12 Deflections of Beams and Shafts CHAPTER OBJECTIVES

12. Deflections of Beams and Shafts CHAPTER OBJECTIVES • Use various methods to determine the deflection and slope at specific pts on beams and shafts: 1. Integration method 2. Discontinuity functions 3. Method of superposition 4. Moment-area method • Use the various methods to solve for the support reactions on a beam or shaft that is statically indeterminate 2005 Pearson Education South Asia Pte Ltd 1

12. Deflections of Beams and Shafts CHAPTER OUTLINE 1. 2. 3. 4. The Elastic Curve Slope and Displacement by Integration *Discontinuity Functions *Slope and Displacement by the Moment-Area Method 5. Method of Superposition 6. Statically Indeterminate Beams and Shafts 7. Statically Indeterminate Beams and Shafts: Method of Integration 2005 Pearson Education South Asia Pte Ltd 2

12. Deflections of Beams and Shafts CHAPTER OUTLINE 8. *Statically Indeterminate Beams and Shafts: Moment-Area Method 9. Statically Indeterminate Beams and Shafts: Method of Superposition 2005 Pearson Education South Asia Pte Ltd 3

12. Deflections of Beams and Shafts 12. 1 THE ELASTIC CURVE • • It is useful to sketch the deflected shape of the loaded beam, to “visualize” computed results and partially check the results. The deflection diagram of the longitudinal axis that passes through the centroid of each x-sectional area of the beam is called the elastic curve. 2005 Pearson Education South Asia Pte Ltd 4

12. Deflections of Beams and Shafts 12. 1 THE ELASTIC CURVE • • Draw the moment diagram for the beam first before creating the elastic curve. Use beam convention as shown and established in chapter 6. 1. 2005 Pearson Education South Asia Pte Ltd 5

12. Deflections of Beams and Shafts 12. 1 THE ELASTIC CURVE • • For example, due to roller and pin supports at B and D, displacements at B and D is zero. For region of -ve moment AC, elastic curve concave downwards. Within region of +ve moment CD, elastic curve concave upwards. At pt C, there is an inflection pt where curve changes from concave up to concave down (zero moment). 2005 Pearson Education South Asia Pte Ltd 6

12. Deflections of Beams and Shafts 12. 1 THE ELASTIC CURVE Moment-Curvature Relationship • x axis extends +ve to the right, alongitudinal axis of beam. • A differential element of undeformed width dx is located. • axis extends +ve upwards from x axis. It measures the displacement of the centroid on xsectional area of element. • A “localized” y coordinate is specified for the position of a fiber in the element. • It is measured +ve upward from the neutral axis. 2005 Pearson Education South Asia Pte Ltd 7

12. Deflections of Beams and Shafts 12. 1 THE ELASTIC CURVE Moment-Curvature Relationship • Limit analysis to the case of initially straight beam elastically deformed by loads applied perpendicular to beam’s x axis and lying in the x- plane of symmetry for beam’s x-sectional area. • Internal moment M deforms element such that angle between x-sections is d. • Arc dx is a part of the elastic curve that intersects the neutral axis for each x-section. • Radius of curvature for this arc defined as the distance , measured from center of curvature O’ to dx. 2005 Pearson Education South Asia Pte Ltd 8

12. Deflections of Beams and Shafts 12. 1 THE ELASTIC CURVE Moment-Curvature Relationship • Strain in arc ds, at position y from neutral axis, is 2005 Pearson Education South Asia Pte Ltd 9

12. Deflections of Beams and Shafts 12. 1 THE ELASTIC CURVE Moment-Curvature Relationship • If material is homogeneous and shows linearelastic behavior, Hooke’s law applies. Since flexure formula also applies, we combing the equations to get = radius of curvature at a specific pt on elastic curve (1/ is referred to as the curvature). M = internal moment in beam at pt where is to be determined. E = material’s modulus of elasticity. I = beam’s moment of inertia computed about neutral axis. 2005 Pearson Education South Asia Pte Ltd 10

12. Deflections of Beams and Shafts 12. 1 THE ELASTIC CURVE Moment-Curvature Relationship • EI is the flexural rigidity and is always positive. • Sign for depends on the direction of the moment. • As shown, when M is +ve, extends above the beam. When M is –ve, extends below the beam. 2005 Pearson Education South Asia Pte Ltd 11

12. Deflections of Beams and Shafts 12. 1 THE ELASTIC CURVE Moment-Curvature Relationship • Using flexure formula, = My/I, curvature is also • Eqns 12 -2 and 12 -3 valid for either small or large radii of curvature. 2005 Pearson Education South Asia Pte Ltd 12

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION • Let’s represent the curvature in terms of and x. • Substitute into Eqn 12 -2 2005 Pearson Education South Asia Pte Ltd 13

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION • • Most engineering codes specify limitations on deflections for tolerance or aesthetic purposes. Slope of elastic curve determined from d /dx is very small and its square will be negligible compared with unity. Therefore, by approximation 1/ = d 2 /dx 2, Eqn 12 -4 rewritten as Differentiate each side w. r. t. x and substitute V = d. M/dx, we get 2005 Pearson Education South Asia Pte Ltd 14

