Structural Analysis of an Aluminum Spiral Staircase EMCH 407 Final Project Presented by: Marcos Lopez and Dillan Nguyen
Abstract An old aluminum spiral staircase at Marcos home has been feeling really unstable lately; see Figure 1. This raises the question of how much weight one step can really hold. The focus of this design project is to model a single step in CATIA and Abaqus and inspect how structural loads are handled when loaded through finite element analysis, particularly examining stress concentrations. This project looks at loads applied in both programs and discerns any differences between the two results. We expect both programs to show more stresses at the end of the step attached to the center pole of the staircase with more displacement on the free end. Since the railing is not load bearing, we decided to not model it, as it was not relevant to the objective. It was found that, however, our predictions were generally correct, the visualization results of the programs showed slight differences. Figure 1: Actual staircase to be analyzed 1
Introduction The subject of our research is the aluminum step. An object that has to endure varying stresses daily, much testing and dedication in perfecting the final model has to be assured. As one step will need to hold the weight of one person, we will analyze the stress it will need to bear due to the load from the person. If the step does not hold the recommended weight, it could ruin the integrity of the entire staircase, holding the manufacturer liable for any injuries or even death, potentially costing the manufacturer millions of dollars. Determining in our research what loading capacity the step will support will answer what the manufacturer wants to set as the maximum weight limit. We figured our material would be made of aluminum, specifically Al6061, as it will be strong and lightweight and the most inexpensive for the manufacturer. We ll strive to have a single step support up to 650 lb, which we believe is sufficient to support most people while also carrying any extra loads. It could also support three people who each weigh about 180 lb for those special cases. The design of the steps of the staircase will be analyzed by using two different CAD software tools, CATIA and Abaqus. By performing a static finite element analysis on the step, the Von Mises stresses and deflection will be determined for a 650 lb load. Furthermore, the results of this study could lead to design modifications that can be implemented upon analysis to optimize the structure but still support the required loads. This sort of optimization would cut down on material, which would reduce production costs. 2
Approach We will be mainly testing for structural failure and improving from there. The 2D model features a 2.24 ft cantilever beam with a distributed load of at least 650 lb in the downward direction at the free end of the beam to ensure the step meets the load requirement. For the sake of thoroughness, we will create one finite element model for testing in both programs. The model will be created and analyzed in CATIA, and then it will be imported into Abaqus for another analysis. The max load the stair can support without deforming plastically will also be determined by applying the material s yield strength. The results obtained from the two software tools will be compared in order to determine the most accurate method. CATIA Modeling The exact dimensions of a single step from the staircase were measured using a ruler. These measurements were used to create a CAD model of the step using CATIA. The step has a length of 26.8 in, and a plate thickness of 0.1 in. Material properties of aluminum were applied to the entire solid model. These can be seen in Table 1 below. Aluminum Material Properties Young's Modulus 1.015e7 psi Poisson ratio 0.346 Density Yield Strength 0.098 lb*in^3 34954.086 psi Table 1: Material properties of aluminum applied to the solid model 3
The final CAD model of the step with the aluminum material applied can be seen in Figure 2. Using this model, a static finite element analysis was performed. Figure 2: CAD model of step in CATIA Abaqus Modeling Choosing to remain consistent across both models, we imported the model into Abaqus as an.stp file. The units for the dimensions of the model were in mm, so we had to adjust any values for the entered properties. We also created and assigned a material with aluminum properties as defined in Table 2. Figure 3 shows the assembled model in Abaqus. Aluminum Material Properties Young's Modulus 70000 MPa Poisson ratio 0.346 Density Yield Strength 2.71e- 9 tonne/mm^3 34954.086 psi Table 2: Material properties of aluminum applied to the solid model 4
Figure 3: Model of step in Abaqus CATIA Finite Element Analysis A linear mesh was applied to the entire CAD model, having an element size of 0.5 in. This element size resulted in 13,348 finite elements for the model. A restraint was also applied to the model on the inner surface of the supporting cylinder. This metal ring is physically clamped to the main column of the spiral staircase, supporting the entire load placed on the step. This restraint can be seen in Figure 4. Figure 4: Applied restraint on the inside of the "metal ring" 5
A distributed load with a magnitude of 650 lbf was then applied to the top surface of the step. This load is almost equivalent to 3 persons with an average weight of 180 lbf, (540 lbf total) and a safety factor of about 1.2. The final element model, with all of the restraints, loads, and mesh can be seen in Figure 5 below. Figure 5: Linear mesh applied to the CAD model. An element size of 0.5 in was applied to the model. Abaqus Finite Element Analysis The software to perform and run these experiments was Abaqus version 6.7-5. After importing the model, the entire step was considered as one part. Since our model will be made of one material, the part will be a solid homogenous section assigned with the material properties previously defined. We instanced the part as independent since it didn t depend on other parts to operate. Then we applied the load and boundary conditions. The cylindrical object connected at the end of the step is supposed to be connected to the pole at the center of the staircase, so we will encastre it, so it will not have any displacements 6
or rotation in any direction. To apply the load of a person stepping on the step, we applied a distributed load of 2891.34 N across the surface in the z- direction. Figure 6 shows these loads and boundary conditions. We then assigned global seeds with approximate global size of 12. Figure 6: Loads and boundary conditions We initially used the default size of 37 for the model, and after running different sizes, we found 12 demonstrated less than 5% difference in results than 24, so we decided 12 provided accurate enough values. The element type is tetrahedron as there are round elements for the meshing. There were 14,524 tetrahedral elements. Figure 7 displays the meshed model. 7
