Summary of Beam Theory 2 - MIT

reelingripebeltUrban and Civil

Nov 15, 2013 (3 years and 8 months ago)

81 views

IV. Successive Integration Method / Shear and Bending Moment Diagrams :


q(x)








(q=
-
dV/dx)

q= loading function


V(x)

=
-

q(x)dx+C
1




(
V=dM/dx)


V=shear force


M(x)

=

V(x)dx+C
1
x +C
2



(
M=d
q
/dx)


M=bending moment


q
(x)

=1/EI

(M(x)dx
+1/2C
1
x
2

+C
2
x+C
3



(
q
=EI(dv/dx))

q
=curvature=
slope of y
-
displacement curve


y(x)

=

q
(x)dx
+1/6C
1
x
3

+1/2C
2
x
2
+C
3
x +C
4




y=vertical displacement

V. Sign Conventions :








VI. Stresses and Strains in Beams :








FLEXURE FORMULA :

s
x

=
-
My/I where : y=vertical distance from NA







s
x
(max)=
-
M
max
y
max
/I, (rectangular) y
max
=h/2







I=moment of inertia of cross section about NA







I
rectangular
=bh
3
/12, I
circular
=
p
r
4
/4





PARALLEL AXIS THEORUM :

I
AA
=I
oo
+Ad
2















STRAIN FORMULA:


e
x
=
-
yM/EI





SHEAR FORMULA

:


t
xy

=VQ/Ib,
t
xy
(max)=V
max
Q
max
/Ib (at NA)





Q=first moment of the area above y about the NA=

A
i
y
i

(area•moment arm)





Q
rectangular
=b/2(h
2
/4
-
y
2
)

t
xy
(max)=3V
max
/2bh


Review : Beam Theory 2 (Cont’d)

(+)

V

V

(
-
)

V

V

(+)

(
-
)

M

M

M

M

compression

tension

compression

tension

y

s
x
(y)

x

t
xy
(y)

s
y
=0

NA

M

M

(+ moment)

A

o

o

A

d

b

h



c

t
xy
(max)

s
x
(max)c

s
x
(max)T