Stiffness of Materials: Aluminum, Brass, and Steel - Lab Report Example

STIFFNESS OF MATERIALS Student’s name & number Course code + course name Professor`s name University name City, state Date Table of Contents i.Moment of inertia 5 ii.Objective of the lab 5 iApparatus 7 iii.Procedure 1a 8 iv.Procedure 1b 9 iAluminium 10 v.Brass 10 vi.Steel 10 vii.The graphs 11 iAluminium 13 viii.Brass 13 ix.Steel 14 LIST OF FIGURES Figure 1: A graph load W (N) against deflection mm for Aluminium 10 Figure 2: A graph load W (N) against deflection mm for brass 11 Figure 3: A graph load W (N) against deflection mm for steel 11 Figure 4: A graph load W (N) against deflection mm for Aluminium with hollow structure 13 Figure 5: A graph load W (N) against deflection mm for Brass with hollow structure 14 Figure 6: A graph load W (N) against deflection mm for Steel with hollow structure 14 1. Abstract In this lab different materials were used as beam to determine the characteristics of such materials. The materials under test included aluminum, brass and steel. However, the lab makes use of the simply supported beam. The deflection of the beam is determined by hanging a known mass at the center of the beam. However, beams with different sections are used to determine the mechanical properties. One time the beam is used as solid section and the other beam is hollow though rectangular section. The magnitude of the load varied so that the relationship between deflection, height, span width and load applied. Using a dial gauge the deflection of materials is determined. The tests were repeated three times. The values obtained were used to compare the theoretical coefficient of elasticity and the one obtained experimentally. Also the width and the thickness of the beams were of great interest in the experiment to calculate the second moment of the beam. Also these values were used to determine the area of the beam. Young modulus is another important mechanical property of the beam. It determines the places where which type of beam can be used. (192 words) 2. Introduction To determine the stiffness of a material, young modulus is used. Important graphs to drawn here are strain and stress graphs. Deflection is used to determine the stiffness as well. i. Moment of inertia This is also another very important characteristic of beams that is used in calculation of young`s modulus. It can be associated with the resistance to rotation of the beam. It is calculated based on thickness and the shape of the beam. The type of material and the length of the materials do not influence moment of inertia of the beam. Moment of inertia of a beam with rectangular section can be determined using the formula below for a solid cross section; I= Equation 1 For hollow cross section I= Equation 2 B is the width of the beam used and the h is the height of the beam. ii. Objective of the lab The general objective of this experiment is to determine the stiffness of different materials when they are subjected to different kinds of loading. The stiffness of the materials is influenced by the geometry of the beam and type of the materials in place. In engineering, the application is mainly determined by the stiffness among many other factors. The apparatus used here is the TE161. It is a simple apparatus that exploits a simple structure as well. The structure used is a beam in a three point bending. The apparatus studies the effects of the geometry of the structure and the material used to make the beam have on the stiffness of the beam. The apparatus is very simple to operate. The operator puts the beam on the apparatus and then the load is applied. To determine the deflection of the material a dial gauge is used to display the deflection at the point of loading. 3. Theory Beams are used in engineering to support the various loads over different lengths. Beam is a structure member that is loaded non-axially making it to be under bending. A beam is said to be under bending when the forces on it are acting in such a way that they induce compressive stress over that part of the beam. The deflection on beams occurs when a point on the beam is displaced on the neutral plane of the beam from its original position when loads of forces are applied. Deflection of the material can be determined using the young`s modulus and moment of inertia of the beam. Other important section properties are included too. Other properties depend on the situation imposed on the beam. Deflection of a simply supported beam can be calculated as shown below for a concentrated point load; = Equation 3 where: Δ = deflection, (mm) P = load, (N) L = length of beam, (mm) E = modulus of elasticity (N/m2) I = moment of inertia of section about the neutral axis, (mm4) Deflection is used to determine the overall stiffness of the beam under testing. Deflection is seen as a function of stiffness of materials in addition to proportion of the piece. Deflection is very important in engineering is that the engineers can determine and calculate the modulus of elasticity for a material in flexure. The equation below is used to determine the stiffness of the material; Stiffness = Equation 4 Where the deflection, p is is the load and stiffness is measured in N/M For a simply supported beam between two points, the stiffness can be found using the equation below; ∆= Equation 5 K is the flexural rigidity K= I*E Equation 6 4. Apparatus and the procedure i Apparatus TE161 Dial gauge Different beams Vernier calipers iii. Procedure 1a Three specimens were used. These specimens were; Mild steel Brass Aluminium 1. For data collection the table below was created Load (w) N Deflection (mm) 0 5 10 15 20 25 Experiment; 1a Material Beam size 2. The apparatus was set up with the knife edges being placed 400 mm apart (the mid span was 200mm from both edges). A flat specimen that was solid was used (nominal 19.05 mm x 3.2mm). The dimensions were determined with very high accuracy. Dimensions were recorded on the table. To ensure high accuracy, Vernier calipers were used for this experiment. 3. On the load hanger, 5N was loaded and the deflection observed was recorded. 4. Another 5N was added on the hanger, the new deflection was recorded on the table. The loaded increment continued until the weight was 25N. The deflections were recorded as the loads were increased. 