Introduction:
In construction applications a column is an element that is used to withstand compressive load. This element is similar to beam but it is used in vertical position and normally horizontal beams are placed on the columns or both ends of beams are rested on two columns on either end of the beam. Columns are usually designed to withstand high compressive loads and they can fail or buckle if the loads are too large and columns are unable to withstand that load. Normally the load at which the column buckles is called critical load and column is typically designed to be used well below this critical load [1].
However, even if the structure is
subjected to small loading well below its critical buckling load, the
continuous application of such loads could eventually fatigue the structure and
build up to buckling failure. Therefore, understanding the buckling
characteristics and designing in safety factors are important. In this
experiment we will see how columns buckles ate different loads when their ends
are fixed or pinned.
Calculations:
The following dimensions of the
column were measured for rectangular column
Width, b = 100 mm
Depth, d = 50 mm
For rectangular shape column the
second moment of area can be calculated as
Second moment of area
I =(bd^3)/12
Putting in values
Second moment of area
I = 0.05*(0.1)^3/12 = 4.16* 〖10〗^(6) m^4
The buckling load can also be calculated using Euler equation where it can be seen that the buckling
load only depends on the cross sectional area, material properties such as Young’s modulus ‘E’ and the way both ends are fixed.
The Euler equation is given by
P =πEI/kL
Where the value of ‘k’ depends how the ends are fixed.
For Pin End Connection
For Pin End Connection
If both ends are pinned then ‘k=1’ will be taken.
P =πEI/kL
P =(3.14*97*〖10〗^9*4.16* 〖10〗^(6))/(1*L)
P =272 N
P =(3.14*97*〖10〗^9*4.16* 〖10〗^(6))/(1*L)
P =272 N
For both fix end connections
P =πEI/kL
P =(3.14*97*〖10〗^9*4.16* 〖10〗^(6))/(0.5*L)
P =1129 N
If one end is pinned and the other end is fixed then ‘k=o.7’ will be taken.
P =πEI/kL
P =(3.14*97*〖10〗^9*4.16* 〖10〗^(6))/(0.7*L)
P =570 N
Discussion:
From the results given in table 2 it can be seen that different columns buckle at different critical loads. The buckling depends on many factors such as the material by which the column is made of and the way by which both ends are fixed or pinned. In this case it is assumed that all columns have similar cross sectional areas and therefore have constant value for second moment of area. The columns made of brass and aluminum will have different values of ‘E’ but same value of second moment of area ‘I’ if the cross sectional area is same. From table 2 readings it is clear that the columns can take larger loads before knuckling when both ends are fixed. Same is true for aluminum and brass columns. The young’s modulus of brass is 97 G Pa while for Aluminum it is 69 MPa. If the second moment of area is same for both columns then the column made of brass should take larger load before buckling given that both ends are fixed for both columns. From table 2 it is clear that the column made of brass buckles at 1129 N load while the column made of Aluminum buckles at 783.5 N when both ends were fixed. From Euler equation it can be seen that the buckling load will be directly proportional to the young’s modulus of the material the column is made of. Therefore brass is more durable and can withstand higher compressive loads.
Conclusions:
For beam design the maximum bending stress was calculated and also keeping in mind the safety factor the design stress was calculated. It was concluded that the material should be used with yield stress of 75 MPa for a safe design of beams. For strut design the aluminum and brass columns were tested and it was seen that the brass column has larger capacity to withstand compressive loads for similar cross sectional area and end fixing. By considering the calculations given in this report more suitable columns can be designed to be used in underground construction for London underground tunnels.
Table 2: Experimental readings for buckling
of struts
Length
(mm)

Material

Second
moment of area (m^{4})

Clamp
condition

Cross
sectional shape

Peak
buckling load 1(N)

Peak
buckling load 2(N)

Avg
buckling load (N)

Deformation
(mm)

650

Steel
700

4.16x10^{6}

Fixed/Fixed


897

896

896 m

14.22

675

Steel 700

Fixed/Pinned

429

430

429.5 m

14.15


700

Steel 700

Pinned/Pinned

177

178

177.5 m

12.82


750

Brass 750

Pinned/Pinned

271

273

272 m

15


725

Brass 750

Pinned/Fixed

570

570

570 m

15.07


700

Brass 750

Fixed/Fixed

1136

1122

1129 m

14.71


750

Aluminium 750

Pinned/Pinned

176

176

176 m

16.53


725

Aluminium 750

Pinned/Fixed

374

373

373.5 m

16.55


700

Aluminium 750

Fixed/Fixed

784

783

783.5 m

16.44

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