Analytical and Experimental Determination of Internal Forces in Members of Three Trusses - Lab Report Example

Lecturer Measuring Forces in Truss Members Using Strain Gages This primary aim of the experiment was to performan analytical and experimental determination of internal forces in members of three trusses, 30-60-90, 60-60-60 and 45-45-90. In this experiment, analytical determination of forces acting through members of each truss under five loading conditions, 250, 500, 750, 1000 and 1250 pounds was done with the load of 1250 being determined as the maximum load that the members would handle without failing through shear failure. Each of the three trusses was then placed in the loading mechanism and subjected to the five loads. After each loading, strains were recorded in each truss member whereby strain was measured by SR-4 strain gages that were attached to each member (two gages per member with one gage on either side of the member). The comparison between analytical and experimental values shows consistency in nature of forces (members under tension or compression). The comparison in the form of absolute and percent error shows high levels of experimental accuracy. Introduction Engineers design and construct various structures including bridges, houses and trusses just to mention a few. Engineering structures, such as a bridge, are supposed to carry a given load (design load) without failure with some allowance for additional load, such as from wind. This calls for engineering design, which involves the determination of the most applicable shape of structure, best material to use and consequent determination of dimensions for every structural member. The determination of best material and material dimensions are aimed to ensure that the structure under consideration does not fail upon load application through the intended use of the structure. An engineering structure is a combination of members so arranged and joined that they facilitate the structure to perform its intended task. When loaded, different members experience different forces in terms of magnitude and direction whereby some members are under tension and others are under compression. It is important to determine the magnitude of force on every structural member and the direction of the force, which aids in designing various structural members so that they can withstand forces in them. This determination is also important in coming up with and selecting the best structural design that promises best economy in terms of the number of elements and dimensions of each element. There are two methods of analyzing forces in structural members, which include the method of joints and the section method. Engineers use any of the two methods to analyze the desired engineering structure based on the load to be applied and the point of application of the load. Engineers can also determine the actual forces acting through structural members using strain gauges, which give the actual magnitude and direction of forces through structural members. In this test, strain gauges were used to measure forces acting through truss members. The truss was also analyzed using the method of joints to give a hypothetical value and nature of forces in members. The experiment started with the determination of the maximum load that a truss under consideration could hold after which a lower than the maximum determined load was placed on the truss. Strain gauges were attached to every structural members before the application of the load and strain was recorded after load application, which was used to calculate forces through respective members. The experiment was aimed at increasing the understanding of forces acting through truss members and increasing the understanding of the use of method of joints to analyze trusses and other engineering structures. The experiment was also aimed at comparing between hypothetical and actual magnitudes and nature of forces acting through truss members and determining whether there is a difference between hypothetical values (using method of joints) and actual values (using strain gages). The experiment was also aimed at facilitating better understanding of the use of strain gages in determining forces through structural members. Finally, the experiment was aimed at increasing the understanding of the importance of analyzing