Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.8 For the uniform beam and loading shown, determine the reaction at each support and the slope at end A. SOLUTION: • Release the “redundant” support at B, and find deformation. • Apply reaction at B as an unknown load to force zero displacement at B. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9-1 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.8 • Distributed Loading: yB w w x 4 2 Lx 3 L3 x 24 EI At point B, x 23 L yB w 4 3 w 2 2 3 2 L 2 L L L L 24 EI 3 3 3 wL4 0.01132 EI © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9-2 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.8 • Redundant Reaction Loading: At x a, Pa 2b 2 y 3EIL For a 23 L and b 13 L yB R 2 R 2 L B L 3EIL 3 3 2 RB L3 0.01646 EI © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9-3 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.8 • For compatibility with original supports, yB = 0 wL4 RB L3 0 yB w yB R 0.01132 0.01646 EI EI RB 0.688wL • From statics, RA 0.271wL RC 0.0413wL © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9-4 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.8 Slope at end A, wL3 wL3 A w 0.04167 24 EI EI A R 2 Pb L2 b 2 0.0688wL L 2 L wL3 L 0.03398 6 EIL 6 EIL 3 EI 3 wL3 wL3 A A w A R 0.04167 0.03398 EI EI © 2006 The McGraw-Hill Companies, Inc. All rights reserved. wL3 A 0.00769 EI 9-5 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Moment-Area Theorems • Geometric properties of the elastic curve can be used to determine deflection and slope. • Consider a beam subjected to arbitrary loading, d d 2 y M dx dx 2 EI D xD C xC d D C M dx EI xD xC M dx EI • First Moment-Area Theorem: D C area under (M/EI) diagram between C and D. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9-6 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Moment-Area Theorems • Tangents to the elastic curve at P and P’ intercept a segment of length dt on the vertical through C. dt x1d x1 tC D xD xC x1 M dx EI M dx = tangential deviation of C EI with respect to D • Second Moment-Area Theorem: The tangential deviation of C with respect to D is equal to the first moment with respect to a vertical axis through C of the area under the (M/EI) diagram between C and D. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9-7 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Application to Cantilever Beams and Beams With Symmetric Loadings • Cantilever beam - Select tangent at A as the reference. with θ A 0, D D A yD t D A • Simply supported, symmetrically loaded beam - select tangent at C as the reference. with θC 0, B B C yB t B C © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9-8 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Bending Moment Diagrams by Parts • Determination of the change of slope and the tangential deviation is simplified if the effect of each load is evaluated separately. • Construct a separate (M/EI) diagram for each load. - The change of slope, D/C, is obtained by adding the areas under the diagrams. - The tangential deviation, tD/C is obtained by adding the first moments of the areas with respect to a vertical axis through D. • Bending moment diagram constructed from individual loads is said to be drawn by parts. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9-9 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.11 SOLUTION: • Determine the reactions at supports. • Construct shear, bending moment and (M/EI) diagrams. For the prismatic beam shown, determine • Taking the tangent at C as the the slope and deflection at E. reference, evaluate the slope and tangential deviations at E. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 10 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.11 SOLUTION: • Determine the reactions at supports. RB RD wa • Construct shear, bending moment and (M/EI) diagrams. wa 2 L wa 2 L A1 2 EI 2 4 EI 1 wa 2 wa 3 a A2 3 2 EI 6 EI © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 11 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 9.11 • Slope at E: E C E C E C wa 2 L wa 3 A1 A2 4 EI 6 EI wa 2 3L 2a E 12 EI • Deflection at E: yE t E C tD C L 3a L A1 a A2 A1 4 4 4 wa 3 L wa 2 L2 wa 4 wa 2 L2 4 EI 16 EI 8 EI 16 EI wa 3 2L a yE 8EI © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 12 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Application of Moment-Area Theorems to Beams With Unsymmetric Loadings • Define reference tangent at support A. Evaluate A by determining the tangential deviation at B with respect to A. A tB A L • The slope at other points is found with respect to reference tangent. D A D A • The deflection at D is found from the tangential deviation at D. EF HB x L EF x tB A L y D ED EF t D A © 2006 The McGraw-Hill Companies, Inc. All rights reserved. x tB A L 9 - 13 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Maximum Deflection • Maximum deflection occurs at point K where the tangent is horizontal. tB A A L K 0 A K K A A A • Point K may be determined by measuring an area under the (M/EI) diagram equal to -A . • Obtain ymax by computing the first moment with respect to the vertical axis through A of the area between A and K. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 14 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Use of Moment-Area Theorems With Statically Indeterminate Beams • Reactions at supports of statically indeterminate beams are found by designating a redundant constraint and treating it as an unknown load which satisfies a displacement compatibility requirement. • The (M/EI) diagram is drawn by parts. The resulting tangential deviations are superposed and related by the compatibility requirement. • With reactions determined, the slope and deflection are found from the moment-area method. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 9 - 15