### Buckling of Strut Lab Report

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 .
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

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

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 (m4) 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