Figure 7: Meshed model We then ran the simulations using automatic time stepping. The CPU was an Intel Core i5-2400 CPU @ 3.10 GHz using only one processor. The runtime for analysis for the model varied between 12-15 seconds. Limitations include not taking into account outside elements like creep. 8
Results and Discussion CATIA In order for the simulation to run, CATIA had to use 0.6 sec of CPU, 2.96 e3 kilobytes of memory, and 9.88 e3 kilobytes of disk. The simulation ran in a computer with an Intel Core i5-2400 CPU @3.10GHz. The simulation outputs a maximum Von Mises Stress of 12305.2 psi, located on the sides of the stair. Because bending occurs on the step, compressive stresses are created on the sides of the steps that are connected to the clamped metal ring. Given that the maximum stress is lower than the material s yield strength (34954.086 psi), no plastic deformation will occur on the part with this applied load. Figure 8: Von Mises stress distribution on the step CAD model An elastic deformation does occur to the step, having a maximum value of 0.153 in. The location of this maximum deformation is located on the outer edge of the step, as expected. This is due to the longer moment arm relative to the clamped 9
metal ring. The longer the moment arm, the larger the actual moment experienced at that point, and thus a greater deformation occurs. A representation of the deformation that occurs on the step can be seen in Figure 9. Figure 9: Elastic deformation distribution on the step CAD model The maximum load that the step could support was also determined by using CATIA s yield strength for aluminum (13,778.58 psi), which is different from our reference (34,954.086 psi). [3] The load was increased from 650 lbf incrementally by 50 lbf until this yield strength was surpassed. Using CATIA s value for the yield strength, it was determined that the maximum load the step could sustain was 730 lbf, creating a maximum Von Mises stress of 13,646.89 psi. Our actual step could sustain a much higher stress since it has a higher yield strength than CATIA s, but we figured 730 lbf was safe enough to list as the manufacturer s recommended limit, rather than pushing it past the lower bounds of the yield strength range in case a different type of aluminum is used. 10
Abaqus Figure 10: Abaqus results We found the average stress to range between 72.28-84.32 MPa. The maximum relative stress can be found at the bottom of the step that ranges between 96.37-132.5 MPa. We can see the stress concentration points occur where the wall of the step meets the cylindrical base. This makes sense because these are the areas where the step is connected to a fixed point, and the downward force causes stresses at the top and bottom, with most of it in the bottom region. 11
Figure 11: Displacement results in Abaqus When examining the displacement of the model, we can see the largest displacements occurred at the free end where there is no support from the center column. We observed in Abaqus that the model produced a maximum displacement of around 0.652 cm. This displacement occurred in the free end of the step as illustrated in Figure 11. The deformation is exaggerated to magnify the displaced elements. This was to be expected since there is no load bearing support in this area. The end of the step that is connected to the vertical column of the staircase experienced minimal to no displacement at all, which is also expected since the column keeps the step fixed in this position. Figure 12: Before & after displacement results 12
Conclusions The results from the finite element models prove our hypothesis that our biggest concern will occur at the points where the step connects to the metal column, specifically towards the bottom. According to the Nanovea document, the yield strength of aluminum is around 34,954.086 psi. [3] The CATIA analysis says our step can sustain up to 750 lb before yielding. Our stress results from Abaqus show that 650 lb will not cause any plastic deformation in the structure, and it will definitely not fracture. Using the found maximum stress values and the actual yield strength, the factor of safety was found to be approximately 1.67. This conclusion is what we expected. Visually, there are only slight differences for the stress concentration points. CATIA shows a larger maximum stress area near the lower joint where the step actually connects to the metal ring. The Abaqus model has a smaller maximum stress area, but this area is closer to the joint. Our results in CATIA and Abaqus show a difference in stress values by about 2000 psi. The initial difference value was much higher. We were unsure what might have caused this difference until we realized that when we imported the model into Abaqus, the dimensions were in mm. With the help of an online reference, we fixed then fixed the material properties from SI to SI (mm) to reflect this change in units. [4] We were able to lower the difference after the conversion. We were satisfied that these stress values were in the same range and of the same order of magnitude. 13
Improvements One of the concerns we encountered was that we also tested our model using a different material from aluminum. We tested the model in Abaqus using steel for the material properties. We expected a vast difference in the stress magnitudes from aluminum to steel as steel has three times the elastic modulus of aluminum s. This was not the case as the stresses were nearly similar. For future results, we will use more of our resources by determining what could be the issue with the course s TA. We also could ve looked at the max principal stresses for the model for more accurate readings in determining if there is structural failure. As for the differences in values from both programs, we could ve also looked into exactly why that occurred. We hypothesize that each program has different methods/formulas in calculating the stresses, which may also be on our end in our decision of element shapes, the number of elements, and whether or not we used quadratic geometric order for the meshing. 14
References Finite element model analysis using computer software: [1] 3DS s CATIA V5 [2] 3DS s Abaqus/CAE 6.7-5 Online references: [3] http://www.nanovea.com/application%20notes/yieldstrengthtesting.pdf [4] http://www.eng- tips.com/viewthread.cfm?qid=296017 15