Asymmetrical Bending BRIEF AND REPORT INSTRUCTIONS The principal objectives of this experiment are: a) To compare theoretical and experimental values for the deflection of beams subjected to asymmetrical bending moments. b) To find the positions of the principal axes in the beam cross-section. c) To demonstrate the reciprocal theorem. Theory See lecture notes on "Asymmetrical Bending (stress and deflection)" found on Moodle, pages 23-28. Using this set of slides, the theoretical slope values can be obtained. You need to include the steps involved in finding these values in the theoretical part of the report. Apparatus (To be read together with watching the "Lab Demonstration Video") (i) (ii) Bending Test Rig with experimental beam fitted. Two weight-hangers and eight 1 kg weights. The end supports of the beam are gimbal-mounted so that the beam and section may rotate freely about any axis through the centroid and so that warping is completely unrestricted by the support. This provides the simply- supported condition to conform to the assumptions used in the theoretical analysis. There is a restriction on axial movement but the error should be small. The central loads are applied through the beam shear centre. The reactions of the supports must also pass through the shear centre by torsional equilibrium. A shear force passing through the shear centre does not produce any direct stresses due to torsion-bending and thus the stress distribution over any cross-section and over the length of the beam is that given by the simple theory of bending. At the loading point and at the supports there will be slight deviations from this stress distribution because the loads and reactions are not ideally distributed. However, by the principle of St. Venant, these deviations become negligible at short distances equivalent to one or two depths of the beam. The effects on the experimental results are therefore small. The loads may be applied horizontally or vertically or in combination to give the effect of a resultant load in any direction applied through the shear centre in a plane perpendicular to the beam axis. The digital gauges are set to measure the horizontal and vertical deflections of the beam centre so that the deflection direction may be found from these readings. in any 2.54cm y Principal Axis 3.81 cm 1 cm * 0.317cm 0.317 cm Principal Axis Beam Section Properties: Iy = 2.866 cm4 Ix = 1.022 cm4 Ixy = 0.98 cm4 E 72.5 GN/m2 1 = 1.22 m 1 = overall beam length Define the following. u V U V || || || || horizontal displacement vertical displacement horizontal load vertical load u, U V Note that the digital gauges give positive readings in the u and v directions but that, in the theory, v is positive in the upwards direction. In the theory we may substitute Sx = U, Sy = -V. Procedure (To be read together with watching the "Lab Demonstration Video" on Moodle) (Note that you should start with the weight hangers already fitted to the vertical and horizontal loading wires, digital gauge readings are set to zero and taking this condition as your zero). 1. Load the vertical hanger in increments of 1 kg up to 8 kg. Note the digital gauge readings u and v at each increment. 2. Transfer the weights one by one to the horizontal weight hanger, noting u and v at each transfer. 3. Unload the horizontal load hanger by increments of 1 kg noting u and v at each increment. Results ди 1. From procedure 1, plot u and v against V on the same graph. Hence find the slopes and av av ди dv 2. From procedure 3, plot u and v against U on the same graph. Hence find the slopes and au au 3. From procedure 2, plot u/v and U/V against V on the same graph. u U Find the value where the curves intersect ν V u U s W V α V B u u U tan α = tan ß: ==: i.e. if then α = ẞ and the overall deflection 8 is in the same direction as the ν ν V overall load W. For this to occur the beam must be bending about one principal axis so that 8 and W are in the u direction of the other principal axis i.e. tan 0 = ==— V U V Hence we find 0 to compare with the theory which states that tan 20 - = 21Xv xy Iy – Ix ди 4. The reciprocal theorem states that = av 5. dv au Compare the experimental values to see whether this is confirmed within reasonable engineering accuracy. Tabulate the experimental and theoretical results, together with the percentage differences for the values of ди ди dv dv au' av' au av and 0 Experimental data: In the physical lab, experimental data need to be taken and recorded in a tabular format as shown indicatively below. Loads (kg) Deflections (mm) V (vertical) u(lateral) v(vertical) Table: Experimental and theoretical values. Deflections (mm) Theoretical Deflections Ratio (unu) U (lateral) V (vertical) u (lateral) v (vertical) u (lateral) v (vertical) U/V u/v U (lateral) 0 0 0 1 Loads (kg) 0 2 0 3 0 4 0 5 0 6 0 7 0 8 1 7 0 0 2 6 3 5 4 4 0 1 5 3 6 2 7 1 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0 0 The above table is only a guide to include both theoretical and experimental deflection values to plot the graphs needed. You need to create additional rows similar to the experimental data sheet given in the lab report. EXAMPLE GRAPHS Graph 1 S3 Deflections due to Asymmetrical Bending 7.00 6.00 u, 5.00 (mm) S3 Procedure 1: Asymmetrical Beam Deflections (u,v) against Vertical ◆lateral u (exp) vertical v (exp) A lateral u (the) × vertical v (the) Loads V y=0.7357x-0.0449 4.00 3.00 2.00 1.00 0.00 y 0.248x-0.0153 0 3 6 7 8 Vertical Load V (kg) Graph 2 S3 Deflections due to Asymmetrical Bending S3 Procedure 2: Ratios of Loads and Deflections against Vertical Loads 8.0 7.0 U№6.0 and u/v 5.0 4.0 3.0 2.0 1.0 UN ―u/v (exp) 0.0 0 2 3 4 5 6 8 Vertical Load V (kg) Graph 3 S3 Deflections due to Asymmetrical Bending S3 Prodecure 3: Asymmetrical Beam Deflections (u,v) against Lateral Loads U 2.50 . lateral u (exp) vertical v (exp) 2.00 A lateral u (the) u, v (mm) 1.50 1.00 0.50 × vertical v (the) --Linear (lateral u (exp)) y = 0.2727x+0.0704 =0.2482x+0.0018 0.00 0 2 3 4 5 6 7 8 9 Lateral Load U (kg)/n

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Asymmetrical Bending BRIEF AND REPORT INSTRUCTIONS The principal objectives of this experiment are: a) To compare theoretical and experimental values for the deflection of beams subjected to asymmetrical bending moments. b) To find the positions of the principal axes in the beam cross-section. c) To demonstrate the re

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