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION • Differentiating again, using w = d. V/dx yields • Flexural rigidity is constant along beam, thus 2005 Pearson Education South Asia Pte Ltd 15

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION • Generally, it is easier to determine the internal moment M as a function of x, integrate twice, and evaluate only two integration constants. • For convenience in writing each moment expression, the origin for each x coordinate can be selected arbitrarily. Sign convention and coordinates • Use the proper signs for M, V and w. 2005 Pearson Education South Asia Pte Ltd 16

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION Boundary and continuity conditions • Possible boundary conditions are shown here. 2005 Pearson Education South Asia Pte Ltd 17

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION Boundary and continuity conditions • If a single x coordinate cannot be used to express the eqn for beam’s slope or elastic curve, then continuity conditions must be used to evaluate some of the integration constants. 2005 Pearson Education South Asia Pte Ltd 18

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION Procedure for analysis Elastic curve • Draw an exaggerated view of the beam’s elastic curve. • Recall that zero slope and zero displacement occur at all fixed supports, and zero displacement occurs at all pin and roller supports. • Establish the x and coordinate axes. • The x axis must be parallel to the undeflected beam and can have an origin at any pt along the beam, with +ve direction either to the right or to the left. 2005 Pearson Education South Asia Pte Ltd 19

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION Procedure for analysis Elastic curve • If several discontinuous loads are present, establish x coordinates that are valid for each region of the beam between the discontinuties. • Choose these coordinates so that they will simplify subsequent algrebraic work. 2005 Pearson Education South Asia Pte Ltd 20

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION Procedure for analysis Load or moment function • For each region in which there is an x coordinate, express that loading w or the internal moment M as a function of x. • In particular, always assume that M acts in the +ve direction when applying the eqn of moment equilibrium to determine M = f(x). 2005 Pearson Education South Asia Pte Ltd 21

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION Procedure for analysis Slope and elastic curve • Provided EI is constant, apply either the load eqn EI d 4 /dx 4 = w(x), which requires four integrations to get = (x), or the moment eqns EI d 2 /dx 2 = M(x), which requires only two integrations. For each integration, we include a constant of integration. • Constants are evaluated using boundary conditions for the supports and the continuity conditions that apply to slope and displacement at pts where two functions meet. 2005 Pearson Education South Asia Pte Ltd 22

12. Deflections of Beams and Shafts 12. 2 SLOPE AND DISPLACEMENT BY INTEGRATION Procedure for analysis Slope and elastic curve • Once constants are evaluated and substituted back into slope and deflection eqns, slope and displacement at specific pts on elastic curve can be determined. • The numerical values obtained is checked graphically by comparing them with sketch of the elastic curve. • Realize that +ve values for slope are counterclockwise if the x axis extends +ve to the right, and clockwise if the x axis extends +ve to the left. For both cases, +ve displacement is upwards. 2005 Pearson Education South Asia Pte Ltd 23

12. Deflections of Beams and Shafts EXAMPLE 12. 1 Cantilevered beam shown is subjected to a vertical load P at its end. Determine the eqn of the elastic curve. EI is constant. 2005 Pearson Education South Asia Pte Ltd 24

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) Elastic curve: Load tends to deflect the beam. By inspection, the internal moment can be represented throughout the beam using a single x coordinate. Moment function: From free-body diagram, with M acting in the +ve direction, we have 2005 Pearson Education South Asia Pte Ltd 25

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) Slope and elastic curve: Applying Eqn 12 -10 and integrating twice yields 2005 Pearson Education South Asia Pte Ltd 26

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) Slope and elastic curve: Using boundary conditions d /dx = 0 at x = L, and = 0 at x = L, Eqn (2) and (3) becomes 2005 Pearson Education South Asia Pte Ltd 27

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) Slope and elastic curve: Thus, C 1 = PL 2/2 and C 2 = PL 3/3. Substituting these results into Eqns (2) and (3) with = d /dx, we get Maximum slope and displacement occur at A (x = 0), 2005 Pearson Education South Asia Pte Ltd 28

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) Slope and elastic curve: Positive result for A indicates counterclockwise rotation and negative result for A indicates that A is downward. Consider beam to have a length of 5 m, support load P = 30 k. N and made of A-36 steel having Est = 200 GPa. 2005 Pearson Education South Asia Pte Ltd 29

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) Slope and elastic curve: Using methods in chapter 11. 3, assuming allowable normal stress is equal to yield stress allow = 250 MPa, then a W 310 39 would be adequate (I = 84. 8(106) mm 4). From Eqns (4) and (5), 2005 Pearson Education South Asia Pte Ltd 30

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) Slope and elastic curve: From Eqns (4) and (5), 2005 Pearson Education South Asia Pte Ltd 31

