What is Non-Linear Structural Analysis?
In simplest terms a non-linear structural analysis is one that considers other secondary effects that may have an impact on the structure's response to the loads applied. In comparison, a Linear Structural Analysis does not consider the possibility of any secondary effects.
Understanding linear structural analysis will facilitate a better understanding of the non-linear method. There are several methodologies that fall into the linear category, but they all have one thing in common. In a linear structural analysis, the structure is assumed to be perfectly elastic and totally linear in its response to forces.
This means that if a given force was applied to a portion of a structure and it deflects a specific amount; doubling the force should generate twice the amount of deflection. This enables, for linear analysis, the principle of superposition of forces. You could analyze the response of the structure to each force individually and add up (superimpose) all of the calculated deflections (or stresses) to arrive at a final response for the structure. The simplicity of this method is very attractive.
The linear methodology assumes many factors which may or may not be accurate. For this reason the linear methodology has always required higher load and safety factors by National codes to account for these assumptions. In Non-Linear methodology, the primary assumptions used in the linear methodology are considered and addressed. The end result is a much more accurate estimation of the structure's response (deflection, stress, etc.), which in turn leads to National Standards Organizations(in general) requiring lower load, strength and safety factors.
With the use of linear methodology, the possibility of the structure becoming unstable had to be evaluated separately from the loading analysis. It is conceivable that a specific linear analysis could show that the structure's components are well below yield point stresses, yet the total structure could be in a failure state. For example, to estimate the stability of a single pole structure to resist vertical loads, Euler's buckling formula was modified by Gere and Carter to take a single pole's taper into account. This in itself is still just an estimate with its own set of assumptions factored in. The principle assumption is that no attachments offer any stability to the pole along its length. Structures consisting of more than one element (i.e. most Transmission Structures) are very difficult to analyze using linear methodologies.
In Non-Linear Structural Analysis, the stability of the structure is determined as part of the complete solution procedure. Structures that are not stable simply do not converge on a final solution.
As loads are initially applied in small amounts to a structure, the structure will respond in an approximately linear manner. At some point, other (secondary) effects begin to influence how the structure responds. This occurs at a point that can be referred to as the Yield Point for the structure, which is similar to the Yield Point of most metals. As the loads are increased on the structure, these secondary effects will become very significant and the structure is said to 'soften' or become less stiff. Once the loads reach what is called a Critical Point, the structure will no longer be able to resist the loads. The result is rapid displacement of the structure components towards the Failure Point. Since the loads placed on the structure in practical applications are held somewhat constant during structure displacement and the structure's load carrying ability is reduced, the ultimate failure of the structure and its materials occurs rapidly. In other words the structure collapses or buckles.
The following are all the possible secondary effects that are commonly associated with the term "Non-Linear Structural Analysis" with commentary on how applicable they are to Transmission and Distribution Line Structures.
- Geometric Non-linearity
- Material Non-Linearity
- Load Non-Linearity
- Boundary Condition Non-Linearity
As a structure deforms/deflects in response to the loads applied to it, the fact that the loads are essentially moved to a different absolute location will change the real loads being applied to the structure. The magnitude and direction of the structure's deflection at loading points will in itself introduce additional loads for the structure to deal with.
For any structure that is expected to stay stable and fixed in one place (all Transmission and Distribution structures), the equations of equilibrium apply to all forces and moments at every point along the structure. So if a vertical force is displaced horizontally, the structure must resist with the appropriate amount of counter-balancing forces and moments in order to completely compensate for this effect. The example shown above in the deformed state is what is commonly called the P-delta effect where the force (P or Fy) times the distance (delta or d) represents the high-level estimation of the bending moments that the structure must resist, over and above the initial loads.
The two-dimensional example above hints at some of the details of this calculation. Of course real structures are three-dimensional and they are exposed to other forces in different directions that are displaced in directions that also create additional moments. Although a single-pole structure is depicted above, these same effects exist in multi-pole, truss and other structure types as well.
It is almost a certainty that all structures will be modeled using Finite Element techniques as part of any non-linear analysis. This is a technique whereby the structure is modeled into many discrete elements which are then combined (based on the structure's geometry) into a larger model for the entire structure. The objective is to create a model that can satisfactorily "estimate" the overall structure's response. Its strength is the ability to model even very complex structures using simple, well understood elements. Generally, the more elements provided in the model, the more accurate will be the result.
The discrete elements are individually modeled (in matrix form) for how they respond to forces in three dimensions.
[F] = [k][d]
F = forces
If an element's properties change with length (e.g. diameter of a pole) or its length is significant, best practices would suggest that an element be broken up into additional smaller elements. This will help provide a better estimate in element and structure responses.
In order to model an entire structure with a collection of element models, they are combined into a much larger model in a way that ensures the structure is statically determinant at every element boundary (sum of the forces and moments are zero). Applying the assumed loads at the element boundaries to the larger structure model will allow the calculation of global structure displacements, which in turn can be used to determine local element displacements and reactions (forces, moments, stresses). At this point you will have the calculation result based on a Linear Analysis. The models used assume a linear elastic response within each element and do not yet have any secondary effects included. It is however a necessary first step to get this initial estimate of structure and element responses.
The geometric non-linear effects (p-delta) can now be estimated with known forces and displacements and then combined with the assumed loads into the structure model and calculated once more. This provides the first estimate of how the structure will respond to both the assumed loads and these secondary effects. The structure may now have deflected further or in different ways than initially estimated. The secondary effects should be re-calculated and re-applied to the structure model at least once more. This process continues until the deflections of the structure do not materially change, or it is obvious that the structure is unstable. If the applied loads are very close to, but not past, the Critical Point, it will take much iteration to reach a solution. If the structure is already past the Critical Point, these iterations will not converge on a solution.
