Aircraft Structures Experiment - Assignment Example

Name Tutor Institution Date Introduction Engineering mechanics describe the deflection of a given structure as the extent to which a given structural element is superseded under a particular load. In most cases, member deflections can be determined using formulas for specific geometry structures. Additionally, other methods such as Macaulay’s method, Unit force method, Castigliano’s method and many more are employed in calculating frame work deflection. However, for this particular experiment framework deflection will be determined by employing the strain energy method. There exist two major requirements that are very crucial towards the study of deflections of given structures. To begin with, where there is the possibility of new materials being employed for instance aluminium alloy, some connection with steel in the way of deflection is necessary. In a number of cases, it is essential that both the top as well as the bottom beam of a given structure be horizontal while loading since it is somehow under these situations that the panel point deflections are obtained. Nonetheless, for this particular experiment case seem different in addition to being explained in the objectives of the experiment part. Immediately the experiment is finished, a comparison between practical test and theory results will be carried out. Objectives of the Experiment An all set outline of the structure will be made available which will then be loaded under the cretin load. This structure will thereafter be loaded starting off from 0 kg up to 1.1 kg and increasing by a load of an equivalent 100 g each time it is loaded. Framework deflection will also be determined by employing a deflection measurement device every time a new load is added on to the structure. A comparison of the results from this experimental data and that of theoretical values will thereafter be determined employing the strain energy method. Apparatus Used All the apparatus meant for this particular experiment were made available. These included deflection measurement machine, load to be applied on the structure, a ruler for measuring lengths of the various framework members and a four bay statically determinate framework composed of rigid spring members. Fig1 Procedure of the Experiment The framework as seen in figure 1 was provided; The initial step was to put a given amount of preload within the load hanger so as to absorb the sagging coming from the framework used in the experiment. Afterwards, the load was applied with increments of 100 grams to 1100 grams while taking deflection measurements at mid point every time a new load was put. Each revolution was equal to one millimetre deflection as determined within the deflection measurement machine. Graphs of deflections versus loads for each point were plot for both theoretical and experimental results in addition to that the slopes of both graphs were measured. Deflection graphs against loads for all points were constructed for both experimental as well as theoretical results. Moreover, slopes of the Graphs were determined. The rigidity of a normal spring was subsequently established. At last, a comparison of both the theoretical and experimental loads was conducted. Theory: Observing the framework illustration employed in this particular experiment, one can without a doubt observe that the structured has been mirrored and that the measurements of members’ lengths were reached to by employing the use of a ruler. On the other hand, triangle of this particular structure is right-angled with sides being in the numeral ratios 3:4:5.And since it is a right-angled triangle; one side is observably 90 degrees. The remaining two sides are roughly 36.86° and 53.13° employing the Pythagoras theory. Therefore sinø = 4/5 and cosø = 3/5. External forces: Reaction forces exerted at the end of this particular structure and as well the load exerted at the central point of it such that the sum of the reaction forces are equivalent to the load exerted. In this case P1 has to be equal to P2 by equilibrium: P1 + P2 = W and P1= P2 (by equilibrium) Therefore P1 = P2 = W/2 To establish forces from each member, the forces ought to be determined both vertically as well as horizontally. P1 cosø = R1 substituting R1 = W/2 and cosø = 3/5 gives P1 x (3/5) = W/2 rearranging for P1 gives P1 = 5W/6 P14 = P1 sinø substituting P1 = 5W/6 and sinø = 4/5 gives P14 = 2W/3 P34 = P1 cosø substituting P1 = 5W/6 and cosø = 3/5 gives P34 = W/2 P3 = P1 sinø + P23 substituting P1 = 5W/6 , sinø = 4/5 and P23 = 0 gives P3 = 2W/3 And P2 = P34 / cosø substituting P34 = W/2 and cosø = 3/5 gives P2 = 5W/6 Strain Energy Strain energy is more often than not built up in an elastic solid once the solid has been distorted by a given load. Without energy losses, for instance friction or yielding, the strain energy is equivalent to the work done on that given solid by the external loads. External Work done by forces on structure = internal Strain Energy Strain energy is a form of potential energy when taking into account work done on a given elastic solid coming from a particular point force F.In case the elastic solid bears the load F, it deforms by way of strains and the material becomes stressed. D is the displacement within the same location and same direction as point force F, D and F are displaced. Work done by force F on elastic solid is the area under force versus displacement curve. Work done = strain energy = ½ px This presupposes that the system is linear-elastic, as a result deflection d can be described as a linear function of W.The overall strain energy within the system is therefore the total individual strain energies from every truss member. Therefore for each spring energy U = ½ px where p=force applied and x=extension According to hook’s law, estimation of a spring’s extension is directly proportional to the additional load so long as the load does not go beyond the elastic limit. According to Hook’s Law: P = k x where p=force applied and x = extension And k=spring constant Therefore; x= p/k and U = P² / 2K In this structure of this particular experiment, the total work done within the system is calculated employing the formula below: W.D. = ½ W = ∑ = 2 Therefore rearranging the formula above and substations of P1, P2 in addition to P3 to obtain the theoretical deflection values d gives: = 4/W Another method that can be employed in analyzing loads on every member is known as the unit load method used in the deflection of beams. If we take into account a simple held up beam and which is put through a single point load as can be observed from the figure below. In the figure, the length at which the bending point can be regarded as constant is specified as distance ‘x’ from the left hand support: According to the simple bending theory   R = radius of curvature and 1/R = curvature of the beam /the change in the slope rate  At this point, if it is presupposed that moment is to the beam progressively more, then this implies that the relation between it and that of change in the slope will appear linear as revealed in the figure below. External load on each one of the members by the bending moment ‘M’= strain energy stored and is provided by dU = (1/2 M x dƟ) At this point differentiating the expression for strain enery with regards to x (distance) gives:  , substituting for   dx=  dx Therefore the sum strain energy in the beam is  According to Castigliano’s first theorem  Where U = the structure’s total strain energy as a result of the applied load W= the acting force where the displacement point acting at the point where the displacement is intended to be.  Linear displacement in the direction of line of action Taking into account the simple held-up beam in the above figure where the mid-span deflection at C is to be determined and which is caused by applied point load P. Now it becomes easy to establish a bending moment diagram that will provide ‘M’ bending moment for the applied load. The second step is to do away with the applied load from this structure. An imaginary unit load is thereafter applied in position going in the direction of the intended deflection. This implies that a vertical load is equivalent to 1 and in this case there will be a new bending moment diagram for the structure ‘m’. If both imaginary load and real load are presupposed to operate at the same time, then bending moment within the beam is provided by: Q = (M + βm) Where M=bending moment as a result of the applied load M=bending moment as a result of imaginary load β is a multiplying factor to reflect the imaginary load value and since it is the imaginary load the value of β is therefore zero. The strain energy within the structure= total energy down the beam: Castigliano’s theorem provides deflection as   =  =  And ;  Given that β=zero then vertical deflection  is given by: This implies that deflection at beam point is provided by: If this theorem was to be employed in this particular experiment, then the summation of every deflection would be found out. Results: Results of the Experiment: Weight (KG) Deflection (mm) 0.1 0.66 0.2 1.19 0.3 1.83 0.4 3.4 0.5 4.33 0.6 5.07 0.7 5.79 0.8 6.56 0.9 7.3 1 7.99 1.1 8.6 Deflection Vs load Theoretical results: Weight (KG) Deflection (mm) 0.1 0.733333 0.2 1.466667 0.3 2.2 0.4 2.933333 0.5 3.666667 0.6 4.4 0.7 5.133333 0.8 5.866667 0.9 6.6 1 7.333333 1.1 8.066667 Deflection Vs load Read More

