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