Deflection of framework report PDF

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University of SalfordSchool of Computing, Science and EngineeringAircraft Structures LaboratoryDeflection of FrameworkSummaryThe aim of this experiment is to calculate the deflection of a statically determinant pinpointed framework, using incremental loads (from 0 to 1 kg). The resulted deflection v...

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University of Salford School of Computing, Science and Engineering Aircraft Structures Laboratory Deflection of Framework

Summary The aim of this experiment is to calculate the deflection of a statically determinant pinpointed framework, using incremental loads (from 0.1 to 1.1 kg). The resulted deflection values are recorded and a graph is plotted based on the experiment results. The procedure used to calculate the theoretical deflection is the Strain Energy Method. Also, it was shown that the error percentage between the theoretical and experimental results is high (>20%). Furthermore, the error source and other methods of calculating the deflection are also taken into consideration.

Contents 1. 2. 3. 4. 5. 6. 7.

Introduction Theory Description of apparatus Experimental procedure Results Discussion Conclusion References

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1. Introduction “Structures are assembled by joining elements at element intersections. In the case of steel structures, the elements are joined by welding or bolting; in reinforced concrete structures, the joint is made monolithic by ensuring proper disposition of the reinforcement; with timber structures glues, nails and various types of ‘connectors’ are used to join members. Two typical types of joints, stiff and pin-joint, are commonly used.” (Bhatt, Structures, 1999) The aim of this laboratory is to study a framework with pin-jointed members. A load is to be applied to the framework in order to measure the deflection and compare it to the theoretical results. In order to determine the load that a structure can resist, graphs are being used as a method.

2. Theory In the engineering field, deflection is defined as the deformation of a beam or a truss from its original position. “Trusses are a common form of structure generally assembled so as to create a series of triangles. It is assumed that the connection between the members is pinned and the only force in the members is an axial force, either tensile or compressive.” (Bhatt, Structures, 1999) To determine the theoretical deflection per unit load it is used the simple pin-pointed method in a framework. It is used a free-body diagram where at each joint the equilibrium in every direction is satisfied as shown in equation (2.1) and (3.1)

( 2.1 ) ∑ Forces ∈ x−direction =0 ( 2.2 ) ∑ Forces ∈ y− direction=0 It is assumed that the moment arm for every force is zero. Therefore, it is necessary to proceed from joint to joint and find the equilibrium. The members therefore exert a pull on the joints they connect. Using the Strain Energy Method, the theoretical value of the deflection over the load (m/kg) is found. Jennings, Structures from theory to practice (2004) defined strain as the changes in the dimensions of the element that come from the stresses acting on an incremental element. Most structures obey Hooke’s Law (equation 2.3), which states that stress is proportional to strain, over a significant part of their loading range.

( 2.3 ) F =−kx ( N ) , where k is the stiffness constant (kgf/mm) and x is the extension (m); The ‘stiffness’ of a structure is given by the slope of the curve of load vs deflection. If the stiffness number is big, there is a large load that produces a small deflection. The Strain Energy that is stored in a spring is given by the equation (2.4). 1 ( 2.4 ) E= Fx (J ) 2 The energy can be also found by calculating the area under the curve given by the force and extension.

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3. Description of apparatus The apparatus used for this laboratory is made of a four-bay, statically determinant pinpointed framework, composed of rigid spring members as shown in Fig. 3.1. The framework has 6 springs on its members, the left side being symmetric to the right side.

Figure 3.1: Pinpointed framework

It is also used a digital Dial Test Indicator (Fig 3.2) and a Vernier Calliper (Fig 3.3). A Dial Test Indicator it is an indicator commonly used in the technology and manufacturing industry. It accurately measures small distances and angles. In this case it is used to measure the distance between to flat surfaces to its nearest millimetres. A Vernier Calliper is the main tool that is used to measure the diameter of an object. It is originated from the ancient China, but it was reinvented by the French mathematician Pierre Vernier. The Vernier Calliper consists of two scales, the Vernier scale V and the Main Scale S. The Main Scale S is fixed but the Vernier Scale or also known as the Auxiliary Scale is movable and slides along the main scale. It can measure an object accurately up to the 1/10th of a millimetre. The total reading of a Vernier Calliper is given by adding up the measurements from the two scales (Main Scale Reading + Vernier Reading).

