Deflection of Pin-Jointed Structures - Assignment Example

TASK#1: DEFLECTION OF PIN-JOINTED STRUCTURES TASK# 2: BEHAVIOR OF COMPRESSION MEMBERS Name Institution Instructor Date Task#1: Deflection of Pin-jointed Structures Introduction Some of the structures that are widely and commonly used involve the use of plane trusses as part of their components. These kinds of trusses also find application in bridges as well as structures of roof support. Plane trusses and pin jointed structures form a very significant component in the structure of most cranes. The design consideration of plane trusses in various structures is a very useful concept and is both experimental and theoretically determined and analyzed. Experimental determination and analysis of deflection in pin jointed structures and plane trusses are very significant design consideration (Kindmann and Kraus, 2011). Deflections in truss joints can also be determined through the use of unit load technique which operates on the basis of virtual work principle. It is important to do a comparison between the results obtained from the experimental deflection determination of the pin jointed structure and the theoretical ones and this is one of the reasons for carrying out this experiment. Accurate results obtained from this experiment are significantly used several practical cases (Ziemian, 2010). Objectives The objectives of this experiment was determination of deflection for pin-jointed structure in loaded state Comparison of the results obtained from this experiment with the ones theoretically obtained through calculation and analysis of virtual work. Theory on deflection of pin-jointed trusses Pin jointed trusses are in their construction and functionality comprises straight component members that form one or more units that are triangular in shape. The truss members in pin jointed trusses are involved in connection at each of their ends normally by pin joints. The pin joints are otherwise referred to as nodes. Reactions and forces from external sources are seen to only act at the pin joints and bring about forces to component members which are compressive or tensile forces. The lying of all nodes and component members in two dimension plane is referred to as plane truss whereas a space truss has got its nodes and component members occurring in three dimensions. Pin jointed trusses in their operations find application in several structures such as the roof supports, bridges, space stations, powers transmission towers among others (Kassapoglou, 2013). Figure 1: Force diagram of pin jointed trusses Methodology The methodology that was employed in the experiment involved procedural steps as follows; 1. The dial gauge was appropriately set and its front tapped lightly as the experiment begun 2. The gauge reading was recorded at no load for the load hanger 3. 20N loads were gently added onto load hanger and the reading recorded. The addition of loads was done in a gentle manner to avoid other impacts apart from the static value. 4. The increments of 20N loads were used to raise the load to 100N in which case the reading of the gauge was recorded at every instance of the increments. 5. The overall load was then reduced down to zero in 20N decrements and the gauge reading recorded in each instance of decrement. 6. The recorded data was then tabulated and the average deflection calculated in each case which corresponded to 20N, 40N, 60N, 80N as well as 100N. Results The results obtained were tabulated as shown and graphs plotted as shown; Load, N Unloading Displacement, mm 20.1 1.0 25.1 1.5 30.1 2.5 31.1 4.5 32.1 6.0 33.1 9.0 34.1 17.0 34.6 43.0 Table 1: Unloading Displacement results Load, N Loading Displacement, mm 0 7.0 10.1 10.0 15.1 13.0 20.1 19.0 25.1 30.0 26.1 33.0 26.6 36.0 Table 2: Loading Displacement results Load(N) Deflection from Loading (mm) Deflection from Unloading (mm) Deflection Average (mm) 0 0 0 0 20 0.031 0.035 0.033 40 0.072 0.067 0.0695 60 0.099 0.098 0.0985 80 0.128 0.129 0.1285 100 0.157 0.157 0.157 Table 3: Loading and Unloading Deflection results Graph of Deflection of joint (G) against Load (N) Discussion Pin jointed trusses could be regarded as being statistically determinate in which case the entire component member forces as well as the reaction from the support are determined (Muttoni, 2011). This determination is achieved through the static equilibrium equation; M + r=2j Where; M = number of truss members r = reaction component number j = joints number in pin jointed truss The deflection that corresponds to the 100N load was established from the graphical best fit as 0.128mm. It was observed from the experiment that the behavior of the truss was linearly elastic. The theoretical deflection value of joint G is computed as; D = (NL) / (AE) Where; D is the deflection value of joint G (mm) N is the load (N) L is the length (mm) A is the area of gross section for the members (mm2) E is the modulus of elasticity value for the truss material (75KN/mm2) D = (100 x 540) / (20 x 75) = 0.036mm Conclusion In conclusion, the deflection of Pin-jointed Structures has clearly demonstrated various structures experience different amounts of deflections in the loaded states. It is therefore a requirement that the deflections be determined and incorporated in the design of structures. It is important for machine elements to be appropriately rigid in their design and construction so as to prevent instances and cases of misalignment. The incorporation of deflection into design and construction of various structures also serves to uphold