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Bending moments and all everything is all cool
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Pure
Ben
ding
4 -3
Pur
e B
endi
ng
Pure
Ben
ding
: Pr
ism
atic
mem
bers
su
bjec
ted
to e
qual
and
opp
osite
cou
ples
ac
ting
in th
e sa
me
long
itudi
nal p
lane
4 -4
Oth
er L
oadi
ng T
ypes
•Pr
inci
ple
of S
uper
posi
tion:
The
nor
mal
st
ress
due
to p
ure
bend
ing
may
be
com
bine
d w
ith th
e no
rmal
stre
ss d
ue to
ax
ial l
oadi
ng a
nd sh
ear s
tress
due
to
shea
r loa
ding
to fi
nd th
e co
mpl
ete
stat
e of
stre
ss.
•Ec
cent
ric
Load
ing:
Axi
al lo
adin
g w
hich
do
es n
ot p
ass t
hrou
gh se
ctio
n ce
ntro
id
prod
uces
inte
rnal
forc
es e
quiv
alen
t to
an
axia
l for
ce a
nd a
cou
ple
•Tr
ansv
erse
Loa
ding
: C
once
ntra
ted
or
dist
ribut
ed tr
ansv
erse
load
pro
duce
s in
tern
al fo
rces
equ
ival
ent t
o a
shea
r fo
rce
and
a co
uple
4 -5
Sym
met
ric M
embe
r in
Pur
e B
endi
ng
MdA
yM
dAz
MdA
F
xz
xy
xx
00
•Th
ese
requ
irem
ents
may
be
appl
ied
to th
e su
ms
of th
e co
mpo
nent
s and
mom
ents
of t
he st
atic
ally
in
dete
rmin
ate
elem
enta
ry in
tern
al fo
rces
.
•In
tern
al fo
rces
in a
ny c
ross
sect
ion
are
equi
vale
nt
to a
cou
ple.
The
mom
ent o
f the
cou
ple
is th
e se
ctio
nbe
ndin
g m
omen
t.•
From
stat
ics,
a co
uple
M c
onsi
sts o
f tw
o eq
ual
and
oppo
site
forc
es.
•Th
e su
m o
f the
com
pone
nts o
f the
forc
es in
any
di
rect
ion
is z
ero.
•Th
e m
omen
t is t
he sa
me
abou
t any
axi
s pe
rpen
dicu
lar t
o th
e pl
ane
of th
e co
uple
and
ze
ro a
bout
any
axi
s con
tain
ed in
the
plan
e.
4 -6
Ben
ding
Def
orm
atio
ns Bea
m w
ith a
pla
ne o
f sym
met
ry in
pur
e be
ndin
g:•
mem
ber r
emai
ns sy
mm
etric
•be
nds u
nifo
rmly
to fo
rm a
circ
ular
arc
•cr
oss-
sect
iona
l pla
ne p
asse
s thr
ough
arc
cen
ter
and
rem
ains
pla
nar
•le
ngth
of t
op d
ecre
ases
and
leng
th o
f bot
tom
in
crea
ses
•a
neut
ral s
urfa
cem
ust e
xist
that
is p
aral
lel t
o th
e up
per a
nd lo
wer
surf
aces
and
for w
hich
the
leng
th
does
not
cha
nge
•st
ress
es a
nd st
rain
s are
neg
ativ
e (c
ompr
essi
ve)
abov
e th
e ne
utra
l pla
ne a
nd p
ositi
ve (t
ensi
on)
belo
w it
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -7
Stra
in D
ue to
Ben
ding
Con
side
r a b
eam
segm
ent o
f len
gth
L.
Afte
r def
orm
atio
n, th
e le
ngth
of t
he n
eutra
lsu
rfac
e re
mai
ns L
. A
t oth
er se
ctio
ns,
mx
mmx
cy
cc
yy
L
yy
LL
yL
or
linea
rly)
ries
(stra
in v
a
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -8
Stre
ss D
ue to
Ben
ding
•Fo
r a li
near
ly e
last
ic m
ater
ial,
linea
rly)
var
ies
(stre
ssm
mx
x
cy
Ecy
E
•Fo
r sta
tic e
quili
briu
m,
dAy
c
dAcy
dAF
m
mx
x 0
0
Firs
t mom
ent w
ith re
spec
t to
neut
ral
plan
e is
zer
o. T
here
fore
, the
neu
tral
surf
ace
mus
t pas
s thr
ough
the
sect
ion
cent
roid
.