5. The loads were removed 6. A graph of load in N against deflection in mm was plotted. On the discussion the graph plotted was well commented, relationship between load and deflection as well as the gradient of the graph. iv. Procedure 1b The specimens used were; Brass Mild steel Aluminium 1. The table below was construction for the collection of the data for the second experiment Load (w) N Deflection (mm) 48I 0 5 10 15 20 25 Experiment; 1a material Material H= I= Beam size L3 2. The dimensions of the cross section of the specimen were measured accurately. Using the dimensions of the cross section, the second moment of area was calculated. The value of second moment of area was recorded on the table 3. The apparatus was set up with the knife edges being placed 400 mm apart (the mid span was 200mm from both edges). 4. The specimen was placed well on the knife edge. The load of 5N was placed on the mid span as the deflection was recorded. The loads were increased in steps of 5N until 25N was reached. 5. The table was completed on the column three and four (48I and respectively) 6. A graph of load (N) w versus was plotted. The gradient of the graph was determined after which the value of E was determined for each specimen On the discussion the results were commented. The experimental values of E were compared to theoretical values. Also, how the second moment of area affects the stiffness is discussed in the discussion section. 5. Results The second moment of area of the beam is I= = =52.0192mm4 i Aluminium Load (w) N Deflection (mm) test 1 Deflection (mm) test 2 Deflection (mm) test 3 Average deflection 0 0 0 0 5 1.68 1.68 1.68 1.68 10 3.27 3.29 3.35 3.303333 15 4.91 5.01 4.99 4.97 25 8.33 8.29 8.28 8.3 Experiment; 1a Material aluminium Beam size 19.05 x 3.2 mm v. Brass Load (w) N Deflection (mm) test 1 Deflection (mm) test 2 Deflection (mm) test 3 Average deflection 0 0 0 0 0 5 1.38 1.37 1.39 1.38 10 2.69 2.69 2.68 2.686667 15 4.02 4.01 4.04 4.023333 25 6.65 6.72 6.69 6.686667 Experiment; 1a Material brass Beam size 19.05x3.2 mm vi. Steel Load (w) N Deflection (mm) test 1 Deflection (mm) test 2 Deflection (mm) test 3 Average Deflection 0 0 0 0 0 5 0.66 0.65 0.66 0.656667 10 1.33 1.33 1.34 1.333333 15 2.01 2.00 2.01 2.006667 25 3.44 3.37 3.35 3.386667 Experiment; 1a Material Beam size 19.05 x 3.2 mm vii. The graphs Figure 1: A graph load W (N) against deflection mm for Aluminium Figure 2: A graph load W (N) against deflection mm for brass Figure 3: A graph load W (N) against deflection mm for steel For the hollow structure The thickness of the Aluminium is 1mm The inner section is therefore 17.05x 1.2mm in dimensions The second moment of area of the hollow section is; I= I= - = = 2.52 mm4 - = 52.0192-2.52 =49.4992 mm4 i Aluminium Load (w) N Deflection (mm) test 1 Deflection (mm) test 2 Deflection (mm) test 3 Average deflection 48I 0 0 0 0 0 0 0 5 1.68 1.68 1.68 1.68 3991.615 6.24E-05 10 3.27 3.29 3.35 3.303333 7848.592 0.000123 15 4.91 5.01 4.99 4.97 11808.53 0.000185 25 8.33 8.29 8.28 8.3 19720.48 0.000308 Experiment; 1a Material aluminium Beam size 19.05 x 3.2 mm viii. Brass Load (w) N Deflection (mm) test 1 Deflection (mm) test 2 Deflection (mm) test 3 Average deflection 48I 0 0 0 0 0 0 0 5 1.38 1.37 1.39 1.38 3278.827 5.12E-05 10 2.69 2.69 2.68 2.686667 6383.418 9.97E-05 15 4.02 4.01 4.04 4.023333 9559.285 0.000149 25 6.65 6.72 6.69 6.686667 15887.26 0.000248 Experiment; 1a Material brass Beam size 19.05x3.2 mm ix. Steel Load (w) N Deflection (mm) test 1 Deflection (mm) test 2 Deflection (mm) test 3 Average Deflection 48I 0 0 0 0 0 0 0 5 0.66 0.65 0.66 0.656667 1560.216 2.44E-05 10 1.33 1.33 1.34 1.333333 3167.948 4.95E-05 15 2.01 2.00 2.01 2.006667 4767.764 7.45E-05 25 3.44 3.37 3.35 3.386667 8046.591 0.000126 Experiment; 1a Material Beam size 19.05 x 3.2 mm The graphs of the hollow structure Figure 4: A graph load W (N) against deflection mm for Aluminium with hollow structure Figure 5: A graph load W (N) against deflection mm for Brass with hollow structure Figure 6: A graph load W (N) against deflection mm for Steel with hollow structure The gradient of the graphs is the; Gradient = Since it is important to get the values of the E we have to find the value of K first as shown; For aluminium; L=400 81227= K=0.06092 Using equation 6 K=E*I 0.06092=49.4992*E E=0.001231N/mm2 = 1231GPa Brass 198650= K=0.148988 K=E*I 0.148988=49.4992*E E=0.00301 N/mm2 = 3010GPa Mild steel 100933== K=0.0757 E=0.001529 N/mm2 =1529GPa 6. Discussion and Conclusion The gradients of figures 1, 2 and 3 are 3.0155, 3.7471 and 7.3748 respectively. The three gradients are all positive. This means that increase in load on the beam, leads to corresponding increase in the deflection. It would mean then that the deflection is directly proportional to load on the beam. However, even though the gradients are all positive, they are not equal. This is to mean that the three different materials deflection (stiffness) is different. Steel has the greatest gradient followed by the brass and lastly the aluminium beam. This means only one thing, steel is the stiffest of the three beams. In short the material type affects the stiffness of the beam. Aluminium is the least stiff on the three. The stiffness is directly related to the modulus of elasticity of the material. When the modulus of elasticity of a material is high, the deflection is less and vice versa. The graphs of the second experiment were all straight lines with positive gradients. The gradients for figures 4, 5 and 6 were 81227, 198650 and 100933 respectively. It is evident that with the increase in load the value also increases. This is because is dependent on the deflection which in turn is directly proportional to the load on the beam. However as one changes the shape of the cross section of the beam, the stiffness of the material is increased. As for the second experiment 1b, the equation 5 and 6 were used to get the value of the modulus of elasticity. The experimental values of the modulus of elasticity for aluminum, brass and mild steel are 1231MPa, 3010MPa, and 1529MPa respectively. The theoretical values are 69 GPa, 105 GPa and 200GPa. There is a great deviation of the experimental results from the theoretical ones. Errors in taking readings is one source of error for this experiment The experiment was successful and the objective achieved. 7. Reference PUPPALA, A. J. (2008). Estimating stiffness of subgrade and unbound materials for pavement design. Washington, D.C., Transportation Research Board. Read More

