2 A Simple Comparison 6 5 4 3 2 1 Stiffness by Definition 2 Degrees of Freedom Direct Stiffness 6 Degrees of Freedom DOFs 3. (20 points) The truss above is pin-connected and composed of two members with constant area A and second moment of area I, and made of material with Poisson's ratio v and Young's modulus E. • Press the “F2” key. These celebrity activists are champions for social justice; Daveed Diggs: 'I worked harder on The Little Mermaid than anything else' Kodak Black has donated $150,000 to charity in five days since. Uniform bars: Consider the problem of calculating the extension due to the application of an axial load P on a uniform bar as shown in the figure. T is a row vector of evenly spaced times. Section Property Calculator An unit structures such as a frame, an arch, a truss, a plate and a shell. Analysis involves assembling the stiffness matrix and fixed-end-forces then solving P=KU through brute force by inverting the stiffness matrix. Frame3DD is free open-source software for static and dynamic structural analysis of 2D and 3D frames and trusses with elastic and geometric stiffness. Literature: Schmertmann, J. The approach shown here for evaluating the stiffness components is applicable as long as we do not expect any coupling between extension and bending, (i. Previous question Next question Transcribed Image Text from this Question. With clever use of an X-Y scatter plot and cell entries, the bridge is drawn to scale when unique span, radius, and point loads are entered. Application of the Stiffness Method for Truss Analysis. 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. c) Derive the transformation matrix and stiffness matrix of two noded truss member with 04 D. You may, however, apply the Matrix commands from the Symbolics menu to matrices on a case by case basis if you prefer. Problems. Control Board Section on the User Interface where the user will enter all of the. The beam matrices are combined into a global matrix, adding coefficients for the beams meeting at each node. The solution of the system of algebraic equations gives an approximation of the solution to the PDE. The differential equation governing the displacement of the body (x) and time (t) is given by dx +5 − 12 x = 0 2 dt dt Find the displacement (x) at time, t = 2, given that x(0) = 4 and y ( 0) = – 5. A-Truss 3D analyse the space truss by using the Stiffness Matrix Method, providing following features: - Easy to use UI with 3D graphics. FEM-Design is the most user friendly FEM software for building analysis on the market. matlab codes for finite element analysis wordpress com. Once you get to advanced structures, you can just put every point in the structure in a 'stiffness' matrix and solve for all displacements, stresses, etc when a force/s is applied. Vibrations in a long floor span and a lightweight construction may be an issue if the strength and stability of the structure and human sensitivity is compromised. Therefore, it is very easy to use and for many years it has been the best choice for structural engineering. The solution includes:. The shape and properties of the generated section can be exported to midas Gen. Apply 1,2 19 For the truss element shown below, if q = [1. 3 Three-Dimensional'Ihlsses 114 4. In the matrix analysis of structures by the displacement method, the structure is first idealized into an assembly of discrete structural components. Matrix equation and solution to calculate displacements. Linear and Quadratic elements, Elimination and Penalty Approach, Properties of global stiffness matrix; Structural and Thermal strains; Treatment for various boundary conditions. • This is useful when the transformation matrix is not square - i. The immediate objective is the development of a load-dependent element tangent stiffness matrix that can be evaluated for given element forces and is a reliable predictor of a type of behavior in the neighborhood of the loads to which those forces correspond. All the joints in the truss are pin jointed. PLANE TRUSS The basic element of a plane truss is three members (bars, angles,tubes,etc) arranged to form a triangle. The stiffness matrix for each beam is generated as before. The matrix K is singular since the boundary conditions of the. Merge individual member stiffness matrices into stiffness matrix, K, for entire frame 9. But in reality there are many members in a truss. Indeed one can observe that applying a uniform displacement corresponding to a rigid body motion yields no strain energy: 37. Stiffness matrix of the member is derived in its local co-ordinate axes and then it is transformed to global co-ordinate system. The joists are braced together; the effects of this bracing on the stiffness may, however, be neglected. flexural stiffness is 53. A one-dimensional truss is loaded by its dead weight. In case of elements carrying distributed loading, the reaction forces that the distributed load would cause on a single beam element, are added to the load vectors of the. Control Board Section on the User Interface where the user will enter all of the. Users can edit the solution to match how. Jones proposed in 1946 a simplified calculation [2] [7]. 6 and you want, stresses that the state department of the dimensions of structure. The transformation of the stiffness matrices into the global coordinate system and the assemblage of the global stiffness matrix can be done similar to 2 dimensional trusses. Apply 1,2 19 For the truss element shown below, if q = [1. This is an ultimate collection of all my structural programs to solve beams, frames updated porticos and trusses using the Stiffness Matrix Method or Displacements method, giving you the results step by step, drawing the beam, frame or truss, cerchqs also an animation of the structure. Then, calculate the vertical deflection of joint 1 and the force in member 2 of the truss. The theory is then applied to the Kagome lattice and the triangular–triangular T–T lattice in order to obtain the collapse mechanisms, states of self-stress and the macroscopic effective stiffness. Matrix Stiffness Method or Direct Stiffness Method or Matrix Displacement Method or Finite Element Method is a structural analysis method that is used as a b. 3 shows that 0 0 1 C 0 0 0 C 0 [ ] y y vert (12). The member stiffness matrix [S M] for an arbitrary truss member with member axes X m and Y m oriented along the member and perpendicular to the member can be obtained by considering Case #1 and Case #7 from. Using the equation shown in (3. The form-finding analysis was done using the program RFEM that employs the Updated Reference Strategy (URS) based on iterative regularization of the stiffness matrix during the process. In the development of the truss equations, we started with Hook’s law and developed the equation for potential energy. %##### %%%%% Version 2. As mentioned in step 1, each 2D truss member is assigned a code vector consisting of 4 numbers. The structural stiffness matrix is formed with the stiffness matrix of each element. you would be probably talking about a spring characteristic when saying “spring constant” during a physics class. This test is Rated positive by 86% students preparing for GATE. ? do i have to assign any support conditions ? or etab will selelct by default. Write the element stiffness matrix for a beam element. Truss, Plate, Plane stress, Plane strain, Axisymmetric and Solid Tresca, von Mises, Mohr-Coulomb and Drucker-Prager Isotropic, kinematic and mixed hardening. Section Property Calculator An unit structures such as a frame, an arch, a truss, a plate and a shell. The method is outlined in the following instructions. 