x M
M ( x) Px
13
Slope and Elastic Curve. Applying equation (4) and integrating twice yields
d w EI 2 Px dx 2 dw Px EI c1 dx 2 Px EIw c1 x c2 6
d w M 2 dx EI
2
(3)
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If EI is constant, rewriting equation (3), we have 2
d w EI 2 M ( x) dx
(4)
Integrating equation (4) twice yields
dw EI M ( x)dx c1 dx
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2
3
dw P 2 2 L x dx 2 EI P 3 2 3 w x 3L x 2 L 6 EI
Maximum slope and deflection occur at A (x = 0), for which


PL A 2 EI
2
PL wA 3EI
5
It is important that we should be able to calculate the deflection of a beam of given section, since for given conditions of span and load it would be possible to adopt a section which would meet a strength criterion but would give an unacceptable deflection. The total deflection of a beam is due to a very large extent to the deflection caused by bending, and to a very much smaller extent to the deflection caused by shear.
19
Px EIw1 c1 x1 c2 183 1源自Likewise for M2
d w2 2 P EI (3a x2 ) 2 3 dx2
16
3
[Example 2] The simply supported beam shown in figure is subjected to the concentrated force P. Determine the maximum deflection of the beam. EI is constant. Elastic Curve. Two P 2a a coordinates must be B A C x1 used, since the x2 moment becomes discontinuous at P. Here we will take x1 and x2, having the same origin at A, so that 0 x1 2a, 2a x2 3a
4
w

w F x
The elastic curve for a beam
The slope is the turning angle of the crosssectional area. The positive slope angle will be measured counterclockwise from the x axis when x is positive to the right. The slope can be determined from dw/dx.
The constants of integration are determined by evaluating the functions for shear, moment, slope, or displacement at a particular point on the beam where the value of the function is known. These values are called boundary conditions. If a single x coordinate cannot be used to express the equation for the beam’s slope or the elastic curve, the continuity conditions must be used to evaluate some of the integration constants.
3
w

w F x
The elastic curve for a beam
The x axis extends positive to the right, along the initially straight longitudinal axis of the beam. The w axis extends positive upward from the x axis. It measures the deflection of the centroid on the cross-sectional area of the element.
2
§15-1 Slope and Deflection of Beams
Introduction Having studied the stresses set up in bending, we now turn to the equally important aspect of beam stiffness. In many structural elements, such as floor joists or aircraft wings, the limiting constraint on the design is stiffness. Any design which is stiff enough will be strong enough.
7

The elastic curve for a beam can be expressed mathematically as w = f (x). To obtain this equation, we must first represent the curvature in terms of w and x. In most calculus books it is shown that this relationship is
12
[Example 1] The cantilevered beam shown in figure is subjected to a vertical load P at its end. Determine the equation of the elastic curve. EI is constant. Moment Function. w From the free-body P B x diagram, with M A x acting in the L positive direction, P we have
EIw M ( x)dx c1 x c2
2
10
Therefore, the equation of the slope and elastic curve for the beam are
dw 1 dx EI
1 w EI
M ( x)dx c
1
2
M ( x)dx
3
2
(a) (b) (c)
14
Using the boundary conditions dw / dx = 0 and w = 0 at x = L, equation (b) and (c) become
PL 0 c1 2 PL 0 c1 L c2 6
Thus, c1= PL2/2 and c2= -PL3/3. Substituting these results into equation (b) and (c), we get
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Moment Function From the free-body diagrams shown in figure,
x1 P/3 M1
P M 1 ( x) x1 3
M 2 ( x) P x2 P ( x2 2 a ) 3 2P (3a x2 ) 3
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2a
x2 P/3
1


1 dw / dx
d w / dx
2 2 2 3/ 2
d w / dx
2
2 2 3/ 2
Substituting into equation (1), we get
1 dw / dx
M EI
(2)
8
Most engineering design codes specify limitations on deflections for tolerance or esthetical purposes, and as a result the elastic deflections for the majority of beams and shafts form a shallow curve. Consequently, the slope of the elastic curve which is determined from dw/dx will be very small, and its square will be negligible compared with unity. Using this simplification, equation (2) can now be written as