Page 175 - Shigley's Mechanical Engineering Design

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150 Mechanical Engineering Design
4–3 Deflection Due to Bending

The problem of bending of beams probably occurs more often than any other loading
problem in mechanical design. Shafts, axles, cranks, levers, springs, brackets, and wheels,
as well as many other elements, must often be treated as beams in the design and analy-
sis of mechanical structures and systems. The subject of bending, however, is one that
you should have studied as preparation for reading this book. It is for this reason that
we include here only a brief review to establish the nomenclature and conventions to be
used throughout this book.
The curvature of a beam subjected to a bending moment M is given by

1 M
= (4–8)
ρ EI
where ρ is the radius of curvature. From studies in mathematics we also learn that the
curvature of a plane curve is given by the equation
2
1 d y/dx 2
= (4–9)
2 3/2
ρ [1 + (dy/dx) ]
where the interpretation here is that y is the lateral deflection of the centroidal axis of
the beam at any point x along its length. The slope of the beam at any point x is
dy
θ = (a)
dx
For many problems in bending, the slope is very small, and for these the denominator
of Eq. (4–9) can be taken as unity. Equation (4–8) can then be written
2
M d y
= (b)
EI dx 2
Noting Eqs. (3–3) and (3–4) and successively differentiating Eq. (b) yields
3
V d y
= (c)
EI dx 3
4
q d y
= (d)
EI dx 4
It is convenient to display these relations in a group as follows:

4
q d y
= (4–10)
EI dx 4
3
V d y
= (4–11)
EI dx 3
2
M d y
= (4–12)
EI dx 2
dy
θ = (4–13)
dx
y = f (x) (4–14)