4 Stiffness Matrix of Beam Elements 179 6. I'll probably use SVG+JS for the GUI. The order of AB is then n×p. b) Analyse the pin jointed truss as shown in fig. Users can edit the solution to match how. Let us now solve for the displacements in a small truss using the above approach (Fig. Fundamentals of the Stiffness Method. analytical solution giving the bearing stiffness. To assemble the Global Stiffness matrix, element-by-element addition is undertaken. A Finite Element Solution of the Beam Equation via MATLAB. Free Structural Frame Calculator Cloud Structural Software. k 1 = A 1 E/L 1 = (86. 3 x 10 5000 x 4-3EI FL y 6 3 The deflection is 2 mm downwards. Figure 7-3(a) Solution: In the case of unit load method the deflection at a point of beam is given as. A one-dimensional truss is loaded by its dead weight. If you are in high school. Minimized stiffness matrix is necessary are deemed to use. now we have more equations to work with PIN ROLLER 0 1 2 2 A Simple Comparison 6 5 4 3 2 1 Stiffness by Definition 2 Degrees of Freedom Direct Stiffness 6 Degrees of Freedom DOFs 3. The reduced theoretical stiffness (K ) is generated by condens-ing the theoretical stiffness matrix (K ) to 2-DOF stiffness matrices. Matrix Stiffness Method or Direct Stiffness Method or Matrix Displacement Method or Finite Element Method is a structural analysis method that is used as a b. The Global Stiffness Matrix. Truss Angle Calculator. by maryam ehsaei. To start, what if you had to different materials that are connected in parallel? To find the stiffness of the overall part refer to the image below. Solving, 3 m + r = 16 and 3 j + e c = 14, Again, this structure is found to be 2 degrees internally statically indeterminate. 2) Displacements are generated for N (=10^5) simulations and a histogram is plotted to determine the. With the option SEPR, the stiffness matrix or load matrix can be printed to the output window if desired. Vibrations in a long floor span and a lightweight construction may be an issue if the strength and stability of the structure and human sensitivity is compromised. Stress in a cylinder. Consistent Mass is calculated with the shape function used to derive the stiffness matrix. For different truss sizes and shapes it is simply a matter of creating and storing separate. The same thing is true for the bridge of the truss. A compound column is built up of two 300 mm x 125 mm R. This element can be used for finite-element analysis of elastic spatial frame structures. 3 shows that 0 0 1 C 0 0 0 C 0 [ ] y y vert (12). Truss Stiffness Matrix Calculator. zip - the download file includes complete open source code. An icon used to represent a menu that can be toggled by interacting with this icon. Element stiffness matrices b. This is a 32 bit Windows 95, 98 & NT Two-Dimensional frame analysis program. It is widely used by civil and structural engineering lecturers and students because of its clear and thorough style and content. 1) The structure stiffness matrix [S] is. The 2D Finite Element Analysis (FEA) tool allows the analysis of any 2D structure using beam elements. The method of correlation between internal forces of optimum elements for weight optimization of trusses. The form-finding analysis was done using the program RFEM that employs the Updated Reference Strategy (URS) based on iterative regularization of the stiffness matrix during the process. The element stiffness matrix for this problem is shown above. In practice, this is applicable to structural problems where stresses remain in the linear elastic range of the used material. The immediate objective is the development of a load-dependent element tangent stiffness matrix that can be evaluated for given element forces and is a reliable predictor of a type of behavior in the neighborhood of the loads to which those forces correspond. 4/8 Truss Analysis using the Direct Stiffness Method (continued). A simple truss. Users can edit the solution to match how. what will be the grade of steel for the truss ? yield or tensile strength. Matrices are used to describe the stiffness of structures and are transposed, inverted, multiplied and added to gain solutions. Use unit load method to find the deflection at the center of the beam shown in figure 7-3(a). Slope Using formula 2E we have 750 x 10 6 (no units) 2 x 53. We will present a more general computational approach in Part 2 of this blog series. the design program provides an economic and efficient design of the main frames and allows the user to utilize the program in different modes to. , determine the following: a) The vector q’ b) The stress in the element c) The K matrix. The SAP name has been synonymous with state-of-the-art analytical methods since its introduction over 30 years ago. In this C Program to find Upper Triangle Matrix, We. Build element geometrical stiffness matrices due to N. A truss is statically determinate if' it has not mare than. The load is always acting on the joints. Now we can calculate a global element stiffness matrix for each element in our structure that encodes the orientation of the member. Truss Force Calculator Software. Pro is a 3D Structural Analysis and Design Software developed by Bentley. Truss Member 11 The transformation matrix given in (11) is valid for all space truss member orientations with the exception of a vertical truss memberas shown in Fig. Let us now solve for the displacements in a small truss using the above approach (Fig. M: Direct stiffness method – introduction. Members are oriented in different directions and hence before forming the global stiffness matrix it is necessary to refer all the member stiffness matrices to the same set of axes. Derivation Of Global Stiff Matrix For A Truss (Part 2) FREE: 7:38: 4.