•Fo
r sta
tic e
quili
briu
m,
IMy
cySM
IMc
cI
dAy
cM
dAcy
ydA
yM
x
mx
m
mm
mx
ngSu
bstit
uti
2
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -9
Bea
m S
ectio
n P
rope
rties
•Th
e m
axim
um n
orm
al st
ress
due
to b
endi
ng,
mod
ulus
sect
ion
iner
tiaof
mom
ent
sect
ion
cISI
SMIMc
m
A b
eam
sect
ion
with
a la
rger
sect
ion
mod
ulus
w
ill h
ave
a lo
wer
max
imum
stre
ss
•C
onsi
der a
rect
angu
lar b
eam
cro
ss se
ctio
n,
Ahbh
hbhcI
S61
361
3121
2
Bet
wee
n tw
o be
ams w
ith th
e sa
me
cros
s se
ctio
nal a
rea,
the
beam
with
the
grea
ter d
epth
w
ill b
e m
ore
effe
ctiv
e in
resi
stin
g be
ndin
g.
•St
ruct
ural
stee
l bea
ms a
re d
esig
ned
to h
ave
a la
rge
sect
ion
mod
ulus
.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
0
Pro
perti
es o
f Am
eric
an S
tand
ard
Sha
pes
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
1
Def
orm
atio
ns in
a T
rans
vers
e C
ross
Sec
tion
•D
efor
mat
ion
due
to b
endi
ng m
omen
t M is
quan
tifie
d by
the
curv
atur
e of
the
neut
ral s
urfa
ce
EIMIMc
EcEc
cm
m1
1
•A
lthou
gh c
ross
sect
iona
l pla
nes r
emai
n pl
anar
whe
n su
bjec
ted
to b
endi
ng m
omen
ts, i
n-pl
ane
defo
rmat
ions
are
non
zero
,y
yx
zx
y
•Ex
pans
ion
abov
e th
e ne
utra
l sur
face
and
co
ntra
ctio
n be
low
it c
ause
an
in-p
lane
cur
vatu
re,
curv
atur
ec
antic
last
i1
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
2
Sam
ple
Pro
blem
4.2
A c
ast-i
ron
mac
hine
par
t is a
cted
upo
n by
a 3
kN
-m c
oupl
e. K
now
ing
E=
165
GPa
and
neg
lect
ing
the
effe
cts o
f fil
lets
, det
erm
ine
(a) t
he m
axim
um
tens
ile a
nd c
ompr
essi
ve st
ress
es, (
b)
the
radi
us o
f cur
vatu
re.
SOLU
TIO
N:
•B
ased
on
the
cros
s sec
tion
geom
etry
,ca
lcul
ate
the
loca
tion
of th
e se
ctio
n ce
ntro
id a
nd m
omen
t of i
nerti
a.2 d
AI
IAAy
Yx
•A
pply
the
elas
tic fl
exur
al fo
rmul
a to
fin
d th
e m
axim
um te
nsile
and
co
mpr
essi
ve st
ress
es.
IMc
m
•C
alcu
late
the
curv
atur
e
EIM1
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
3
Sam
ple
Pro
blem
4.2
SOLU
TIO
N:
Bas
ed o
n th
e cr
oss s
ectio
n ge
omet
ry, c
alcu
late
th
e lo
catio
n of
the
sect
ion
cent
roid
and
m
omen
t of i
nerti
a.
33332
1011
430
0010
4220
1200
3040
210
9050
1800
9020
1m
m,
mm
,m
mA
rea,
AyA
Ayy
mm
3830
001011
43
AAyY
49-
3
23
1212
3121
23
1212
m10
868
mm
1086
8
1812
0040
3012
1800
2090
I
dA
bhd
AI
I x
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
4
Sam
ple
Pro
blem
4.2
•A
pply
the
elas
tic fl
exur
al fo
rmul
a to
find
the
max
imum
tens
ile a
nd c
ompr
essi
ve st
ress
es.
