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Structural Mechanics
Practical Work
Finite Element Method - Scale and PrattStructure
Mayur Srivatsav V S
Jerol Soibam
November 2015
ISAE - ENSMA
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1 Introduction
A Scale structure is a series of rectangular frames which achieves stability bythe rigid connection of the vertical members. In this structure, shear is transferredfrom the chords by the bending moments at the joints and by the bending mo-ments in the vertical members. Though the diagonal members are absent, bendingin all members results in chord sizes and vertical members significantly larger in
cross sectional area. This structure has a major advantage when the mechanical re-quirements are extensive and require room to accommodate adequate working space.
APratt structure is identified by its diagonal members, and these diagonal mem-bers are slant towards the center of the span. The diagonal members are subject toTension forces, and the vertical members are subject to the compressive forces. Thetension in the diagonal members eliminates the risk of buckling, and thus thinnerdiagonal members can be used which results in weight reduction of the structure, inturn making the design more economical.
In this study, we investigate both the structures experimentally, and using theFinite Element Method. For the Finite Element study (i.e. Numerical method),we make use ofFEMGEN and Abaqus. The material properties, boundary con-ditions, degrees of freedom, and the physical dimensions are fed as input parametersin the software to obtain the results, then we compare the results obtained fromboth the methods to get a broad understanding about these structures.
2 Scale Structure
The geometrical representation of a Scale Structure is shown below:
Figure 1: Trellis Echelle
We study the structure in two methods: Experimental method
Simulation on Abaqus
2.1 Experimental study of the Scale structure
We consider three cases for loading the structure,
Case 1: Loading at node 6 or 14
Case 2: Loading at node 10
Case 3: Loading at node 6 and 14
The deflection is measured at center node. The load cases are considered withmasses 1kg, 2kg, 3kg, 4kg, and 5kg.
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The experimental setup is shown below:
Figure 2: Treillis Echelle experimental setup
Figure 3: Deflection vs Load
Once the Deflection vs Load graph has been plotted, a Linear equation can be gener-ated using the experimental values, using which we determine the deflection at anygiven load. From the graph we see that the lines representing the linear equations
do not start from the origin. This signifies the error in the experiment.
This is caused due to various factors:
Human errors while performing the experiment affects the measurements.
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Energy losses at the supports due to Friction.
We are assuming the reactions in only x and y directions and neglect theTorsional forces in the z direction. Hence there is some error as we are dealingwith a 3 Dimensional structure.
In this case we determined the deflection for a load of 50kg since we carried out the
Abaqus simulation for the same load.
2.2 Numerical Study of the Scale structure
The Scale structure is simulated on the ABAQUSsoftware and the vertical displace-ments are noted for the three cases (loading at node 6, node 10, node 6 and 14).The simulation is carried out by considering the load as 50 kg.
Figure 4: Displacement at node 6
Figure 5: Deflection of beam for loading at node 6
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Figure 6: Deflection of beam for loading at node 10
Figure 7: Deflection of beam for loading at node 6 and 14
Figure 8: Comparison of Experimental and Simulation results
2.3 Discussion
A brief comparison between the experimental results and simulation results showsus that there is a huge deflection error. The deflection obtained experimentally at
Node 10 for 50kg is 4.48 mm, and the deflection obtained using Abaqus is 9.21mm. It is seen that the experimental value is almost half that of the simulationvalue. The same holds good for the nodes 6 and 14 as well. This error is becauseof the dimensional correction performed during the Simulation i.e. the length ofthe vertical member is taken as 70mm in the software, whereas the actual length is105mm. In order to minimize this error, we perform dimensional corrections in sucha way that the beam characteristics remain unaffected.
Deflection is given by,
=F l3
EI; 1= 2
where subscript 1 refers to the Original scale structure and 2 refers to,
FL3
EIz1=
FL3
EIz2
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L31
l31
=L3
2
l32
=105
l2=
70
10
l2= 15mm
We now use the modified dimensions and run the simulation again, and obtain thefollowing results:
Figure 9: Displacement at node 6
Figure 10: Displacement at node 10
Figure 11: Displacement at node 6 and 14
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Figure 12: Comparison of Experimental and Simulation results
The above table represents the comparison between the Original scale struc-ture and the modified one. We now observe that the value of deflection obtainedexperimentally is in close agreement with the value obtained using Abaqus.
2.4 Calculations
Once the results are obtained through Numerical analysis, the next stage is tocalculate the Normal and Shear Stress by making use of the minimum and maximumstresses obtained from the Abaqus simulation. This has to be done for both the cases.
