# Summary of Beam Theory 2 - MIT

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15 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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

q(x)

(q=
-
dV/dx)

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
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