MECHANICS LAB AM 317 EXP 1 BEAM DEFLECTIONS

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1 MECHANICS AB AM 317 EX 1 BEAM DEFECTIONS

2 I. OBJECTIVES I.1 To observe, evaluate and report on the load-deflection relationship of a simply supported beam and a cantilever beam. I.2 To determine the modulus of elasticity of the beam and what the material the beam is made of using beam deflection theory. I.3 To verify the principle of superposition and Mawell s Reciprocity Theorem. II. INTRODUCTION AND BACKGROUND The deflections of a beam are engineering concerns as they can create an unstable structure if they are large. eople don t want to work in a building in which the floor beams deflect an ecessive amount, even though it may be in no danger of failing. Consequently, limits are often placed upon the allowable deflections of a beam, as well as upon the stresses. When loads are applied to a beam their originally straight aes become curved. Displacements from the initial aes are called bending or fleural deflections. The amount of fleural deflection in a beam is related to the beam s crosssectional area moment of inertia (I), the single applied concentrated load (), length of the beam (), the modulus of elasticity (E), and the position of the applied load on the beam. The amount of deflection due to a single concentrated load, is given by: 3 δ = 1.1 kei where k is a constant based on the position of the load, and on the end conditions of the beam. For deflection of specific loading conditions refer to Table I. The bending stress at any location of a beam section is determined by the fleure formula: My σ = 1.2 I where: M - internal moment at the section y - distance from the neutral ais to the point of interest I - moment of inertia of the cross-sectional area about the neutral ais R. Ehrgott (Created) 2/11 01/28/01

3 The largest stress at the same section follows from this relation, Eq. (1.2), by taking y at an etreme fiber at distance c, which leads to: Mc σ ma = 1.3 I III. AARATUS III.1 Simply-supported and Cantilever rectangular beams III.2 Weights & Hangers III.3 Micrometer III.4 Ruler or Tape measure III.5 Dial gauges IV. ROCEDURE ART A - SIMY SUORTED BEAM IV.1A Record the beam dimensions and calculate the area moment of inertia (I) using: 3 bh I = IV.2A Calculate the maimum permissible loads for mid-span and quarterspan loading using Eq. 1.3, where maimum allowable stress is 18,000 psi. IMORTANT - Check these calculated maimum loads with instructor before proceeding with the eperiment. IV.3A oad the beam at the mid-span in 5-lb increments, until the maimum load limit is reached. Record the deflection at the point of loading at each incremental load. Small divisions on the dial gauge are inch. One full revolution of the dial is 0.1 inch (100 small divisions). IV.4A Repeat the above procedure at the quarter-span. R. Ehrgott (Created) 3/11 01/28/01

4 ART B - CANTIEVER BEAM IV.1B Record the beam dimensions and calculate the area moment of inertia (I) using Eq. (1.4). IV.2B Calculate the safe loads at the mid-span and end of the cantilever beam using Eq. (1.3), and a maimum allowable stress of 18,000 psi. IV.3B oad the beam at the mid-span in 2-lb increments, until the maimum load limit is reached. Record the deflection at the point of loading at each increment. IV.4B Repeat the above procedure at the free end of the beam. Care must be taken so that the displacement does not eceed the maimum travel of the dial gauge. ART C - THE RINCIE OF SUEROSITION δ 3 = δ 1 +δ 2 δ 1 δ 2 IV.1C Choose a convenient reference point on either beam. All deflections will be measured at this point. IV.2C lace a single concentrated load at some point other than the reference point and measure the resulting deflection ( δ 1 ) at the reference point. IV.3C Remove the first load, and place a second load at another point on the beam and measure the resulting deflection ( δ 2 ) at the reference point. IV.4C Apply both loads simultaneously, as placed in IV.2C and IV.3C, and measure the resulting deflection ( δ 3 ) at the reference point. The sum of the single deflection ( δ 3 ) should closely approimate the total deflection ( δ 1 + δ 2 ). δ = δ 1 δ 2 R. Ehrgott (Created) 4/11 01/28/01