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Fig. 3.2: Digital Dial Test Indicator

Fig. 3.3: Vernier Calliper

4. Experimental Procedure The first step in this experiment is to measure the diameter and the length of the members of the framework using the Vernier Calliper. An average of these readings is done, in order to be used further in the calculations. To measure the deflection accurately, a small amount of load is applied to reduce the slack from the springs. After this, the load can be applied and the values of the deflection can be recorded using the Dial Test Indicator. The total load is 1.1 kg, made of 11 increments of 100g. Every increment is applied at one time, increasing the load constantly. There are 11 values of deflection for each load that is applied. A graph is plotted using the deflection and the applied load. The slope of the graph represents the stiffness of the structure and the area under the curve represents the strain energy. This will be later used to compare the theoretical and experimental values of deflection.

5. Results 5.1

Theoretical Results

The theoretical results for the experiment are found using the pin-jointed and Strain Energy method which is applied on the Statically Determinant Pin-Pointed Structure showed in Fig 5.1.1.

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Fig 5.1.1: Statically Determinant Pin-Pointed Structure

The load (W) is applied downward. There are two upward reactions (R1 and R2). By symmetry, it is assumed that R1=R2. R1 + R2=W Resolving it upward and downward: R = R =W /2. Therefore, 1 2 The force in each point is calculated to satisfy the equilibrium equations (2.1) and (2.2). Figure 5.1.2 shows the forces acting at point 1. By measurements: L23=41cm; L12=30.5cm; L13=51cm; Therefore, Cos θ= 30.5/51; Sin θ= 41/51.

Fig. 5.1.2: Point 1

At point 1: 51 W 30.5 P = ; ∴ P 1= W . 61 51 1 2 41 41 P1 ; ∴ P 14 = W .  Horizontal Forces: P14 = P1 sinθ= 51 61 At point 2:  Vertical Forces: P12 =0 ;  Horizontal Forces: P23=0. At point 3: 1 51 30.5 W= W ;  Vertical Forces: P34 = P1 cosθ= x 61 51 2 51 41 41  Horizontal Forces: P3= P1 sinθ+ P23 = P1 sinθ= x W = W . 61 61 51 At point 4: 51  Vertical Forces: P34 = P2 cosθ ; ∴ P 2= W . 61 

Vertical Forces: R1= P1 cosθ=

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For the whole system, the work done is given by the formula: P21 P22 P32 P2 ; Work done=∑ =2 + + 2K 2K 2K 2K 2 2 512 W 2 412 W 2 51 W ∴Work done=2 + + 2 2 2 61 × 2× 0.5 61 × 2× 0.5 61 ×2 × 0.5 ∴Work done=2 W 2 (1.845 ) The work done is also given by formula: 1 Work done= Wδ 2 1 W =2 W 2 x(1.845) Therefore, 2 δ ∴ =1.845 x 4 =7.38 mm / kg W

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5.2

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Experimental Results

Table 5.2.1 shows the load and deflection data recorded from the experimental procedure. Using this table, a graph can be plotted to get the slope of the cure, which represents the δ/W value. The area under the curve gives the Strain Energy stored in the system. Graph 5.2.1 is not linear as expected because of some errors that will be discussed further.

Load (kg) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Deflection (mm) 0.56 1.23 2.12 3.15 4.06 5.13 6.01 7.07 7.93 8.77 9.72

Table 5.2.1: Experimental Results

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Deflection vs Load Graph 10 f(x) = 9.38 x − 0.56

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Deflection (mm)

8 7 6 5 4 3 2 1 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Load (kg) Graph 5.2.1: Deflection vs Load

The slope of the graph is calculated using a feature of Microsoft Excel, and it is 9.38 mm/kg.

6 Discussion It is noticed that there is a big difference between the theoretical and experimental values. The theoretical value of deflection per unit load is 7.38 mm/kg, while the experimental value is 9.38. 9.38−7.38 x 100=21.32 % . The percentage of error is 9.38 This percentage of error is really high and is vital when comes to build a framework structure. The deflection need to be eliminated in order to prevent adverse effects.

