What is the difference between linear and nonlinear analysis?

Linear Analysis

What is linear analysis? A proportional analysis. For example, if I say that a moment M is generating a deflection of D, then what will 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 principles of superposition are 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 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 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, y with stress and m is the stiffness of material. The equation of the same straight line becomes:

σ = E?

Therefore, 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 conservative results and sometimes inaccurate as well. (I will justify inaccurate in Nonlinear analysis)

Whatever we learn in under-graduation is linear analysis. You calculate the forces, you design the section and you are done. We do not consider any cracking effects, nor 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 greater than the specified strength and specified strength is the strength without considering any factors of safety.

Nonlinear analysis

Material Non-linearity

When the materials move into the zone beyond its yield strengths, it no longer behaves in a linear fashion. Want to learn more about the difference between ductility and elasticity? I have written a blog on my page Structural Madness for detailed description explaining the difference between the two.

There are many things that happen when material go into this plastic zone:

• Permanent deformations: This means that when the material is unloaded it will not go back to its 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. As a reference attached is an image of a coupling beam moment-rotation plot and we can clearly see how much the beam is "walking" towards one side. This type of walking occurs more often in a flexible structure as it is susceptible to more deformations.

• Cracking: Generally cracking 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 approximate stiffness 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 its 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 non-linearity but for beams it is called backbone curve (aka F-D relationship). In case of linear design, we do not check for anything greater than the capacity of the member. But in nonlinear analysis we do monitor the rotations of the member and make sure that they are within acceptable limits derived from testing as well as building codes like ASCE 41-13.

• Energy Dissipation: In linear analysis, energy dissipation is in the form of strain energy and viscous damping, while in case of nonlinear analysis it is in the form of inelastic energy, small percentage of strain energy and significant contribution from damping. Below is an image showing proportion of different types of energy dissipation from a building analysis in PERFORM 3D.

Here 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. 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 known 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 but we use the modified stiffness of structure that is revised because of either cracking or material going into plastic state. 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 can create a mathematical model which to around 90% of the accuracy can give us results which again depends on modelling assumptions and the detail to which it is done. It ideally should give 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 that is reduced by a ductility factor "R". 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 also known as MCE level event) and check the beam, many times structural engineers find that the beam is failing. Now with increased load we expect some rotations, but failure of beam is just not acceptable.

Geometric Non-linearity

The most famous geometric non-linearity is P-Delta analysis. A force follower approach.

P-Delta analysis is quite a traditional form of force follower analysis. It is also called "Geometric Non-linearity" because as the deflection increases you again must test the additional forces generated by P-delta effects. A force follower analysis is the one in which, when a member deforms, the force follows the deformed member and creates furthermore 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 comes 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 its 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 must 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 had not considered the P-delta demands and if we provided insufficient shear resistance, then the building might collapse, like 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.

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.

Do read ATC-72 -1 report for better understanding of nonlinear modelling and analysis of buildings.

Now how does a computer program deal 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. So, all in all, to create and test one nonlinear model, it will take you anywhere from a week to a month. For analysis of such structures a computer can take from minutes to weeks depending on the size and complexity of the model.

The other question that ponders is what does a computer software do differently to perform nonlinear analysis?

Here is what will happen:

1. 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?
2. Then the building is loaded with incremental loads.
3. The program will go on increasing the loads very rapidly till it reaches the limit of linearity.
4. As soon as it hits the non-linearity of a single element, it will start iterating the model.
5. Load the structure calculate the strains and deflections and stiffness.
6. Loss in stiffness -> Yes? Iterate the same step with updated stiffness
7. . 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 non-linearity then first learn it and then perform analysis. If you do it without understanding the concepts of non-linearity, plasticity and numerical methods of nonlinear analysis, then 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 a general post, I think I did my best to explain you in a "Nutshell".