Construction Characteristics of Trusses - Report Example

Student Name: xxxx Professor: xxxx Title: Trusses Experiment Report Date: xxxx Abstract A truss is a structure comprised of two- force members only, assembled to behave as a single object. A ‘two- force member’ is a structural component where a force is applied only to two points [Ple13]. Trusses are commonly comprised of triangular units made of straight members whose ends are connected at joints, called nodes. External forces and their reactions are considered to act on these nodes only, with moments in straight members excluded as the nodes are taken to be pinned joints [Nor08]. This arrangement of members means trusses have significantly lower weight that other structural elements over similar span. This allows trusses to support loads over longer spans, providing a practical and economic solution to projects that require long spanning elements, such as bridges and industrial complex buildings. Trusses are classified as being statically determinate or indeterminate depending on the number of members in the structure. A statically determinate truss can be fully analyzed using the basic equations of equilibrium as it has the optimum number of members required to maintain stability. However, the failure of one of these members results in the failure of the whole structure. A statically indeterminate truss has extra members over the optimum number required to maintain stability. This introduces some redundancy in the structure, meaning that the failure of one member does not necessarily mean collapse of the whole structure. Due to the extra member(s), the equations of static equilibrium are not sufficient to fully analyze the member forces. This report explores the load distribution among the structural members of cantilever statically determinate and indeterminate trusses by experimental analysis. Keywords: statistically determinate, static equilibrium, truss, equilibrium Table of Contents Abstract 1 1.Statically Determinate Truss 1 1.1Introduction 1 1.2Material and Methodology 1 1.2.1Material 1 1.2.2Methodology 3 1.3Results and Discussion 3 2.Indeterminate Truss 8 2.1Introduction 8 2.2Material and Methodology 8 2.2.1Material 8 2.2.2Methodology 9 2.3Results and Discussion 9 3.Comparison of Determinate and Indeterminate Trusses 12 3.1Load- Deflection Relationship 12 3.2Internal forces 13 4.Conclusion 13 Work Cited 14 1. Statically Determinate Truss 1.1 Introduction The redundant member of the frame structure is not engaged in this experiment. The truss is subjected to an increasing load at its unsupported, right end. The apparatus is assembled to allow measurement of longitudinal strain in the truss members and the deflection of one of the nodes at the top of the frame. The strain measured is used to determine the normal forces in the members. These forces are the compared to the analysis of the frame structure done by the method of joints. The final results obtained are then compared and any differences in the values obtained experimentally and theoretically are discussed. 1.2 Material and Methodology 1.2.1 Material TecQuipment® determinate truss study device shown in Figure 1 below is used to conduct the experiment. This apparatus allows the recording of normal strains in the different members of the truss as well as the deflection at the top node on the right-hand-side of the truss. Figure 1: Schematic representation of the truss frame, including the member numbers The characteristics of the truss are as follows; Diameter of rod – 6 mm Modulus of elasticity of material making rods, E – 210 GPa 1.2.2 Methodology When a structure is subjected to an external load (P), it develops internal normal forces (Fi) in its members to counteract the external load. The main aim of this experiment is to determine the relationship that exists between these two entities. The normal strain (εi) recorded in each member (i) is used to obtain the experimental results for the internal normal forces. This relationship is defined as; Equation 1: Relationship between Fi and εi The method of joints is used to calculate the theoretical forces in the members, as the truss is statically determinate. The results shown below are attained when a load of 250 N is applied at the node farthest from the support. Figure 2: Results of Method of Joints Analysis for a 250 N load on frame structure 1.3 Results and Discussion The experimental results for the deflections and the concurrent strains εi in the different members resulting from the incremental loads applied are as shown below; Table 1: Experimental results for member strains and node deflection of frame Load Member Strains Digital Indicator Reading 1 2 3 4 5 6 7 8 (N) (με) (με) (με) (με) (με) (με) (με) (με) (mm) 0 0.0 0.0 0.2 0.2 -0.1 _ 0.3 -0.3 0.000 50 9.3 -8.9 -8.7 -17.5 -0.3 _ 12.9 13.1 0.026 100 17.9 -17.8 -17.5 -34.5 -0.5 _ 25.4 25.4 0.044 150 27.4 -27.3 -27.0 -52.6 -0.6 _ 38.6 38.3 0.059 200 37.2 -6.6 -36.0 -70.3 -0.7 _ 51.7 51.7 0.075 250 44.9 -45.0 -44.4 -86.3 -1.1 _ 63.2 62.7 0.086 The true strains in the members are obtained by deducting the strain reading from the zero load to all the readings of a given sensor. The true strains are as shown below; Table 2: Experimental results for true member strains and node deflection of frame Load True Member Strains Digital Indicator Reading 1 2 3 4 5 6 7 8 (N) (με) (με) (με) (με) (με) (με) (με) (με) (mm) 0 0.0 0.0 0.2 0.2 -0.1 _ 0.3 -0.3 0.000 50 9.3 -8.9 -8.7 -17.5 -0.3 _ 12.9 13.1 0.026 100 17.9 -17.8 -17.5 -34.5 -0.5 _ 25.4 25.4 0.044 150 27.4 -27.3 -27.0 -52.6 -0.6 _ 38.6 38.3 0.059 200 37.2 -6.6 -36.0 -70.3 -0.7 _ 51.7 51.7 0.075 250 44.9 -45.0 -44.4 -86.3 -1.1 _ 63.2 62.7 0.086 The results in the tables, as well as the method of joints, are the same with regard to the state of stress of the members in the frame; Members 2, 3 and 4 are compression members Members 1, 7 and 8 are tension members Member 5 has no normal strain on it (zero- force member) Plotting a graph of both the recorded and true internal strain with incremental load for both a compression and tension member gives the following; Figure 3: Load- Strain curve for Compression Member Figure 4: Load- Strain Curve for Tension Member The graphs for both members demonstrate a linear relationship between the strains and forces in the members and the incremental load applied to the frame. This confirms the hypothesis made by analyzing the truss using the method of joints, which is valid for materials that obey Hooke’s Law and for small deformations. The relationship between the deflection of the node of the truss and the incremental load applied adopts an almost linear trajectory, as shown below; Figure 5: Load- Deflection Curve for node of Truss Frame The internal forces in the truss members from the experiment are calculated using Equation 1 above, and the deviation from the theoretical values determined by method of joints determined as below; Table 3: Comparison of Experimental and Theoretical Analysis of Statically Determinate Truss Member Experimental Force (Fexp) Theoretical Force (Ftheo) Deviation (N) (N) (%) 1 266.2 250.0 6.48 2 -267.1 -250.0 6.84 3 -246.6 -250.0 1.36 4 -512.2 -500.0 2.44 5 -6.3 0.0 _ 6 _ _ _ 7 373.2 353.6 5.54 8 374.5 353.6 5.91 The results show a good correlation between the experimental and theoretical results, with small deviations less than 7%. Therefore, the framework is congruent with the pin joint theory. It can also be noted that member 5 is subjected to a slight compression force in the experiment despite the method of joints depicting it as a zero- member. There should be no internal forces due to the fact that the member is attached to a roller support, which allows for movement in the y- direction. The discrepancies in the results obtained could be attributed to the fact that the truss joints are not truly pin joints, and that there may be development of moments in the structure in the real life scenario. Defections in the geometric properties (dimensions or assembly) of the individual members of the frame could also result in development of moments and internal forces. Imperfect connection of the roller support, as well as friction between the wheel and the support may cause a reaction force that subjects member 5 to a small loading. 2. Indeterminate Truss 2.1 Introduction The redundant member of the frame is engaged in this experiment and the parameters studied in the first experiment are also considered. The statically indeterminate truss is loaded on the right end with a load on increments of 50N upto a maximum load of 250N. The apparatus used allows the measurement of longitudinal strain in the members and the deflection of the top right node of the frame that develops as a result of the application of the external load. The normal forces in the members, calculated based on the measured strain are compared to the ones obtained theoretically by the method of forces. Any discrepancies that may exist between the two sets of values are discussed. 