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) Slope and elastic curve: Since 2 A = (d /dx)2 = 0. 000488 << 1, this justifies the use of Eqn 12 -10 than the more exact 12 -4. Also, since it is for a cantilevered beam, we’ve obtained larger values for and than would be obtained otherwise. 2005 Pearson Education South Asia Pte Ltd 32

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) SOLUTION 2 Using Eqn 12 -8 to solve the problem. Here w(x) = 0 for 0 x L, so that upon integrating once, we get the form of Eqn 12 -19 2005 Pearson Education South Asia Pte Ltd 33

12. Deflections of Beams and Shafts EXAMPLE 12. 1 (SOLN) Solution II Shear constant C’ 1 can be evaluated at x = 0, since VA = P. Thus, C’ 1 = P. Integrating again yields the form of Eqn 12 -10, Here, M = 0 at x = 0, so C’ 2 = 0, and as a result, we obtain Eqn 1 and solution proceeds as before. 2005 Pearson Education South Asia Pte Ltd 34

12. Deflections of Beams and Shafts EXAMPLE 12. 4 Beam is subjected to load P at its end. Determine the displacement at C. EI is a constant. 2005 Pearson Education South Asia Pte Ltd 35

12. Deflections of Beams and Shafts EXAMPLE 12. 4 (SOLN) Elastic curve Beam deflects into shape shown. Due to loading, two x coordinates will be considered, 0 x 1 2 a and 0 x 2 a, where x 2 is directed to the left from C since internal moment is easy to formulate. 2005 Pearson Education South Asia Pte Ltd 36

12. Deflections of Beams and Shafts EXAMPLE 12. 4 (SOLN) Moment functions Using free-body diagrams, we have Slope and Elastic curve: Applying Eqn 10 -12, 2005 Pearson Education South Asia Pte Ltd 37

12. Deflections of Beams and Shafts EXAMPLE 12. 4 (SOLN) Slope and Elastic curve: Applying Eqn 10 -12, 2005 Pearson Education South Asia Pte Ltd 38

12. Deflections of Beams and Shafts EXAMPLE 12. 4 (SOLN) Slope and Elastic curve: The four constants of integration determined using three boundary conditions, 1 = 0 at x 1 = 0, 1 = 0 at x 1 = 2 a, and 2 =0 at x 2 = a and a discontinuity eqn. Here, continuity of slope at roller requires d 1/dx 1 = d 2/dx 2 at x 1 = 2 a and x 2 = a. 2005 Pearson Education South Asia Pte Ltd 39

12. Deflections of Beams and Shafts EXAMPLE 12. 4 (SOLN) Slope and Elastic curve: Solving, we obtain 2005 Pearson Education South Asia Pte Ltd 40

12. Deflections of Beams and Shafts EXAMPLE 12. 4 (SOLN) Slope and Elastic curve: Substituting C 3 and C 4 into Eqn (4) gives Displacement at C is determined by setting x 2 = 0, 2005 Pearson Education South Asia Pte Ltd 41

12. Deflections of Beams and Shafts *12. 3 DISCONTINUITY FUNCTIONS • A simplified method for finding the eqn of the elastic curve for a multiply loaded beam using a single expression, formulated from the loading on the beam , w = w(x), or the beam’s internal moment, M = M(x) is discussed below. Discontinuity functions Macaulay functions • Such functions can be used to describe distributed loadings, written generally as 2005 Pearson Education South Asia Pte Ltd 42

12. Deflections of Beams and Shafts *12. 3 DISCONTINUITY FUNCTIONS Discontinuity functions Macaulay functions • x represents the coordinate position of a pt along the beam • a is the location on the beam where a “discontinuity” occurs, or the pt where a distributed loading begins. • Integrating Macaulay functions, we get • The functions describe both uniform load and triangular load. 2005 Pearson Education South Asia Pte Ltd 43

12. Deflections of Beams and Shafts *12. 3 DISCONTINUITY FUNCTIONS Discontinuity functions Macaulay functions 2005 Pearson Education South Asia Pte Ltd 44

12. Deflections of Beams and Shafts *12. 3 DISCONTINUITY FUNCTIONS Discontinuity functions Singularity functions • Used to describe the pt location of concentrated forces or couple moments acting on a beam. • A concentrated force P can be considered as a special case of distributed loading, where w = P/e such that its width is , → 0. 2005 Pearson Education South Asia Pte Ltd 45

12. Deflections of Beams and Shafts *12. 3 DISCONTINUITY FUNCTIONS Discontinuity functions Singularity functions • Similarly, couple moment M 0, considered +ve counterclockwise, is a limitation as → 0 of two distributed loadings. Hence, • Integration of the two functions yields 2005 Pearson Education South Asia Pte Ltd 46

12. Deflections of Beams and Shafts *12. 3 DISCONTINUITY FUNCTIONS Discontinuity functions Singularity functions 2005 Pearson Education South Asia Pte Ltd 47

