Bending of Beams MECHENG242 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 2 .0 Bending of Beams 2 . 4 Deflections in Beams 2 . 5 Buckling (Refer: B,C & A –Sec’s 7.1-7.4) (Refer: B,C & A –Sec’s 10.1, 10.2) x x M xz M xz x P P 1 P 2 ☻
2.2 Stresses in Beams Mxz Mxz 2.2.1 The Engineering Beam Theory (Refer: B, C & A–Sec 6.3, 6.4, 6.5, 6.6) 2.2.1 The Engineering Beam Theory Compression z y Mxz C D x y y’ No Stress NA Neutral Axis A B dx Tension dq R sx=0 on the Neutral Axis. In general we must find the position of the Neutral Axis. C’ D’ y’ A’ B’
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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
2.0 Bending of Beams
2.4 Deflections in Beams
2.5 Buckling
(Refer: B,C & A –Sec’s 7.1-7.4)
(Refer: B,C & A –Sec’s 10.1, 10.2)
x
x
Mxz Mxz
x
P
P1
P2
☻
Bending of BeamsMECHENG242 Mechanics of Materials
Mxz Mxz
2.2 Stresses in Beams(Refer: B, C & A–Sec 6.3, 6.4, 6.5, 6.6)2.2.1 The Engineering Beam Theory
x
yMxz Mxz
A B
C D
Compression
Tension
No StressNA
Neutral Axis
z
y
y’ y’
A’ B’
C’ D’ y’
R
x
x=0 on the Neutral Axis. In general we must find the position of the Neutral Axis.
Bending of BeamsMECHENG242 Mechanics of Materials
Mxz Mxz
A B
C Dy’
A’ B’
C’ D’ y’
R
Mxz Mxz
Assumptions Rx'B'AAB
Plane surfaces remain plane
Beam material is elastic0zy 0x and only
Bending of BeamsMECHENG242 Mechanics of Materials
1
Geometry of Deformation:
Rx'B'AABCD0
x LL
'yR'D'C
R
R'yRx
R'y
Hookes Law: zyxx E1
0zy
Ex
x and 'y
RE
x
CDCD'D'C
Bending of BeamsMECHENG242 Mechanics of Materials
'yRE
x
1
x
y
y’
NA
Neutral Axis 0+ve-ve
Linear Distribution of x
(Eqn )1
x
Note:E is a Material Property
is Curvature
R1
x
x
y
Mxz Mxz
x
Bending of BeamsMECHENG242 Mechanics of Materials
z
y
x
Equilibrium:
x
z
y
y’
Ax
Mxz
Area, A
Let xxx FA
But 0Fx 0dA
A x
0dA'yRE
A
AdA'y First Moment of Area
,0dA'yIfA
Then y’ is measured from the centroidal axis of the beam cross-section.
“Neutral Axis” coincides with the XZ plane through the centroid.
y’y’
NA
Neutral Axis
Centroid
Bending of BeamsMECHENG242 Mechanics of Materials
2
Equilibrium:0M'yF xzxx 0Mz
A x dA'y
xzA2 MdA'y
RE
as 'y
RE
x
1
Let A
2Z dA'yI =The 2nd Moment of Area about Z-axis
xzz MIRE
R
EI
Mz
xz
THE SIMPLE BEAM THEORY:
1 2&RE
'yIM x
z
xz
xz
y
y’
Ax
Mxz
Area, AA xzM xzM
Bending of BeamsMECHENG242 Mechanics of Materials
RE
'yIM x
z
xz
E'yx
zIxzM
R
- Applied Bending Moment - Property of Cross-Sectional Area - Stress due to Mxz - Distance from the Neutral Axis - Young’s Modulus of Beam Material - Radius of Curvature due to Mxz
- N.m - m4 - N/m2 or Pa- m - N/m2 or Pa- m
z
y
y’y’NA
Neutral Axis
xo
'yI
Mz
xzx
Bending of BeamsMECHENG242 Mechanics of Materials
z
y
o
(Refer: B, C & A–Appendix A, p598-601)2.2.2 Properties of Area
y’
A
x
Mxz
x
z
y
o
RE
'yIM x
z
xz
y’ is measured from the Centroidal or Neutral Axis, z. Iz is the 2nd Moment of Area about the Centroidal or Neutral Axis, z.
Position of Centroidal or Neutral Axis:
y’
Centroidal Axis
zoy’
Area, Ay
n
,0dA'y.e.iA
(Definition)
y A
dA'yyA
A
dA'yA1y
AA
Bending of BeamsMECHENG242 Mechanics of Materials
y
n
Example:
zo
Centroidal Axis
A
dA'yA1y
200
10
20
120
(Dimensions in mm)
y mm6.89
20120102001y
000,144000,250400,41y
400,4000,394
mm55.89
m106.89 3
12510200 6020120
125
60
Bending of BeamsMECHENG242 Mechanics of Materials
Example:
z
y
o
2nd Moment of Area:
2
d
2d
2z yb'yI
Definition:
A
2Z dA'yI
A
2y dAzI,Also
z’
y’
y’y2
d
2d
2b
2b
2d
2d
3
3yb
12bd3
12dbI,Also
3
y
A
o
y
z
Bending of BeamsMECHENG242 Mechanics of Materials
The Parallel Axis Theorem:
z
y
o
d
0
2n yb'yI
Definition:2
zn yAII
Example:
y’y2
d
2d
2b
2b
d
0
3
3yb
3bd3
12bdI
3
z
n y
ny
2nz yAII
23
2dbd
3bd
z
o
y
Bending of BeamsMECHENG242 Mechanics of Materials
Example: (Dimensions in mm)
z
y
o
20010
20
120
89.6
30.4
89.6
20
2030.4
20010
1
23
3bdI
3
1,z
36.8920 3
46 mm1079.4
3bdI
3
2,z
34.3020 3
46 mm1019.0
23
3,z yA12bdI 2
3
4.351020012
10200 46 mm1028.3
2zn yAII
• What is Iz?• What is maximum x?
35.4
Bending of BeamsMECHENG242 Mechanics of Materials
Example: (Dimensions in mm)
z
y
o
20010
20
120
89.6
30.4
89.6
20
2030.4
20010
35.4
1
23
3,z2,z1,zz IIII
46z mm1026.8I 46 m1026.8
2zn yAII
• What is Iz?• What is maximum x?
Bending of BeamsMECHENG242 Mechanics of Materials
y
NA x
'yI
Mz
xzx
Maximum Stress:
89.6
40.4 Mxz
Maxz
xzMax,x y
IM
36
xzMax,x 106.89
1026.8M
(N/m2 or Pa)
Bending of BeamsMECHENG242 Mechanics of Materials
Example:
The Perpendicular Axis Theorem:
z
y
o
222 'z'yR
32d4
Az’
y’ R A
2
A
2
A
2 dA'zdA'ydAR
yzx IIJ
The Polar 2nd Moment of Area (About the X-axis)
R
R
A
2x dARJ
2d
0
2 dRR2R
From Symmetry, yz II
yyzx I2IIJ
2JI x
y
o
y
z
64d4
Bending of BeamsMECHENG242 Mechanics of Materials
2.2.3 SummaryThe Engineering Beam Theory determines the axial stress distribution generated across the section of a beam. It is applicable to long, slender load carrying devices.
RE
'yIM x
z
xz
Calculating properties of beam cross sections is a necessary part of the analysis.
• Neutral Axis Position, y
• 2nd Moments of Area, Iy, Iz, Jx
Properties of Areas are discussed in Appendix A of B, C & A.