49
49
mm
1086
8m
038
.0m
kN3m
m10
868
m02
2.0
mkN3
IcMIc
MIMc
BB
AAm
MPa
0.76
A
MPa
3.13
1B
•C
alcu
late
the
curv
atur
e
49-
m10
868
GPa
165
mkN3
1EIM
m7.
47
m10
95.20
11-
3
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
5
Ben
ding
of M
embe
rs M
ade
of S
ever
al M
ater
ials
•C
onsi
der a
com
posi
te b
eam
form
ed fr
om
two
mat
eria
ls w
ith E
1an
dE 2
.
•N
orm
al st
rain
var
ies l
inea
rly.
yx
•Pi
ecew
ise
linea
r nor
mal
stre
ss v
aria
tion.
yE
Ey
EE
xx
22
21
11 Neu
tral a
xis d
oes n
ot p
ass t
hrou
gh
sect
ion
cent
roid
of c
ompo
site
sect
ion.
•El
emen
tal f
orce
s on
the
sect
ion
are
dAyE
dAdF
dAy
EdA
dF2
22
11
1
121
12
EEn
dAny
EdAy
nEdF
•D
efin
e a
trans
form
ed se
ctio
n su
ch th
at
xx
x
nIMy
21
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
6
Exa
mpl
e 4.
03
Bar
is m
ade
from
bon
ded
piec
es o
f st
eel (
E s=
29x1
06ps
i) an
d br
ass
(Eb
= 15
x106
psi).
Det
erm
ine
the
max
imum
stre
ss in
the
stee
l and
br
ass w
hen
a m
omen
t of 4
0 ki
p*in
is
app
lied.
SOLU
TIO
N:
•Tr
ansf
orm
the
bar t
o an
equ
ival
ent c
ross
se
ctio
n m
ade
entir
ely
of b
rass
•Ev
alua
te th
e cr
oss s
ectio
nal p
rope
rties
of
the
trans
form
ed se
ctio
n
•C
alcu
late
the
max
imum
stre
ss in
the
trans
form
ed se
ctio
n. T
his i
s the
cor
rect
m
axim
um st
ress
for t
he b
rass
pie
ces o
f th
e ba
r.
•D
eter
min
e th
e m
axim
um st
ress
in th
e st
eel p
ortio
n of
the
bar b
y m
ultip
lyin
g th
e m
axim
um st
ress
for t
he tr
ansf
orm
ed
sect
ion
by th
e ra
tio o
f the
mod
uli o
f el
astic
ity.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
7
Exa
mpl
e 4.
03
•Ev
alua
te th
e tra
nsfo
rmed
cro
ss se
ctio
nal p
rope
rties
4
3121
3121
in06
3.5
in3in
.25.2
hb
IT
SOLU
TIO
N:
•Tr
ansf
orm
the
bar t
o an
equ
ival
ent c
ross
sect
ion
mad
e en
tirel
y of
bra
ss.
in25.2
in4.0in
75.093
3.1
in4.0
933
.1ps
i10
15ps
i10
2966
T
bs
b
EEn
•C
alcu
late
the
max
imum
stre
sses
ksi
85.11
in5.
063
in5.1in
kip
404
IMc
m
ksi
85.11
933
.1m
ax
max
ms
mb
nks
i22
.9
ksi
85.11
max
max
sb
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
8
Rei
nfor
ced
Con
cret
e B
eam
s•
Con
cret
e be
ams s
ubje
cted
to b
endi
ng m
omen
ts a
re
rein
forc
ed b
y st
eel r
ods.
•Th
e st
eel r
ods c
arry
the
entir
e te
nsile
load
bel
ow
the
neut
ral s
urfa
ce.
The
uppe
r par
t of t
he
conc
rete
bea
m c
arrie
s the
com
pres
sive
load
.
•In
the
trans
form
ed se
ctio
n, th
e cr
oss s
ectio
nal a
rea
of th
e st
eel,
A s, i
s rep
lace
d by
the
equi
vale
nt a
rea
nAs
whe
ren
= E
s/Ec.
