Analysis of Forces and Strains in a Truss under Controlled Loading Done Using a Specialized Machine - Coursework Example

Running Head: TRUSS ANALYSIS Truss Analysis Name Institution Contents Abstract 4 FORCES IN A STATICALLY DETRMINATE TRUSS 4 EXPERIMENTAL OBJECTIVES 4 THEORY 5 METHODOLOGY 6 MATERIALS 6 METHODOLOGY 7 RESULTS AND DISCUSION 7 INDETERINATE TRUSS 15 INTRODUCTION 15 MATERIALS AND METHODOLOGY 16 MATERIALS 16 METHODOLOGY 16 RESULTS AND DISCUSSION 17 RESULTS 17 DISCUSSION 18 CONCLUSION 21 References 23 Table of Figures Figure 1: Free body diagram for the truss 10 Figure 2: Comparison of true and recorded strain for member 3 under similar loading 12 Figure 3: Comparison of true and recorded strain for member 7 under similar loading 13 Figure 4: Graph of deflection against loading 14 Figure 5: Theoretical internal forces in each member of the truss. 19 Figure 6: Comparison of experimental and theoretical internal forces in the truss members. 19 Figure 7: Graph of deflection against loading for both determinate and indeterminate trusses. 20 Figure 8: Comparison of forces in members. 21 Table of tables Table 1: Strain readings and frame deflections 7 Table 2: True strain readings and frame deflections from the experiment. 8 Table 3: The theoretical and experimental values of forces for each member 11 Table 4: The nature of the various members in the truss. 11 Table 5: Strain gauge and deflection readings from each member for each loading 17 Table 6: True strain readings for each member for every loading. 17 Abstract This report covers the in-depth analysis of forces and strains in a truss under controlled loading done using a specialized machine. The analysis also determines the forces in a truss when it is statically determinate as well as when it is statically indeterminate. Each type of loading compares both the experimental readings of strain in each member from the gauge as well as determining the actual or true strains in each member. Comparison on these strains is also done within the report. Theoretical as well as experimental magnitude of force in both the statically determinate as well as the statically indeterminate truss arrangements are calculated and comparison made in each case. The report also gives an iterative view from the authors view on the type of readings from some of the members within the structure. Various graphical aids are used to carry out the various force comparisons in the experiment. TRUSSES FORCES IN A STATICALLY DETRMINATE TRUSS EXPERIMENTAL OBJECTIVES To analyses forces in a statically determinate truss using mechanical engineering principles learnt in class. To equip the learner with hands on techniques on how to carry out truss analysis using laboratory machines. To observe the difference between the margins in experimental and theoretical force values through comparison. THEORY A truss is typically a combination of members joined together using joints to form a series of triangles with static stability. Conventionally, the joints are considered as frictionless and are pinned type joints (Hibbeler, 1999). Analysis of forces in each member of the trusses requires and experimental procedure that puts into consideration various assumptions considered necessary to accomplish the analysis. The assumptions are as follows; The members are joined by frictionless pin-joints at their ends and there is no moment transfer from one member to another at this joint. The pin-joints is the only point where application of external forces is done. For statically determinate trusses, Newton`s law applies to the whole structure although each joint in the structure applies similar principles. However, before an analysis to determine the forces at each member, some conditions needs to be met. One of the most important conditions is that all the horizontal, vertical as well as moments at any node is always equals to zero. Thus with this condition, it possible to determine the magnitude of force at each end of a member or at each member (Megson, 2005). METHODOLOGY MATERIALS In order to carry out force analysis in determinate trusses, TecQuipment® device is required. The device is composed of sensors and provisions for attaching steel rods to form the desired truss structure. The measured deflection upon loading is measured by reading the digital dial gauge. The properties of the metal rod used as truss are as follows; Metal rod diameter D = 6 mm. Modulus of elasticity of the metal rod E = 210 GPa. Picture 1: A pictorial view of the apparatus fitted with truss members. METHODOLOGY The experiment was set up by carrying out an initial inspection of the equipment for any damages or loose connections. All the connections both mechanical and electrical were tightened and then the equipment was placed on a flat surface i.e. bench (Kassimali, 1999). The truss structure was then developed on the equipment by fixing the metal rod members in position to give result to the structure under consideration. The joints of the members were pinned by use of the joints provided. The dial gauge was then fixed and set to zero at no load. Loading was then subsequently done from 0N to 250N at an interval of 50N and strain readings for each member from the dial gauge recorded in a table. The deflection in the dial gauge was