Secant Equation for Buckling and Yielding in Eccentrically Loaded Structures

[fonseh]

Homework Statement

In the following example , the secant equation circle part) is used to find the max stress when the whole thing is subjected to force P before buckling , am i right ?
Why the author wrote it as the yielding equation ? And why the author use 2 equation for yy axis ?

Homework Equations

The Attempt at a Solution

IMO , there green circle part is worn g, and the equation not exist at all , because the P is loaded eccentrically from yy axis

 

[PhanthomJay]

Homework Statement

In the following example , the secant equation circle part) is used to find the max stress when the whole thing is subjected to force P before buckling , am i right ?
Why the author wrote it as the yielding equation ? And why the author use 2 equation for yy axis ?

Homework Equations

The Attempt at a Solution

IMO , there green circle part is worn g, and the equation not exist at all , because the P is loaded eccentrically from yy axis

The author appears to have made an error when calculating the Euler buckling load for the major y y-axis by using K = 2 when K should be 1, but anyway, weak axis buckling about x x-axis controls for Euler buckling. But about the major y y axis, the loading is eccentrically applied, which means that there is also bending stress about that axis , and the secant formula is used to be sure that under bending and axial stresses the column will not fail by yielding under the combined bending and axial stresses about the yy axis before it buckled about the xx axis . It doesn't , so weak axis bucking about xx axis controls design.

 

[fonseh]

which means that there is also bending stress about that axis , and the secant formula is used to be sure that under bending and axial stresses the column will not fail by yielding under the combined bending and axial stresses about the yy axis before it buckled about the xx axis .

i still don't understand , what is the usage of secant formula , can you explain further ?

 

[fonseh]

secant formula is used to be sure that under bending and axial stresses the column will not fail by yielding under the combined bending and axial stresses about the yy axis before it buckled about the xx axis

So , what we get from the secant equation is yielding stress , it's not buckling stress ?

 

[fonseh]

The author appears to have made an error when calculating the Euler buckling load for the major y y-axis by using K = 2 when K should be 1, but anyway, weak axis buckling about x x-axis controls for Euler buckling. But about the major y y axis, the loading is eccentrically applied, which means that there is also bending stress about that axis , and the secant formula is used to be sure that under bending and axial stresses the column will not fail by yielding under the combined bending and axial stresses about the yy axis before it buckled about the xx axis . It doesn't , so weak axis bucking about xx axis controls design.

Do you mean we use both the euler equation and secant equation to find out the max load first , then we take the smallest Pmax of them as the critical load . The larger Pmax is considered as yielding stress ?

 

[fonseh]

secant formula is used to be sure that under bending and axial stresses the column will not fail by yielding under the combined bending and axial stresses about the yy axis before it buckled about the xx axis

http://www.continuummechanics.org/eccentriccolumnbuckling.html

in here , i was told that the for eccentrical loaded column , it will break by yielding first before buckling ...

 

[PhanthomJay]

http://www.continuummechanics.org/eccentriccolumnbuckling.html

in here , i was told that the for eccentrical loaded column , it will break by yielding first before buckling ...

the column will fail by buckling first about the weak xx axis before it ffails by yielding about the major yy axis, because the critical load is less about xx than it is about yy. You should note that there is no eccentricity about the xx axis. So secant formula does not apply about that axis. Understood? This is complex stuff. Get back to the basics.

 

[fonseh]

the column will fail by buckling first about the weak xx axis before it ffails by yielding about the major yy axis, because the critical load is less about xx than it is about yy. You should note that there is no eccentricity about the xx axis. So secant formula does not apply about that axis. Understood? This is complex stuff. Get back to the basics.

So , secant formula only apply for yy axis ?
Secant formula is used to find the yielding stress about yy axis as stated my the author ?

 

[fonseh]

the column will fail by buckling first about the weak xx axis before it ffails by yielding about the major yy axis, because the critical load is less about xx than it is about yy. You should note that there is no eccentricity about the xx axis. So secant formula does not apply about that axis. Understood? This is complex stuff. Get back to the basics.

why for the yy axis , it's fail by yielding , not fail by buckling ?

 

[fonseh]

so , when the load is applied eccentricaly , the structure will fail by yielding ?

 

[PhanthomJay]

So , secant formula only apply for yy axis ?
Secant formula is used to find the yielding stress about yy axis as stated my the author ?

yes , both correct

why for the yy axis , it's fail by yielding , not fail by buckling ?

because you have bending stresses and axial compression stresses that combine to reach yield stress at a load less than the critical buckling load

so , when the load is applied eccentricaly , the structure will fail by yielding ?

yes, it bends under the eccentric moment and compresses under the axial load, and will yield at a load less than the buckling load about that axis. Failure by Euler buckling applies when the load is thru the centroid.