•To
det
erm
ine
the
loca
tion
of th
e ne
utra
l axi
s,
0
02 2
21d
An
xA
nx
b
xd
An
xbx
sss
•Th
e no
rmal
stre
ss in
the
conc
rete
and
stee
l
xs
xcx
nIMy
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -1
9
Sam
ple
Pro
blem
4.4
A c
oncr
ete
floor
slab
is re
info
rced
with
5/
8-in
-dia
met
er st
eel r
ods.
The
mod
ulus
of
ela
stic
ity is
29x
106p
si fo
r ste
el a
nd
3.6x
106p
si fo
r con
cret
e. W
ith a
n ap
plie
d be
ndin
g m
omen
t of 4
0 ki
p*in
for 1
-ft
wid
th o
f the
slab
, det
erm
ine
the
max
imum
st
ress
in th
e co
ncre
te a
nd st
eel.
SOLU
TIO
N:
•Tr
ansf
orm
to a
sect
ion
mad
e en
tirel
yof
con
cret
e.
•Ev
alua
te g
eom
etric
pro
perti
es o
f tra
nsfo
rmed
sect
ion.
•C
alcu
late
the
max
imum
stre
sses
in
the
conc
rete
and
stee
l.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
0
Sam
ple
Pro
blem
4.4 SO
LUTI
ON
:•
Tran
sfor
m to
a se
ctio
n m
ade
entir
ely
of c
oncr
ete.
22
854
66
in95.4
in2
06.8
06.8ps
i10
6.3ps
i10
29
s
cs
nA
EEn
•Ev
alua
te th
e ge
omet
ric p
rope
rties
of t
he
trans
form
ed se
ctio
n.
42
23
31in4.
44in
55.2in
95.4in
45.1in
12
in45
0.1
04
95.42
12 I
xx
xx
•C
alcu
late
the
max
imum
stre
sses
.
42
41
in44
.4in
55.2in
kip
4006.8
in44
.4in
1.45
inki
p40
IM
cn
IMc
scks
i30
6.1
c
ksi
52.18
s
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
1
Stre
ss C
once
ntra
tions
Stre
ss c
once
ntra
tions
may
occ
ur:
•in
the
vici
nity
of p
oint
s whe
re th
e lo
ads a
re a
pplie
d
IMc
Km
•in
the
vici
nity
of a
brup
t cha
nges
in
cro
ss se
ctio
n
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
2
Pla
stic
Def
orm
atio
ns •Fo
r any
mem
ber s
ubje
cted
to p
ure
bend
ing
mx
cyst
rain
var
ies l
inea
rly a
cros
s the
sect
ion
•If
the
mem
ber i
s mad
e of
a li
near
ly e
last
ic m
ater
ial,
the
neut
ral a
xis p
asse
s thr
ough
the
sect
ion
cent
roid
IMy
xan
d
•Fo
r a m
ater
ial w
ith a
non
linea
r stre
ss-s
train
cur
ve,
the
neut
ral a
xis l
ocat
ion
is fo
und
by sa
tisfy
ing
dAy
MdA
Fx
xx
0
•Fo
r a m
embe
r with
ver
tical
and
hor
izon
tal p
lane
s of
sym
met
ry a
nd a
mat
eria
l with
the
sam
e te
nsile
and
co
mpr
essi
ve st
ress
-stra
in re
latio
nshi
p, th
e ne
utra
l ax
is is
loca
ted
at th
e se
ctio
n ce
ntro
id a
nd th
e st
ress
-st
rain
rela
tions
hip
may
be
used
to m
ap th
e st
rain
di
strib
utio
n fr
om th
e st
ress
dis
tribu
tion.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
3
Pla
stic
Def
orm
atio
ns •W
hen
the
max
imum
stre
ss is
equ
al to
the
ultim
ate
stre
ngth
of t
he m
ater
ial,
failu
re o
ccur
s and
the
corr
espo
ndin
g m
omen
t MU
is re
ferr
ed to
as t
he
ultim
ate
bend
ing
mom
ent.
•Th
em
odul
us o
f rup
ture
in b
endi
ng, R
B, is
foun
d fr
om a
n ex
perim
enta
lly d
eter
min
ed v
alue
of M
U
and
a fic
titio
us li
near
stre
ss d
istri
butio
n.
