Bending stress in Beams annotated
Erbil Polytechnic University Strength of Materials-I (STM303)
Erbil Technic al College
Dr. Feras Khlef
Civil Engineering Techniques Dept. First Semester 2021
Lecture 17
1. Erbil Polytechnic University Strength of Materials-I (STM303)
Erbil Technical College Dr. Feras Khlef
Civil Engineering Techniques Dept. First Semester 2021
Lecture 17
Bending stress in Beams
Th e b e nding m o m e nt “M” c a uses th e
m a terial within th e b otto m p ortion of th e
b ar to stretc h a nd th e m a terial within th e
to p p ortion to c o m press. C onse qu e ntly,
b etw e e n th ese tw o re gions th ere must b e
a surfa c e, c alle d th e n e utral surfa c e.
2. Erbil Polytechnic University Strength of Materials-I (STM303)
Erbil Technical College Dr. Feras Khlef
Civil Engineering Techniques Dept. First Semester 2021
Lecture 17
m ax = th e m aximum norm al stress in th e
m e m b er, whic h o c c urs a t a p oint on th e
cross-se ction al are a farth est a w a y fro m
th e n e utral axis.
M = th e result a nt intern al m o m e nt, d etermin e d
fro m th e m etho d of se ctions a nd th e
e qu a tions of e quilibrium, a nd c alc ula te d
a b out th e n e utral axis of th e cross se ction
c = p erp e ndic ular dist a n c e fro m th e n e utral
axis to a p oint farth est a w a y fro m th e
n e utral axis. This is wh ere m ax a cts.
I = m o m e nt of in ertia of th e cross-se ction al
are a a b out th e n e utral axis
!"#
$
%
&
'
(
)* +
, -./0-- #1 2 +
###-,34(0(5(3(
)
6
78709. /.:; ;<82.
=02./;> ;?:- @ = + A +
B
<09C:91
D
&
-291
E &
;
#. ;
.DF G
=01;.:H0 I+ J +
K8-:.:H0 I +
LM +
3. Erbil Polytechnic University Strength of Materials-I (STM303)
Erbil Technical College Dr. Feras Khlef
Civil Engineering Techniques Dept. First Semester 2021
Lecture 17
Ex:
N87O2.0 P;?:727
<09C:91 QRST
-./0-- :9 .U0 <0;7 -U8V9 +
"
W8> +
&XLL
.
& F
YP
QPZ
-.0O- [ [
! !/;V I+.. +
!:;1/;7 XST S/
& QMST + +
7
]
^ _8/ `77 + -0>0N.
I+ :. +
a 7;b _/87 .U0 %
78709. !:;1/;7 [8
J79?%QcS=+[d
%
e
f7;g%P7 h;-0>b
"= &
A W8>:C /0N.;912>;/ h/8-- &
-0N.:89 +
i c8N;.0 = +
A + ;. .U0 N;7O +
QRR77
h/8-- &
W0N.:89 f09- j
[
#[A N8
k%:-87
&
3 RR77
l , m
+
e N;>N2>;.0 h[ ;9C hN N87O
"h1
:-8 n
.U0 ..7/? %
&LQMSo +
7
&pL4 +
77
h.N-
[09-:89
m &R
4. C& h87O2.0 "9 +
A+
_8/ .U0 N/8--&-0N.:89
"9 +;%<" @/0N1.;-912->0;/
+
B
?20
% Y88BI 3 -8
LQ
=
L%M+pMRX
B 77
n. qLpR
rB #7 +
&sa7
.09-:89
a77 +
"=A +
h
j
QM`R`B *
Lp4
#70? &
.8
%Z=L77&f
[
"= n
A
aV; +
_
h
N87O +
t_0"#:M877"
&
"= +
A +
5. Erbil Polytechnic University Strength of Materials-I (STM303)
Erbil Technical College Dr. Feras Khlef
Civil Engineering Techniques Dept. First Semester 2021
Lecture 17
Ex: h0;-0 Q )
I8? N/8--&-0N.:89
h89-:C0/ .U0 W;70 <0;7 QRR77
V:.U .U0 W;70
c8;C:91
P890b +
&MMQMS4
G
-8
"S877
" 77
&
c8N;.0 H+ A +
fap8
h87O +
_:<0/ "u879
&
VVV+:-.8-:.:H0&-:___
j
:. :.
h
[_ QRU7
Q8U7
&
"= +A +
[09-:89
_:<0/
%
'8/ .U:- <8? N/8-- &
W0N +
2-0 K;/;>>0> ;?:- .U08/07
"= +
A +
@ "O;/.. AO;/. + CQ B
"
+(#.vAiC!j.
