Plastic Failure Mechanisms When X Bar has Area A or 2A

[Dell]

in the following question i am asked to find all the possible mechanisms, in the given state, and then when the "X" bar has an area of A and not 2A

the answers are:

when area of X=2A
i) plastic hinge at D
ii) yielding of Y and X bars
iii) plastic hinge at C, yielding of "Y" bar




when area of X=A
i) plastic hinge at D
ii) yielding of Y and X bars

i understand why these are correct, but why can i not have a situation where "X" yields and Y does not?

also in the second case, where area=A, why is the 3rd option not a possibility??

 

[Mapes]

in the following question i am asked to find all the possible mechanisms, in the given state, and then when the "X" bar has an area of A and not 2A
View attachment 25629

the answers are:

when area of X=2A
i) plastic hinge at D
ii) yielding of Y and X bars
iii) plastic hinge at C, yielding of "Y" bar




when area of X=A
i) plastic hinge at D
ii) yielding of Y and X bars

i understand why these are correct, but why can i not have a situation where "X" yields and Y does not?

also in the second case, where area=A, why is the 3rd option not a possibility??

Please show your attempt, and then you'll likely get feedback. (1) Can you draw a schematic where X yields and Y does not? (2) Note that failure of the plastic hinges is a function of stress in the bar (X or Y).

 

[Dell]

i don't really have any calculations or anything like that, just looking at the structure and seeing what will happen if certain points fail, could a plastic hinge not develop at C and then X yield?? or on the other hand X yields and then a hinge at D,??

2)i realize that the [plastic hinge at C is a function of the stress, but why can we never have a situation where there is a plastic hinge at C if the 2 bars have the same cross sectional area??

i know that:

X/3 + 2Y/3 = P
-=-->X=3P-2Y
assuming a plastic hinge at C
P=Y/2+2/3*Mp/L

why can Y not be sigma(y)*A [Xp]

meaning P=Sigma(y)/2+2/3*Mp/L

making X=3*Sigma(y)/2+2*Mp/L-2*Sigma(y)*A
X=2Mp/L-Sigma*A/2

i don't see why this cannot be in the case where the area is A but can be when the area is 2A

 

[Mapes]

i don't really have any calculations or anything like that, just looking at the structure and seeing what will happen if certain points fail, could a plastic hinge not develop at C and then X yield?? or on the other hand X yields and then a hinge at D,??

What does "could a plastic hinge not develop" mean? On the second possibility: how could X yield before Y yielding or before D failing?

2)i realize that the [plastic hinge at C is a function of the stress, but why can we never have a situation where there is a plastic hinge at C if the 2 bars have the same cross sectional area??

I'm not following you here. There is always a plastic hinge at C; what do you mean be "why can we never have a situation where there is a plastic hinge at C"?

 

[Dell]

I'm not following you here. There is always a plastic hinge at C

why will we always have a plastic hinge at C ?? what if Mp>>yield stress? then the bars X,Y will yield long before point C.

in the answer sheet it says that the 3rd option( plastic hinge at C, yielding of "Y" bar) is impossible for a case where X,Y have the same cross sectional area, i cannot understand why this is.

 

[pongo38]

"i know that:
X/3 + 2Y/3 = P
-=-->X=3P-2Y"
comment 1: X =3P-2P applies with the plastic hinge at where? (and only there).

"assuming a plastic hinge at C
P=Y/2+2/3*Mp/L"
comment 2: I would like to see a free body diagram accompanying your development of this equation, as it could be wrong.