also recorded at each loading. RESULTS AND DISCUSION The following results were obtained from the dial gauge readings and strain gauge readings at each member. Load (N) Strain 1 (µε) Strain 2 (µε) Strain 3 (µε) Strain 4 (µε) Strain 5 (µε) Strain 6 (µε) Strain 7 (µε) Strain 8 (µε) Digital Indicator Reading (mm) 0 0 0 0 0 0 0 0 0 0 50 11.0 -10.5 -9.3 -21.6 0.9 0 14.5 14.8 0.020 100 19.9 -18.0 -15.8 -37.2 1.4 0 25.0 25.4 0.025 150 31.9 -28.6 -25.3 -59.2 2.0 0 39.6 40.2 0.052 200 39.4 -36.1 -32.4 -74.4 2.3 0 49.7 49.9 0.053 250 50.5 -47.1 -43.1 -96.3 2.3 0 64.2 65.0 0.045 Table 1: Strain readings and frame deflections Calculation of true strain from the experiment readings True strain readings from the experiment can be determined by reducing the strain values read at the strain gauges for every reading by the value recorded at zero newton loading. Thus, in this case, the values in the above table are true strain readings because there is no strain reading or deflection measured at zero loading. Therefore, true strain readings are given by; Load (N) Strain 1 (µε) Strain 2 (µε) Strain 3 (µε) Strain 4 (µε) Strain 5 (µε) Strain 6 (µε) Strain 7 (µε) Strain 8 (µε) Digital Indicator Reading (mm) 0 0 0 0 0 0 0 0 0 0 50 11.0 -10.5 -9.3 -21.6 0.9 0 14.5 14.8 0.020 100 19.9 -18.0 -15.8 -37.2 1.4 0 25.0 25.4 0.025 150 31.9 -28.6 -25.3 -59.2 2.0 0 39.6 40.2 0.052 200 39.4 -36.1 -32.4 -74.4 2.3 0 49.7 49.9 0.053 250 50.5 -47.1 -43.1 -96.3 2.3 0 64.2 65.0 0.045 Table 2: True strain readings and frame deflections from the experiment. Calculation of cross sectional area for the members Since each member used in the experiment are uniform, we only need to determine the cross sectional area for one of the members. Diameter = 6mm Calculation of experimental force (N) Experimental force is obtained from the stress values. Stress is given by; and therefore force is given by Modulus of elasticity and therefore; (Equation 1) Where; E = Young`s modulus σ = Stress in the member ε = Strain readings F = force in the member. Therefore, using equation 1, the experimental force in each members was calculated. Determination of theoretical forces in each member Using the principle of moments as discussed early and equating the value of moments at the joints to zero gives the theoretical magnitude of force for each member. Figure 1: Free body diagram for the truss The values of experimental forces as well as theoretical magnitude of forces for each member determined from the above procedure is as tabulated in the table below. Member Experimental Force (N) Theoretical Force (N) 1 299.8489515 250.0 2 -279.6611013 -250.0 3 -255.9106893 -250.0 4 -571.7911689 -500.0 5 13.6564869 0.0 6 0 - 7 381.1941126 354.0 8 385.944195 354.0 Table 3: The theoretical and experimental values of forces for each member From the values in table 1 and 2 above, the following nature of the members can be deduced. Member Nature 1 Tension 2 Compression 3 Compression 4 Compression 5 Tension 6 Zero Force member 7 Tension 8 Tension Table 4: The nature of the various members in the truss. Graphical comparison of strains in member 3 Figure 2: Comparison of true and recorded strain for member 3 under similar loading Graphical comparison of strains in member 7 Figure 3: Comparison of true and recorded strain for member 7 under similar loading Graph of deflection against load Figure 4: Graph of deflection against loading Comparison The difference between the experimental and theoretical values of force are found to have a small margin. This proves the fact that experimental framework complies with the pin joint theory. The small deviation also depicts the high level of accuracy of strain gauges as sensors in the experiment. Member 5 in the experiment by inspection is expected to be a zero force member turns out to be under compression although the force is little in magnitude. This occurrence can be attributed to some geometric errors in the structure. Member 5 moreover is attached to a roller and actually records an internal force which is a big reason why it should be a zero force member. INDETERINATE TRUSS INTRODUCTION The main difference between the statically determinate and the statically indeterminate trusses is the inclusion of the redundant member in the experiment in the case of statically indeterminate truss (Hibbeler, 1999). In statically indeterminate trusses, loading is gradually done at the far right end of the truss structure. Longitudinal strain is thus measured along the truss members and subsequently the forces on each member can be determined from the strain measured (Megson, 2005). MATERIALS AND METHODOLOGY MATERIALS Similarly, this experiment was performed on a TecQuipment® by use of metal rods as truss members like in the case of statically determinate trusses. Normal strains are measured in each member as well as the deflection at the right hand side node of the top trusses is measured by use of strain-gauges and dial gauge respectively. The properties of the metal rods used as members are as follows: The metal rods diameter D =6mm. Modulus of elasticity of the rods E = 210 GPa. METHODOLOGY The redundant member of the truss is hand-tightened on the truss structure and caution should be noted that tightening should never be done by use of any tool. The redundant member in