the accuracy of structural dimensions. A comparison between the theoretical and experimental evaluation and analysis of virtual work had demonstrated the significance of experimentations when it comes to the selection of materials for various construction applications. Task# 2: Behavior of compression members Introduction The buckling theory is majorly concerned with the application of compression loads and the behavior of structural members in response to the compression loads. Structural elements that are under tensile stresses are able to withstand such kind of stresses to a certain given extent. This extent is dependent on the material used in the fabrication of the structural element. Structures that are fabricated from steel have the capability of tensile stresses up to the values of yield stress. At this value of yield stress, the elastic behavior ends and it marks the beginning of plastic behavior. When structural elements are under compressive stresses, they are likely to experience failure when the level of stress that they experience is still below the value of yield stress for the materials used the fabrication of the component elements of the structure. Objectives To perform an experimental investigation on the manner in eccentrically loaded struts behave having initial curvature To use South-well plot to carry out an experimental determination of the Euler buckling load Methodology The methodology used in the experiment involved procedural steps as follows; 1. The strut was set up in a frame as demonstrated in figure 2. 2. Knife edge brackets which are two in number were covered in the groove that corresponded to eccentricity of zero value 3. The counterweight was adjusted on the loading arm in such a way that the arm weight did not act on the strut 4. 0.98N (100g) load hanger was used together with a member to make an addition of its weight to the ones that had already been used 5. The scale i.e. was subjected to adjustments to a reference reading which was zero 6. The addition of weight on to the hanger then continued while the corresponding deflections were being measured. The addition of weight took place in this manner; 20N+5N+5N+1N+1N+1N+1N+0.5N. The damaging of the strut was prevented in the process weight addition through the maintenance of the stop bracket i.e. (HST1502) within the range of elastic. 7. The load that was actually acting on the strut was calculated. The magnification of weights was done by the lever arm. 8. The steps above were then repeated using two knife edge brackets i.e. (HST.1505) which were located in the grove and corresponded to the highest eccentricity. In this case, the loading sequence that was used involved; 10N+5N+5N+5N+1N+0.5N The illustration of the setup that was used in the experiment is as follows; Figure 2: Behavior of compression members experimental set up Results The results were obtained and presented in a tabulated form as shown; Load on Pan Actual load on Strut (P) Lateral deflection measured (y*) Displacement, (mm) Load, N y*/P 20 20.1 1.0 0 0 0.05 5 25.1 1.5 7.0 10.1 0.06 5 30.1 2.5 10.0 15.1 0.08 1 31.1 4.5 13.0 20.1 0.14 1 32.1 6.0 19.0 25.1 0.19 1 33.1 9.0 30.0 26.1 0.27 1 34.1 17.0 33.0 26.6 0.50 0.5 34.6 43.0 36.0 26,7 1.24 Table 4: Loading and Lateral Deflection results Load on Pan Actual load on Strut (P) Lateral deflection measured (y*) y*/P 20 20.1 1.0 0.05 5 25.1 1.5 0.06 5 30.1 2.5 0.08 1 31.1 4.5 0.14 1 32.1 6.0 0.19 1 33.1 9.0 0.27 1 34.1 17.0 0.50 0.5 34.6 43.0 1.24 Table 5: Loading and Lateral Deflection results Graph of y* against y*/P with curvature plus eccentricity Discussion The yield stress level at which the failure of a structure occurs is dependent on its cross section and length. Slender structures are likely to bow under high levels of stress which is referred to as buckling. For the elements that are thin and long, their actual shape may be recovered when the effect of the compressive forces is done away with. A straight strut buckles and becomes unstable when the forces of axial compression that it experiences attains critical buckling load (Ziemian, 2010). The theoretical value of potential energy is obtained as PE = (3.142)2 x (EI/L2) Where; PE = potential energy E = is the modulus of elasticity value for the truss material (75KN/mm2) L = is the length (mm) I = is the second moment of area (mm4) Gradient (M): M=P (N) = (20-10)/ (1.3-0.9) = 25N e = 8 (from graph) Weight magnification: Load on the strut = Force on the hanger x (1000/750) Conclusion In conclusion, the experimental determination of the behavior of compression members for structures has demonstrated the consideration of eccentricity as well as curvature is significant. This is in line with the theoretical explanation that also points out the role played by the two quantities in design and construction of structures. The Euler buckling load is also significant as demonstrated from the results and their analysis. It has been determined that buckling load must not be exceeded in any design of structures. References KASSAPOGLOU, C. (2013). Design and analysis of composite structures with applications to aerospace structures. Chichester [England], John Wiley & Sons Inc. KINDMANN, R., & KRAUS, M. (2011). Steel structures design using FEM. Berlin, Germany, Wilhelm Ernst & Sohn. MUTTONI, A. (2011). The art of structures ; introduction to the functioning of structures in architecture. Abingdon, Oxford, UK, EPFL Press/Routledge. ZIEMIAN, R. D. (2010). Guide to stability design criteria for metal structures. Hoboken, N.J., John Wiley & Sons. Read More