Finding the collapse load P

[Dell]

finding the collapse load "P"

i am taking a course on solid mechanics, and we are learning plastic deformations in transverse loading, in this question i am asked to find an expression for the "collapse load P" as a function of its placement on the beam "x"

i do not remember learning the term "collapse load" but from what i could assume, i would say that this means the the force P is enough to cause a collapse in the beam no matter where it is placed, - P will be a function of X, at X=0 P will be 0, then it will get larger and then smaller again till X=L, P=0 again,

first of all is this a correct assumption??

now to solving the problem:

using castiglianos 2nd theorem i found
(A is the leftmost point)

Ay=0.5*P/L³*[2L³-3Lx²+x³]

MA=0.5*P/L²*x*[2L²-3Lx+x²]

now as far as i can tell the maximum moment, which will be equal to the plastic moment, will either be felt at point A and will be equal to MA, or will be found at point X and will be equal to Ay*x-MA
--how do i know which is the correct one?


P can be minimum when the moment found is maximum.

i found both of these functions' maximums

for MA i get maximum when X=0.4040L
for Ay*x-Ma i get X=0.6364L

neither of these is correct, what am i doing wrong??

 

[KataKoniK]

Dear fellow scientist,

Thank you for your question. Your assumption about the meaning of "collapse load" is correct. It refers to the maximum load that can cause the beam to fail or collapse, regardless of its placement on the beam.

To solve this problem, you will need to use the plastic moment equation, which is given by Mp = σy * Z, where σy is the yield strength of the material and Z is the plastic section modulus. The plastic section modulus is a measure of a beam's resistance to bending and is calculated by integrating the cross-sectional area of the beam.

To determine the collapse load, you will need to find the point on the beam where the plastic moment is equal to the moment at that point. This will give you the maximum load that the beam can withstand without collapsing.

To find this point, you can use the principle of superposition. This means that the total moment at a point is equal to the sum of the moments caused by each individual load acting on the beam. So, you can calculate the moment at a point using the equation you found for Ay and MA, and then add them together to get the total moment.

Once you have the total moment, you can set it equal to the plastic moment equation and solve for P. This will give you the collapse load as a function of x.

I hope this helps and good luck with your course!