This is an example on how to obtain values of normal stress in a given cross section of a member subjected to eccentric axial loading.

1. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Block A.
How to obtain normal stress due to eccentric axial loading
Step by step example.

2. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
1. Find and mark on the cross section the point where te
eccentric axial load has been applied (cm).
2. Obtain the point where the neutral line interscts the Y
axis. Represent the neutral line on the figure (cm).
3. Normal stress on the centroid is 25 N/mm2. Find the
value of the axial load (kN).
4. Find the value and the point where the maximum positive
normal stress will appear (kN/cm2).
5. Find the value and the point where the maximum
negative normal stress will appear (kN/cm2).
6. The member is fix supported to the ground and free at
the other end. Represent on the 3D schematic the
neutral line, the area of the cross section subjected to
tension, the one subjected to compression, the expected
deformation and the point where the maximum normal
stress as an absolute value appears.
 The cross section in the figure is subjected to an eccentric axial load which has been applied on line a-a’. Point
A belongs to the neutral line. Answer the following:

3. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
1. Find and mark on the cross section the point where the eccentric axial load
has been applied (cm).
We start at the general case of normal stress due to vending (acording to the sign
convention used througout this course). The cordinates of the point where the axial
force has been applied can be obtained after manipulating this formula:
yz
x
z y
M zM yN
A I I


  



2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm

4. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
1. Find and mark on the cross section the point where the eccentric axial load
has been applied (cm).
We start at the general case of normal stress due to vending (acording to the sign
convention used througout this course). The cordinates of the point where the axial
force has been applied can be obtained after manipulating this formula:
If we substitute in the previous formula Mz and My with the moments that a point
load applied eccentrically would produce and use “P” as the value of the load that
has been applied, the formula would be this:
yz
x
z y
M zM yN
A I I


  
y z
x
z y
P e y P e zP
A I I

   
  


2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm

5. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
1. Find and mark on the cross section the point where the eccentric axial load
has been applied (cm).
We start at the general case of normal stress due to vending (acording to the sign
convention used througout this course). The cordinates of the point where the axial
force has been applied can be obtained after manipulating this formula:
If we substitute in the previous formula Mz and My with the moments that a point
load applied eccentrically would produce and use “P” as the value of the load that
has been applied, the formula would be this:
And obtaining P as a common factor:
yz
x
z y
M zM yN
A I I


  
y z
x
z y
P e y P e zP
A I I

   
  


2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm
1 y z
x
z y
e y e z
P
A I I

  
    
 

6. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
ey and ez are the coordinates of the point where load P has been applied. The
subindexes indicate the axis related to each of these eccentricities. The following
schematic shows an example:



2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm

7. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
ey and ez are the coordinates of the point where load P has been applied. The
subindexes indicate the axis related to each of these eccentricities. The following
schematic shows an example:
We’ve been told that normal stress at point A is zero (because it belongs to the
neutral axis). Substituting everything we know so far in the formula:
(0, 10)
0 ( 10)1
0
116 24318,67 8198,67
y z
A
e e
P 
  
    
 



2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm

8. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
If P where zero there wouldn’t be any problem to be solved, therefore the value to
be found verifies this:
    
      
 
0 ( 10)1 8198,67
0 7.067
116 24318,67 8198,67 1160
y z
z
e e
e cm



2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm

9. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
If P where zero there wouldn’t be any problem to be solved, therefore the value to
be found verifies this:
We also know that P is on line a-a’, therefore the ey value can be obtained with the
geometrical relationship between them:
    
      
 
0 ( 10)1 8198,67
0 7.067
116 24318,67 8198,67 1160
y z
z
e e
e cm
1
; 14,13
2
z y ye e e cm 



2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm

10. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
If P where zero there wouldn’t be any problem to be solved, therefore the value to
be found verifies this:
We also know that P is on line a-a’, therefore the ey value can be obtained with the
geometrical relationship between them:
This is the point where the load has been applied:
    
      
 
0 ( 10)1 8198,67
0 7.067
116 24318,67 8198,67 1160
y z
z
e e
e cm
1
; 14,13
2
z y ye e e cm 



2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm

11. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
2. Obtain the point where the neutral line intersects the Y axis.
Represent the neutral line on the figure (cm).
Once the eccentricities have been obtained, we just have to substitute the “z”
coordinate with zero in the formula (because we are looking for a point that
belongs to the neutral line). The only unknown left will be the “y” coordinate we are
looking for.



2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm

12. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
2. Obtain the point where the neutral line intersects the Y axis.
Represent the neutral line on the figure (cm).
Once the eccentricities have been obtained, we just have to substitute the “z”
coordinate with zero in the formula (because we are looking for a point that
belongs to the neutral line). The only unknown left will be the “y” coordinate we are
looking for.



