^{1}

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The manuscript reviews the history and quo of the theory of Timoshenko’s method in stability analysis of compressive levers first, taking an example to explain the
m-simulation method and putting forward the 3
^{rd}-7
^{th} boundary conditions demonstrating their superiorities in improving the precision through examples, followed by proposing and applying the join conditions in the stability analysis of combined axial force compressive levers gaining success. Through a brief example showing the effect of some related theories in a simple structural stability analysis, its application prospect is discussed.

As is known, the exact static method in structural stability analysis is difficult to push the popularization for the huge calculation amount etc. making the energy method of approximation including several simulated deformation method ([

The accuracy of the m-simulation method depends on the degree of the designed m-curve approaching to the critical state of the object. The author found in practice that in addition to the 2 BC.s put forward by Timoshenko, some other relative values on certain specific sections in the lever could also be predetermined with qualitative even quantitative; thus, they could be made use in designing the trial function

In order to make the text concise and clear, below agreed to use “A ≥ B” instead of “proposition B would be derived from proposition A” and agreed upon in the formula that “l” to be the length measurement of the lever; “z” to be a variable with no dimension and “x” to be the one with the dimension of length; that is

For the convenience in reading comprehension, it is necessary to review the basic principle of m-simulation method especially for the English readers as it is narrated in Chinese ( [

As is known, the theory of energy method comes from the principle of minimum potential energy, namely

In the presence of prismatic cantilever column at the top of compression, using

Or

And

The formula calculating the critical load of the pressure P on the top of the lever could be driven respectively as≥

Or

Although the more precision result could be calculated with formula (2-A) (

(only the prismatic cantilever column with a pressure on the top), restricted (2-A) of popularization. On the contrary, formula (2-B) has spread around the world such as the literatures of references [

Making the result be determined by the unique function m; thus the y-designing in (2-B) could be replaced by the m-designing in (3-A). For the function m is a simulation of the critical state, it is called the m-simulation method ( [

In case of several axial loads or ladder cross-section levers, (3-A) becomes

Besides can easily satisfy the 2 BC.s suggested by Timoshenko ensuring an accuracy of certain degree, the application of m-simulation method also simplifies the calculation and reveals the direction for the boundary theory developing as well, see the example below.

Example 1

A pressure lever as

Method 1: Choose the simulation m-curve as

That is

BC.2 on A (see

Taking (3-A) gives

The error is about 1.3% comparing with the exact solution

a good precision, in addition to the method satisfies the 2 BC.s put forward by Timoshenko, it satisfies the volume of 0, the moment at the top, as well, hereinafter referred to as the 3^{rd} BC.: See

Method 2: Choose the m curve as

^{th} BC. on the bottom (see

Method 3: Take the simulation m curve as

The error is about 0.13%; the precision is about 10 times higher than that of method 1. Obviously, in addition to the main reason satisfying the 3 BC.s mentioned in method 1, it satisfies BC. 4 also, see

Method 4: Choose

Due to the chosen m-curve is exactly the same with that of the lever in critical state; the results are the same also.

Discussion: Although the m-simulation method has certain guarantee of accuracy, the above methods of 1 and 2 satisfying all the first 3 BC.s mentioned, made different errors, some one even larger than 10%, such as method 2. However, the accuracy of method 3 and 4 are very high, this is because they not only satisfy the first 3 BC.s, but also satisfy the shear at the bottom section, the 4^{th} BC. on A being 0 as well, see ^{th}-6^{th} BC.s, see

tisfied, a more accuracy of energy method with the trial function different from

Method 5: Choose

The error is about 0.014% and the accuracy increased significantly about 10 times as that of method 3. Obviously, the main reason is that the designed trial function satisfies all the fist 5 BC.s mentioned; verify it please.

Method 6: In order to satisfy all the first 6 BC.s in

Suppose

BC.5:

BC.4:

BC.3:

BC.2:

The error is about 0.00017%, more than 80 times as accurate comparing with method 5.

Brief summary:

Despite of method 4 gives the exact solution; the trigonometric trial function would not be discussed below (see the section under). In all the others except method 4, the most accurate result belongs to method 6, for all the BC.s for Lever 1 in

deformation function of

of (2-A) and (2-B) gaining different results and the one with (2-A) getting more accurate. The reason is that different methods were applied in calculating the function of m: the former add the load on the assumed curve calculating the m-function with static method; whereas the latter obtained the m-function applying the differential relationship of

The method 6 in the above example would not be called the m-simulation method being not begins from supposing the function m; as it makes more convenience, it is recommended here and would be called the

We can see from example 1 that the

As mentioned earlier, there are many BC.s at some specific sections in a prismatic compressive cantilever; they will be introduced in

The stability analysis for Lever 2 is taken as one of the most classic example in the course of energy method for stability analysis in multiple versions of textbook [

Example 2

A prismatic cantilever as in

Analysis

As the gravity q is the only factor considered in the stability analysis, formula (2-A) or (2-B) could not be applied directly. The result should be gained by formula (1).

