Computing Project 4: Truss Analysis [ CEE/CNE 210—Statics ]

Question # 40677
  • Engineering
    5 months ago

    CEE/CNE 210—Statics SSEBE Mechanics Group
    Arizona State University
    Computing Project 4: Truss Analysis
    Computing Project 4 builds upon CP3 to develop a program to perform truss analysis. A truss consists of straight, slender bars pinned together at their end points. Truss members are considered to be two force, axial members. Thus, the force caused by each truss member - and the internal force in each member - acts only along it’s axis. In other words, the direction of each member force is known and only the magnitudes must be determined.
    To analyze a truss we study the forces acting at each individual pin joint. This is known as the Method of Joints. We will call each pin joint a node and the slender bars connecting the nodes will be called members. The previous project computed a unit vector to describe the vector direction of every member of a truss structure. To analyze the structure a few other key inputs must be included like the support reactions and external loads applied to the structure. With all of this information, you will need to make the correct changes to the provided planar (2-D) truss template program to be able to analyze a space (3-D) truss.
    What you need to do
    For a planar truss, every node has 2 degrees of freedom, the e1 and e2 directions. Therefore, for every planar truss problem, the total number of degrees of freedom (DOF) in the structure is equal to 2 times the number of nodes. We will consider the first degree of freedom for each node as the component acting in the e1 direction. So for any given node, i, the corresponding degree of freedom is (2·i)-1. For the same node, i, the corresponding value for the second degree of freedom, the component in the e2 direction, is 2-i. This numbering notation can be modified for a space truss. The difference with the space truss is that every node has 3 degrees of freedom, one degree for each of the e1, e2 and e3 directions. The degree of freedom indices are extremely crucial in understanding how to set up the matrices for the truss analysis.
    For this computing project, you will first need to understand the planar truss program and the inputs that are needed for that program. The first input is the spatial coordinates (x, y, z) of the nodal locations for a truss. It is convenient to label each node with a unique number (also known as the “node number”). Each row of the nodal coordinate array should contain the x and y coordinates of the node. We will use the matrix name of “x” for all nodal coordinates. Please note that “nNode” is an integer value that corresponds to the number of nodes in the truss and must be adjusted for every new truss problem. For Node 1 this matrix array input looks like:
    x(1,:) = [0,0];
    Once the coordinates of the nodes are in the program, you will need to input how those nodes are connected by the members of the truss. In order to describe how the members connect the nodes you will also need to label each member with a “member number”. This connectivity array should contain only the nodes that are joined by a member, with each row containing first the node at the start of the member and then the node at the other end of the member. The first node that is input into the Member array will be called the start node (SN) and the second node that is input is known as the end node (EN). For any member it doesn’t matter which end you call the start node and which is the end node. We will use the matrix name of “Member” for the connectivity array. Also note that “nMembers” is a single integer value that must equal the total number of members for each truss. The nodes and members can be numbered in any way you want, but it can sometimes be advantageous to use some sort of logical systematic order if you can discern one. For Member 1 this matrix array input should look like:

