Column THEORY

1. Sub- Solid Mechanics
Eccentrically Loaded Columns
Lect. 8 Direct and Bending Stresses
Sanjivani Rural Education Society’s
Sanjivani College of Engineering, Kopargaon-423603
An Autonomous Institute, Affiliated to Savitribai Phule Pune University, Pune
ISO 9001:2015 Certified, Approved by AICTE, Accredited by NAAC (A Grade) & NBA
Department of Civil Engineering
Prepared by:
Dr. Ghumare S. M.
Asso Professor, Civil Engg. Department

2. Axial, Uniaxial and Biaxial Bending
0
0
, , ,
1.
2
:
,
.
0 b
(Uni-axial Bending)
(Bi-axial Bending)
σ = σ ± σ
When Eccentricity is about any one axis (x-x & y-y)
When Eccentricity is about both axis (x- & y-y
,
x )
y
x
bx by
xx yy
bx
M
M
P
and y or y
A I I
Total Stress
P
and
A
  
 
  
  , ,
0 bx by
σ = σ ± σ ± σ
y
x
by
xx yy
M
M
y or y
I I
Total Stress
 

3. 0
, ,
, ,
,
0 bx by
When Eccentricity is
about both axis(x-x
Bi-axial B
Total Stress σ = σ ± σ ± σ
& y
en
-
g
y
d n
)
i
x
bx
xx
y
by
yy
P
Direct Stress
A
M
Bending Stress y
I
M
Bending Stress y
I







4. Prob.1 A masonry pillar of size 500mmx 400mm is subjected to a
eccentric load of 500KN as shown in Fig. Find the stresses at all the
corners. The load has eccentricity 25mm about x-x axis and 30mm
about y-y axis.
Given: Rect. masonry pillar = 500mmx 400mm, P= 500KN
3 3
4
3 3
4
2
500 400
,
12 12
,
400 500
,
12 12
400 500 ,
30 , 2
, , ,
5 ,
Find Corner Stresses
xx
xx
yy
y
B
y
A D
C
BD X
I
I mm
DB X
I
I mm
A x mm
e mm e mm
   
 

 

 
 
Given
Eccentricity
@ x-x axis,ey= 25mm
and
@y-y axis, ex= 30mm

5. 3
3
3
2
0
2
500 10 25 .
500 10 30 .
500 10
,
@ ,
@
/ ,
(500 400
,
,
)
, /
,
To find Direct & Bending stresses
x y
y x
x
bx
xx
y
by
yy
Moment x x A
M P e x x N mm
M P e x x N mm
P x
Direct Stress N mm
A x
M
Bending Stress y N
xis
Mome
mm
I
M
n
B
t y y Ax
ending Stress y
I
is



  
  
  
 



 2
/
N mm

Problem1 continue…..

6. 0
0
0
0
0
,
,
( )
,
,
To find corner stresses
Us
:
ing bx by
A bx by
B bx by
C bx by
D bx by
At each corner of Colu
mpa
mpa
mp
n
a
p
m
m a
   
   
   
   
   
  
   
   
   
   
Problem 1 continue…..

7. Prob.2 A Column 400x350mm is subjected to a eccentric load of
50KN as shown in Fig. Find the maximum and minimum stress
intensities induced at each corner of the column. Take eccentricity
about both the axis as ex=ey=75mm.
Given: Column = 400 x 350mm, P= 50KN, ex=ey=75mm.
3 3
9 4
3 3
9 4
2
400 350
,
12 12
14.4 10 ,
350 400
,
12 12
25.6 10
400 350 480000 ,
7
, ,
5 7 ,
,
, 5
Find Corner Stresses
xx
xx
yy
y
A D
y
y
B
x
C
BD X
I
I X mm
DB X
I
I X mm
A x mm
e mm e mm
   
 

 

 
 
Given
Eccentricity:
ex= ey= 75mm

8. 3
3
3
2
0
2
50 10 75 .
50 10 75 .
50 10
, / ,
@ ,
@ ,
(400 350)
, / ,
, /
To find Direct & Bending stresses
x y
y x
x
bx
xx
y
by
yy
M P e x x N mm
M P e x x N mm
P x
Direct Stress N mm
A x
M
Bending Stres
Moment x x Ax
s y N mm
I
M
Bending Stress y N m
I
is
Moment y y Axis



  
  
  

 



2
m
Problem 2 continue…..

9. 0
0
0
0
0
,
,
( )
,
,
To find corner stresses
Us
:
ing bx by
A bx by
B bx by
C bx by
D bx by
At each corner of Colu
mpa
mpa
mp
n
a
p
m
m a
   
   
   
   
   
  
   
   
   
   
Problem 2 continue…..

10. Prob.3 A masonry pillar of size 5m x 3m supports a vertical
load of 750KN as shown in Fig. Find the stresses at all the
corner of the pier. Eccentricity ex= 1.1m, ey= 1 m
Given: Masonry pillar = 5m x 3m, P= 600KN, ex= 1.1m, ey= 1m

11. Thank you