Graphs of deflections versus loads for each point were plot for both theoretical and experimental results in addition to that the slopes of both graphs were measured. Deflection graphs against loads for all points were constructed for both experimental as well as theoretical results. Moreover, slopes of the Graphs were determined. The rigidity of a normal spring was subsequently established. At last, a comparison of both the theoretical and experimental loads was conducted. Theory: Observing the framework illustration employed in this particular experiment, one can without a doubt observe that the structured has been mirrored and that the measurements of members’ lengths were reached to by employing the use of a ruler.

On the other hand, triangle of this particular structure is right-angled with sides being in the numeral ratios 3:4:5.And since it is a right-angled triangle; one side is observably 90 degrees. The remaining two sides are roughly 36.86° and 53.13° employing the Pythagoras theory. Therefore sinø = 4/5 and cosø = 3/5. External forces: Reaction forces exerted at the end of this particular structure and as well the load exerted at the central point of it such that the sum of the reaction forces are equivalent to the load exerted.

In this case P1 has to be equal to P2 by equilibrium: P1 + P2 = W and P1= P2 (by equilibrium) Therefore P1 = P2 = W/2 To establish forces from each member, the forces ought to be determined both vertically as well as horizontally. P1 cosø = R1 substituting R1 = W/2 and cosø = 3/5 gives P1 x (3/5) = W/2 rearranging for P1 gives P1 = 5W/6 P14 = P1 sinø substituting P1 = 5W/6 and sinø = 4/5 gives P14 = 2W/3 P34 = P1 cosø substituting P1 = 5W/6 and cosø = 3/5 gives P34 = W/2 P3 = P1 sinø + P23 substituting P1 = 5W/6 , sinø = 4/5 and P23 = 0 gives P3 = 2W/3 And P2 = P34 / cosø substituting P34 = W/2 and cosø = 3/5 gives P2 = 5W/6 Strain Energy Strain energy is more often than not built up in an elastic solid once the solid has been distorted by a given load.

Without energy losses, for instance friction or yielding, the strain energy is equivalent to the work done on that given solid by the external loads. External Work done by forces on structure = internal Strain Energy Strain energy is a form of potential energy when taking into account work done on a given elastic solid coming from a particular point force F.In case the elastic solid bears the load F, it deforms by way of strains and the material becomes stressed. D is the displacement within the same location and same direction as point force F, D and F are displaced.

Work done by force F on elastic solid is the area under force versus displacement curve. Work done = strain energy = ½ px This presupposes that the system is linear-elastic, as a result deflection d can be described as a linear function of W.The overall strain energy within the system is therefore the total individual strain energies from every truss member. Therefore for each spring energy U = ½ px where p=force applied and x=extension According to hook’s law, estimation of a spring’s extension is directly proportional to the additional load so long as the load does not go beyond the elastic limit.

According to Hook’s Law: P = k x where p=force applied and x = extension And k=spring constant Therefore; x= p/k and U = P² / 2K In this structure of this particular experiment, the total work done within the system is calculated employing the formula below: W.D. = ½ W = ∑ = 2 Therefore rearranging the formula above and substations of P1, P2 in addition to P3 to obtain the theoretical deflection values d gives: = 4/W Another method that can be employed in analyzing loads on every member is known as the unit load method used in the deflection of beams.

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