12. Deflections of Beams and Shafts *12. 3 DISCONTINUITY FUNCTIONS Procedure for analysis Elastic curve • Sketch the beam’s elastic curve and identify the boundary conditions at the supports. • Zero displacement occurs at all pin and roller supports, and zero slope and zero displacement occurs at fixed supports. • Establish the x axis so that it extends to the right and has its origin at the beam’s left end. Load or moment function • Calculate the support reactions and then use the discontinuity functions in Table 12 -2 to express either the loading w or the internal moment M as a function of x. 2005 Pearson Education South Asia Pte Ltd 48

12. Deflections of Beams and Shafts *12. 3 DISCONTINUITY FUNCTIONS Procedure for analysis Load or moment function • Calculate the support reactions and then use the discontinuity functions in Table 12 -2 to express either the loading w or the internal moment M as a function of x. • Make sure to follow the sign convention for each loading as it applies for this equation. • Note that the distributed loadings must extend all the way to the beam’s right end to be valid. If this does not occur, use the method of superposition. 2005 Pearson Education South Asia Pte Ltd 49

12. Deflections of Beams and Shafts *12. 3 DISCONTINUITY FUNCTIONS Procedure for analysis Slope and elastic curve • Substitute w into EI d 4 /dx 4 = w(x) or M into the moment curvature relation EI d 2 /dx 2 = M, and integrate to obtain the eqns for the beam’s slope and deflection. • Evaluate the constants of integration using the boundary conditions, and substitute these constants into the slope and deflection eqns to obtain the final results. • When the slope and deflection eqns are evaluated at any pt on the beam, a +ve slope is counterclockwise, and a +ve displacement is upward. 2005 Pearson Education South Asia Pte Ltd 50

12. Deflections of Beams and Shafts EXAMPLE 12. 5 Determine the eqn of the elastic curve for the cantilevered beam shown. EI is constant. 2005 Pearson Education South Asia Pte Ltd 51

12. Deflections of Beams and Shafts EXAMPLE 12. 5 (SOLN) Elastic curve The loads cause the beam to deflect as shown. The boundary conditions require zero slope and displacement at A. 2005 Pearson Education South Asia Pte Ltd 52

12. Deflections of Beams and Shafts EXAMPLE 12. 5 (SOLN) Loading functions Support reactions shown on free-body diagram. Since distributed loading does not extend to C as required, use superposition of loadings to represent same effect. By sign convention, the 50 -k. Nm couple moment, the 52 -k. N force at A, and portion of distributed loading from B to C on the bottom of the beam are all –ve. 2005 Pearson Education South Asia Pte Ltd 53

12. Deflections of Beams and Shafts EXAMPLE 12. 5 (SOLN) Loading functions Therefore, The 12 -k. N load is not included, since x cannot be greater than 9 m. Because d. V/dx = w(x), then by integrating, neglect constant of integration since reactions are included in load function, we have 2005 Pearson Education South Asia Pte Ltd 54

12. Deflections of Beams and Shafts EXAMPLE 12. 5 (SOLN) Loading functions Furthermore, d. M/dx = V, so integrating again yields The same result can be obtained directly from Table 12 -2. 2005 Pearson Education South Asia Pte Ltd 55

12. Deflections of Beams and Shafts EXAMPLE 12. 5 (SOLN) Slope and elastic curve Applying Eqn 12 -10 and integrating twice, we have 2005 Pearson Education South Asia Pte Ltd 56

12. Deflections of Beams and Shafts EXAMPLE 12. 5 (SOLN) Slope and elastic curve Since d /dx = 0 at x = 0, C 1 = 0; and = 0 at x = 0, so C 2 = 0. Thus 2005 Pearson Education South Asia Pte Ltd 57

12. Deflections of Beams and Shafts *12. 4 SLOPE & DISPLACEMENT BY THE MOMENT-AREA METHOD Theorem 2 • The vertical deviation of the tangent at a pt (A) on the elastic curve w. r. t. the tangent extended from another pt (B) equals the moment of the area under the ME/I diagram between these two pts (A and B). • This moment is computed about pt (A) where the vertical deviation (t. A/B) is to be determined. 2005 Pearson Education South Asia Pte Ltd 58

12. Deflections of Beams and Shafts *12. 4 SLOPE & DISPLACEMENT BY THE MOMENT-AREA METHOD Theorem 2 2005 Pearson Education South Asia Pte Ltd 59

12. Deflections of Beams and Shafts *12. 4 SLOPE & DISPLACEMENT BY THE MOMENT-AREA METHOD Procedure for analysis M/EI Diagram • Determine the support reactions and draw the beam’s M/EI diagram. • If the beam is loaded with concentrated forces, the M/EI diagram will consist of a series of straight line segments, and the areas and their moments required for the moment-area theorems will be relatively easy to compute. • If the loading consists of a series of distributed loads, the M/EI diagram will consist of parabolic or perhaps higher-order curves, and we use the table on the inside front cover to locate the area and centroid under each curve. 2005 Pearson Education South Asia Pte Ltd 60