Method 1 Take the deflection curve as

Comparing with the exact result

Brief summary: The above table providing 6 and 7 BC.s for Lever 1 and Lever 2 respectively. For the application to Lever 1 has been introduced in section 1, below will introduce their application to Lever 2 only.

Method 2 Take the deflection curve as

Then

and

Equaling

terature [

the first 3 BC.s are satisfied, the accuracy is poor. Yet, the calculation is much simpler than that in method 1.

Method 3 Due to the load uniformly distributed along the stem, it is not difficult to find in critical situation, that the shear at the top and bottom sections are all 0 (being called BC.4, see

bottom sections being all 0. For the corresponding slope in

Making

That is

BC. on A:

Equaling

the error is about 0.36%. Not only improved the precision greatly and simplify the calculation comparing with the one in method 1 and 2. We can see that BC.4 has a great impact to the result.

Method 4 According BC.4, we can also suppose

BC. on B:

As the m- function has the same variable factor of

Method 5 Suppose

BC. on A:

BC. on B:

BC. on A:

Then

And

Equaling

Comparing with exact one

thod 3 (4). So there is good effect, satisfy all the 7 BC.s is the main reason. It also provides the scope of the

rough of

Method 6 Suppose

In order to meet

BC.4:

BC. on B:

BC. on A:

Then

And

Equaling

Comparing with exact one

comparing with that of method 5. Visible,

In order to simplifying the narrative, the follow methods would omit some steps in calculation, only the key dates would be provided in

Method 11 Suppose

Comparing with the exact solution, the error is about 2.08%. When compare with the traditional method 2 having been incorporated into many versions of textbooks, it improves in both the precision and the amount of calculation greatly.

Brief summary

1) The analysis for 2 kinds of cantilever pressure levers considering only the top load or the own weight are introduced in the above 2 sections. If only considering the precision requirement, it seems to reinvent the wheel, for the exact results had been given by P. Timoshenko in ( [

2) There are 11 kinds of methods in example 2 and method 5 - 10 satisfied all the 7 BC.s; the related dada are provided in

The combined axial force compressive lever refers to that with uniform or ladder sections supported loads on multi sections or only on the top being one of the common structure components. In this case, there are certain relationships between some physical quantities on the up and down of the sections loaded, hereinafter called Join Conditions, simply JC.; the precision can be improved in applying them.

In order to simplify the narrative, save for “

The analysis for a lever under the action of 2 loads would be introduced first.

Example 3

Calculate the critical load

Method 1 Suppose the simulative loads as

Upper segment:

Lower segment:

BC. on A:

JC. on C:

By taking formula (3-B), we have

The results of method 2 and 3 are finished by computer and the results of 3 methods are shown in

For the stability analysis of levers with ladder cross-section, many references analyzed with static method

Brief summary: The precision of method 2 is the highest being with a linear S-simulation function.

[

Example 4

Calculate the critical load

There will show you 3 methods with the trial function of S-simulation function as Figures 4(b)-(d). All the results are calculated by computer (

Example 5

Calculate the critical load

Admittedly, how the theory could (directly or indirectly) apply to actual can reflect its value. A brief example

Brief summary: Obviously the precision of method 2 with linear S-simulation function is the best.

preliminary applying the Boundary theory on a simple structural analysis will be showed below.

Example 6

Calculate the critical loads

Analysis:

In practical engineering, in mechanical analysis, in addition to the precision requirement the simplified calculation is very important to the promotion. Therefore, the first selection algorithm is the most simple m-simula- tion method. As the structure is symmetry, the antisymmetry critical load is lesser;

Method 1

Suppose the simulation load as

Brief summary: The above 3 examples show the precision guarantee of applying BC.s and JC.s in the stability analysis for combination of axial force. We can also see that the highest accuracy belongs to the one with linear

distribution as is shown in

Method 2

Based on the result of method 1, the m diagram would be adding a cubic parabola as in

That is

firmed ( [

BC. on C:

Then

This is the result of a straight line adding a cubic parabola. In the above equation, constant

lected automatically by the software designed. Comparing with the exact result

0.49%. Such a simple dealing with (only make the analyzing satisfy the BC. 4, the 0 shear on the bottom section C of the column approximately), that brings the effect of error in half of that in method 1. It indicates the boundary theory of broad application prospects. In addition, the programming is simple, and can be used in the stability analysis of high-rise frame structures and the application of the software is very convenient, especially in input ( [

Since 1961, when the Energy Method was put forward by Timoshenko [^{rd}-7^{th} BC.s and JC.s put forward here would improve the precision obviously. The related theory and the high precision trial functions would add some advantage factors for the stability theory to further develop.

In spite of wide application of the matrix displacement method [

In addition, due to the result of author’s subjective and objective limitations, the errors are inevitable; the software also has some defects; so, we sincerely hope readers to give us more criticism and help, especially in such as variety of software development and application of all-round cooperation. We certainly hope this article can cause the reader’s interest, for energy method to expand the application scope, simplify the calculation, improve the precision and so on, and put forward some new more effective method.

R.Song,S. X.Wu,11, (2015) An Expansion of Boundary Theory and the Application of Joint Condition. Open Journal of Applied Sciences,05,844-859. doi: 10.4236/ojapps.2015.512082