Ic
MR
UB
•R B
may
be
used
to d
eter
min
e M
Uof
any
mem
ber
mad
e of
the
sam
e m
ater
ial a
nd w
ith th
e sa
me
cros
s sec
tiona
l sha
pe b
ut d
iffer
ent d
imen
sion
s.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
4
Mem
bers
Mad
e of
an
Ela
stop
last
ic M
ater
ial
•R
ecta
ngul
ar b
eam
mad
e of
an
elas
topl
astic
mat
eria
l
mom
ent
elas
ticm
axim
umY
YY
m
mY
x
cIM
IMc
•If
the
mom
ent i
s inc
reas
ed b
eyon
d th
e m
axim
um
elas
tic m
omen
t, pl
astic
zon
es d
evel
op a
roun
d an
el
astic
cor
e.
thic
knes
s-
half
core
elas
tic1
22
3123
YY
Yy
cyM
M
•In
the
limit
as th
e m
omen
t is i
ncre
ased
furth
er, t
he
elas
tic c
ore
thic
knes
s goe
s to
zero
, cor
resp
ondi
ng to
a
fully
pla
stic
def
orm
atio
n.
shap
e)se
ctio
ncr
oss
onon
ly(d
epen
dsfa
ctor
shap
e
mom
ent
plas
tic23 Yp
Yp
MMk
MM
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
5
Pla
stic
Def
orm
atio
ns o
f Mem
bers
With
a
Sin
gle
Pla
ne o
f Sym
met
ry•
Fully
pla
stic
def
orm
atio
n of
a b
eam
with
onl
y a
verti
cal p
lane
of s
ymm
etry
.
•R
esul
tant
sR1
and
R 2of
the
elem
enta
ry
com
pres
sive
and
tens
ile fo
rces
form
a c
oupl
e.
YY
AA
RR
21
21
The
neut
ral a
xis d
ivid
es th
e se
ctio
n in
to e
qual
ar
eas.
•Th
e pl
astic
mom
ent f
or th
e m
embe
r,
dA
MY
p21
•Th
e ne
utra
l axi
s can
not b
e as
sum
ed to
pas
s th
roug
h th
e se
ctio
n ce
ntro
id.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
6
Res
idua
l Stre
sses
•Pl
astic
zon
es d
evel
op in
a m
embe
r mad
e of
an
elas
topl
astic
mat
eria
l if t
he b
endi
ng m
omen
t is
larg
e en
ough
.
•Si
nce
the
linea
r rel
atio
n be
twee
n no
rmal
stre
ss a
nd
stra
in a
pplie
s at a
ll po
ints
dur
ing
the
unlo
adin
g ph
ase,
it m
ay b
e ha
ndle
d by
ass
umin
g th
e m
embe
r to
be
fully
ela
stic
.
•R
esid
ual s
tress
es a
re o
btai
ned
by a
pply
ing
the
prin
cipl
e of
supe
rpos
ition
to c
ombi
ne th
e st
ress
es
due
to lo
adin
g w
ith a
mom
ent M
(ela
stop
last
icde
form
atio
n) a
nd u
nloa
ding
with
a m
omen
t -M
(ela
stic
def
orm
atio
n).
•Th
e fin
al v
alue
of s
tress
at a
poi
nt w
ill n
ot, i
n ge
nera
l, be
zer
o.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
7
Exa
mpl
e 4.
05, 4
.06 A
mem
ber o
f uni
form
rect
angu
lar c
ross
sect
ion
is
subj
ecte
d to
a b
endi
ng m
omen
t M =
36.
8 kN
-m.
The
mem
ber i
s mad
e of
an
elas
topl
astic
mat
eria
l w
ith a
yie
ld st
reng
th o
f 240
MPa
and
a m
odul
us
of e
last
icity
of 2
00 G
Pa.
Det
erm
ine
(a) t
he th
ickn
ess o
f the
ela
stic
cor
e, (b
) th
e ra
dius
of c
urva
ture
of t
he n
eutra
l sur
face
.