5 ART D - MAXWE S RECIROCITY THEOREM 1 2 δ 21 δ 12 IV.1D Choose two non-symmetrical reference points on either beam. IV.2D Apply a concentrated load ( 1 ) at one point and measure the resulting deflection ( δ 21 ) at the other point. IV.3D Remove the load from the first reference point and place a different load ( 2 ) on the second reference point. Measure the resulting deflection ( δ 12 ) at the first reference point. The loads and the deflections should satisfy the following relationship: = 1.6 1δ 12 2δ 21 V. REORT V.1 lot the curve of load versus deflection for two loading configurations of the simply supported beam. Show loads as ordinates and deflections as abscissas. Include this plot in the Results section with proper title, ais labels and figure number. Raw data goes in the Appendi. V.2 lot load versus deflection for the two loading configurations of the cantilever beam. Include this plot in the Results section. Data goes in the Appendi. V.3 Referring to the configuration in Table I and selecting the appropriate equations, determine the values of modulus of elasticity for each loading conditions. Create a table of the results you obtained for the modulus of elasticity for the simply supported beam (two load cases) and the cantilever beam (two load cases). Raw data goes in the Appendi. V.4 Determine what material the beams are made from by comparing the modulus of elasticity you calculated to values referenced in a Materials or Strength of Materials tetbook. R. Ehrgott (Created) 5/11 01/28/01

6 V.5 Verify the validity of the principle of Superposition and Mawell s Reciprocity theorem by performing all necessary calculations. resent these results in the Results section. VI. SEECTED REFERENCES Hibbeler, R.C., Statics and Strength of Materials, 4th edition, earson, opov, E.., Introduction to Mechanics of Solids, 2nd edition, rentice Hall, Stevens, Karl K., Statics and Strength of Materials, rentice Hall, R. Ehrgott (Created) 6/11 01/28/01

7 Table I Deflection Equations for Cantilever and Simply-Supported Beams Cantilever Beam: Deflection y as a function of y() y ma = y ma 3 = 3EI y() a 0 a 2 y( ) = (3a ) 6EI a 2 a y( ) = (3 a) 6EI Simply-Supported Beam: Deflection y as a function of y() /2 y ma = 2 y ma 3 = 48EI y() a b a b y( ) = ( b ) 6EI 3 3 = a = y = 4 256EI R. Ehrgott (Created) 7/11 01/28/01

8 Table II Simply Supported Beam Data ength (inches) Cross-Section Height h (inches) Cross-Section Width b (inches) Moment of Inertia I (in. 4 ) /2 /4 V.D. V.D. M.D. M.D. Safe oad = Safe oad = R. Ehrgott (Created) 8/11 01/28/01

9 Table III Cantilever Beam Data ength (inches) Cross-Section Height h (inches) Cross-Section Width b (inches) Moment of Inertia I (in. 4 ) /2 V.D. V.D. M.D. M.D. Safe oad = Safe oad = R. Ehrgott (Created) 9/11 01/28/01

10 oad (lb) ¼ Span Deflection (in.) Table IV Simply Supported Beam Data Modulus of Elasticity, E Mid-Span Deflection (in.) Average E = Average E = Modulus of Elasticity, E oad (lb) Mid-Span Deflection (in.) Table V Cantilever Beam Data Modulus of Elasticity, E End Deflection (in.) Average E = Average E = Modulus of Elasticity, E R. Ehrgott (Created) 10/11 01/28/01

11 Table VI Superposition Beam Used: oint ocation oad (lb) Deflection (in.) 1 2 Both Eperimental Deflection (in.): Theoretical Deflection (in.) (Equations Table 1): % Error (Ref. to Eperimental Value): Beam Used: Table VII Mawell s Reciprocity oint ocation oad (lb) Deflection (in.) 1 2 Eperimental Theoretical 1δ 12 = 2δ 21 = % Error = R. Ehrgott (Created) 11/11 01/28/01

1.1 To observe, evaluate and report on the load deflection relationship of a simply supported beam and a cantilever beam.

I. OBJECTIVES 1.1 To observe, evaluate and report on the load deflection relationship of a simply supported beam and a cantilever beam. 1.2 To determine the modulus of elasticity of the beam and what the