Application of the Finite Element Method Using MARC and Mentat 5-1 Chapter 5: Analysis of a Truss 5.1 Problem Statement and Objectives A truss will be analyzed in order to predict whether any members will fail due to either material yield or buckling. The geometrical, material, and loading specifications for the truss are given in Figure 5.1. Each member of the truss has a solid circular cross section. Geometry: Material: Steel Area of members 1 and 2: 30 cm2 Yield Strength: 250 MPa Area of members 3 and 4: 20 cm2 Modulus of Elasticity: 200 GPa Area of member 5: 25 cm2 Poisson’s Ratio: 0.3 Loading: Vertical Load: P=18kN Figure 5.1 Geometry, material, and loading specifications for the truss. 1 m 1 m 3 m 3 m P 30o 1 5 2 3 4Application of the Finite Element Method Using MARC and Mentat 5-2 5.2 Analysis Assumptions 1. No friction is present in any of the truss pin joints. Thus, each truss member is an ideal two-force member. Also, there is no friction between the ground and the rollers. 2. Deflections are small enough that geometrically linear analysis is valid. 3. Because the geometry, material properties, and loading conditions are all symmetric about the vertical plane of symmetry that passes through member 5, the response of the structure (i.e., displacements, strains, and stresses) will also be symmetric about this plane. Therefore, a symmetric model may be used. 5.3 Mathematical Idealization Other than the assumptions above, no additional simplifications need be made for a truss. Each truss member can be represented by a two-noded linear truss finite element. This model should yield the correct analytical values for displacements and stresses. 5.4 Finite Element Model The finite element model of this structure will be developed using 3D linear two-noded truss finite elements. The present analysis can be greatly simplified by taking advantage of the vertical plane of symmetry in the truss. Hence, it is necessary to model only one-half of the truss, as shown in Figure 5.2. The boundary conditions on the symmetry plane are those that occur naturally on this plane, as can be verified by obtaining a solution using the entire truss. Taking advantage of symmetry reduces the modeling effort, the amount of computer memory, and the amount of CPU time required to obtain a solution. Admittedly, the savings are small in this simple problem. When a reduced model is developed due to symmetry, note that loads and truss members that lie within the plane of symmetry should be treated in a special way. If a load is within a plane of symmetry, then only one-half that load should be applied to the symmetric model. Similarly, if a truss member lies in a plane of symmetry, then only one-half of the cross-sectional area of that member should be assigned to the element representing the member. In other words, member 5 will be modeled as having one-half of its actual cross-sectional area.Application of the Finite Element Method Using MARC and Mentat 5-3 5.4 Model Validation The structure under consideration is statically determinate. Therefore, simple hand calculations can be performed to calculate the reactions and internal forces in the truss members. These results should be almost identical to the finite element results, with the only difference being due to round off errors. Additionally, it is always good practice to examine the predicted deformed shape of the structure to ensure that boundary conditions are properly satisfied and that the structural deflections are in the expected directions. In the present model, the deflection of the center joints should be downward, in the direction of the point load. At the same time, the joints on rollers should move along the rolling direction. A good way to verify these motions is to superpose a properly scaled deformed mesh on the undeformed mesh. 5.5 Post Processing The axial stress in any member can be estimated by assuming that the axial stress is uniformly distributed over the cross-section. The resultant force at any cross-section is known from the analysis, so the axial stress in the ith member can be estimated using the relation: σiiiPA= where Ai is the cross-sectional area and Pi is the axial force in the ith member. Figure 5.2 One-half domain to be modeled. 1 m 1 m 3 m P/2 30o 1 5 3Application of the Finite Element Method Using MARC and Mentat 5-4 Alternatively, iσ can be calculated using the one-dimensional Hooke’s law iiEεσ= , where iiiLL∆=ε is the strain in the ith member. The change in length iL∆ can be calculated using the distance formula: ( ) ( )[ ]( ) ( )[ ]{}iiLvyvyuxuxL −+−+++−+=∆212112221122 where jjyx , and jjvu , (j = 1, 2) are the nodal coordinates and nodal displacements, respectively, of the ith member. It follows that ()jjjjvyux ++ , is the position of the jth node of the ith member after deformation. Note that j = 1, 2 refers to the local (element) node number. The axial stress in each member should be compared to the yield stress to determine if material failure will occur due to the applied loading. Note that this is a uniaxial state of stress. When a truss member carries compressive load, the possibility of buckling should be examined. For the purpose of buckling analysis, each member in the truss can be considered a pinned column. Hence, Euler’s formula can be used to predict if a member will buckle: PEILcr=π22 where I is the second moment of the cross-sectional area and L is the original length of the member.Application of the Finite Element Method Using MARC and Mentat 5-5 PROCEDURE FOR ANALYSIS OF THE TRUSS 1. Add points to define geometry. 1a. Add points. <ML> MAIN MENU / MESH GENERATION <ML> MAIN MENU / MESH GENERATION / PTS ADD Enter the coordinates at the command line, one point per line with a space separating each coordinate. > 0.0 0.0 0.0 > 3.0 1.0 0.0 > 3.0 2.0 0.0 The points may not appear in the Graphics window because Mentat does not yet know the size of the model being built. When the FILL command command in the static menu is executed, Mentat calculates a