Chapter 5 Internal forces in bending;
Properties of matirials
Properties of matirials 5.1 Introduction
5.2 Types of beams and reactions
5.3 Shearing forces and bending moments
5. Shear!"orce and #ending!$oment diagrams
5.5 %elations among shear force& bending moment and
loads
 
  Chapter 5 Internal forces in bending;
Properties of matirials
Properties of matirials 5.1 Introduction
5.2 Types of beams and reactions
5.3 Shearing forces and bending moments
5. Shear!"orce and #ending!$oment diagrams
5.5 %elations among shear force& bending moment and
loads
 
5.1.1 Concept of bending
1. #ending)  $embers sub*ected to e+ternal forces and moments
perpendicular to the a+le&and the a+le become to cur(e.
#ending deformation
5.1 Introduction
3. ,+ample
5.1 Introduction
Symmetric bending!!!special case of plane bending
 M 
-+le
/on!symmetric bending 0 /o longitudinal symmetric
surface or e+ternal forces not in longitudinal
symmetric plane.
stresses& deformations& strength& and stiffness
5.1 Introduction
  Chapter 5 Internal forces in bending;
Properties of matirials
Properties of matirials 5.1 Introduction
5.2 Types of beams and reactions
5.3 Shearing forces and bending moments
5. Shear!"orce and #ending!$oment diagrams
5.5 %elations among shear force& bending moment and
loads
 
inged supporter
roll!supporter
"i+ed end
  Three reactiona& 4ero
 
5.3.2 Three types of statically determined beams
Simplly supported beam
q6 x 7— istributed
 forceCantile(er bean
 
CH.5 INTERNAL FORCES IN BENDING 
,+.5.1 Shon is the circular cylinder tan8 of   L95m inside diameter
 D91m thic8nesst 91:mm density of steel s9 <.=g>cm? density of li@uid
l9 1g>cm?; height of the li@uid surface h9:.=m e+tensional length a91m Plot mechanical s8etch of the tan8.
Solution
q
 
 L
 L  g V 
rad7"55.().(!$* ! ==θ 
22
 g  A g  A 22((   γ  γ     +=
2
2
).(# ( !.!( %"!! &." -).(# !.5
 
  Chapter 5 Internal forces in bending;
Properties of matirials
Properties of matirials 5.1 Introduction
5.2 Types of beams and reactions
5.3 Shearing forces and bending moments
5. Shear!"orce and #ending!$oment diagrams
5.5 %elations among shear force& bending moment and
loads
 
,+.5.1 Anoing F  a l . "ind internal
forces on the section x  from
 A.
 
 A B F 
 y AC 
to the a+is.
 
Internal force acting on the cross!section.
3. Con(ention of sense of internal forces "s and $) F s) "s is positi(e hen moment of "s to the arbitrary
internal point is cloc8!ised.
 
don!con(e+.
F s6B
 M 6 – 7
 
F 
l 
 A
 B
F sA+=-F  Ay=-2F 
 M  A+=M e-F  Ay×=Fl 
 
 M A1
F sA+
! Solution)  1. %eactions
. l ! 
 
F 
l 
 A
 B
F sD-
 M  D-
 
F 
 A
 B
F sD32F+F  By=F 
 M A1 M e3F A'42l+F  By 4l !  D-
$ethod 2) 
 
CH.5 INTERNAL FORCES IN BENDING 
* +
* +
↑
− + ↵
ql  q
a b
(3(
 x(
 x2
2
q x a F ql q x a M qlx  
2
 
CH.5 INTERNAL FORCES IN BENDING 
1 !"#$%& '()*'(+,-.* '()'(+%
2/0/0 1230"#$%&4567 '()/08567'(+/0, '()/0*'(+/09 :;<=>&?=.%
/@ABCDE 
 
(
( !=
ql ql F 
 
  Chapter 5 Internal forces in bending;
Properties of matirials
Properties of matirials 5.1 Introduction
5.2 Types of beams and reactions
5.3 Shearing forces and bending moments
5. Shear!"orce and #ending!$oment diagrams 
5.5 %elations among shear force& bending moment and
loads
 
R
/0R
shearing forces
76 x  M  M  =
bending mpment
 
,+.5.1 The cantile(er. "ind
and $ diagrams.
l 
−−=
+−=
 
F  x F  &    =76
F s6 x 7
FL
−−=
+−=
 
,+.5.2 Anoing F  a " l . Set up "s& $ ,@us.
Plot "s& $ diagrams. F  Ay
F  B Solution %eactions
l 
F  Ay
F s
l 
 
F  a
 A   B
 
,+.5.2 Anoing q l . Set up "s& $ ,@us..
Plot "s& $ diagrams. F  Ay F  By
Solution %eactions
76
76
 x  1
F  Ay
Simple cross!
 
q
l 
iscussion)
AG„onx…
 
no„†‡x…ˆ
 
  Chapter 5 Internal forces in bending;
Properties of matirials
Properties of matirials 5.1 Introduction
5.2 Types of beams and reactions
5.3 Shearing forces and bending moments
5. Shear!"orce and #ending!$oment diagrams
5.5 %elations among shear force& bending moment and
loads
 
76 76
76 76
76 76
sC 
 
5.5.2 Plot "s& $ diagrams using relations among "s& $ and @
1%DŠ8 2%D‹Œhihi7‰ nosKA%
3%‘q "sJ$„ @’“‰x…pG
2%Irno "sGgNe$Gtue
76 76
76 76
76 76
 
,+.5.5.1 Plot "s& $ diagrams.F  Ay F  By

q
ql/)
) / %ql 
12)
 
qa2>=
qa2
qa2qa2
section "s  $
 
  Chapter 5 Internal forces in bending;
Properties of matirials
Properties of matirials 5.1 Introduction
5.2 Types of beams and reactions
5.3 Shearing forces and bending moments
5. Shear!"orce and #ending!$oment diagrams
5.5 %elations among shear force& bending moment and
loads
 
1. "rames& Plane frames&
 
        1   .
,+a.' raing $!diagram of the frame -#C shon.
Solution) 617 "ind reactions 
627 ist bending e@uations
Portion #C&   M(x 2 )= -a+.x 2=.(x 2!a)
 .a
 .a
$!.a
 
,+. < Plot "/& "s&& $ diagrams of the frame.
 F (
5.' Internal forces in frames and cur(e beams
 
F01sF s  =
 Fs#F      −=
MFRsin;
F3
Xe
section)
5.' Internal forces in frames and cur(e beams
 
1 1>2
3>2 1
Solution %eactions
 A Ay   ==
Sec. "s  $
s   ==
 M 6qa27
 
+)* $
 
L¨§AR
¡RšA G
qx ) x( Q   −=
 
¢F ©¥ qª«EšGHBŠI/0G%  P