The theory of the laboratory is based on the fact that all members take equal forces. Also, all the springs have the same tension when the load is applied. This assumption should be changed in order that the experimental and theoretical values to be closer. Apart from more obvious possible sources of error due to inaccuracies in reading the apparatus, overall dimensions and material proprieties, significant errors may arise from ignoring the thickness of members. In this experiment the thickness of the members is ignored completely. It is noticed that the framework has two types of members with different diameters: 2.78 and 8.28 mm. This has a huge impact on the overall percentage of error. Several factors can affect the magnitude of forces acting on each member. The movement of supports or the slippage within joint have an impact on the structure, making the deflection bigger. The structure can be also affected by the temperature and different stresses. Temperature effects occur mainly because of sunshine and the daily temperature cycle, industrial activity or fire, but in this case the temperature is not a major factor, the experiment taking place in a laboratory. The material of the members can have small cracks and will not be possible to 7

support the load of which it was designed to; therefore there will be a difference between the theoretical and experimental results. There are other ways of measuring the deflection on a framework structure and to avoid the high percentage of error found above. “Although graphical methods have been superceded by other methods, even a sketch of a Williot or Williot-Mohr diagram can provide an insight into how a truss deflects, or can form a basis for checking results computed by other means. In a Williot diagram, deflections are assumed to be small enough for second order effects to be neglected, and are plotted all on the same diagram”. (Jennings, Structures from theory to practice, 2004) One method for structural analysis is the direct stiffness method (matrix stiffness method). All the properties of the members are compiled into a single matrix equation which governs the behaviour of the entire structure. By solving this equation the displacements and the forces can be easily found. “In the direct stiffness method, the stiffness equations for a structure (Ku = f) are formulated and (given the applied forces f) are directly solved for the displacements u. It is then possible to evaluate the member forces from the multiplication: p= kAu. Such a technique is cumbersome using hand calculation for the statically determinate trusses. However, on a computer, it has the advantages of being straightforward and versatile; one aspect of the versatility being that it can be used to solve statically indeterminate structures just as easily as statically determinate ones.”(Jennings, Structures from theory to practice, 2004) “As soon as digital computers became available, the benefit of direct stiffness methods became apparent, as recognised by Livesley (1953) for framed structures and Argyris and Kelsey (1960) for aircraft structures. Key requirements for effective computational methods are simplicity and efficiency. Furthermore, with the speed of modern computers, efficiency is only important when equations to be solved have very many variables. A computer program based on this technique is able to analyse a plane truss.” (Jennings, Structures from theory to practice, 2004) Energy methods are used to obtain the equilibrium equations for different framework structures. Otto Mohr extended the basic concept of virtual work of Jean Bernoulli to frameworks. “The structure is subjected to a single concentrated load and the deflection of the loaded point is required. Let F be the force in a member due to the external load, and the corresponding extension be FL/ (AE), where L= length of member, A=Area of cross-section of the member and E= Young’s modulus. Thus the equation becomes: Virtual work by external loads= applied load x deflection under the load= ∑ F 2 L/( AE ) 2 ❑ ∴ Deflection under the load = { ∑ F L/( AE)} load at the joint ” (Bhatt, Structures, 1999) “In the above case, because there was only one load acting on the truss and the deflection required was also that at the loaded joint, the virtual extension was simply made equal to the real extension. If the deflection is required at some joint other than the loaded joint or if several loads act on the structure then the strategy adopted is slightly different. Let the extensions of the members under the action of external forces be FL/(AE) and the desired deflection at a specific joint be ∆. Consider another load system, where the truss is loaded by a single external force of unity at the joint and in the direction in which the actual deflection ∆ is required. Let the force developed in the ith member be f i . Then if the forces in the new system are subjected to deflections and extensions caused by the actual loads, then

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clearly 1 x ∆=∑ ¿ ∑ f i

{ FLAE } i

, where the summation is over all members of the truss.”

(Bhatt, Structures, 1999) Deflection at any joint can be also found using the direct integration method. The curvature of the member at any section under the action of external forces is M/EI and the desired deflection at a specific joint is ∆. M is the bending moment at a section in a member, I is the second moment of area and E is the Young’s Modulus. Therefore, the equation for the deflection is ∆=∑∫ m i { M /EI }i dx , where m i is the bending moment developed in the ith member.

7 Conclusion The structural performance of a frame depends on the ability of the members to resist bending and the joints to maintain the angles between attached members. It was found that the percentage error from the theoretical and experimental results is high. This comes from inaccuracies when reading the apparatus, overall dimensions and material properties, ignored thickness of the members, supports or joints movement and from using a method of calculation of deflection that has inaccurate assumptions. There are couple of ways for calculating the deflection of a framework structures such as the stiffness method, the direct integration or the virtual work method.

References A. Jennings (2004). Structures from theory to practice. Spon Press London P. Bhatt (1999). Structures. Longman Limited

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