1. 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. Discontinuity functions 3. Method of superposition 4. Moment-area method 005 Pearson Education South Asia Pte Ltd 1
1. Deflections of Beams and Shafts 1.1 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. 005 Pearson Education South Asia Pte Ltd
1. Deflections of Beams and Shafts 1.1 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. 005 Pearson Education South Asia Pte Ltd 3
1. Deflections of Beams and Shafts 1.1 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). 005 Pearson Education South Asia Pte Ltd 4
1. Deflections of Beams and Shafts 1.1 THE ELASTIC CURVE 005 Pearson Education South Asia Pte Ltd 5
1. Deflections of Beams and Shafts 1.1 THE ELASTIC CURVE Moment-Curvature Relationship x axis extends +ve to the right, along longitudinal axis of beam. A differential element of undeformed width dx is located. ν axis extends +ve upwards from x axis. It measures the displacement of the centroid on x- sectional 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. 005 Pearson Education South Asia Pte Ltd 6
1. Deflections of Beams and Shafts 1.1 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-ν plane of symmetry for beam s x-sectional area. Internal moment M deforms element such that angle between x-sections is dθ. 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 ρ, ρ measured from center of curvature O to dx. 005 Pearson Education South Asia Pte Ltd 7
1. Deflections of Beams and Shafts 1.1 THE ELASTIC CURVE Moment-Curvature Relationship Strain in arc ds, at position y from neutral axis, is But ds' ds ε = ds ds = dx = ρdθ and ε = ds' = ( ρ y) [ ( ρ y ) d θ ρ d θ ] 1 ε = ρ y ρdθ s or ( 1-1) dθ 005 Pearson Education South Asia Pte Ltd 8
1. Deflections of Beams and Shafts 1.1 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 ε=σ/e, σ=-my/i 1 M = ( 1 - ρ EI ) ρ = radius of curvature at a specific pt on elastic curve (1/ρ 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. 005 Pearson Education South Asia Pte Ltd 9
1. Deflections of Beams and Shafts 1.1 THE ELASTIC CURVE Moment-Curvature Relationship EI is the flexural rigidity and is always positive. Sign for ρ depends on the direction of the moment. As shown, when M is +ve, ρ extends above the beam. When M is ve, ρ extends below the beam. 005 Pearson Education South Asia Pte Ltd 10
1. Deflections of Beams and Shafts 1.1 THE ELASTIC CURVE Moment-Curvature Relationship Using flexure formula, σ = My/I, curvature is also 1 = ρ σ Ey ( 1-3) Eqns 1- and 1-3 valid for either small or large radii of curvature. 005 Pearson Education South Asia Pte Ltd 11
1. Deflections of Beams and Shafts 1. SLOPE AND DISPLACEMENT BY INTEGRATION Let s represent the curvature in terms of ν and x. 1 = ρ 1 + d υ dx 3 ( dυυ ) dx Substitute into Eqn 1- d υ dx M = 3 1 ( ) EI d + υ dx ( 1-4 ) 005 Pearson Education South Asia Pte Ltd 1
1. Deflections of Beams and Shafts 1. SLOPE AND DISPLACEMENT BY INTEGRATION Most engineering codes specify limitations on deflections for tolerance or aesthetic purposes. Slope of elastic curve determined from dν/dx is very small and its square will be negligible compared with unity. Therefore, by approximation 1/ρ = d ν /dx, Eqn 1-4 rewritten as d υ M ( 1-5 ) dx = EI Differentiate each side w.r.t. x and substitute V = dm/dx, we get d d υ EI = V x 1-6 dx dx ( ) ( ) 005 Pearson Education South Asia Pte Ltd 13
1. Deflections of Beams and Shafts 1. SLOPE AND DISPLACEMENT BY INTEGRATION Differentiating again, using w = dv/dx yields d dx EI d υ dx = w ( x ) ( 1-7 ) Flexural rigidity is constant along beam, thus 4 d EI υ 4 dx = 3 d EI υ V 3 dx = d υ EI = dx w ( x ) ( 1-8 ) ( x ) ( 1-9 ) M ( x ) ( 1-10 ) 005 Pearson Education South Asia Pte Ltd 14