2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm
( ,0)
1 14,3 7,067 (0)
0 ; 14.86
116 24318,67 8198,67
EN y
y
P y cm
  
      
 

13. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
2. Obtain the point where the neutral line intersects the Y axis.
Represent the neutral line on the figure (cm).
Once the eccentricities have been obtained, we just have to substitute the “z”
coordinate with zero in the formula (because we are looking for a point that
belongs to the neutral line). The only unknown left will be the “y” coordinate we are
looking for.



2
4
4
116
24318,67
8198,67
z
y
A cm
I cm
I cm
( ,0)
1 14,3 7,067 (0)
0 ; 14.86
116 24318,67 8198,67
EN y
y
P y cm
  
      
 
Notice that the position of the neutral line doesn’t depend on the value
nor the sign of the axial load, just .

14. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
3. Normal stress on the centroid is 25 N/mm2. Find the value of the axial
load (kN).
In the general formula, let’s substitute all known values: eccentricity, coordinates of
the centroid (where normal stress is 25 N/mm2) and the normal stress itself and I’ll
be able to obtain “P”.
1 y z
x
z y
e y e z
P
A I I

  
    
 

15. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
3. Normal stress on the centroid is 25 N/mm2. Find the value of the axial
load (kN).
In the general formula, let’s substitute all known values: eccentricity, coordinates of
the centroid (where normal stress is 25 N/mm2) and the normal stress itself and I’ll
be able to obtain “P”.
1 y z
x
z y
e y e z
P
A I I

  
    
 

  
    
 
    
2(0,0) 2 4 4
3
1 14,3 0 7,067 0
2500
116 24318,67 8198,67
290 10 290
G
cm cmN P
cm cm cm cm
P N kN

16. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
3. Normal stress on the centroid is 25 N/mm2. Find the value of the axial
load (kN).
In the general formula, let’s substitute all known values: eccentricity, coordinates of
the centroid (where normal stress is 25 N/mm2) and the normal stress itself and I’ll
be able to obtain “P”.
1 y z
x
z y
e y e z
P
A I I

  
    
 

  
    
 
    
2(0,0) 2 4 4
3
1 14,3 0 7,067 0
2500
116 24318,67 8198,67
290 10 290
G
cm cmN P
cm cm cm cm
P N kN
4. Find the value and the point where the maximum positive normal stress will appear (kN/cm2).
The maximum positive normal stress (tensile) will take place in the furthest point from the neutral line in, the tension zone of
the cross section (shaded in blue) thus coordinates (+20,+10)

17. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
3. Normal stress on the centroid is 25 N/mm2. Find the value of the axial
load (kN).
In the general formula, let’s substitute all known values: eccentricity, coordinates of
the centroid (where normal stress is 25 N/mm2) and the normal stress itself and I’ll
be able to obtain “P”.
1 y z
x
z y
e y e z
P
A I I

  
    
 

  
    
 
    
2(0,0) 2 4 4
3
1 14,3 0 7,067 0
2500
116 24318,67 8198,67
290 10 290
G
cm cmN P
cm cm cm cm
P N kN
4. Find the value and the point where the maximum positive normal stress will appear (kN/cm2).
The maximum positive normal stress (tensile) will take place in the furthest point from the neutral line in, the tension zone of
the cross section (shaded in blue) thus coordinates (+20,+10)

    
    
 
2
(20,10) 2 4 4
1 14,3 ( 20 ) 7,067 ( 10 )
290 8,360 /
116 24318,67 8198,67
G
cm cm cm cm
kN kN cm
cm cm cm

18. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
5, Normal stress on the centroid is 25 N/mm2. Find the value of the axial
load (kN).
The maximum negative normal stress (compression) will take place in the furthest
point from the neutral line in, the compressed zone of the cross section (shaded in
red) thus coordinates (-20,-10)


 
 
    
   
 
 
( 20, 10) 2 4 4
2
( 20, 10)
1 14,3 ( 20 ) 7,067 ( 10 )
290
116 24318,67 8198,67
3,365 /
G
G
cm cm cm cm
kN
cm cm cm
kN cm

19. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
6. The member is fix supported to the ground and free at the other end. Represent on the 3D schematic the neutral line,
the area of the cross section subjected to tension, the one subjected to compression, the expected deformation and the
point where the maximum normal stress as an absolute value appears.

20. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
6. The member is fix supported to the ground and free at the other end. Represent on the 3D schematic the neutral line,
the area of the cross section subjected to tension, the one subjected to compression, the expected deformation and the
point where the maximum normal stress as an absolute value appears.

21. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Normal stress in a member subjected to an eccentric axial load
The edge with the
greatest normal
stress as an
absolute value
6. The member is fix supported to the ground and free at the other end. Represent on the 3D schematic the neutral line,
the area of the cross section subjected to tension, the one subjected to compression, the expected deformation and the
point where the maximum normal stress as an absolute value appears.

22. Structural Systems – Academic Year 2016/2017 Instructor: Maribel Castilla Heredia @maribelcastilla
Block A.
How to obtain normal stress due to eccentric axial loading
Step by step example.