engineering structures to determine magnitude and nature of forces acting through structural members. Background Before embarking on the experiment, there is a need to have a clear understanding of the strain gage including the working principle of this fundamental engineering tool. This is so because the strain gage is the core of this experiment. A strain gage is defined as a “resistor in which the resistance changes with strain” (Anon). In order to measure force acting along a member, a strain gage is glued to the members where force is to be measured. Most strain gages are made of a resistive foil, and they are known as foil-type strain gages (DEU). Figure 1 shows the arrangement of a strain gage. Figure 1: schematic representation of the strain gage showing the various parts of a strain gage. The strain gage shown in this figure is a foil-type strain gage. As shown in figure 1, a foil-type strain gage consists of a foil that is mounted on a backing material. The whole assembly is then attached to the member whose force is being determined and it operates on the principle of change in foil resistance when the foil is subjected to stress (DEU). Strain gages are then assembled into a Wheatstone Bridge as a full bridge (consisting of four active strain gages), half bridge (consisting of two active strain gages) or quarter bridge (consisting of single strain gage) (DEU). In this experiment, a full bridge consisting of four strain gages was used. A stabilized direct current (DC) source is powers and excites the Wheatstone Bridge such that changes in resistance occur when the gage is stressed, which causes an unbalance in the Wheatstone Bridge (Anon). Through appropriate amplification and calibration, the strain gage is able to measure forces through members however small the force maybe. Figure 2 below demonstrates how a strain gage measures the magnitude and nature of forces in structural members. Figure 2: demonstration of the working principle of the strain gage. As figure 2 shows, tensile or compressive forces on the member causes member tension and compression respectively, and in each case, there is proportional increase in foil resistance. The strain gage is arranged in such a way that it is insensitive to lateral forces acting within the structural member under consideration. this is aimed at increasing gage accuracy by eliminating chances of additional forces acting laterally. Methods and Procedures Experimental procedure Three trusses, 45-45-90, 30-60-90 and 60-60-60, were used in the experiment whereby forces in their respective members were determined. All the three trusses had members and pins made of 1018 steel whose yield stress was given as 36,000 psi and modulus of elasticity given as 30,000,000 psi. For each of the trusses, cross sectional areas and lengths for each member and pins were determined and recorded. For each truss, the maximum load that the truss could handle was calculated. Minimum applicable load for every truss was then determined. the difference between the minimum and maximum loads (considering a safety factor for the maximum load) was then determined and a series of five loads falling within the minimum and maximum loads set out, which would afterwards be used for theoretical and actual determination of forces in members. The maximum load was determined by dividing the maximum applicable load based on calculations divided by the factor of safety. Each of the trusses was then analyzed using the method of joints to determine theoretical magnitude and nature of forces acting in each of the truss members upon application of theoretical loads ranging from the minimum applicable load to the maximum load that the truss could handle, which was previously determined. Strain gages were properly inspected to ensure that they were not broken. The truss was then setup in the Tinius-Olsen Testing Machine making sure that loads and reactions are applied at truss joints and that they do not interfere with the members. Strain gages were then connected to the strain indicator, later balanced, and calibrated using the instructional manual. The five loads that were previously set out and used for theoretical determination of forces in members were then applied to each truss starting with the smallest load (that gave a reasonable strain reading). After every load application, the