1. Deflections of Beams and Shafts 1. 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. 005 Pearson Education South Asia Pte Ltd 15
1. Deflections of Beams and Shafts 1. SLOPE AND DISPLACEMENT BY INTEGRATION Positive deflection v is upward. Positive slope angle θ will be measured counterclockwise when x is positive to the right. Positive slope angle θ will be measured clockwise when x is positive to the left. 005 Pearson Education South Asia Pte Ltd 16
1. Deflections of Beams and Shafts 1. SLOPE AND DISPLACEMENT BY INTEGRATION 005 Pearson Education South Asia Pte Ltd 17
1. Deflections of Beams and Shafts 1. SLOPE AND DISPLACEMENT BY INTEGRATION Boundary and continuity conditions Possible boundary conditions are shown here. 005 Pearson Education South Asia Pte Ltd 18
1. Deflections of Beams and Shafts 1. 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. 0 x1 a, a x (a+b) θ1 (a) = θ (a) v1 (a) = v (a) 0 x1 a, 0 x b θ1 (a) = - θ (b) v1 (a) = v (b) 005 Pearson Education South Asia Pte Ltd 19
1. Deflections of Beams and Shafts 1. 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 ν 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. 005 Pearson Education South Asia Pte Ltd 0
1. Deflections of Beams and Shafts 1. 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. 005 Pearson Education South Asia Pte Ltd 1
1. Deflections of Beams and Shafts 1. 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). 005 Pearson Education South Asia Pte Ltd
1. Deflections of Beams and Shafts 1. SLOPE AND DISPLACEMENT BY INTEGRATION Procedure for analysis Slope and elastic curve Provided EI is constant, apply either the load eqn EI d 4 ν/dx 4 = w(x), which requires four integrations to get ν = ν(x), ( ) or the moment eqns EI d ν /dx = 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 it conditions that apply to slope and displacement at pts where two functions meet. 005 Pearson Education South Asia Pte Ltd 3
1. Deflections of Beams and Shafts 1. 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 t +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. 005 Pearson Education South Asia Pte Ltd 4
1. Deflections of Beams and Shafts EXAMPLE 1.1 Cantilevered beam shown is subjected to a vertical load P at its end. Determine the eqn of the elastic curve. EI is constant. 005 Pearson Education South Asia Pte Ltd 5
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) Elastic curve: Load tends to deflect the beam. By inspection, the internal moment can be represented throughout h t the beam using a single x coordinate. Moment function: From free-body diagram, with M acting in the +ve direction, we have M = Px 005 Pearson Education South Asia Pte Ltd 6
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) Slope and elastic curve: Applying Eqn 1-10 and integrating twice yields EI EI d υ = dx dυ = dx Px Px 3 + C 1 Px EI + 6 ( 1 ) ( ) υ = + C x + C ( 3 ) 1 C 005 Pearson Education South Asia Pte Ltd 7
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) Slope and elastic curve: Using boundary conditions dν/dx =0atx=L x = L, and ν =0 at x = L, Eqn () and (3) becomes PL 0 = + C 3 PL 0 = + C1L + C 6 1 005 Pearson Education South Asia Pte Ltd 8
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) Slope and elastic curve: Thus, C = = 3 1 PL / and C PL /3. Substituting these results into Eqns () and (3) with θ = dν/dx, we get P EI P 6EI ( θ = L x ) υ = ( 3 3 x + 3 L x L ) Maximum slope and displacement occur at A (x = 0), θ A PL PL = υa = EI 3EI 3 005 Pearson Education South Asia Pte Ltd 9
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) Slope and elastic curve: Positive result for θ A indicates counterclockwise rotation and negative result for A indicates that ν A is downward. Consider beam to have a length of 5 m, support load P = 30 kn and made of A-36 steel having E st = 00 GPa. 005 Pearson Education South Asia Pte Ltd 30