Figure 13: Maximum and Minimum Stress
top = 306MPa; bottom= 306MPa
tensile = top+ bottom
2 = 0
shear = top bottom
2 = 306Mpa
The elastic limit for the structure is 180 Mpa, but for a load of 50kg, the Stress is306 Mpa. We now calculate the allowable load for the given Elastic Limit.
306Mpa 50kg
180Mpa ?(w)
w=180 50
306
w= 29.41kg
To calculate the Maximum Bending Moment,
Mmax = shear Iz
y
The Moment of Inertia is given by,
Iz = bl3
12 =
2 153
12
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Iz = 562.5mm4
Mmax=306 562.5
7.5
Mmax = 22.95Nm
2.5 Comments
From the above calculations it is seen that the maximum load the Scale structurecan withstand is 29.41 kg for the given maximum stress. Application of loads higherthan this value will lead to the failure of the structure.
3 Pratt Structure
The geometrical representation of a Pratt Structure is shown below.
Figure 14: Treillis Pratt
We study the structure in two methods:
Experimental method
Simulation on Abaqus
3.1 Experimental study of the Pratt structure
The experimental setup is shown below:
Figure 15: Treillis Pratt experimental setup
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We consider three cases for loading the structure,
Case 1: Loading at node 6 or 14
Case 2: Loading at node 10
Case 3: Loading at node 6 and 14
The deflection is measured at center node. The load cases are considered withmasses 2kg, 4kg, 6kg, 8kg, and 10kg.
Figure 16: Deflection vs Load
Once the Deflection vs Load graph has been plotted, a Linear equation can be gener-ated using the experimental values, using which we determine the deflection at anygiven load. From the graph we see that the lines representing the linear equationsdo not start from the origin. This signifies the error in the experiment.
This is caused due to various factors:
Human errors while performing the experiment affects the measurements.
Energy losses at the supports due to Friction.
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We are assuming the reactions in only x and y directions and neglect theTorsional forces in the z direction. Hence there is some error as we are dealingwith a 3 Dimensional structure.
In this case we determined the deflection for a load of 50kg since we carried out theAbaqus simulation for the same load.
3.2 Numerical study of the Pratt structure
The Pratt structure is simulated on the ABAQUS software and the vertical dis-placements are noted for the three cases (loading at node 6, node 10, node 6 and14). The simulation is carried out by considering the load as 50 kg.
Figure 17: Displacement at node 6
Figure 18: Deflection of beam for loading at node 6
Figure 19: Deflection of beam for loading at node 10
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Figure 20: Deflection of beam for loading at node 6 and 14
Figure 21: Comparison of Experimental and Simulation results
The above table represents the comparison between the experimental and sim-ulation results. It is seen that there is a slight difference between the two results.This difference is due to the following factors:
The simulation is carried out in 2 Dimension where as the Pratt truss is a3 Dimensional structure. Hence the Torsional effects in the z direction isneglected.
Though the experiment is conducted in a strict and methodical manner, thereare some errors caused due to improper fixing of the beam, frictional losses atthe supports, improper loading.
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3.3 Calculations
Once the results are obtained through Numerical analysis, the next stage is to cal-culate the Normal and Shear Stress by making use of the minimum and maximumstresses obtained from the Abaqus simulation. This has to be done for both thecases.
Figure 22: Maximum and Minimum Stress
The mean Stress is given as follows,
top = 39.8MPa; bottom= 42.7MPa
tensile= top+ bottom
2 = 1.45MPa
shear= top bottom
2 = 41.25Mpa
To calculate the Maximum Bending Moment at 10 mm,
M= shear Iz
y
The Moment of Inertia is given by,
Iz = bl3
12 =
2 103
12
Iz = 166.66mm4
M=41.25 106 166.67 1010
5 103
M= 1.32Nm
The elastic limit for the structure is 180 Mpa, but for a load of 50kg, the Stress is41.25 Mpa. We now calculate the allowable load for the given Elastic Limit.
w=180 50
41.25
w= 218.18 kg
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4 Conclusion
From theoretical and practical observations, it is seen that the Bending mo-ment for the Scale structure is much higher compared to that of the Prattstructure, under identical loading conditions. Hence we can say that the Prattstructure is stronger and more reliable than the Scale structure.
This increased strength in the Pratt structure is due to the diagonal memberswhich is subject to Tension forces only, and the vertical members handle thecompressive forces.
The tension in the diagonal members eliminates the risk of buckling, and thusthinner diagonal members can be used which results in weight reduction of thestructure, in turn making the design more economical.
From the experiment it is observed that the Pratt structure can bear a max-imum load of 218.18 kg, and the Scale structure can bear a maximum loadof 29.4 kg beyond which Plastic deformation occurs, eventually leading to the
failure of the structure. Thus we can conclude that the Pratt structure isbetter, safer, and more reliable than the Scale structure in all aspects.