2.2 Material and Methodology 2.2.1 Material TecQuipment® determinate truss study device shown in Figure 1 below is used to conduct the experiment. This apparatus allows the recording of normal strains in the different members of the truss as well as the deflection at the top node on the right-hand-side of the truss. Figure 6: Schematic representation of the indeterminate truss, including the member numbers The characteristics of the truss are as follows; Diameter of rod – 6 mm Modulus of elasticity of material making rods, E – 210 GPa 2.2.2 Methodology When a structure is subjected to an external load (P), it develops internal normal forces (Fi) in its members to counteract the external load. The main aim of this experiment is to determine the relationship that exists between these two entities. The normal strain (εi) recorded in each member (i) is used to obtain the experimental results for the internal normal forces (Equation 1). The work method provides a way to theoretically calculate the force developed in the members using the equations below; Equation 2: Method of Work Equation Where displacement along the redundant member when removed from the truss Fi force in member i Equation 3: Displacement equation Equation 4: Internal member forces equation 2.3 Results and Discussion The experimental results for the deflections and the concurrent strains εi in the different members resulting from the incremental loads applied are as shown below; Table 4: Experimental results for member strains and node deflection of frame Load Member Strains Digital Indicator Reading 1 2 3 4 5 6 7 8 (N) (με) (με) (με) (με) (με) (με) (με) (με) (mm) 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.000 50 14 -6 -10 -16 4 -6 14 9 -0.030 100 24 -11 -16 -27 6 -10 25 15 -0.046 150 37 -16 -26 -40 10 -16 37 22 -0.055 200 51 -21 -36 -55 13 -22 51 29 -0.059 250 65 -27 -46 -69 17 -28 65 36 -0.058 The true strains in the members are obtained by deducting the strain reading from the zero load to all the readings of a given sensor. The true strains are as shown below; Table 5: True Strains Load True Member Strains 1 2 3 4 5 6 7 8 (N) (με) (με) (με) (με) (με) (με) (με) (με) 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 50 14 -6 -10 -16 4 -6 14 9 100 24 -11 -16 -27 6 -10 25 15 150 37 -16 -26 -40 10 -16 37 22 200 51 -21 -36 -55 13 -22 51 29 250 65 -27 -46 -69 17 -28 65 36 The results in the tables, as well as the method of work, are the same with regard to the state of stress of the members in the frame; Members 2, 3, 4 and 6 are compression members Members 1, 5, 7 and 8 are tension members The internal forces in the truss members from the experiment are calculated using Equation 1 above, and the deviation from the theoretical values determined by method of joints determined as below; Table 6: Comparison of Experimental and Theoretical Analysis of Statically Indeterminate Truss Member Experimental Force (Fexp) Theoretical Force (Ftheo) Deviation (N) (N) (%) 1 387 375.6 3.04 2 -159 -125.2 27 3 -273 -250.4 9.03 4 -412 -375.6 9.69 5 102 125.2 18.5 6 -167 -177.0 5.65 7 387 354.1 9.29 8 216 177.0 22.03 This table shows that there is relatively good correlation between the experimental and theoretical results, with a maximum deviation of about 27%. This means that the method of work does not produce accurate results for all members of the indeterminate truss. The high deviation could also be attributed to the process followed when applying the method of work, which requires the elimination of the redundant member under consideration. 3. Comparison of Determinate and Indeterminate Trusses 3.1 Load- Deflection Relationship The Load- Deflection curves for both trusses is as shown below; Figure 7: Load- Deflection Curves for the Different Truss Types The deflection of the indeterminate truss is clearly lower than that of the determinate one by about 20% for the same load. The presence of the redundant member increases the overall stiffness of the structure, meaning that the displacement due to load is reduced, keeping the frame close to its original shape. 3.2 Internal forces The internal member forces for both trusses are represented as below; Figure 8: Comparison of Internal Member Forces The graph shows that, on average, the members of the determinate truss develop more internal forces, with the exception of member 5 which is regarded as a zero- member and member 6 which is disregarded altogether. However, it is important to note that the difference in the internal forces is not large enough to sufficiently reduce the member sizes, and coupled with the additional member, the cost of the indeterminate truss may be prohibitive. Fabrication and assembly of indeterminate truss members also poses a challenge to their usage. 4. Conclusion The experiment has proven there’s a good correlation between the experimentally measured and theoretically obtained internal member forces for both trusses. The comparison of results has shown the advantages of each type of frame, though the determinate structure is more realistic for application, despite the increased safety factor provided by the indeterminate frame. The experiments have also proven the strain gauges are effective transducers for the measurement of forces in the frame. Work Cited Ple13: , (Plesha, Gray, & Costanzo, 2013), Nor08: , (Norton, 2008), 1 Read More