12. Deflections of Beams and Shafts *12. 4 SLOPE & DISPLACEMENT BY THE MOMENT-AREA METHOD Procedure for analysis Elastic curve • Draw an exaggerated view of the beam’s elastic curve. • Recall that pts of zero slope and zero displacement always occur at a fixed support, and zero displacement occurs at all pin and roller supports. • If it is difficult to draw the general shape of the elastic curve, use the moment (M/EI) diagram. • Realize that when the beam is subjected to a +ve moment, the beam bends concave up, whereas -ve moment bends the beam concave down. 2005 Pearson Education South Asia Pte Ltd 61

12. Deflections of Beams and Shafts *12. 4 SLOPE & DISPLACEMENT BY THE MOMENT-AREA METHOD Procedure for analysis Elastic curve • An inflection pt or change in curvature occurs when the moment if the beam (or M/EI) is zero. • The unknown displacement and slope to be determined should be indicated on the curve. • Since moment-area theorems apply only between two tangents, attention should be given as to which tangents should be constructed so that the angles or deviations between them will lead to the solution of the problem. • The tangents at the supports should be considered, since the beam usually has zero displacement and/or zero slope at the supports. 2005 Pearson Education South Asia Pte Ltd 62

12. Deflections of Beams and Shafts *12. 4 SLOPE & DISPLACEMENT BY THE MOMENT-AREA METHOD Procedure for analysis Moment-area theorems • Apply Theorem 1 to determine the angle between any two tangents on the elastic curve and Theorem 2 to determine the tangential deviation. • The algebraic sign of the answer can be checked from the angle or deviation indicated on the elastic curve. • A positive B/A represents a counterclockwise rotation of the tangent at B w. r. t. tangent at A, and a +ve t. B/A indicates that pt B on the elastic curve lies above the extended tangent from pt A. 2005 Pearson Education South Asia Pte Ltd 63

12. Deflections of Beams and Shafts EXAMPLE 12. 7 Determine the slope of the beam shown at pts B and C. EI is constant. 2005 Pearson Education South Asia Pte Ltd 64

12. Deflections of Beams and Shafts EXAMPLE 12. 7 (SOLN) M/EI diagram: See below. Elastic curve: The force P causes the beam to deflect as shown. 2005 Pearson Education South Asia Pte Ltd 65

12. Deflections of Beams and Shafts EXAMPLE 12. 7 (SOLN) Elastic curve: The tangents at B and C are indicated since we are required to find B and C. Also, the tangent at the support (A) is shown. This tangent has a known zero slope. By construction, the angle between tan A and tan B, B/A, is equivalent to B, or 2005 Pearson Education South Asia Pte Ltd 66

12. Deflections of Beams and Shafts EXAMPLE 12. 7 (SOLN) Moment-area theorem: Applying Theorem 1, B/A is equal to the area under the M/EI diagram between pts A and B, that is, 2005 Pearson Education South Asia Pte Ltd 67

12. Deflections of Beams and Shafts EXAMPLE 12. 7 (SOLN) Moment-area theorem: The negative sign indicates that angle measured from tangent at A to tangent at B is clockwise. This checks, since beam slopes downward at B. Similarly, area under the M/EI diagram between pts A and C equals C/A. We have 2005 Pearson Education South Asia Pte Ltd 68

12. Deflections of Beams and Shafts EXAMPLE 12. 8 Determine the displacement of pts B and C of beam shown. EI is constant. 2005 Pearson Education South Asia Pte Ltd 69

12. Deflections of Beams and Shafts EXAMPLE 12. 8 (SOLN) M/EI diagram: See below. Elastic curve: The couple moment at C cause the beam to deflect as shown. 2005 Pearson Education South Asia Pte Ltd 70

12. Deflections of Beams and Shafts EXAMPLE 12. 8 (SOLN) Elastic curve: The required displacements can be related directly to deviations between the tangents at B and A and C and A. Specifically, B is equal to deviation of tan A from tan B, 2005 Pearson Education South Asia Pte Ltd 71

12. Deflections of Beams and Shafts EXAMPLE 12. 8 (SOLN) Moment-area theorem: Applying Theorem 2, t. B/A is equal to the moment of the shaded area under the M/EI diagram between A and B computed about pt B, since this is the pt where tangential deviation is to be determined. Hence, 2005 Pearson Education South Asia Pte Ltd 72

12. Deflections of Beams and Shafts EXAMPLE 12. 8 (SOLN) Moment-area theorem: Likewise, for t. C/A we must determine the moment of the area under the entire M/EI diagram from A to C about pt C. We have Since both answers are –ve, they indicate that pts B and C lie below the tangent at A. This checks with the figure. 2005 Pearson Education South Asia Pte Ltd 73

12. Deflections of Beams and Shafts 12. 5 METHOD OF SUPERPOSITION • • • The differential eqn EI d 4 /dx 4 = w(x) satisfies the two necessary requirements for applying the principle of superposition The load w(x) is linearly related to the deflection (x) The load is assumed not to change significantly the original geometry of the beam or shaft. 2005 Pearson Education South Asia Pte Ltd 74