value of the applied load and the resulting strain reading were recorded. since each truss member had two strain gages attached to it on opposite sides of the respective member, readings from the two strain gages were recorded and the average value from the two strain gages determined as the gauge reading for the respective truss member. The experiment started with the determination of maximum load that the trusses could handle. Cross-sectional area for each member was determined using equation 1. A = b x h ----------------------------------------------------------------- (1) Equation 2 shows allowable axial lead for each member τ (allowable) x (A) P = P (allowable) -------------------------------- (2) The applied load should not exceed critical bending steel for steel, which is shown in equation 3 (π2 x Ea x I) /L2 = FCR ------------------------------------------------- (3) Whereby L = Length of the Member; E is the elastic modulus of steel; and I is the moment of inertia of the respective member, which is determined using equation 4 I = b (h) 3/12 ------------------------------------------------------- (4) Experimental stress for each member is then determined using equation 5 σ exp = Ea x ε ------------------------------------------------------ (5) Analytical determination of forces through members was then done by taking each joint and working on the argument that summation of horizontal forces at the joint is zero and likewise for vertical forces as shown in equations 6 and 7. Ʃfy = 0 ------------------------------------------------------------ (6) Ʃfx = 0 ------------------------------------------------------------ (7) Experimental force values were then compared with theoretical forces using absolute and percent errors as shown in equations 8 and 9 Absolute error -  ----------------------------------- (8) Percent error =  * 100% ---------------------------- (9) Results and Discussion Analytical determination of theoretical forces acting in members using the method of joints was done prior to this experiment in class, and respective values for the three trusses were recorded. These were compared with actual values of forces in members as determined using the strain gauge. Maximum allowable load for every truss was found to be 1250 pounds taking into consideration factor of safety. Load above this value would result to shear failure of members or pins since shear failure was found to be the limiting failure mode or one that would result to truss failure at the minimal loading. Table 1 summarizes allowable forces for the three trusses Table 1: allowable force table for the three trusses, 30-60-90, 60-60-60 and 45-45-90 yield Force (lb) Allowable yield Force(lbs) Bucking Allowable bucking shear force allowable shear force 9000 4500 3727 1242 2454 1227 15517 7759 4462 1487 2454 1227 12730 6365 5150 2428 2454 1227 Experimental Gage Values All the trusses had three members, member AB, member AC and member BC. The following tables summarize gage reading for the three members and for the five loads that were applied. Table 2: Experimental gauge values for truss 30-60-90 (X 10-6) P (lb) AB Member AC Member BC Member F B Average F B Average F B Average 250 -32 31 -0.5 -48 -30 -39 20 -37 -8.5 500 -32 33 0.5 -91 -31 -61 15 -39 -12 750 5 -18 -6.5 -130 -67 -98.5 -15 4 -5.5 1000 14 -12 1 -172 -92 -132 -12 7 -2.5 1250 14 -11 1.5 -216 114 -51 -11 7 -2 Table 3: Experimental gage readings for truss 60-60-60 (X 10-6) P (lb) AB Member AC Member BC Member F B Average F B Average F B Average 250 18 -8 5 -39 -4 -21.5 -123 84 -19.5 500 33 -11 11 -61 -22 -41.5 -188 113 -37.5 750 41 -9 16 -83 -43 -63 -251 133 -59 1000 56 -9 23.5 -100 -72 -86 -289 132 -78.5 1250 72 -12 30 -118 -96 -107 -304 107 -98.5 Table 4: Experimental gage readings for truss 45-45-90 (X 10-6) P (lb) AB Member AC Member BC Member F B Average F B Average F B Average 250 67 -18 24.5 -63 13 -25 -68 17 -25.5 500 107 -16 45.5 -120 25 -47.5 -115 17 -49 750 145 -11 67 -181 40 -70.5 -152 5 -73.5 1000 181 -5 88 -242 56 -93 -180 -15 -97.5 1250 215 2 108.5 -300 67 -116.5 -204 -38 -121 Tables 3 and 4 show that member AB is under tension for all the loads while members AC and BC are under compression for all the loads. For table 1, members AC and BC are under compression for all loads while member AB is under compression for loads 250 and 750 pounds