I_BEAM DEFLECTIONS - USE OF DISCONTINUITY FUNCTIONS FOR...

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Unformatted text preview: USE OF DISCONTINUITY FUNCTIONS FOR FINDING BEAM DEFLECTIONS 1 Introduction From the theory of the bending of beams, we know that the bending moment M and the curvature 1 /ρ are related by the equation 1 ρ = dθ dx = M EI , (1) where the convention for a positive bending moment is shown in Figure 1(b) below. In equation (1), θ = du dx (2) is the slope of the beam, u is the vertically upward deflection and EI is the flexural rigidity. F L x F M V (L-x) (a) (b) Figure 1 These equations permit us to find the deflection of the beam if M ( x ) is known as a function of x . For example, Figure 1(a) shows a beam built in at x = 0 and loaded by a concentrated force F at the other end x = L . The free-body diagram Figure 1(b) then shows that M ( x ) = − F ( L − x ) (3) and hence, substituting into (1), we have EI dθ dx = − F ( L − x ) . 1 Integrating with respect to x and using (2) then gives EIθ = EI du dx = − F parenleftBigg Lx − x 2 2 parenrightBigg + A , (4) where A is an arbitrary constant. One more integration gives EIu = − F parenleftBigg L x 2 2 − x 3 6 parenrightBigg + Ax + B , (5) where B is another arbitrary constant. The arbitrary constants in equations (4, 5) are determined from the kine- matic conditions describing the way the beam is supported. In the present example, the beam is built in at x = 0, which means that both the slope and the deflection are zero at this point — i.e. u = 0 ; θ = 0 ; at x = 0 . Applying these conditions to equations (4, 5), we obtain A = 0 ; B = 0 (6) and hence the final expression for the deflection is u = − F EI parenleftBigg L x 2 2 − x 3 6 parenrightBigg , from (5, 6). 1.1 Types of Support In beam problems, the only kinds of support allowed are a built-in support, which prevents both deflection and rotation (slope) and a simple support which prevents deflection but permits rotation. Pin joints can also be re- garded as simple supports in this context. A summary of the boundary conditions in a beam problem is therefore Built-in support at x = a u ( a ) = 0 ; θ ( a ) = 0 . (7) 2 Simple support at x = b u ( b ) = 0 . (8) A determinate beam must have either a single built-in support or two simple supports, in which case equations (7, 8) will provide exactly two conditions for the two arbitrary constants A,B . If the problem is indetermi- nate, there will more than two boundary conditions, but there will also be an equivalent number of additional unknown reaction forces associated with the indeterminacy. 2 Solution starting from the applied loads To use the method presented so far, we need to start by drawing a free-body diagram to determine the moment M as a function of x . An alternative method that avoids this step is to start from the loading on the beam w ( x ) and use the equilibrium relations dV dx = − w (9) dM dx = V , (10) where V is the shear force and a positive value of w corresponds to a down- ward load on the beam....
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This note was uploaded on 10/05/2011 for the course MECHENG 211 taught by Professor ? during the Fall '07 term at University of Michigan.

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I_BEAM DEFLECTIONS - USE OF DISCONTINUITY FUNCTIONS FOR...

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