Transcript Document

2.0 Bending of Beams
☻2.1
☻2.2
Revision – Bending Moments
Stresses in Beams
sx
sx
P
x
Mxz
2.3 Combined Bending and Axial Loading
P1
Mxz
(Refer: B,C & A –
Sec’s 6.11, 6.12)
P2
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)
MECHENG242 Mechanics of Materials
Bending of Beams
2.3 Combined Bending and Axial Loading
(Refer: B, C & A–Sec 6.11, 6.12)
2.3.1 Superposition
How does axial stress distribution look?
P1
L
P1.L
Mxz
P2
y
x
x
P1
z
P2
x
P2
A
Fxx
Qxy
Fxx=P2
Fxx
y
+ve
0
Normal Force
Diagram
P1
y
i) Axial Loading
s x ,Axial
MECHENG242 Mechanics of Materials
x
Fxx P2


A
A
x
s x,Axial
Bending of Beams
ii) Bending
P1
L
y
x
Mxz
P2
P1.L
P2
Fxx
x
z
P1
Bending Mxz
0
Moment
Diagram
P1
y
x
P2
A
x
-ve
-P1.L
s x ,Bending
Mxz
Px  L 

 y'  
 y'
Iz
Iz
y
NA
x
s x,Bending
MECHENG242 Mechanics of Materials
Bending of Beams
Using SUPERPOSITION:
s x,Total  s x,Axial  s x,Bending
 sx 
Note:
P2 P1 x  L

 y'
A
Iz
s x  s x,Max
 s x ,Max
z
x
at x=0 and y’=+yMax
P2 P1L


 yMax
A
Iz
y
P1
y
A
(at the fixed end)
P2
s x,Max
y
+
NA
x
=
NA
P
sx  2
A
P1x  L
sx  
 y'
Iz
MECHENG242 Mechanics of Materials
P2 P1x  L
sx 

 y'
A
Iz
Bending of Beams
2.3.2 Eccentric Axial Loading
y
y
y
d
P
C
L
x
z
o
e
z
o
P
P
x
b
P

sx 
d
P
y
P Mxz

 y'
A
Iz
MECHENG242 Mechanics of Materials
2
b
2
2
2
Mxz=Pe
P x
C
L
 sx 
Pe   y'
P

bd bd3
12


Bending of Beams

P
Pe 
sx 

 y'
3
bd bd
12


y
y
x
C
L
y
NA
C
L
MECHENG242 Mechanics of Materials
x
C
L
x
Bending of Beams