and under tension for other loadings. A value of zero shows that there is no force in the respective member, which means that the member is a redundant member at the given load. For the 30-60-90 truss, member AB has a gauge reading of 0.5 and -0.5 for loads 250 and 500 pounds respectively. These values are close to zero, which suggests that the member might have been redundant at the given loads. Theoretical Values of Forces in the Members using the Method of Joints The method of joints is an analytical method used to determine forces acting in truss members. The method of joints determines forces acting in members through a given joint. Accordingly, the method of joints divides the structure using the available joints and analyzes forces through the given joint. In order to calculate the theoretical forces acting through the members effectively, there is a need to have a visual representation of the trusses, which are as follows: 30-60-90 60-60-60 45-45-90 For all the three trusses, load was applied at point C while supports are at A and B. For the 30-60-90 truss, Consider joint B Ʃfy = 0 Reaction at B is zero. Therefore, CB = 0 Ʃfx = 0 Therefore, AB = 0 Since force is acting directly through member AC, force through member AC = -P For the 45-45-90 truss: Consider joint B, which is the same as joint A Ʃfy = 0 0.5P + BCsin45 = 0 => BC = (-0.5P)/ (sin45) = -0.7071P Ʃfx = 0 -AB – BCcos45 = 0 => AB = - BCcos45 = 0.5Pcos45/sin45 = 0.5P Therefore, AB = 0.5P AC = BC = -0.7071P For the 60-60-60 truss, forces through member AC and BC were obtained by multiplying the applied load by -0.577 while force through member AB was obtained by multiplying the applied force by 0.289. Theoretical force values for all the members for the three trusses and for the various applied loads are summarized in the tables that follow. Table 5: Truss experimental forces and stress for truss 30-60-90 AB AC BC Load Stress Force Stress Force Stress Force 250 -15 -3.75 -1170 -292.5 -255 -63.75 500 15 3.75 -1830 -457.5 -360 -90 750 -195 -48.75 -2955 -738.75 -165 -41.25 1000 30 7.5 -3960 -990 -75 -18.75 1250 45 11.25 -1530 -382.5 -60 -15 Table 6: Theoretical forces for members of truss 30-60-90 Applied Load (P) AB (0) AC (-P) BC (0) 250 0 -250 0 500 0 -500 0 750 0 -750 0 1000 0 -1000 0 1250 0 -1250 0 Tables 5 and 6 show some differences in values for members AB and BC with theoretical values showing that this member is redundant while experimental values show that the member was having some force. There is also a significant difference in force value for member AC at 1250 lb load. These differences may have resulted from high sensitivity of the gages such that they measured slight forces in members AB and BC. Gage inaccuracy may have caused the low value recorded for AC at 1250lb load. Table 7: Truss experimental forces and stress for truss 60-60-60 P (lb) AB AC BC Stress Force Stress Force Stress Force 250 150 37.5 -645 -161.25 -585 -146.25 500 330 82.5 -1245 -311.25 -1125 -281.25 750 480 120 -1890 -472.5 -1770 -442.5 1000 705 176.25 -2580 -645 -2355 -588.75 1250 900 225 -3210 -802.5 -2955 -738.75 Table 8: Theoretical forces for members of truss 60-60-60 AC (-0.577P) BC (-0.577P) AB (0.289P) 250 -144.25 -144.25 72.25 500 -288.5 -288.5 144.5 750 -432.75 -432.75 216.75 1000 -577 -577 289 1250 -721.25 -721.25 361.25 Tables 7 and 8 show consistency in force nature in the given all the members and for all loads. However, there are some differences in force magnitude especially for member AB at all loads. Incorrect mounting of the gage or gage problem may have caused these differences. Table 9: Truss experimental forces and stress for truss 45-45-90 P (lb) AB AC BC Stress (Psi) Force (lb) Stress Force Stress Force 250 735 183.75 -750 -187.5 -765 -191.25 500 1365 341.25 -1425 -356.25 -1470 -367.5 750 2010 502.5 -2115 -528.75 -2205 -551.25 1000 2640 660 -2790 -697.5 -2925 -731.25 1250 3255 813.75 -3495 -873.75 -3630 -907.5 Table 10: Theoretical forces for members of truss 45-45-90 P (lb) AC (-0.7071P) BC (-0.7071P) AB (0.5P) 250 -176.775 -176.775 125 500 -353.55 -353.55 250 750 -530.325 -530.325 375 1000 -707.1 -707.1 500 1250 -883.875 -883.875 625 Tables 9 and 10 show consistency in force magnitude and nature for all members at all loads. This signifies high levels of gage accuracy and proper mounting. Accurate and percent errors Percent