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) Slope and elastic curve: Using methods in chapter 11.3, assuming allowable normal stress is equal to yield stress σ allow = 50 MPa, then a W310 39 would be adequate (I =848(10 84.8(10 6 )mm 4 ). From Eqns (4) and (5), θ A PL PL = υa = EI 3EI 3 005 Pearson Education South Asia Pte Ltd 31
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) Slope and elastic curve: From Eqns (4) and (5), θ A = 30 ( 3 N/kN) 5 m( 10 mm/m) [ ( 3) ] N/mm ( 84.8( 6 10 ) 4 mm ) kn 10 00 10 3 = 0.01 rad υ A 3 3 3 ( N/kN) 5 m( 10 mm/m) [ ( 3 ) ] 8 6 4 N/mm ( 84.8( 10 ) mm ) 30 kn 10 = 3 00 10 = 73.7 mm 005 Pearson Education South Asia Pte Ltd 3
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) Slope and elastic curve: Since θ = A (dν/dx) = 0.000488 << 1, this justifies the use of Eqn 1-10 than the more exact 1-4. 005 Pearson Education South Asia Pte Ltd 33
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) SOLUTION Using Eqn 1-8 to solve the problem. Here w(x) = 0 for 0 x L, so that upon integrating once, we get the form of Eqn 1-19 4 d υ EI = 4 dx EI 3 0 d υ = C' 1 = dx 3 V 005 Pearson Education South Asia Pte Ltd 34
1. Deflections of Beams and Shafts EXAMPLE 1.1 (SOLN) Solution II Shear constant C 1 can be evaluated at x = 0, since V A = P. Thus, C 1 = P. Integrating again yields the form of Eqn 1-10, EI d 3 υ 3 dx = P d υ EI = Px + C' = dx Here, M = 0 at x = 0, so C = 0, and as a result, we obtain Eqn 1 and solution proceeds as before. M 005 Pearson Education South Asia Pte Ltd 35
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1. Deflections of Beams and Shafts EXAMPLE 1.4 Beam is subjected to load P at its end. Determine the displacement at C. EI is a constant. 005 Pearson Education South Asia Pte Ltd 43
1. Deflections of Beams and Shafts EXAMPLE 1.4 (SOLN) Elastic curve Beam deflects into shape shown. Due to loading, two x coordinates will be considered, 0 x 1 a and 0 x a, where x is directed to the left from C since internal moment is easy to formulate. 005 Pearson Education South Asia Pte Ltd 44
1. Deflections of Beams and Shafts EXAMPLE 1.4 (SOLN) Moment functions Using free-body diagrams, we have P M1 = x1 M = Px Slope and Elastic curve: Applying Eqn 10-1, for d υ1 P 0 x1 a EI = x 1 dx 1 dυ1 P EI = x1 + C dx 4 1 P EI υ + 1 3 1 = x1 + C1x1 C 1 ( ) () 1 005 Pearson Education South Asia Pte Ltd 45
1. Deflections of Beams and Shafts EXAMPLE 1.4 (SOLN) Slope and Elastic curve: Applying Eqn 10-1, for d υ 0 x a EI = = Px dx EI dυ dx P = x + C 3 P EI 3 + 6 ( 3 ) 3 υ = x + C x + C ( 4 ) C4 005 Pearson Education South Asia Pte Ltd 46
1. Deflections of Beams and Shafts EXAMPLE 1.4 (SOLN) Slope and Elastic curve: The four constants of integration determined using three boundary conditions, ν 1 = 0 at x 1 = 0, ν 1 = 0 at x 1 = a, and ν =0 at x = a and a discontinuity eqn. Here, continuity of slope at roller requires dν 1/dx 1 = dν /dx at x 1 = a and x = a. υ υ 1 = 0 at x1 = 0; 0 = 0 + 0 + C P 0 at + C 1 ( a) + C1( a) 1 = x1 = a; 0 = 005 Pearson Education South Asia Pte Ltd 47
1. Deflections of Beams and Shafts EXAMPLE 1.4 (SOLN) Slope and Elastic curve: + + 3 0 0 C C P t ( ) ( ) ( ) ( ) + + + + = = = 1 4 3 3 ; 6 0 ; 0 C a P C a P a d a d C a C a a x at υ υ υ Solving we obtain ( ) ( ) + = + = 3 1 1 1 4 ; C a C a dx dx Solving, we obtain 7 Pa 3 4 3 1 6 7 0 3 Pa C Pa C C Pa C = = = = 005 Pearson Education South Asia Pte Ltd 48
1. Deflections of Beams and Shafts EXAMPLE 1.4 (SOLN) Slope and Elastic curve: Substituting C 3 and C 4 into Eqn (4) gives υ P = 6EI x 3 7 + Pa 6EI x Pa EI 3 Displacement at C is determined by setting x =0 0, υ C = Pa 3 EI 005 Pearson Education South Asia Pte Ltd 49