this case is part of the ideal structure of the truss. Loading at the right hand side of the structure is done gradually from 0N to a maximum load of 250N checking for the stability of the structure at each loading. Record the strain readings in each member as well as the corresponding deflection recorded by the dial gauge and recorded the results in a table. RESULTS AND DISCUSSION RESULTS The following results were obtained from the strain gauge readings as well as from the digital dial gauge. Load (N) Strain 1 (µε) Strain 2 (µε) Strain 3 (µε) Strain 4 (µε) Strain 5 (µε) Strain 6 (µε) Strain 7 (µε) Strain 8 (µε) Digital Indicator Reading (mm) 0 0 0 0 0 0 0 0 0 0 50 14.4 -7.3 -10.2 -17.8 3.6 -4.7 14.0 9.4 0.037 100 27.3 -11.9 -16.8 -30.8 7.3 -8.4 25.0 15.5 0.047 150 41.6 -16.8 -24.7 -45.7 11.7 -13.4 38.0 22.2 0.042 200 55.1 -21.8 -33.0 -59.6 16.0 -18.1 50.0 28.6 0.072 250 72.0 -28.3 -43.7 -77.7 21.2 -24.4 65.4 36.6 0.068 Table 5: Strain gauge and deflection readings from each member for each loading The true strain readings in each member can be found by reducing the strain readings in each member by the value recorded at zero loading. The following table shows the true strain readings. Load (N) Strain 1 (µε) Strain 2 (µε) Strain 3 (µε) Strain 4 (µε) Strain 5 (µε) Strain 6 (µε) Strain 7 (µε) Strain 8 (µε) Digital Indicator Reading (mm) 0 0 0 0 0 0 0 0 0 0 50 14.4 -7.3 -10.2 -17.8 3.6 -4.7 14.0 9.4 0.037 100 27.3 -11.9 -16.8 -30.8 7.3 -8.4 25.0 15.5 0.047 150 41.6 -16.8 -24.7 -45.7 11.7 -13.4 38.0 22.2 0.042 200 55.1 -21.8 -33.0 -59.6 16.0 -18.1 50.0 28.6 0.072 250 72.0 -28.3 -43.7 -77.7 21.2 -24.4 65.4 36.6 0.068 Table 6: True strain readings for each member for every loading. DISCUSSION Calculation of true strain readings from experimental strain readings. The true strain readings are determined by reducing the recorded strain readings for each member and loading by their respective zero loading strain recorded. Since the rods used are similar to those used in the statically determinate truss experiment, the cross sectional area of the rod will the same. Determination of internal forces in the trusses. Using the force method analysis, the internal forces in each member was determined for a loading magnitude of 250N. The forces were determined for each member and their magnitudes are as indicated in the truss diagram below. Figure 5: Theoretical internal forces in each member of the truss. The experimental internal forces in each member of the trusses at a loaf of 250N is calculated in a similar way as in the statically determinate trusses. A comparison of the experimental and theoretical forces in each truss member is summarized below. Comparison of the theoretical and experimental internal forces in each member. Member Experimental Force (N) Theoretical Force (N) 1 427.507416 375.0 2 -168.0341649 -125.0 3 -259.4732511 -250.0 4 -461.3517531 -375.0 5 125.8771836 125.0 6 -144.8775132 -176.8 7 388.3192362 353.5 8 217.3162698 176.8 Figure 6: Comparison of experimental and theoretical internal forces in the truss members. The deviation between the theoretical and experimental magnitude of internal force is within the allowed magnitude in all the members except member 5. The high deviation in member 5 is attributed to the high internal force recorded in this member. Comparison between the determinate and indeterminate trusses. The graph below compares the variation of deflection with loading in both determinate and indeterminate trusses. The graph depicts the importance of using the redundant member in trusses as a way of increasing structural stiffness of the truss. Figure 7: Graph of deflection against loading for both determinate and indeterminate trusses. Comparison of internal forces in both determinate and indeterminate trusses. The graph below shows a comparison between the internal forces in both statically determinate and statically indeterminate trusses. Figure 8: Comparison of forces in members. It is clear from the chart above that the magnitude in internal forces is always larger in statically indeterminate trusses as compared to those in statically determinate trusses. CONCLUSION The data and analysis of the data from the experiment has depicted that there is little variation between the experimental and theoretical magnitudes of force. The good correlation in this comparison is a clear sign of the efficiency of the experimental method. The comparison between the statically determinate and statically indeterminate trusses has also depicted the importance of using both structures as well as the desirable properties of each truss arrangement. Indeterminate truss is rarely used although it possess more safety properties as well as energy conservation capabilities as compared to the more practically used determinate trusses. References 1. Belvin, W. K., & United States. (1987). Modeling of joints for the dynamic analysis of truss structures. Washington, D.C.: National Aeronautics and Space Administration, Scientific and Technical Information Branch. 2. Megson, T. H. G. (2005). Structural and stress analysis. Amsterdam: Elsevier Butterworth-Heineman. 3. Hibbeler, R. C. (1999). Structural analysis. Upper Saddler River, N.J: Prentice Hall. 4. Kassimali, A. (1999). Structural analysis. Pacific Grove: PWS Pub. Read More

