Daunting differential equations, Alex had a challenging desire to help engineers working in the front line of structural design understand the stability problem through a series of articles! This series of articles will involve the knowledge of calculation length and overall stability.
However, if there is a slight gust of wind or if we accidentally touch it when passing by, we will fall.
According to this method, the N diagram and M diagram can be drawn, and the value at the bottom of M diagram is FH, This internal force calculation based on the initial shape is also called first-order analysis.
Let’s take a look at the compression bar experiment.
Many students are confused when it comes to stability.
From the sudden change from linear state to bending state when the hinged axial compression bar at both ends is unstable, to the overall large deformation instability of portal frame and scaffold, the actual essential characteristics are the above two.
Each of us has such experience.
From these figures, we can see that the unstable components or structures have undergone great deformation, so the first feature of instability is large deformation.
Due to sudden lateral displacement due to early instability, the failure occurs when the axial force is less than f · A.
The internal force of the second-order analysis is greater than that of the first-order analysis.
The second feature is the sudden change of state, such as the pressure bar, brace and scaffold above have changed from straight to curved, and the web of portal frame has changed from vertical ground to lateral bulge, so instability also has an image name called buckling, which means straight to curved.
The wonderful derivation and unexpected conclusions in the stability problem all reflect the solid foundation and outstanding wisdom of the previous masters, but I never moved the idea of writing stability problems myself.
set the failure force as F, and define one ψ=， Named stability factor, ψ ≤ 1, find ψ Value is the final problem to be solved for all stability problems.
In the past two years, friends and students have constantly suggested Alex to write about the stability problem, combined with his own understanding and engineering experience, and explain the physical meaning of the stability problem in a language as easy as possible.
We try to avoid differential equations, Feel the principle and explain the problem from simple and intuitive physical concepts.
In the first section of today, let’s talk about some concepts.
First, let’s review the strength and stiffness problems.
This infinite series is convergent, and the final total horizontal displacement is Δ T means, then the bending moment at the rod bottom MZ = FH + P Δ t.
In this case, the first-order analysis can be used instead of the second-order analysis..
Let’s start with the figure above to give you an intuitive feeling of instability and failure.
Additional bending moment p Δ 1 will cause a new displacement increment Δ 2 (because M = EI) ψ，ψ Is the curvature, and the curvature is integrated twice along the rod length to obtain the lateral displacement), Δ 2 will produce a new moment increment P Δ 2.
The first-order internal force and the second-order internal force are a cantilever rod, and the top is affected by the vertical force P and the horizontal force F, as shown in the figure below: We all know that the conventional calculation assumes that the structural deformation is very small, so we think that there is no big error in the calculation of internal force on the initial shape without deformation, and then the superposition method is used to directly add the coordinates of each point of the internal force diagram under various working conditions to obtain the total internal force.
This is an infinite iterative process, but fortunately Δ 1、 Δ 2、 Δ 3…
From today on, I will take the students step by step from the basic principles and simple components to the understanding of the system level.
After a period of observation and understanding, it is found that most of the books on structural stability on the market are written in the mathematical language of differential equation, emphasizing the tightness of logic.
Although structural design is complicated, it generally belongs to three types of problems: strength, stiffness and stability.
The content is more detailed and comprehensive, so that everyone will no longer be confused when encountering stability related problems in the design work, and can not only make qualitative judgment, but also calculate stability problems quantitatively.
The strength problem and stiffness problem are very intuitive.
Stiffness problem refers to excessive deformation, as shown in the following figure: It is assumed that the beams and columns of the frame are made of very elastic materials, such as rubber materials.
Of course, in the early 20th century, the engineering community has not formed a deep understanding of instability.
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However, since the equipment pipelines will be crushed and the furniture and people will overturn at this time, it is obvious that such large deformation is not allowed in the design.
However, for most structural engineers engaged in design work, these books are relatively difficult, need a good foundation in mathematics and mechanics to understand, and spend a lot of time, which makes many engineers daunted and unable to read.
The strength problem is whether the components will be damaged.
For example, the steel will yield, for example, the concrete will break or crush.
I mainly felt that my foundation was shallow compared with the previous masters, Can’t control such a difficult theoretical system.
This ring is made of steel after the collapse of Quebec Bridge, and the reason for the collapse of the bridge is due to the compression and instability of the lower chord near the anchor arm of the main pier.
Everyone has no problem in understanding, but the stability problem is not so easy to understand.
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If there are old people at home, they will teach us: it’s so unstable, so each of us, even the illiterate old people, has a natural concept of stability, Without profound knowledge of calculus and mechanics, the concept of stability can be understood from life.
When the member stiffness is very large Δ 1 is small, because Δ 1 generated by iteration Δ T is not big, then P Δ T is negligible compared with FH.
The language of the series of articles shall be as easy to understand as possible, with pictures and texts as much as possible, highlighting the physical significance, helping everyone to establish an intuitive concept and deeply understand the stability problem.
At the same time, it is not necessary to use the unfriendly mathematical tools such as differential equations.
Before we talk about stability design, we first clarify the position of stability in structural design.
The origin of the stability coefficient and the stability problem are very important, so how serious consequences will occur without considering the stability calculation? As we all know, when graduating from engineering in Canada, each student will issue an engineer’s ring.
For each section, we draw vertical and horizontal lines with a pencil and divide them into small squares, as shown in the figure below, Then, if any small cell of any small beam section is damaged, we say that the beam is damaged in strength, like the small cell in the lower right corner in the photo.
For example, if we put a tea cup by the table and hang it in the air for one third, we can’t fall without touching it.
Although its deformation is very exaggerated, the strength damage described above will not occur.
When calculated according to the strength, the axial force that the compression bar can bear is f · a, that is, the strength of the material × Section area, but for steel structures, in order to save materials, slender members are generally used.
The differences and relations among strength, stiffness and stability.
But the actual stress process is different, when the cantilever column top moves laterally under the action of horizontal force Δ After 1, the force P produces the initial additional bending moment P on the bottom of the compression bar Δ 1.
I was very interested in the stability problem very early.
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We use our imagination to cut a beam into countless small beam segments with a knife, so as to expose countless small sections.
This internal force calculation based on the deformed shape is also called second-order analysis.