12. Deflections of Beams and Shafts EXAMPLE 12. 16 Steel bar shown is supported by two springs at its ends A and B. Each spring has a stiffness k = 45 k. N/m and is originally unstretched. If the bar is loaded with a force of 3 k. N at pt C, determine the vertical displacement of the force. Neglect the weight of the bar and take Est = 200 GPa, I = 4. 6875 10 -6 m. 2005 Pearson Education South Asia Pte Ltd 75

12. Deflections of Beams and Shafts EXAMPLE 12. 16 (SOLN) End reactions at A and B are computed and shown. Each spring deflects by an amount 2005 Pearson Education South Asia Pte Ltd 76

12. Deflections of Beams and Shafts EXAMPLE 12. 16 (SOLN) If bar is considered rigid, these displacements cause it to move into positions shown. For this case, the vertical displacement at C is 2005 Pearson Education South Asia Pte Ltd 77

12. Deflections of Beams and Shafts EXAMPLE 12. 16 (SOLN) We can find the displacement at C caused by the deformation of the bar, by using the table in Appendix C. We have 2005 Pearson Education South Asia Pte Ltd 78

12. Deflections of Beams and Shafts EXAMPLE 12. 16 (SOLN) Adding the two displacement components, we get 2005 Pearson Education South Asia Pte Ltd 79

12. Deflections of Beams and Shafts 12. 6 STATICALLY INDETERMINATE BEAMS AND SHAFTS • • • A member of any type is classified as statically indeterminate if the no. of unknown reactions exceeds the available no. of equilibrium eqns. Additional support reactions on beam that are not needed to keep it in stable equilibrium are called redundants. No. of these redundants is referred to as the degree of indeterminacy. 2005 Pearson Education South Asia Pte Ltd 80

12. Deflections of Beams and Shafts 12. 7 STATICALLY INDETERMINATE BEAMS AND SHAFTS: METHOD OF INTEGRATION • • • For a statically indeterminate beam, the internal moment M can be expressed in terms of the unknown redundants. After integrating this eqn twice, there will be two constants of integration and the redundants to be found. The unknowns can be found from the boundary and/or continuity conditions for the problem. 2005 Pearson Education South Asia Pte Ltd 81

12. Deflections of Beams and Shafts EXAMPLE 12. 17 Beam is subjected to the distributed loading shown. Determine the reactions at A. EI is a constant. 2005 Pearson Education South Asia Pte Ltd 82

12. Deflections of Beams and Shafts EXAMPLE 12. 17 (SOLN) Elastic curve: Beam deflects as shown. Only one coordinate x is needed. For convenience, we will take it directed to the right, since internal moment is easy to formulate. 2005 Pearson Education South Asia Pte Ltd 83

12. Deflections of Beams and Shafts EXAMPLE 12. 17 (SOLN) Moment function: Beam is indeterminate to first degree as indicated from the free-body diagram. We can express the internal moment M in terms of the redundant force at A using segment shown below. 2005 Pearson Education South Asia Pte Ltd 84

12. Deflections of Beams and Shafts EXAMPLE 12. 17 (SOLN) Moment function: Slope and elastic curve: Applying Eqn 12 -10, 2005 Pearson Education South Asia Pte Ltd 85

12. Deflections of Beams and Shafts EXAMPLE 12. 17 (SOLN) Slope and elastic curve: The three unknowns Ay, C 1 and C 2 are determined from the boundary conditions x = 0, = 0; x = L, d /dx = 0; and x = L, = 0. Applying these conditions yields 2005 Pearson Education South Asia Pte Ltd 86

12. Deflections of Beams and Shafts EXAMPLE 12. 17 (SOLN) Slope and elastic curve: Solving, Using the result for Ay, the reactions at B can be determined from the equations of equilibrium. Show that Bx = 0. By = 2 w 0 L/5 and MB= w 0 L 2/15 2005 Pearson Education South Asia Pte Ltd 87

12. Deflections of Beams and Shafts 12. 8 STATICALLY INDETERMINATE BEAMS AND SHAFTS: MOMENT-AREA METHOD • Draw the ME/I diagrams such that the redundants are represented as unknowns. • Apply the 2 moment-area theorems to get the relationships between the tangents on elastic curve to meet conditions of displacement and/or slope at supports of beam. • For all cases, no. of compatibility conditions is equivalent to no. of redundants. 2005 Pearson Education South Asia Pte Ltd 88

12. Deflections of Beams and Shafts 12. 8 STATICALLY INDETERMINATE BEAMS AND SHAFTS: MOMENT-AREA METHOD Moment diagrams constructed by method of superposition • Since moment-area theorems needs calculation of both the area under the ME/I diagram and centroidal location of this area, the method of superposition can be used to combine separate ME/I diagrams for each of the known loads. • This will be relevant if the resultant moment diagram is of a complicated shape. 2005 Pearson Education South Asia Pte Ltd 89

12. Deflections of Beams and Shafts 12. 8 STATICALLY INDETERMINATE BEAMS AND SHAFTS: MOMENT-AREA METHOD Moment diagrams constructed by method of superposition • Most loadings on beams are a combination of the four loadings as shown. 2005 Pearson Education South Asia Pte Ltd 90

