Equação do calor

Chapter 9

Deflections of Beams

9.1 Introduction in this chapter, we describe methods for determining the equation of the deflection curve of beams and finding deflection and slope at specific points along the axis of the beam

9.2 Differential Equations of the Deflection Curve consider a cantilever beam with a concentrated load acting upward at the free end the deflection in the

y

v

is the displacement

direction

the angle of rotation

of the axis

(also called slope) is the angle between the x axis and the tangent to the deflection

curve point m1 is located at distance x point m2 is located at distance x + dx slope at

m1

is

slope at

m2

is

denote O'

+d

the center of curvature and

the radius of curvature, then d =

ds

and the curvature

is

1

=

1
C=

d
C
ds

the sign convention is pictured in figure slope of the deflection curve dv C dx for

=

tan

or ds j dx

small

=

dv tan-1 C dx cos j 1

=

1
C=

d
C
dx

and

=

1
C=

d
C=
dx

tan

j,

then

d 2v
CC
d x2

=

dv
C
dx

if the materials of the beam is linear elastic
=

1
C=

M
C
EI

[chapter 5]

then the differential equation of the deflection curve is obtained d C dx d2v
= CC dx2 =

M
C
EI

it can be integrated to find
∵

dM
CC
dx

=

then

d 3v
CC
dx3

V
=C
EI

V

and v dV CC dx =

-q

d 4v
CC
dx4

=

q
-C
EI

2

sign conventions for M,

V

and

q are shown

the above equations can be written in a simple form
EIv"

=

M

EIv"' =

V

EIv""

=

-q

this equations are valid only when Hooke's law applies and when the slope and the deflection are very small for nonprismatic beam [I = I(x)], the equations are d 2v
E Ix CC dx2 =

M

d d 2v
C (EIx CC) dx dx2

dM
CC
dx

=

d 2v d2 CC (EIx CC) d x2 d x2

=

=

dV
CC
dx

V

=