column theory

1. Sub- Solid Mechanics
Eccentrically Loaded Columns
Lect. 5 Direct & 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. Axially and Eccentrically loaded columns
Axially loaded column Eccentrically loaded column

3. Direct and bending stresses for eccentrically loaded short
column
Eccentrically loaded column
When eccentric load acts on column, it causes the bending
moment which produces tension on one side and
compression on opposite side
e.g. Load acts on Dams, chimneys, etc.
Eccentric load: A load whose line of action does not
coincide with the axis of the column called as Eccentric Load.
Or The load which does not pass through the axis of the
column called as Eccentric Load.
Eccentricity: It is the distance between axis of the column
and line of action of the load / force.

4. Direct and Bending Stress
0
0
0
,
,
,
,
,
,
,
,
,
Eccentric load produces the
Direct stress as well as
bending stres
=
s.
= b
b
P Eccentric Load on Column
e Eccentricity
A bxd Area of Column
b Width of Column
d Depth of Column
P M
y
A I
P M
y
A I
where
Direct Stress
  
 




 



 


b Bending Stress
 

5. Direct and Bending Stress
3
3
, , / 2 &
12
.
2
max 0 b min 0 b
max min
max min
2 2
max min
σ =σ + σ , σ =σ - σ ,
P M P M
σ = + y, σ = - y
A I A I
P 6Pe P 6Pe
σ = + y, σ = - y,
A db A d
Maximum Stress: Minimum Stress:
In this c
b
P 6e P 6e
σ = 1+ ,
as
σ =
e
1-
A b A b
P
A
yy
b
db
M P e y b I
M P e b
y
db
I

   
   
   
  
 
    
 
.
6e
b

6. Stress variation with Direct and Bending Stress
(3 Conditions)

7. Eccentricity about x-x and y-y axis
0 bx by
0 bx by
y
x
x y
xx yy
x y
x y y x
y x
x
x
y
x yy
σ =σ ± σ ± σ ,
σ =σ + σ + σ ,
M
M
P
σ = + y + y ,
A I I
d b
y = , y = ,
2 2
M =P e , M =P e
Max & Min. Stresses Case-I
Case-II ,e =0, P on x axis,
Eccentricity about y-y axi
,
P e P e
P d b
σ = + . + . ,
A I 2 I 2
P e
P
s
b
σ = + . ,

8. C
P e
P
σ = + . ,
ase-III ,e =0, P on axis
A I 2
y
xx
x
d
y 
Case III- Eccentricity about x-x axis

9. Core of Section
(Kern or Kernel of Section)
It is the portion or area of the cross section in
which if load is acts on this area, tension will
not develop anywhere and stresses produced
are totally compressive throughout the
section.

10. Core of Section for Rectangular Section
Using No tension Condition  
0 b
 

0 ,
3
. ,
.
. ,
2
12 6
,
6
b
xx
x
xx
x
P M
y
A I
P e
P D
BxD
BxD
D
e
 
  
 


11. Core of Section for Rectangular Section
,
. . ,
6
6
xx
Eccentricity for
Rectangular Setion
D
e =
6
xx
yy
D
i e e
B
e
 
 

12. Core of Section for Circular Section
Using No tension Condition
 
0 b
 

2 4
. ,
.
. ,
2
.
8
4 64
,
8
. xx yy
P M
y
D
e
D
i e e
A I
P e
P D
D D
e
 
 
 

 

13. Core of Section for Hollow Circular Section
Using No tension Condition
 
0 b
 

   
2 2 4 4
2 2
. ,
.
. ,
4 4
,
2
8
6
P M
y
A I
P e
P D
D d D d
D d
e
D
 

 
 
 


14. Problem.1 Find the Core of section for rectangular section of
Size 300 x 200mm. Show it with sketch. Explain no tension
condition.
200
33.33 ,
6 6
300
50 ,
6 6
:
xx
yy
D
e m
Sol
m
B
e
u
mm
tion
  
  

15. No tension condition
0
i.e Direct and bending stresses are same Then stresses
prduced at the base will be throughout compressive and
no tension will deveoped in bases.
b
When  

Stress Dia. For
No tension condition.

16. Problem.1 Find the Core of section for circular section of
size250 diameter. Show it with sketch.
250
31.33 ,
8 8
:
xx yy
Solution
For Circular Sectio
e
n
D
e mm
   

17. Next lecture numerical on
Eccentrically loaded columns
Thank you