Linear Analysis

What is linear analysis? A proportional analysis. For example if I say that a moment M is generating a deflection of D, and what would be the moment acting on the beam if the deflection is 2D? It will be 2M. Quite simple right? This analysis is called linear analysis. All the principle of superposition are also valid.

Let us say dead load is causing a beam deflection the beam by 1″ and live load is causing a deflection of 0.5″ and if I ask you what will by the sum of deflection cause by the two loads? It will be 1 + 0.5 = 1.5″. This is pretty simple, principle of superposition.

This all can happen because the stresses are proportional to strains. Take an example of mathematical equation of a straight line.

y=mx

Now if I say that the value of slope is known and I give a particular value of x, can you figure out the value y? Of course yes. And this can be done in a single step. No repetition is required. Now replace x with strain and y with stress and m is the stiffness of material. The equation of the same straight line becomes:

σ=Es ϵ

So this is why linear analysis is simple. If you know the deformation for 1 unit of load and if you wish to find out the deformation for 5 units of load, you just multiply the deformation by 5 and you have your results. This will reduce the time and effort put into analysis. It will give you sometimes conservative results and sometimes inaccurate as well. (I will justify inaccurate in Nonlinear analysis)

Whatever we lean in under grad is linear analysis. You calculate the forces, you design the section and you are done. We do not consider any cracking effects not do we look for strength loss.

We are still doing linear analysis because we also consider material safety factors and specified properties. The actual strength of material is actually greater than the specified strength and specified strength is the strength without considering any factors of safety.

Nonlinear analysis

1. Material Nonlinearity

When the materials move into the zone beyond it’s yield strengths, it no longer behaves in a linear fashion. There are many things that happen when material go into this zone:

• Permanent deformations: This means that when the material is unloaded it will not go back to it’s original shape or position. For example if you take a plastic bag and stretch it, after a certain point even if you release the bag you will see the permanent stretch marks. This is called permanent deformation.
• Cracking: Generally this occurs in linear design as well, but we neglect the cracking of concrete, even though we still consider the reduced stiffness of members while doing seismic design, but still it is an assumed value. While in nonlinear analysis we monitor the cracking and so concrete will crack and member will start losing its stiffness.
• Beam rotations: When a beam is subjected to moments greater than it’s capacity, it no longer resists the moments, instead it rotates and forms a plastic hinge and start dissipating energy. This is a part of material nonlinearity but for beams it is called backbone curve (aka F-D relationship). In case of linear design we do not case for anything greater than the capacity of the member.
• Energy Dissipation: In linear analysis, energy dissipation is in the form of strain energy, while in case of nonlinear analysis it is in the form of inelastic energy in addition to strain energy dissipation.

 These were a few generalized things that came to my mind while looking at nonlinear analysis.

This is what happens in nonlinear analysis. If a member goes beyond its capacity (elastic limit), it will experience some sort of strain hardening or cracking and it will start losing its stiffness which also means that the total stiffness of the structure or building is also changing. Thus what you do is, you load the structure and see if it went into nonlinear stage, if it does then we see how much the material has cracked also know as softening of structure. If the loss in stiffness is significant and the results or the energy balance do not converge, we iterate the same process and do the analysis again. This cycle will go on till the desired accuracy is achieved. Thus a nonlinear analysis takes longer than a linear analysis because of such loses in stiffness and its iterative nature. But this was talking about a nonlinear static analysis.

As I mentioned before, a linear analysis cannot give a complete picture as what can happen to the structure if an earthquake hits. Today we have the ability to create a mathematical model which to around 90% of the accuracy can give us results which again depends on modelling assumptions and the detail at which it is done. But it gives us an idea whether everything is okay or not. But to everyone’s utmost surprise, the linear dynamic analysis gives a far off result. For example, in case of a beam which is subjected to earthquake shakes. It will experience some force but that force is limited. And we design the beam to that limited force. When we check the same beam for actual earthquake (The one which is not limited) and see check the beam, many times structural engineers find that the beam is actually getting shattered. Now with increased load we definitely expect some rotations but shattering of beam is just not acceptable.

So this is the benefit of nonlinear analysis over linear analysis.

2. Geometric Nonlinearity

The most famous geometric nonlinearity is P-Delta analysis. A force follower approach. (I am copying the data from my other answer over here)

P Delta analysis is quite a traditional form of force follower analysis. It is also called “Geometric Nonlinearity” because as the deflection increases you again have to test the additional forces generated by P-delta effects. A force follower analysis is the one in which, when a member loses its stability the force follows the deformed member and creates further more instability very quickly. A P-Delta analysis is not as simple as it sounds and its effects will be very adverse if neglected. These effects will be more severe in case of soft lateral force resisting systems  like moment frames as compared to stiff systems like core wall systems and braced frames.

Talking about P-Delta, P delta is a term coined from P that is load and delta is the lateral deformation. These lateral deformations are more lethal in case of earthquakes and not so much in case of wind.

What is the significance of it’s study? Is it just limited to design of columns? Something like this:

What it does is, it generates additional shear forces and bending moments in columns because of the deformed shape. The moments generated will be equal to the load acting on the column times the horizontal displacement. Now we have to check the column capacity particularly in case of slender columns so that they do not fail in case of these additional moments along with the axial loads. This can be checked with P-M interaction diagram of the column cross section.