and absolute errors enable better comparison between experimental and theoretical values, which gives an idea of the experimental accuracy. Tables 11 to 13 provide a summary of absolute and percent errors. Table 11: Absolute and percent error for truss 30-60--90 Member AB Load (P) Theoretical Experimental Absolute error % error 250 0 -3.75 3.75 N/A 500 0 3.75 3.75 N/A 750 0 -48.75 48.75 N/A 1000 0 7.5 7.5 N/A 1250 0 11.25 11.25 N/A Member AC Theoretical Experimental Absolute error % error -250 -292.5 42.5 17.0 -500 -457.5 42.5 8.5 -750 -738.75 11.25 1.5 -1000 -990 10 1 -1250 -382.5 867.5 69 Member BC Theoretical Experimental Absolute error % error 0 -63.75 63.75 N/A 0 -90 90 N/A 0 -41.25 41.25 N/A 0 -18.75 18.7 N/A 0 -15 15 N/A Table 12: Absolute and percent error for truss 60-60-60 Member AB Load (P) Theoretical Experimental Absolute error % error 250 72.250 37.500 34.75 0.481 500 144.500 82.500 62 0.429 750 216.750 120.000 96.75 0.446 1000 289.000 176.250 112.750 0.390 1250 361.250 225.000 136.250 0.377 Member AC Load (P) Theoretical Experimental Absolute error % error 250 -144.25 -161.250 17 11.785 500 -288.5 -311.250 22.75 7.886 750 -432.75 -472.500 39.75 9.185 1000 -577 -645.000 68 11.785 1250 -721.25 -802.500 81.25 11.265 Member BC Load (P) Theoretical Experimental Absolute error % error 250 -144.25 -146.250 2 0.014 500 -288.50 -281.250 7.25 0.025 750 -432.75 -442.500 9.75 0.023 1000 -577 -588.750 11.75 0.020 1250 -721.25 -738.750 17.5 0.024 Table 13: Absolute and percent error for truss 45-45-90 Member AB Load (P) Theoretical Experimental Absolute error % error 250 125 183.75 58.75 0.47 500 250 341.25 91.25 0.365 750 375 502.5 127.5 0.340 1000 500 660 160 0.320 1250 625 813.75 188.750 0.302 Member AC Load (P) Theoretical Experimental Absolute error % error 250 -176.775 -187.500 10.725 6.067 500 -353.55 -356.250 2.700 0.764 750 -530.325 -528.750 1.575 0.297 1000 -707.1 -697.500 9.600 1.358 1250 -883.875 -873.750 10.125 1.146 Member BC Load (P) Theoretical Experimental Absolute error % error 250 -176.775 -191.250 14.475 0.082 500 -353.55 -367.500 13.95 0.039 750 -530.325 -551.250 20.925 0.039 1000 -707.1 -731.250 24.15 0.034 1250 -883.875 -907.500 23.625 0.027 Tables 11 to 13 show high levels of accuracy of the experiment except for the 60-60-60 truss at all loads where percentage error is above 10%. This shows a significant difference between theoretical and experimental force values although there is consistence in nature of force (extension or compression). Poor mounting of the gages or incorrect gages, especially considering that the problem is affecting the entire truss, could most probably attribute this difference. Conclusion The primary objective of the experiment was to increase the understanding of the use of strain gage to get actual values of forces in members as well as the use of the method of joints to get theoretical values of forces in truss members. Through this experiment, the use of strain gages was clearly understood. Truss analysis found a maximum of 1250 pounds as the maximum load that would not result to shear failure of truss member. Strain gages facilitated the determination of experimental forces, which were compared with theoretical values of forces in members found using the method of joints. Percent error shows high levels of experimental accuracy except for the 60-60-60 truss at all loads where percentage error is above 10%. Considering that all the values for this truss show high levels of percent error, it is highly possible that poor mounting of the gages resulted to erroneous experimental values. Nevertheless, the objectives of the experiment were achieved with a particular focus on the use of strain gages to measure forces in structural members. This experiment shows that analytical determination of forces in truss members using the method of joints holds true in that experimental results were in agreement with theoretical values. Therefore, engineers should continue to use method of joints without worry. Works Cited Anon. Introduction to Strain Gages. n.d. Web. March 23, 2014. http://www.facstaff.bucknell.edu/mastascu/elessonshtml/Sensors/StrainGage.htm DEU. The Strain Gauge. Automatic Control, Robot and Mechatronics Labs of Mechanical Engineering Department. 2007. Web. March 23, 2014. http://web.deu.edu.tr/mechatronics/TUR/strain_gauge.htm Read More