The main aim of this experiment is to determine the relationship that exists between these two entities. The normal strain (εi) recorded in each member (i) is used to obtain the experimental results for the internal normal forces. This relationship is defined as; Equation 1: Relationship between Fi and εi The method of joints is used to calculate the theoretical forces in the members, as the truss is statically determinate. The results shown below are attained when a load of 250 N is applied at the node farthest from the support.

Figure 2: Results of Method of Joints Analysis for a 250 N load on frame structure 1.3 Results and Discussion The experimental results for the deflections and the concurrent strains εi in the different members resulting from the incremental loads applied are as shown below; Table 1: Experimental results for member strains and node deflection of frame Load Member Strains Digital Indicator Reading 1 2 3 4 5 6 7 8 (N) (με) (με) (με) (με) (με) (με) (με) (με) (mm) 0 0.0 0.0 0.2 0.2 -0.1 _ 0.3 -0.3 0.000 50 9.3 -8.9 -8.7 -17.5 -0.3 _ 12.9 13.1 0.

026 100 17.9 -17.8 -17.5 -34.5 -0.5 _ 25.4 25.4 0.044 150 27.4 -27.3 -27.0 -52.6 -0.6 _ 38.6 38.3 0.059 200 37.2 -6.6 -36.0 -70.3 -0.7 _ 51.7 51.7 0.075 250 44.9 -45.0 -44.4 -86.3 -1.1 _ 63.2 62.7 0.086 The true strains in the members are obtained by deducting the strain reading from the zero load to all the readings of a given sensor. The true strains are as shown below; Table 2: Experimental results for true member strains and node deflection of frame Load True Member Strains Digital Indicator Reading 1 2 3 4 5 6 7 8 (N) (με) (με) (με) (με) (με) (με) (με) (με) (mm) 0 0.0 0.0 0.2 0.2 -0.1 _ 0.3 -0.3 0.000 50 9.3 -8.9 -8.7 -17.5 -0.3 _ 12.9 13.1 0.026 100 17.9 -17.8 -17.5 -34.5 -0.5 _ 25.4 25.4 0.044 150 27.4 -27.3 -27.0 -52.6 -0.6 _ 38.6 38.3 0.059 200 37.2 -6.6 -36.0 -70.3 -0.7 _ 51.7 51.7 0.075 250 44.9 -45.0 -44.4 -86.3 -1.1 _ 63.2 62.7 0.086 The results in the tables, as well as the method of joints, are the same with regard to the state of stress of the members in the frame; Members 2, 3 and 4 are compression members Members 1, 7 and 8 are tension members Member 5 has no normal strain on it (zero- force member) Plotting a graph of both the recorded and true internal strain with incremental load for both a compression and tension member gives the following; Figure 3: Load- Strain curve for Compression Member Figure 4: Load- Strain Curve for Tension Member The graphs for both members demonstrate a linear relationship between the strains and forces in the members and the incremental load applied to the frame.

This confirms the hypothesis made by analyzing the truss using the method of joints, which is valid for materials that obey Hooke’s Law and for small deformations. The relationship between the deflection of the node of the truss and the incremental load applied adopts an almost linear trajectory, as shown below; Figure 5: Load- Deflection Curve for node of Truss Frame The internal forces in the truss members from the experiment are calculated using Equation 1 above, and the deviation from the theoretical values determined by method of joints determined as below; Table 3: Comparison of Experimental and Theoretical Analysis of Statically Determinate Truss Member Experimental Force (Fexp) Theoretical Force (Ftheo) Deviation (N) (N) (%) 1 266.2 250.0 6.48 2 -267.1 -250.0 6.84 3 -246.6 -250.0 1.36 4 -512.2 -500.0 2.44 5 -6.3 0.0 _ 6 _ _ _ 7 373.2 353.6 5.54 8 374.5 353.6 5.91 The results show a good correlation between the experimental and theoretical results, with small deviations less than 7%.

Therefore, the framework is congruent with the pin joint theory. It can also be noted that member 5 is subjected to a slight compression force in the experiment despite the method of joints depicting it as a zero- member. There should be no internal forces due to the fact that the member is attached to a roller support, which allows for movement in the y- direction.

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