Afte
r the
load
ing
has b
een
redu
ced
back
to z
ero,
de
term
ine
(c) t
he d
istri
butio
n of
resi
dual
stre
sses
, (d
) rad
ius o
f cur
vatu
re.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
8
Exa
mpl
e 4.
05, 4
.06
•Th
ickn
ess o
f ela
stic
cor
e:
666
.0m
m60
1m
kN28
.8m
kN8.36
1
22
3123
22
3123
YY
Y
YY y
cycy
cyM
M
mm
802
Yy
mkN
8.28
MPa
240
m10
120m
1012
0
1060
1050
36
36
23
332
232
YY
cIM
mm
bccI
•M
axim
um e
last
ic m
omen
t:•
Rad
ius o
f cur
vatu
re: 33
3
96
102.1
m10
40102.1
Pa10
200
Pa10
240
YYYY
YY
yyE
m3.33
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -2
9
Exa
mpl
e 4.
05, 4
.06
•M
= 36
.8 k
N-m
MPa
240m
m40
YYy
•M
= -3
6.8
kN-m
Y36 2
MPa
7.30
6m
1012
0m
kN8.36
IMc
m
•M
= 0
6
3
6
96
105.
177
m10
40105.
177
Pa10
200
Pa10
5.35
core
,el
astic
the
ofed
geA
t the
xYxx
yE
m22
5
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -3
0
Ecc
entri
c A
xial
Loa
ding
in a
Pla
ne o
f Sym
met
ry•
Stre
ss d
ue to
ecc
entri
c lo
adin
g fo
und
by
supe
rpos
ing
the
unifo
rm st
ress
due
to a
cen
tric
load
and
line
ar st
ress
dis
tribu
tion
due
a pu
re
bend
ing
mom
ent
IMy
APx
xx
bend
ing
cent
ric
•Ec
cent
ric lo
adin
g
PdM
PF
•V
alid
ity re
quire
s stre
sses
bel
ow p
ropo
rtion
al
limit,
def
orm
atio
ns h
ave
negl
igib
le e
ffec
t on
geom
etry
, and
stre
sses
not
eva
luat
ed n
ear p
oint
s of
load
app
licat
ion.
MEC
HA
NIC
SO
F M
ATE
RIA
LSB
eer
•Jo
hnst
on •
DeW
olf
4 -3
1
Exa
mpl
e 4.
07
An
open
-link
cha
in is
obt
aine
d by
be
ndin
g lo
w-c
arbo
n st
eel r
ods i
nto
the
shap
e sh
own.
For
160
lb lo
ad, d
eter
min
e (a
) max
imum
tens
ile a
nd c
ompr
essi
ve
stre
sses
, (b)
dis
tanc
e be
twee
n se
ctio
nce
ntro
id a
nd n
eutra
l axi
s
SOLU
TIO
N:
•Fi
nd th
e eq
uiva
lent
cen
tric
load
and
be
ndin
g m
omen
t
•Su
perp
ose
the
unifo
rm st
ress
due
to
the
cent
ric lo
ad a
nd th
e lin
ear s
tress
du
e to
the
bend
ing
mom
ent.
•Ev
alua
te th
e m
axim
um te
nsile
and
co
mpr
essi
ve st
ress
es a
t the
inne
r an
d ou
ter e
dges
, res
pect
ivel
y, o
f the
su
perp
osed
stre
ss d
istri
butio
n.
•Fi
nd th
e ne
utra
l axi
s by
dete
rmin
ing
the
loca
tion
whe
re th
e no
rmal
stre
ss
is z
ero.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -3
2
Exa
mpl
e 4.
07
psi
815
in19
63.0
lb16
0in
1963
.0
in25.0
20
2
22
APcA
•N
orm
al st
ress
due
to a
ce
ntric
load
•Eq
uiva
lent
cen
tric
load
an
d be
ndin
g m
omen
t
inlb
104
in6.0lb
160
lb16
0Pd
MP
psi
8475
in10
068
.in
25.0in
lb10
4in
1006
8.3
25.0
43
43
441
441 IM
ccI m
•N
orm
al st
ress
due
to
bend
ing
mom
ent
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -3
3
Exa
mpl
e 4.
