Analysis of a 2D Truss Analytically and Finite Element Method in MATLAB and Abaqus
Abstract
Trusses have been an important part of structural mechanics and their analysis is always been under constant process of improvement and development. In this study a simple truss was analyzed using different techniques including method of joints and finite element method. For FEA, MATLAB and Abaqus are utilized. Truss in this problem very simple and consists of 6 members. It has a roller and a pin support along side with the concentrated force at one node. Model is solved and the results are exported for further analysis. Maximum von misses stress and displacement are found out to be 1.898 MPa and 0.1762 mm at node 1 and 5 respectively. All of the three results are compared in the discussion section and they have less than 1% of absolute error. It is determined that any of the above-mentioned techniques can be used to solve a truss problem but using FEA and especially a software like Abaqus can reduce computation effort and time in most of the cases.
Analytical Method
Method of joints is used for solving the truss here
Then the equations of static equilibrium were applied in order to get the support reactions Ax, Bx and By.
⇒ΣFx=0 →+
Bx+Ax=0
⇒ΣFy=0 ↑+
By-L=0⇒By=10 kN
⇒ΣMA=0 ↺+
-2 mBx-4 m10 kN=0
Bx=20 kN
And by putting it in Eq. 1, Ax can obtained as,
Ax=-20 kN
FEA using MATLAB
A function named TrussAnalysis was used and is given in the appendix along with the input file. All of the required data is given in form Nodes, elements, properties, boundary conditions and the loads. This function solves FEA like someone does with hands but much fast and accurately. Here is the 10-by-10 global stiffness matrix assembly from MATLAB:
And then multiplying it with the displacement vector as shown below in Equation, forces and unknown variables are obtained.
f=Kd
Further all of the displacements and nodal forces are listed in table below
FEA using ABAQUS
FEA stands for Finite Element Analysis which is a method in which a part is discretized into multiple elements and then those elements are solved individually to get the overall results. It is a very common and popular technique for structural and thermal analysis.
Discussion
It can be seen that that from all of the above-mentioned methods results obtained are very close. In method of Joints the reaction force at A and B are found to be exactly the same as by MATLAB and Abaqus. Further in the case of odal displacement it is seen that Abaqus and Analytical results are closer to each other and MATLAB results are bit different. But in case of elemental stresses both MATLAB and Abaqus are very close with an A.E. percentage error of less than 1. Practically, there is no difference in the readings at all. All of the above results are checked through multiple online available tools and they are valid.
Conclusion
All of the techniques used in this study have their own pros and cons. Like for a quick force analysis of truss method of sections or joints is the way to go. But if one needs more in-depth analysis like each elemental stress and even the minor details then using a software package like Abaqus or Ansys Workbench is more suitable. MATLAB is more suitable to those who are closer to the programming area. These methods reduce the computation time and provide mental relief in case of lengthy and complex problems. Other than that, they provide us with graphical plots which are more helpful and efficient in getting quick and sound analysis.
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