1. Deflections of Beams and Shafts EXAMPLE 1.4 (SOLN) If several different loadings act on the beam the method of integration becomes more tedious to apply, because separate loadings or moment functions must be written for each region of the beam. 8 constants t of integration 005 Pearson Education South Asia Pte Ltd 50
1. Deflections of Beams and Shafts *1.3 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 x a n = { 0 for x < { n ( x a) for x a ( 1-11) n 0 005 Pearson Education South Asia Pte Ltd 51 a
1. Deflections of Beams and Shafts *1.3 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. The functions describe both uniform load and triangular load. 005 Pearson Education South Asia Pte Ltd 5
1. Deflections of Beams and Shafts *1.3 DISCONTINUITY FUNCTIONS Discontinuity functions Macaulay functions 005 Pearson Education South Asia Pte Ltd 53
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1. Deflections of Beams and Shafts Singularity functions 005 Pearson Education South Asia Pte Ltd 56
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1. Deflections of Beams and Shafts *1.3 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 1- to express either the loading w or the internal moment M as a function of x. 005 Pearson Education South Asia Pte Ltd 58
1. Deflections of Beams and Shafts *1.3 DISCONTINUITY FUNCTIONS Procedure for analysis Load or moment function Calculate the support reactions and then use the discontinuity functions in Table 1- 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. 005 Pearson Education South Asia Pte Ltd 59
1. Deflections of Beams and Shafts *1.3 DISCONTINUITY FUNCTIONS Procedure for analysis Slope and elastic curve Substitute w into EI d 4 ν/dx 4 = w(x) or M into the moment curvature relation EI d ν/dx = 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. 005 Pearson Education South Asia Pte Ltd 60
1. Deflections of Beams and Shafts EXAMPLE 1.5 Determine the eqn of the elastic curve for the cantilevered beam shown. EI is constant. 005 Pearson Education South Asia Pte Ltd 61
1. Deflections of Beams and Shafts EXAMPLE 1.5 (SOLN) Elastic curve The loads cause the beam to deflect as shown. The boundary conditions require zero slope and displacement at A. 005 Pearson Education South Asia Pte Ltd 6
1. Deflections of Beams and Shafts EXAMPLE 1.5 (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. 005 Pearson Education South Asia Pte Ltd 63
1. Deflections of Beams and Shafts EXAMPLE 1.5 (SOLN) Loading functions 1 0 Therefore, w = 5 kn x 0 58 kn m x 0 8 KN / m x 0 + 50 kn m x 5 m + 8 kn / m x 5 m 0 The 1-kN load is not included, since x cannot be greater than 9 m. Because dv/dx = w(x), then by integrating, neglect constant of integration since reactions are included in load function, we have V = 0 1 5 x 0 58 x 0 8 x 0 1 1 + 50 x 5 + 8 x 5 1 005 Pearson Education South Asia Pte Ltd 64
1. Deflections of Beams and Shafts EXAMPLE 1.5 (SOLN) Loading functions Furthermore, dm/dx = V, V so integrating again yields M 0 1 1 1 = 58 x 0 + 5 x 0 x 0 () 8 x 0 + 50 x 5 + () 8 5 ( ) 0 58 + 5x 4x + 4 x 5 + 50 5 kn m = x The same result can be obtained directly from Table 1-. 005 Pearson Education South Asia Pte Ltd 65
1. Deflections of Beams and Shafts EXAMPLE 1.5 (SOLN) Slope and elastic curve Applying Eqn 1-10 10 and integrating twice, we have d υ 0 EI = 58 + 5 x 4 x + 50 x 5 + 4 x 5 dx dυ 4 1 4 3 EI = 58 x + 6 x x 3 + 50 x 5 + x 5 + C dx 3 3 6 3 1 4 EI υ = 19 x + x x + 5 x 5 3 3 1 4 + x 5 + C 1 x + C 3 1 005 Pearson Education South Asia Pte Ltd 66
1. Deflections of Beams and Shafts EXAMPLE 1.5 (SOLN) Slope and elastic curve Since dν/dx = 0atx x = 0, C 1 = 0; and ν = 0atx x = 0, so C = 0. Thus 1 6 3 1 4 υ = ( 19x + x x EI 3 3 1 + 5 x 5 + x 5 3 4 ) m 005 Pearson Education South Asia Pte Ltd 67
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