Picture 1: A pictorial view of the apparatus fitted with truss members. METHODOLOGY The experiment was set up by carrying out an initial inspection of the equipment for any damages or loose connections. All the connections both mechanical and electrical were tightened and then the equipment was placed on a flat surface i.e. bench (Kassimali, 1999). The truss structure was then developed on the equipment by fixing the metal rod members in a position to give results to the structure under consideration.

The joints of the members were pinned by use of the joints provided. The dial gauge was then fixed and set to zero at no load. Loading was then subsequently done from 0N to 250N at an interval of 50N and strain readings for each member from the dial gauge were recorded in a table. The deflection in the dial gauge was also recorded at each loading. 

Calculation of cross-sectional area for the members Since each member used in the experiment is uniform, we only need to determine the cross-sectional area for one of the members. Diameter = 6mm Calculation of experimental force (N) Experimental force is obtained from the stress values. Stress is given by; and therefore the force is given by Modulus of elasticity and therefore; (Equation 1) Where; E = Young`s modulus σ = Stress in the member ε = Strain readings F = force in the member.

Therefore, using equation 1, the experimental force in each member was calculated. Determination of theoretical forces in each member Using the principle of moments as discussed early and equating the value of moments at the joints to zero gives the theoretical magnitude of force for each member. Figure 1: Free body diagram for the truss The values of experimental forces as well as the theoretical magnitude of forces for each member determined from the above procedure is as tabulated in the table below.

Member Experimental Force (N) Theoretical Force (N) 1 299.8489515 250.0 2 -279.6611013 -250.0 3 -255.9106893 -250.0 4 -571.7911689 -500.0 5 13.6564869 0.0 6 0 - 7 381.1941126 354.0 8 385.944195 354.0 Table 3: The theoretical and experimental values of forces for each member From the values in table 1 and 2 above, the following nature of the members can be deduced. Member Nature 1 Tension 2 Compression 3 Compression 4 Compression 5 Tension 6 Zero Force member 7 Tension 8 Tension Table 4: The nature of the various members in the truss.

Graphical comparison of strains in member 3 Figure 2: Comparison of true and recorded strain for member 3 under similar loading Graphical comparison of strains in member 7 Figure 3: Comparison of true and recorded strain for member 7 under similar loading Graph of deflection against load Figure 4: Graph of deflection against loading Comparison The difference between the experimental and theoretical values of force are found to have a small margin.

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