Bending of Beams MECHENG242 Mechanics of Materials www.engineering.auckland.ac.nz/mechanical/ MechEng242 2 .0 Bending of Beams Now we consider the elastic deformation of beams (bars) under bending loads. 1.0 Axial Forces ☻ M M
Bending of BeamsMECHENG242 Mechanics of Materials
www.engineering.auckland.ac.nz/mechanical/MechEng242
2.0 Bending of BeamsNow we consider the elastic deformation of beams (bars) under bending loads.
1.0 Axial Forces☻
MM
Bending of BeamsMECHENG242 Mechanics of Materials
Application to a Bar
Normal Force:
FnFn
Shear Force:
FtFt
Bending Moment:
MtMt
Torque or Twisting Moment:
Mn
Mn
S.B.
K.J.
Bending of BeamsMECHENG242 Mechanics of Materials
Examples of Devices under Bending Loading:
Car Chassis
YachtExcavator
Atrium Structure
Bending of BeamsMECHENG242 Mechanics of Materials
2.2 Stresses in Beams (Refer: B,C & A –Sec’s 6.3-6.6)
2.3 Combined Bending and Axial Loading (Refer: B,C & A –Sec’s 6.11, 6.12)
2.1 Revision – Bending Moments (Refer: B,C & A – Sec’s 6.1,6.2)
2.0 Bending of Beams
2.4 Deflections in Beams (Refer: B,C & A –Sec’s 7.1-7.4)
2.5 Buckling (Refer: B,C & A –Sec’s 10.1, 10.2)
x
x
Mxz Mxz
x
P
P1
P2
Bending of BeamsMECHENG242 Mechanics of Materials
2.1 Revision – Bending Moments
Last year Jason Ingham introduced Shear Force and Bending Moment Diagrams.
(Refer: B, C & A – Chapter 6)
3m 3m
12 kN
RECALL…
Q
0
(SFD)
0
M (BMD)
Bending of BeamsMECHENG242 Mechanics of Materials
(Refer: B, C&A – Sections 1.14, 1.15, 1.16, 6.1)
x
y
PA B
RByRAyMxz Mxz
Radius of Curvature, R
Deflected Shape
Consider the simply supported beam below:
Mxz MxzWhat stresses are generated within, due to bending?
Bending of BeamsMECHENG242 Mechanics of Materials
Axial Stiffness
Load (W)
Extension (u)
Bending Moment
(Mxz)
Curvature (1/R)
Flexural Stiffness
PA B
RByMxzRAy
Mxz
BendingRecall: Axial Deformation
W
u
Bending of BeamsMECHENG242 Mechanics of Materials
x
y
Mxz=Bending Moment
Mxz Mxz
Beam
x (Tension)
x (Compression)
x=0
(i) Bending Moment, Mxz (ii) Geometry of Cross-section
x is NOT UNIFORM through the section depth
x DEPENDS ON:
Axial Stress Due to Bending:
Unlike stress generated by axial loads, due to bending:
Bending of BeamsMECHENG242 Mechanics of Materials
“Happy” Beam is +VE “Sad” Beam is -VE
x
yMxz=Bending Moment
+VE (POSITIVE)
Sign Conventions: Qxy=Shear Force
Mxz Mxz
Qxy Qxy
-ve x
+ve x
Bending of BeamsMECHENG242 Mechanics of Materials
;0Fy ;0Mz
PQxy
x
yExample 1: Bending Moment Diagrams P
RAy=P
A B
L
Mxz=P.L
P.L
P Qxy
Mxz
xP
Mxz
QxyMxz Mxz
Qxy Qxy
Q & M are POSITIVE xLPMxz
Bending of BeamsMECHENG242 Mechanics of Materials
;PQxy
x
yP
P
BP.L
x
xLPMxz L
Qxy 0
Mxz 0
A Mxz
Qxy
To find x and deflections, need to know Mxz.
Shear Force Diagram (SFD)
Bending Moment Diagram (BMD)
+veP
-ve
-P.L
Bending of BeamsMECHENG242 Mechanics of Materials
x
yExample 2: Macaulay’s Notation
Qxy
Mxz
A BC
a bP
baaPRBy
ba
bPRAy
x
babP
;0Mz xzM
axPxba
PbMxz
Where can only be +VE or ZERO. ax
Pa
A
axP 0xba
Pb
Bending of BeamsMECHENG242 Mechanics of Materials
x
y
x
A BC
a bP
baPa ba
Pb
(i) When
:ax
(ii) When
:ax axPxba
PbMxz
1
2
axPxba
PbMxz
0
A BC
+ve baPab
Mxz
0
BMD: Eq. 1Eq. 2
Bending of BeamsMECHENG242 Mechanics of Materials
;0Fy ;0Mz
x
y
Example 3: Distributed Load
RAy=wL
A B
Lx
Qxy
Mxz
Mxz
Qxy
Mxz=wL2
2
wL
wL2
2
Distributed Load w per unit length
wL
2wLM
2
xz
xLwQxy
wx
wx 0Qxy
xwL 02xwx
Bending of BeamsMECHENG242 Mechanics of Materials
;0x@
2wL
2wxwLxM
22
xz xLwQxy
-ve
-wL2
2
x
Mxz
0BMD: L
2wLM
2
xz
;Lx@ 0Mxz
;2Lx@ 8
wLM2
xz
Bending of BeamsMECHENG242 Mechanics of Materials
Summary – Is anything Necessary for RevisionGenerating Bending Moment Diagrams is a key skill you must revise. From these we will determine:
• Stress Distributions within beams,
• and the resulting Deflections
Apart from the revision problems on Sheet 4, you might try these sources:
• B, C & A Worked Examples, pg 126-132 Problems, 6.1 to 6.8, pg 173
• Jason Ingham’s problem sheets: www.engineering.auckland.ac.nz/mechanical/EngGen121