07
•M
axim
um te
nsile
and
com
pres
sive
st
ress
es
8475
815
8475
815 00
mc
mt
psi
9260
t
psi
7660
c
•N
eutra
l axi
s loc
atio
n
inlb
105
in10
068
.3ps
i81
5
0
43
0
0
MIAP
y
IM
yAP
in02
40.0
0y
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -3
4
Sam
ple
Pro
blem
4.8
The
larg
est a
llow
able
stre
sses
for t
he c
ast
iron
link
are
30 M
Pa in
tens
ion
and
120
MPa
in c
ompr
essi
on.
Det
erm
ine
the
larg
est
forc
eP
whi
ch c
an b
e ap
plie
d to
the
link.
SOLU
TIO
N:
•D
eter
min
e an
equ
ival
ent c
entri
c lo
ad a
nd
bend
ing
mom
ent.
•Ev
alua
te th
e cr
itica
l loa
ds fo
r the
allo
wab
le
tens
ile a
nd c
ompr
essi
ve st
ress
es.
From
Sam
ple
Prob
lem
2.4
,
•Th
e la
rges
t allo
wab
le lo
ad is
the
smal
lest
of
the
two
criti
cal l
oads
.4
923
m10
868
m03
8.0
m10
3
IYA
•Su
perp
ose
the
stre
ss d
ue to
a c
entri
c lo
ad a
nd th
e st
ress
due
to b
endi
ng.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -3
5
Sam
ple
Pro
blem
4.8 •
Det
erm
ine
an e
quiv
alen
t cen
tric
and
bend
ing
load
s.
mom
ent
bend
ing
028
.0load
cent
ricm
028
.001
0.0
038
.0
PPd
MPd
•Ev
alua
te c
ritic
al lo
ads f
or a
llow
able
stre
sses
.
kN6.79
MPa
120
1559
kN6.79
MPa
3037
7
PP
PP
BA
kN0.
77P
•Th
e la
rges
t allo
wab
le lo
ad
•Su
perp
ose
stre
sses
due
to c
entri
c an
d be
ndin
g lo
ads P
PP
IM
cAP
PP
PI
Mc
AP
AB
AA
1559
1086
802
2.0
028
.010
3
377
1086
802
2.0
028
.010
3
93
93
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -3
6
Uns
ymm
etric
Ben
ding
•A
naly
sis o
f pur
e be
ndin
g ha
s bee
n lim
ited
to m
embe
rs su
bjec
ted
to b
endi
ng c
oupl
es
actin
g in
a p
lane
of s
ymm
etry
.
•M
embe
rs re
mai
n sy
mm
etric
and
ben
d in
th
e pl
ane
of sy
mm
etry
.
•W
ill n
ow c
onsi
der s
ituat
ions
in w
hich
the
bend
ing
coup
les d
o no
t act
in a
pla
ne o
f sy
mm
etry
.
•Th
e ne
utra
l axi
s of t
he c
ross
sect
ion
coin
cide
s with
the
axis
of t
he c
oupl
e
•C
anno
t ass
ume
that
the
mem
ber w
ill b
end
in th
e pl
ane
of th
e co
uple
s.
•In
gen
eral
, the
neu
tral a
xis o
f the
sect
ion
will
no
t coi
ncid
e w
ith th
e ax
is o
f the
cou
ple.
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -3
7
Uns
ymm
etric
Ben
ding
•
neut
ral a
xis p
asse
s thr
ough
cen
troid
dAy
dAcy
dAF
mx
x
0or0
•
defin
es st
ress
dis
tribu
tion
iner
tiaof
mom
ent
II
cI
dAcy
yM
M
zm
mz
Mor
Wis
h to
det
erm
ine
the
cond
ition
s und
er
whi
ch th
e ne
utra
l axi
s of a
cro
ss se
ctio
n of
arb
itrar
y sh
ape
coin
cide
s with
the
axis
of t
he c
oupl
e as
show
n.