12. Deflections of Beams and Shafts 12. 8 STATICALLY INDETERMINATE BEAMS AND SHAFTS: MOMENT-AREA METHOD 2005 Pearson Education South Asia Pte Ltd 91

12. Deflections of Beams and Shafts 12. 8 STATICALLY INDETERMINATE BEAMS AND SHAFTS: MOMENT-AREA METHOD 2005 Pearson Education South Asia Pte Ltd 92

12. Deflections of Beams and Shafts EXAMPLE 12. 20 Beam is subjected to couple moment at its end C as shown. Determine the reaction at B. EI is constant. 2005 Pearson Education South Asia Pte Ltd 93

12. Deflections of Beams and Shafts EXAMPLE 12. 20 (SOLN) M/EI Diagram: Free-body diagram as shown. By inspection, beam is indeterminate to first degree. To get a direct solution, choose By as the redundant. Using superposition, the M/EI diagrams for By and M 0, each applied to the simply supported beam are shown. 2005 Pearson Education South Asia Pte Ltd 94

12. Deflections of Beams and Shafts EXAMPLE 12. 20 (SOLN) Elastic curve: Elastic curve as shown. Tangents at A, B and C has been established. Since A = B = C = 0, then tangential deviations shown must be proportional, 2005 Pearson Education South Asia Pte Ltd 95

12. Deflections of Beams and Shafts EXAMPLE 12. 20 (SOLN) Elastic curve: From ME/I diagram, we have 2005 Pearson Education South Asia Pte Ltd 96

12. Deflections of Beams and Shafts EXAMPLE 12. 20 (SOLN) Elastic curve: Substituting into Eqn (1), we have Equations of equilibrium: Reactions at A and C can now be determined from the eqns of equilibrium. Show that Ax = 0, Cy = 5 M 0/4 L, and Ay = M 0/4 L. 2005 Pearson Education South Asia Pte Ltd 97

12. Deflections of Beams and Shafts EXAMPLE 12. 20 (SOLN) Equations of equilibrium: From figure shown, this problem can also be worked out in terms of the tangential deviations, 2005 Pearson Education South Asia Pte Ltd 98

12. Deflections of Beams and Shafts 12. 9 STATICALLY INDETERMINATE BEAMS AND SHAFTS: METHOD OF SUPERPOSITION • First, identify the redundant support reactions on the beam. • Remove these reactions from the beam to get a primary beam that is statically determinate and stable and subjected to external load only. • Add to this beam with a series of similarly supported beams, each with a separate redundant, then by principle of superposition, the final loaded beam is obtained. • After computing the redundants, the other reactions on the beam determined from the eqns of equilibrium. • This method of analysis is sometimes called the force method. 2005 Pearson Education South Asia Pte Ltd 99

12. Deflections of Beams and Shafts 12. 9 STATICALLY INDETERMINATE BEAMS AND SHAFTS: METHOD OF SUPERPOSITION Procedure for analysis Elastic curve • Specify unknown redundant forces or moments that must be removed from the beam in order to make it statically determinate and stable. • Use principle of superposition, draw the statically indeterminate beam and show it to be equal to a sequence of corresponding statically determinate beams. • The first beam (primary) supports the same external loads as the statically indeterminate beam, and each of the other beams “added” to the primary beam shows the beam loaded with a separate single redundant force or moment. 2005 Pearson Education South Asia Pte Ltd 100

12. Deflections of Beams and Shafts 12. 9 STATICALLY INDETERMINATE BEAMS AND SHAFTS: METHOD OF SUPERPOSITION Procedure for analysis Elastic curve • Sketch the deflection curve for each beam and indicate symbolically the displacement or slope at the pt of each redundant force or moment. Compatibility equations • Write a compatibility eqn for the displacement or slope at each pt where there is a redundant force or moment. • Determine all the displacements or slopes using an appropriate method explained in chapter 12. 2 12. 5. • Substitute the results into the compatibility eqns and solve for the unknown redundants. 2005 Pearson Education South Asia Pte Ltd 101

12. Deflections of Beams and Shafts 12. 9 STATICALLY INDETERMINATE BEAMS AND SHAFTS: METHOD OF SUPERPOSITION Procedure for analysis Compatibility equations • If a numerical value for a redundant is +ve, it has the same sense of direction as originally assumed. • Similarly, a –ve numerical value indicates the redundant acts opposite to its assumed sense of direction. Equilibrium equations • Once the redundant forces and/or moments have been determined, the remaining unknown reactions can be found from the eqns of equilibrium applied to the loadings shown on the beam’s free-body diagram. 2005 Pearson Education South Asia Pte Ltd 102

12. Deflections of Beams and Shafts EXAMPLE 12. 22 Determine the reactions on the beam shown. Due to loading and poor construction, the roller support at B settles 12 mm. Take E = 200 GPa and I = 80(106) mm 4. 2005 Pearson Education South Asia Pte Ltd 103

