A mechanical equipment support structure is made from three metal members, each 10 mm by 10 mm. The vertical members, CA and DB, are fixed to the ground. The horizontal member, AB, is pinned to the vertical members at A and B. If the equipment is expected to placed 10 cm from the the left, as shown, what is the maximum load F? Assume all columns are solid steel bars with a stiffness, E, of 200 GPa and a allowable stress of 400 MPa.

Basic W8x31 I-Beam

With the slender vertical columns, 10 mm by 10 mm, it would be expect this structure to fail due to column buckling. However, it could also fail due to beam bending or column crushing. These other possible failure modes will be checked after load P is determined for just column buckling.

Both columns, CA and DB, can be considered as fixed-pinned columns that have a critical buckling as

     P_cr = 2.046 π²EI / L²

Moment of inertia for both columns is,

     I = ab³ / 12 = (0.01)(0.01)³ / 12 = 8.333x10^-10 m⁴

The allowable buckling load for each column is

      P_CA = 2.046 π² (200x10⁹) (8.333x10^-10)/0.25²
             = 53.85 kN

     P_BD = 2.046 π² (200x10⁹) (8.333x10^-10)/0.35²
            = 27.47 kN

Now, the critical buckling loads can be used to find F. First, the P_CA will be used to find F, and then P_BD. Taking moments about A gives,

      (10)F = P_BD(30) = 27.47(30)
      P = 82.41 kN

Next, using P_BD and taking moments about B gives,

       (20)F = P_AC(30) = 53.85(30)
       F = 80.78 kN

Thus the largest load F before either column will be buckle when F = 80.78 kN.

Check column normal stress (crushing stress):

     σ = P/A = 80.78/0.01² = 807 MPa

There is a problem. This is higher than the allowable of 400 MPa. It will fail due to crushing before buckling. The maximum column load is,

      P = (400 MPa) (0.01 m) (0.01 m) = 40 kN

This means the largest F (moments about point B) will be

     20(F) = 40(30)

     F = 60 kN

Check beam bending stress:

The largest bending moment due to a load of F = 60 kN (maximum due to column crushing) will be at the location of the load.

      M_max = 40(10) = 400 kN-m

The bending stress will be

     σ = My/I
        = (400 kN-m) (0.005 m) / 8.333x10^-10 m⁴
        = 2,400,000 MPa

There is another serious problem. This is much higher than the allowable of 400 MPa. It will fail due to beam bending before buckling or column crushing. The maximum moment is,

     400 MPa = M (0.005 m) / 8.333x10^-10 m⁴
     M = 66.66 N-m

This moment will be generated a reaction at A of

     P_A = (66.66 N-m)/ (0.1 m) = 666.66 N

Thus, the maximum load F will be

     F (20) = 666.6 (30)

     F = 999.9 N = 1.0 kN

Thus, the structure will fail due to beam bending before column buckling or column crushing. It is important to check all failure modes.