    Member(1,:) = [1,2];
    The support reactions for the truss must then be input. The variable “nReactions” is an integer that corresponds to the total number of supports. Note: a pin support counts as one reaction for the nReaction count even though it has two unknowns. The support reactions will be input into the “Reaction” matrix. The first input value is the node number of where the support reaction is. The second and third inputs are for the fixity conditions at that node. If the support reaction provides resistance in a direction then a 1 should be used otherwise a 0 is used to denote that the truss is free to move in that direction. For example, if a roller support was used at node 4 and had resistance in the e2 direction but was free to move in the e1 direction it would be input as follows:
    Reaction(2,:) = [4,0,1];
    Finally, the external loads must be input. The variable “nLoads” is a single integer value that corresponds to the number of loads placed on the truss. The matrix name of “Load” will be used to store all external loads. The first input is the node number where the load acts. The second and third inputs are the magnitude of the force in the e1 and e2 direction at that node. For a load placed at node 3, that has a magnitude of 1200 N in the –e2 direction should be input like:
    Load(1,:) = [3,0,-1200];
    Next, the unit vectors (direction vectors) must be computed for every member. The direction of the unit vector will depend on the node of the connection you chose to be the start node and the node that was selected as the end node. Implement your CP3 code (the part that computes the unit vectors) into the code. Use the matrix name “UnVec” for the matrix that will store all the positive unit vector values (the vectors pointing from SN to EN). Use the matrix name “OppUnVec” to store all the negative unit vector values (the vectors pointing from EN to SN).
    Once all the unit vectors are created for every member, the coefficient matrix is formed. We will call this matrix “C” and it will have the size of (DOF, Members). For each member the SN and EN must be deter-mined. These nodal values have components in the member unit vector array that correspond to a degree of freedom value in the truss. This component from the unit vector array is stored in the matching degree of freedom row and corresponding member column in the “C” matrix. Note: the negative value of the unit vector should be stored for every end node.
    The load vector is then created. The load vector has the same number of rows as DOF. The external loads have the same corresponding degree of freedom value as the nodal coordinates, so a loop is used to create this vector to store the load values in the correct row. The array name of “F” will be used to store all external forces.
    Once the coefficient matrix has been created, this matrix must be partitioned into two separate matrices. The first matrix contains the rows that correspond to a degree of freedom value that has a support reaction, and the second matrix will contain the rows that are not restrained. For every support reaction using the corresponding degree of freedom, that row from the coefficient matrix should be placed in the “Crest” matrix. The “Crest” matrix should have the same number of rows as the number of degrees of freedom that are restricted in the system (i.e. support reactions – 2 for a pin and 1 for a roller).

    When the reaction force matrix has been created then the member force matrix is created using all the rows not in the “Crest” matrix. This matrix can be labeled “Cfree”. This member force matrix should be a square matrix of the size (Members, Members) for any statically determinate truss. The load vector must also be partitioned to contain only the degree of freedom values that are not restricted by a support. The new load vector (called “Ffree”) should have the same number of rows as the number of members.
    Once the free coefficient matrix has been created then an array named “T” solves for the individual member forces using

    {T} = - [ Cfree ]^-1 { Ffree }

    Then an array named “Reactions” solves for the support reactions using
    {Reactions} = - [Crest]  {T}
    Once the reactions are calculated a “SupportReaction” matrix is created that contains all the values of the support reactions in the corresponding degree of freedom spot for each node. The first input for the support reaction matrix is the node number of the support and the second and third input are the corresponding reaction value for the two degrees of freedom for that node. Note: if there is no resistance for a degree of freedom then the value should be 0.
    The template file provided to you will work for any planar truss once you implement the unit vector calculation code. You need to get the planar truss code working and understand it. Then make all the necessary changes to create a space truss program. The changes do not need to be very extensive. Note: when your code is graded there will be at least two different truss problems run in your code to ensure your code is robust.
    Also provided is a truss data input file. Use this file to create several input sections that can be easily copied and pasted into the input section of your truss analysis code. This allows you to create and store several trusses (2-D and 3-D) that can be analyzed quickly and easily. Be sure to test your code with a few different 2-D trusses and several 3-D trusses.
    Write a report only after you have a working 3-D truss analysis code. Document your work and the results (in accord with the specifications given in the document CEE210 Guidelines for Computing Project Reports). Include figures, plots, and results. Discuss your discoveries and explorations. All test cases need to be included in your report with detailed documentation of the problem (i.e. a figure and tables) and the results (i.e. a table of the member forces and reactions). For every truss you test, there needs to be back-up calculations to verify that the code is working correctly (either hand calcs or copies of existing problem solutions with proper source citations). The back-up calculations should be placed in the Appendix of the report. Also include a copy of your MATLAB .m program code and you truss input data .m cod in the appendix.

    Upload your report to Blackboard prior to the deadline. Also upload your MATLAB .m truss analysis program file and your truss data input .m file to Blackboard as well. Make sure your .m and report files include your first and last name in the file name.


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