12. Deflections of Beams and Shafts EXAMPLE 12. 22 (SOLN) Principle of superposition By inspection, beam is indeterminate to the first degree. Roller support at B is chosen as the redundant. Principle of superposition is shown. Here, By is assumed to act upwards on the beam. 2005 Pearson Education South Asia Pte Ltd 104

12. Deflections of Beams and Shafts EXAMPLE 12. 22 (SOLN) Compatibility equation With reference to pt B, we require Using table in Appendix C, displacements are 2005 Pearson Education South Asia Pte Ltd 105

12. Deflections of Beams and Shafts EXAMPLE 12. 22 (SOLN) Compatibility equation Thus Eqn (1) becomes Expressing E and I in units of k. N/m 2 and m 4, we have 2005 Pearson Education South Asia Pte Ltd 106

12. Deflections of Beams and Shafts EXAMPLE 12. 22 (SOLN) Equilibrium equations: Applying this result to the beam, we then calculate the reactions at A and C using eqns of equilibrium. 2005 Pearson Education South Asia Pte Ltd 107

12. Deflections of Beams and Shafts EXAMPLE 12. 24 Determine the moment at B for beam shown. EI is constant. Neglect the effects of axial load. 2005 Pearson Education South Asia Pte Ltd 108

12. Deflections of Beams and Shafts EXAMPLE 12. 24 Principle of superposition: Since axial load if neglected, a there is a vertical force and moment at A and B. Since only two eqns of equilibrium are available, problem is indeterminate to the second degree. Assume that By and MB are redundant, so that by principle of superposition, beam is represented as a cantilever, loaded separately by distributed load and reactions By and MB, as shown. 2005 Pearson Education South Asia Pte Ltd 109

12. Deflections of Beams and Shafts EXAMPLE 12. 24 Compatibility equations: Referring to displacement and slope at B, we require Using table in Appendix C to compute slopes and displacements, we have 2005 Pearson Education South Asia Pte Ltd 110

12. Deflections of Beams and Shafts EXAMPLE 12. 24 Compatibility equations: 2005 Pearson Education South Asia Pte Ltd 111

12. Deflections of Beams and Shafts EXAMPLE 12. 24 Compatibility equations: Substituting these values into Eqns (1) and (2) and canceling out the common factor EI, we have Solving simultaneously, we get 2005 Pearson Education South Asia Pte Ltd 112

12. Deflections of Beams and Shafts CHAPTER REVIEW • The elastic curve represents the centerline deflection of a beam or shaft. • Its shape can be determined using the moment diagram. • Positive moments cause the elastic curve to concave upwards and negative moments cause it to concave downwards. • The radius of curvature at any pt is determined from 1/ = M/EI. • Eqn of elastic curve and its slope can be obtained by first finding the internal moment in the member as a function of x. 2005 Pearson Education South Asia Pte Ltd 113

12. Deflections of Beams and Shafts CHAPTER REVIEW • If several loadings act on the member, then separate moment functions must be determined between each of the loadings. • Integrating these functions once using EI(d 2 /dx 2) = M(x) gives the eqn for the slope of the elastic curve, and integrating again gives the eqn for the deflection. • The constants of integration are determined from the boundary conditions at the supports, or in cases where several moment functions are involved, continuity of slope and deflection at pts where these functions join must be satisfied. 2005 Pearson Education South Asia Pte Ltd 114

12. Deflections of Beams and Shafts CHAPTER REVIEW • Discontinuity functions allow us to express the eqn of elastic curve as a continuous function, regardless of the no. of loadings on the member. • This method eliminates the need to use continuity conditions, since the two constants of integration can be determined solely from the two boundary conditions. • The moment-area method if a semi-graphical technique for finding the slope of tangents or the vertical deviation of tangents at specific pts on the elastic curve. 2005 Pearson Education South Asia Pte Ltd 115

12. Deflections of Beams and Shafts CHAPTER REVIEW • The moment-area method requires finding area segments under the M/EI diagram, or the moment of these segments about pts on the elastic curve. • The method works well for M/EI diagrams composed of simple shapes, such as those produced by concentrated forces and couple moments. • The deflection or slope at a pt on a member subjected to various types of loadings can be determined by using the principle of superposition. The table in the back of the book can be used for this purpose. 2005 Pearson Education South Asia Pte Ltd 116

12. Deflections of Beams and Shafts CHAPTER REVIEW • Statically indeterminate beams and shafts have more unknown support reactions than available eqns of equilibrium. • To solve them such problems, we first identify the redundant reactions, and the other unknown reactions are written in terms of these redundants. • The method of integration or moment-area theorems can be used to solve for the unknown redundants. • We can also determine the redundants by using the method of superposition, where we consider the continuity of displacement at the redundant. 2005 Pearson Education South Asia Pte Ltd 117

12. Deflections of Beams and Shafts CHAPTER REVIEW • The displacement due to the external loading is determined with the redundant removed, and again with the redundant applied and external loading removed. • The tables in Appendix C of this book can be used to determine these necessary displacements. 2005 Pearson Education South Asia Pte Ltd 118