•Th
e re
sulta
nt fo
rce
and
mom
ent
from
the
dist
ribut
ion
of
elem
enta
ry fo
rces
in th
e se
ctio
nm
ust s
atis
fyco
uple
appl
ied
MM
MF
zy
x0
•
coup
le v
ecto
r mus
t be
dire
cted
alo
ng
a pr
inci
pal c
entro
idal
axi
s
iner
tiaof
prod
uct
IdA
yz
dAcy
zdA
zM
yz
mx
y
0or0
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -3
8
Uns
ymm
etric
Ben
ding Su
perp
ositi
on is
app
lied
to d
eter
min
e st
ress
es in
th
e m
ost g
ener
al c
ase
of u
nsym
met
ric b
endi
ng.
•R
esol
ve th
e co
uple
vec
tor i
nto
com
pone
nts a
long
th
e pr
inci
ple
cent
roid
al a
xes.
sin
cos
MM
MM
yz
•Su
perp
ose
the
com
pone
nt st
ress
dis
tribu
tions
yy
zzx
Iy
MI
yM
•A
long
the
neut
ral a
xis,
tan
tan
sin
cos
0
yz
yz
yy
zzx
IIzy
Iy
MI
yM
Iy
MI
yM
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -3
9
Exa
mpl
e 4.
08SO
LUTI
ON
:
•R
esol
ve th
e co
uple
vec
tor i
nto
com
pone
nts a
long
the
prin
cipl
ece
ntro
idal
axe
s and
cal
cula
te th
e co
rres
pond
ing
max
imum
stre
sses
.si
nco
sM
MM
My
z
•C
ombi
ne th
e st
ress
es fr
om th
e co
mpo
nent
stre
ss d
istri
butio
ns.
yy
zzx
Iy
MI
yM
A 1
600
lb-in
cou
ple
is a
pplie
d to
a
rect
angu
lar w
oode
n be
am in
a p
lane
fo
rmin
g an
ang
le o
f 30
deg.
with
the
verti
cal.
Det
erm
ine
(a) t
he m
axim
um
stre
ss in
the
beam
, (b)
the
angl
e th
at th
e ne
utra
l axi
s for
ms w
ith th
e ho
rizon
tal
plan
e.
•D
eter
min
e th
e an
gle
of th
e ne
utra
l ax
is.
tan
tan
yz IIzy
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -4
0
Exa
mpl
e 4.
08•
Res
olve
the
coup
le v
ecto
r int
o co
mpo
nent
s and
cal
cula
teth
e co
rres
pond
ing
max
imum
stre
sses
. psi
5.60
9in
9844
.0in
75.0in
lb80
0
alon
goc
curs
todu
est
ress
nsile
larg
est t
eTh
e
psi
6.45
2in
359
.5in
75.1in
lb13
86al
ong
occu
rs to
due
stre
ssns
ilela
rges
t te
The
in98
44.0
in5.1in5.3
in35
9.5
in5.3in5.1
inlb
800
30si
nin
lb16
00in
lb13
8630
cos
inlb
1600
42
41
43
121
43
121
yy
z
zz
z
yz
yz
Iz
MAD
M
Iy
MAB
M
IIMM
•Th
e la
rges
t ten
sile
stre
ss d
ue to
the
com
bine
d lo
adin
g oc
curs
at A
.5.
609
6.45
22
1m
axps
i10
62m
ax
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -4
1
Exa
mpl
e 4.
08
•D
eter
min
e th
e an
gle
of th
e ne
utra
l axi
s.
143
.3
30ta
nin
9844
.0in
359
.5ta
nta
n44
yz II o 4.72
MEC
HA
NIC
SO
F M
ATE
RIA
LS
4 -4
2
Gen
eral
Cas
e of
Ecc
entri
c A
xial
Loa
ding
•C
onsi
der a
stra
ight
mem
ber s
ubje
ct to
equ
al
and
oppo
site
ecc
entri
c fo
rces
.
•Th
e ec
cent
ric fo
rce
is e
quiv
alen
t to
the
syst
em
of a
cen
tric
forc
e an
d tw
o co
uple
s.
PbM
PaMP
zy
forc
ece
ntric
•B
y th
e pr
inci
ple
of su
perp
ositi
on, t
he
com
bine
d st
ress
dis
tribu
tion
is
yy
zzx
Iz
MI
yM
AP
•If
the
neut
ral a
xis l
ies o
n th
e se
ctio
n, it
may
be
foun
d fr
om
APz
IMy
IMyy
zz