Just make sure that the load point lies inside the P-M interaction boundary of the column.

In addition to this, the P-Delta effects has one more adverse effects, specifically in tall buildings. As we know, in case of earthquake a building deforms. And this deformation is huge and the structure is already in its inelastic zone with concrete cracking. This means that the structure is already losing its stiffness. Now the P-Delta shear (The force that is generated at the top and bottom of the column because of P-delta moments), generates an additional demand for lateral shear resistance of the structural system. This additional demand is in addition to the earthquake shear demands. Which means that if we have not consider the P-delta demands and if we provided in sufficient shear resistance, than the building might collapse, similar to this:

As you can see, it is very severe.

Now, the effect of P-Delta shear demands is more in case of moment resisting frames as compared to shear core systems. The reason is, moment frame is already moment governed and so it is a soft system. A soft system tends to drift more in case of lateral load and more drift means more “delta” which means more shear and moment demands because of the P-delta effects. While in case of shear core, the structural system itself is very stiff and as the name suggests, a shear core system is resisting shear forces so it will not impact the structural system to a great extent.

Refer to chapter 2.3 in the following guidelines for more understanding of P-Delta effects as they will show you some charts of strength deterioration of the system.

http://peer.berkeley.edu/tbi/wp-…

Now how does a computer program deals with everything? Do we have to do something special to do nonlinear analysis? Or all computer program does that by default?

By default, a computer program is set for linear analysis. Quick and easy method and for most of the small structures it will be more than good approach.

Can the same model be used for nonlinear analysis? No, you will have to add a ton of information into the computer model to do nonlinear analysis. You will have to add stress strain curve for concrete, for steel. You will have to define backbone curves for beams. You will have to define P-M-M back bone curves for columns. You will have to define fiber elements for shear walls. You will be defining P-delta columns. You will be defining the limit states. The back bone curves for coupling beams for different aspect ratios are different. So all in all, to create and test one nonlinear model, it will take you about a month. Analysis will take a day. And processing the results will take another day.

What does program do is. It will start with the initial stiffness of the building which is right because before a building is loaded how can there be any cracks and loss in stiffness? Then the building is loaded with incremental loads. And it will go on increasing the loads till it reaches the limit of linearity. As soon as it hits the nonlinear zones, it will start iterating the model. Load the structure calculate the strains and deflections and stiffness. Loss in stiffness -> Yes? Iterate the same step. Loss in stiffness -> No? Go to the next load step and so on.

Nonlinear analysis is a complex task. It is the best example of “Half knowledge” is dangerous. If you do not know anything about nonlinearity then first learn it and then perform analysis. If you do it before that you will set up incorrect model and you will not be able to interpret the results.

I hope I gave you some idea about nonlinear analysis. There is much more to this. I cannot even describe how vast this topic gets. But for the purpose of a general answer, I think I did my best to explain you in a Nutshell.

http://www.thestructuralmadness.com/p/home.html

非线性问题分类及求解

线性linear，指量与量之间按比例、成直线的关系，在空间和时间上代表规则和光滑的运动；非线性non-linear则指不按比例、不成直线的关系，代表不规则的运动和突变。

非线性问题分类

 当材料是线弹性体，结构受到载荷作用时，其产生的位移和变形是微小的，不足以影响载荷的作用方向和受力特点。静力平衡方程表示为：

其基本方程的特点如下：

• a.材料的应力与应变，即本构方程为线性关系。

• b.结构应变与位移微小、即几何方程保持线性关系。

• c.结构的平衡方程属于线性关系，且平衡方程建立于结构变形前，即结构原始状态的基础之上。

• d. 结构的边界(约束)条件为线性关系。

• 不同时满足上述条件的工程问题称为非线性问题。

  习惯上将不满足条件a的称为材料非线性;不能够满足条件b、c的称为几何非线性;不满足条件d的称为边界非线性 。对于兼有材料非线性和几何非线性的问题称为混合非线性问题 。 对于上述非线性问题总可归结为两大类，即材料非线性和几何非线性。

  END

非线性问题求解

 　　非线性问题用有限单元法求解的步骤和线性问题基本相同，不过求解时需要多次反复迭代，基本三大步骤如下：

　(1) 单元分析

非线性问题与线性问题的单元刚度矩阵不同，仅为材料非线性时， 使用材料的非线性物理(本构)关系。 仅为几何非线性时， 在计算应变位移转换矩阵[B]时， 应该考虑位移的高阶微分的影响。 同时， 具有材料和几何非线性的问题，受到两种非线性特性的藕合作用。

(2) 整体刚度矩阵集成

整体刚度矩阵集成、平衡方程的建立以及约束处理，与线性问题求解相似 。

(3) 非线性平衡方程求解

对于几何非线性问题，平衡方程必须建立在变形后的位置，严格来讲是建立在结构的几何位置及变形状态上，简称为位形状态。因而，非线性问题的平衡方程表为

求解的方法按照载荷的处理方式可分为全量法和增量法两大类。