3. Theory Beams are used in engineering to support the various loads over different lengths. Beam is a structure member that is loaded non-axially making it to be under bending. A beam is said to be under bending when the forces on it are acting in such a way that they induce compressive stress over that part of the beam. The deflection on beams occurs when a point on the beam is displaced on the neutral plane of the beam from its original position when loads of forces are applied. Deflection of the material can be determined using the young`s modulus and moment of inertia of the beam.

Other important section properties are included too. Other properties depend on the situation imposed on the beam. Deflection of a simply supported beam can be calculated as shown below for a concentrated point load; = Equation 3 where: Δ = deflection, (mm) P = load, (N) L = length of beam, (mm) E = modulus of elasticity (N/m2) I = moment of inertia of section about the neutral axis, (mm4) Deflection is used to determine the overall stiffness of the beam under testing.

Deflection is seen as a function of stiffness of materials in addition to proportion of the piece. Deflection is very important in engineering is that the engineers can determine and calculate the modulus of elasticity for a material in flexure. The equation below is used to determine the stiffness of the material; Stiffness = Equation 4 Where the deflection, p is is the load and stiffness is measured in N/M For a simply supported beam between two points, the stiffness can be found using the equation below; ∆= Equation 5 K is the flexural rigidity K= I*E Equation 6 4.

Apparatus and the procedure i Apparatus TE161 Dial gauge Different beams Vernier calipers iii. Procedure 1a Three specimens were used. These specimens were; Mild steel Brass Aluminium 1. For data collection the table below was created Load (w) N Deflection (mm) 0 5 10 15 20 25 Experiment; 1a Material Beam size 2. The apparatus was set up with the knife edges being placed 400 mm apart (the mid span was 200mm from both edges). A flat specimen that was solid was used (nominal 19.05 mm x 3.2mm). The dimensions were determined with very high accuracy.

Dimensions were recorded on the table. To ensure high accuracy, Vernier calipers were used for this experiment. 3. On the load hanger, 5N was loaded and the deflection observed was recorded. 4. Another 5N was added on the hanger, the new deflection was recorded on the table. The loaded increment continued until the weight was 25N. The deflections were recorded as the loads were increased. 5. The loads were removed 6. A graph of load in N against deflection in mm was plotted. On the discussion the graph plotted was well commented, relationship between load and deflection as well as the gradient of the graph. iv. Procedure 1b The specimens used were; Brass Mild steel Aluminium 1.

The table below was construction for the collection of the data for the second experiment Load (w) N Deflection (mm) 48I 0 5 10 15 20 25 Experiment; 1a material Material H= I= Beam size L3 2. The dimensions of the cross section of the specimen were measured accurately. Using the dimensions of the cross section, the second moment of area was calculated. The value of second moment of area was recorded on the table 3. The apparatus was set up with the knife edges being placed 400 mm apart (the mid span was 200mm from both edges). 4. The specimen was placed well on the knife edge.

The load of 5N was placed on the mid span as the deflection was recorded.

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