"
w^.A^Cuxv C+
_.
&
y
&
".n-
+
Q e i
= &
w . z j
C:__
[CM%8A
w ;
z
k
^
[8
M
6. QRR
&
C+
_.
&
y
&
4
+
(4
+
#>:
)(
+
"^
&
"-8
"
"9 +
A +
%""" f88 *
QRgLpR&QppL*Q
v
wQRrMQ`rY&v`*Q`RY`LQz
A
Q&&QLp`pYYYY
U:7
M
E/09
+
+ $
&LMS4
6
+
a"[
{
%
Y& $
##
LQ%QLp`pYYYY
77
n.
`7"g%P79 QML4pM
_.O+"[O;_
"= +
A
%
QLp`pYYYY
_99S&:.71+9+NNd+&Y||)n-"YC>`":;x
7. Erbil Polytechnic University Strength of Materials-I (STM303)
Erbil Technical College Dr. Feras Khlef
Civil Engineering Techniques Dept. First Semester 2021
Lecture 17
Ex: h;-0 Y { " &
W0N.:89
u8877
W;70 <0;7 ;9C
c8;C:91 -8 77
&
y
&
h89C:.:89
&RLM77
3 "-87
&
a_; kQMS= &
7
/&u88+7bd##A
A= &
A +
:9 .U0 7:CC>0
;7
&
Wb7 j
u879
^
LLpR77
h[ "N0 &MpR77
}Q887d
"9 +
9;.:89
)!
+-:
" A C
w
LQRYjQQ&`RYYv@QR*Q`~`pB%LX`Y`RR`R
77M
#9;
n"%#:g(C<IaK"
m
N87O/0--:89 _:<0/
LpRZ
"
m &
LpR%QL
"
v
.09-:870.0/
8. Erbil Polytechnic University Strength of Materials-I (STM303)
Erbil Technical College Dr. Feras Khlef
Civil Engineering Techniques Dept. First Semester 2021
Lecture 17
Ex:
a:7
jttM"0N.:7D3?V8•
W0N.:89 :- 98. 1b7 +
€887
"
c8N;.0 = +
A +
% ".
€8/9 &
A9 +
;
j &L Au;u
"-
4Q
; j ;/0
3 %
@QRRtQRrMR~Bv@QRtRYLQLRB '8V0b
&
-8
d
@
Qt~QRrvpQR
•u88 B
j
[
3hN
G
:-&77;+-2d8_*8&:"":"‚:;
6
n
l N
%
QQR&Lpp%`p
77
< } <
*
h[ % Lpp77
€8/9 &
h
j
"99 +
_.O+/.:&AO;/.&Cb..7;?::-1505:.+:9k+O87-0/
[09-:89
_:<0/
9. Erbil Polytechnic University Strength of Materials-I (STM303)
Erbil Technical College Dr. Feras Khlef
Civil Engineering Techniques Dept. First Semester 2021
Lecture 17
Ex:
V891
f##
i
[a"
)
=
d
&
&
A +
:-
+"[d:8&""88+088
v
": "[P &
Y A. C j
"=
+11k_pk.u88B&&cu8;28B@>pp&>88'x.
f88-0 B@ `p &
>8. z
%Y
&r`````r77 n.
( h87O/0--:89 _:<0/
hN h89_O:/U0-0-1:89
hEPK[ q
h;-.
f/00S %
&QMM4 B N
qs +
&
Yr`````r A +
[09-:89
{[:;
N
j
4LppL
ƒ ƒ
& &
q
[09-:89
_u11.09-: QMMR`ƒ@Lp&pB _:<0/
Yr`````r&&LpX+XLMLYr