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MEEN20040 Mechanics of Solids I Deflection of beams in Bending Name: HOOR MM M ALMHANA Student Number: 21209980 Introduction The deflection in beams in bending depends on • Young's modulus E Load applied • Length of the beam • Cross sectional geometry, 2nd moment of area I Date: 29/02/2024 Distribution of the applied load For a beam in three point bending the deflection is given by the formula FL³ ဝ် - where 48.E.I S is the deflection of the beam, F is the applied force L is the length of the beam E is Young's modulus I is the 2nd moment of area bh³ For a rectangular cross section this is given by I 12 b: beam width, h: beam thickness 12.FL³ 8 = 48..b.h³.E b d = FL³ 4..b.h³.E δ F # 1. Deflection depending on applied force Use Aluminium bar. Cross section b× h = 20 × 5mm. Length = 50cm Load body with weight 5, 10, 15 and 20 N Measure deflection at each load Weight of load body [N] Deflection [mm] δαμ 5 10 15 20 0.86 1.68 2.55 2.78 F = m.g Plot deflection versus mass and comment on the influence of the applied load on the deflection of the beam. This is acceptable as m is linearly related to applied load 2 deflection (mm)_ 2. Deflection depending on bar length L Use Aluminium bars. Cross section b× h = 20 × 4 mm Length of bars is 20, 30, 40 and 50 cm. Apply 10 N load for each bar and measure deflection. Length [cm] 20 30 40 δαι 50 Deflection [mm] 0.22 0.79 1.97 5.74 L³ in cm³ 8000 27000 64000 125000 Plot deflection versus Ls and comment on the influence of the length on the deflection of the beam Linear Relationship 3 L³ (m³) 3 3. Deflection depending on profile width b Use Aluminium bars. Thickness is 5 mm. Length 50 cm Width of bars varies as 10, 15, 20, 25 mm. Length = 50 cm. Apply 10 N load for each bar and measure the deflection. 8 x 1½ b Profile bxh [mm] 10 × 5 Deflection [mm] 3.7 15 × 5 20 x 5* 2.33 1.68 25 × 5 1.43 1/b [1/mm] 0.1 0.0666 0.05 0.04 Plot deflection versus 1/b and comment on the influence of the profile width on the deflection of the beam 4. Deflection depending on profile thickness h Use Aluminium bars. Width is 20 mm. Length is 50 cm Height: 3, 4, 5, 6, and 8 mm. Apply 10 N load and measure, the deflection for each bar -13 Ꮵ Profile Deflection [mm] 1/h3 [1/mm³] 20 × 3 20 × 4* 7.25 3.74 20 × 5* 20 × 6 20 × 10 1.68 1-15 0-2 0.037 0.0145 0.008 0.00487 0.00203 0.001 Plot deflection versus 1/h³ and comment on the influence of the thickness on the deflection of the beam 5/n Name: MEEN20040 Mechanics of Solids I Deflection of beams in Bending Introduction The deflection in beams in bending depends on • Young's modulus E Load applied • Length of the beam Student Number: Date: • • Cross sectional geometry, 2nd moment of area I Distribution of the applied load For a beam in three point bending the deflection is given by the formula 8 = FL³ 48.E.I where S is the deflection of the beam, F is the applied force L is the length of the beam E is Young's modulus I is the 2nd moment of area For a rectangular cross section this is given by I b: beam width, h: beam thickness 12.FL³ δε 48..b.h³.E h FL³ δ 4..b.h³.E b 1 = bh³ 12 δ F 1. Deflection depending on applied force Use Aluminium bar. Cross section b × h = 20 × 5mm. Length = 50cm Load body with weight 5, 10, 15 and 20 N Measure deflection at each load δα F F = m.g Weight of load body [N] 5 10 15 20 Deflection [mm] Plot deflection versus mass and comment on the influence of the applied load on the deflection of the beam. This is acceptable as m is linearly related to applied load 2 2. Deflection depending on bar length L Use Aluminium bars. Cross section b× h = 20 × 4 mm Length of bars is 20, 30, 40 and 50 cm. Apply 10 N load for each bar and measure deflection. Length [cm] Deflection [mm] 20 30 L³ in cm³ 8000 δα 40 50 27000 64000 125000 Plot deflection versus L³ and comment on the influence of the length on the deflection of the beam 3 3. Deflection depending on profile width b Use Aluminium bars. Thickness is 5 mm. Length 50 cm Width of bars varies as 10, 15, 20, 25 mm. Length = 50 cm. Apply 10 N load for each bar and measure the deflection. Profile b × h [mm] 10×5 15 × 5 20 × 5* 25 × 5 Deflection [mm] 1/b [1/mm] 0.1 0.0666 0.05 0.04 1 δα b Plot deflection versus 1/b and comment on the influence of the profile width on the deflection of the beam 4 4. Deflection depending on profile thickness h Use Aluminium bars. Width is 20 mm. Length is 50 cm Height: 3, 4, 5, 6, and 8 mm. Apply 10 N load and measure the deflection for each bar Profile 1 δα h³ 20 × 3 20 × 4* 20 × 5* 20 × 6 20 × 10 Deflection [mm] 1/h³ [1/mm³] 0.037 0.0145 0.008 0.00487 0.00203 Plot deflection versus 1/h³ and comment on the influence of the thickness on the deflection of the beam 5