Buckling deformation of axially loaded beam with P=30kN exceeding critical load

[a_hargy]

Homework Statement

I am trying to determine the deformation of a beam loaded as shown in the attached picture.

where:
P=30kN
L=1m
A=0.000225m^2
E=2.22*10^5MPa
v=0.31
I=4.21875*10^9m^4

Homework Equations

Pcr=pi^2*EI/L
=9.2435kN

where Pcr is the critical load

The Attempt at a Solution

I assume that the deformation of the beam in the vertical direction is not simpily PL/AE as P>Pcr?
I was thinking I could calculate the horizontal deflection of the beam as it buckles and use trig to find the vertical deformation but I cannot find a way to calculate this maximum horizontal deflection. Any ideas?

Thanks in advance.
Adam

 

[nvn]

a_hargy: I am currently doubtful regarding your units. Can you check your units in the given data, and ensure you are listing correct units and values?

Also, always leave a space between a numeric value and its following unit symbol. E.g., 30 kN, not 30kN. See the international standard for writing units[/color] (ISO 31-0[/color]).

Also, are you missing an exponent in your relevant equation? Check your formula. Post corrections.

 

[a_hargy]

nvn: Sorry about that, I typed it in a bit of a rush.

P = 30 kN
L = 1 m
A = 0.000225 m² (square bar 15 mm x 15 mm)
E = 2.22*10⁵ MPa
v = 0.31
I = 4.21875*10^-9 m⁴

Using Euler's formula to find the critical load:
Pcr = (pi²*E*I) / L²
For which my results are 9.2435 kN proving P > Pcr

 

[nvn]

Excellent work, a_hargy.

Unfortunately, you will not be able to compute the exact deformation, because your analysis shows the column buckles. The column might completely collapse. Do you have an exact wording of the given problem statement?

 

[a_hargy]

So there is no way of calculating the deflection of the beam since P > Pcr?

Its actually part of a much larger question. I probably should have mentioned this to begin with but I was fairly confident I could complete the other parts.

The whole question is: For the 2D three bar truss structure shown in figure 3.1 (attached) calculate the following:
a) The unknown displacements, reactions and element stresses of the structure and;
b) The maximum axial tensile and compressive stresses.

 

[nvn]

Did you omit something from the problem statement? Where are the member cross-sectional dimensions given?

You basically cannot compute deflection if P > Pcr. Pcr needs to be much greater than P. If Pcr is much greater than P, then you can use P*L/(E*A).

 

[a_hargy]

Yes sorry the areas of the bars are:
AB = 20 mm x 20 mm
BC = 25 mm x 25 mm
AC = 15 mm x 15 mm

The next question is to analyse the same structure in Strand7 which I have already completed and it shows a displacement at point C. I am confident that I have calculated the deflection of bar AB and BC correctly, I am just stuck on calculating the deflection at point C

 

[nvn]

a_hargy: You are doing well, and your force in member AC currently looks correct. If you pretend member AC does not buckle, you can perhaps use P*L/(E*A). See post 6.