All non-linear solutions must be solved iteratively as there is no way to combine geometric non-linear effects into the structure or element models to solve in one calculation. Other secondary effects can be combined at each iteration step of the solution procedure discussed here, for efficiency. Given the large amount of matrix manipulation and the iterative procedure required, computer software tools are commonly used.
When the term "Material Non-Linearity" is used in Structural Analysis, it always refers to the stress-strain characteristics of the material or, how the material responds physically to external forces, or force over a specific area (Stress=Force/Area).
Perfectly elastic materials are said to have a constant Modulus of Elasticity because they respond in a linear manner to changes in Stress. This is recognized by a constant slope of the material's Stress/Strain curve.
A non-linear material will have a stress/strain response that varies with the level of stress applied.
When one thinks of non-linear materials, one often thinks about metals and the fact that they begin to permanently stretch after a certain stress level. This is the case for most metals after or near their Yield Strength. There are many other materials that exhibit non-linear behaviors as well. Their exact behavior may be influenced by its current deformed state and possibly past history of deformation activity. Other influencing factors may include: pre-stressing, temperature, time, moisture, and electromagnetic fields.
With respect to Transmission and Distribution Structural Analysis, most common materials used in their construction (e.g. Wood, Concrete and Steel Poles, H- structures, steel truss towers…) can be assumed to respond in a linear manner to external forces, at the element level. This assumption is used to enable the efficient analysis of the structure, followed by confirmation of this assumption by the Designer. This confirmation can be done by determining each element's maximum stress level and comparing it with the Yield Point for the material used in the element. If a certain material is used in a structure that has some non-linear properties, the most common practice would be to use updated element models in the iterative non-linear analysis procedure.
The non-linearity of wire and cable attachment material needs to be considered in order to accurately determine the initial loads applied to the structure. These loads may need to be re-calculated as part of Force Non-Linearity (below).
While rock can be considered to be very solid, other types of soil will exhibit plastic behavior if they are excessively loaded past their Yield Point. This non-linear, plastic behavior is dependent on the soil's cohesive and non-cohesive properties, plus how much stress is applied to the soil in order to resist movement of the structure. Limiting soil stresses below the Yield Point is the common practice so that these factors can be ignored.
If the assumed loads that are applied to the structure in the analysis can change due to movement or deflection of the structure, this can be an important secondary effect. The loads are said to be non-linear as they are neither constant nor predictable separately from the structure. When Load Non-Linearity is considered during Structural Analysis, the loads applied to the structure are normally adjusted iteratively throughout the Non-Linear Analysis.
These effects are very applicable to Transmission and Distribution Lines, but primarily to Distribution Lines as their span lengths are shorter. Over a shorter span, the deflection of the structure(s) will have more of an impact than over a longer span. The primary application that creates Load Non-Linearity for Line Structures is from the various wires under tension that may be attached between structures. The tension forces that are applied to the structures will change in both magnitude and direction as their supporting structures deflect. Since the structures deflect in response to all applied loads, determining the final structure deflection and internal loads is therefore an iterative process.
The following example will demonstrate. A structure with the tension T of one wire attached to another structure a distance of approximately L away is under study. The structure is subjected to wind load on itself and the apportioned wind load from the wire.
In this initial condition it is assumed that the structure has no deflection when the forces are initially applied:
- Fx = T
- Fy = Wind
- Wire Length = L , based on the span length between the structures and the installation tension.
The structure deflects in response to the loads that were applied. It moves away from the wind that was applied and pulled slightly towards the tension in the wire. The wire length is now shorter than it was before, as the structure has now deflected towards the wire exerting the tension. This reduces the tension T to T – dt. The direction of T has changed slightly as well with the introduction of a small angle Θ to the horizontal.
The loads have now changed as the angle and magnitude of the Tension T are now different.
- Fx = (T – dt)* Cos Θ
- Fy = Wind – (T – dt) * SinΘ
- Wire Length = L - dl
The effect of the angle Θ on T producing some Y direction forces will tend to help support the structure against the wind forces. The structure will no longer need to support the entire Wind load, so it does not need to deflect as much as it did initially. The reduction in tension of the wire will reduce the amount of pole deflection in that direction, as well.
The adjustments to the net forces acting on the structure will result in less deflections overall.
This process of force adjustments is iterative until the structure's deflections stabilize.
Boundary Condition Non-Linearity
This category of non-linearity refers to the possibility that the points where the structure is fastened, may move, deflect or rotate. Where most structures are fastened to the soil or rock through embedment or other fasteners, it would be rare that this aspect of non-linearity would need to be considered for aerial structures used in Transmission and Distribution lines. This is based on the assumption that the foundation for the structure has been designed to be adequate to support the assumed loads that the structure requires and that the possibilities of foundation movement (e.g. washouts) have been considered.
The Non-Linear Structural Analysis of Transmission and Distribution structures is an advancement in analysis techniques for Utilities in general. The technique has been taught in Universities and used in other engineering disciplines for many years. It is a tried and proven methodology that Utilities can use to get the best estimation of structural response.
As noted above, Material Non-Linearity and Boundary Condition Non-Linearity of Structures are not normally a concern for Utility Designers. Geometric and Load Non-Linearity are the two aspects of Non-Linear Analysis that are the most important to consider in the design of Transmission and Distribution structures.