Secant Equation for Buckling and Yielding in Eccentrically Loaded Structures

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In summary, the author writes the secant equation to determine the max stress when the whole thing is subjected to force P before buckling. However, the author makes an error when calculating the Euler buckling load for the major y y-axis by using K = 2 when K should be 1. 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. The author uses the secant equation to be sure that under bending and axial stresses the column will not fail by yielding under the combined bending and axial
  • #1
fonseh
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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
 

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  • #2
fonseh said:

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.
 
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  • #3
PhanthomJay said:
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 ?
 
  • #4
PhanthomJay said:
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 ?
 
  • #5
PhanthomJay said:
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 ?
 
  • #6
PhanthomJay said:
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 ...
 
  • #7
fonseh said:
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.
 
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  • #8
PhanthomJay said:
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 ?
 
  • #9
PhanthomJay said:
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 ?
 
  • #10
so , when the load is applied eccentricaly , the structure will fail by yielding ?
 
  • #11
fonseh said:
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
fonseh said:
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
fonseh said:
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.
 
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Related to Secant Equation for Buckling and Yielding in Eccentrically Loaded Structures

1. What is the Secant Equation for Buckling and Yielding in Eccentrically Loaded Structures?

The Secant Equation for Buckling and Yielding in Eccentrically Loaded Structures is a mathematical equation used to determine the critical load at which a structure will buckle or yield due to an eccentric (off-center) load. It takes into account both the strength of the material and the structure's geometry.

2. How is the Secant Equation derived?

The Secant Equation is derived by considering the equilibrium of forces acting on a structure under an eccentric load. It takes into account the bending moment, axial load, and geometric properties of the structure to determine the critical load at which buckling or yielding will occur.

3. When is the Secant Equation typically used?

The Secant Equation is commonly used in structural engineering and design to determine the maximum safe load that a structure can withstand. It is particularly useful for eccentrically loaded structures, where the load is not applied directly through the centroid of the structure.

4. What are the assumptions made in the Secant Equation?

The Secant Equation makes certain assumptions about the behavior of the structure, including that it is made of a homogenous material, has a constant cross-sectional area, and is loaded in the elastic range. It also assumes that the structure will fail by either buckling or yielding, and not by any other failure mode.

5. Are there any limitations to the use of the Secant Equation?

Yes, the Secant Equation has some limitations and may not accurately predict the behavior of structures under certain conditions. It does not account for the effects of imperfections in the structure, such as initial deflections or residual stresses. It also does not consider the effects of dynamic loading or temperature changes, which can affect the structural response.

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