13.1. Introduction

Prior to analysis, a structure must be idealized for a suitable mathematical representation. Since it is practically impossible (and most often unnecessary) to model every single detail, assumptions must be made. Hence, structural idealization is as much an art as a science.

Structural idealization is the process of simplifying a complex structure into a mathematical model. This involves making various assumptions about how the structure behaves under loads. The purpose of these assumptions is to make the analysis simpler while still capturing the essential behavior of the structure. For example, it may not be necessary to simulate every bolt in a large bridge, so engineers might simplify the configuration into simplified beams or trusses.

Imagine trying to represent a crowded city map. It would be impractical to include every single tree or bench; you could simplify the map to only show the streets and major buildings. Similarly, in structural engineering, simplifications help focus on the overall behavior of structures instead of getting lost in unnecessary details.

Some of the questions confronting the analyst include:

1. Two dimensional versus three dimensional; Should we model a single bay of a building, or the entire structure?
2. Frame or truss, can we neglect flexural stiffness?
3. Rigid or semi-rigid connections (most important in steel structures).
4. Rigid supports or elastic foundations (are the foundations over solid rock, or over clay which may consolidate over time)?
5. Include or not secondary members (such as diagonal braces in a three-dimensional analysis)?
6. Include or not axial deformation (can we neglect the axial stiffness of a beam in a building)?
7. Cross-sectional properties (what is the moment of inertia of a reinforced concrete beam)?
8. Neglect or not haunches (those are usually present in zones of high negative moments)?
9. Linear or nonlinear analysis (linear analysis can not predict the peak or failure load, and will underestimate the deformations)?
10. Small or large deformations (In the analysis of a high-rise building subjected to wind load, the moments should be amplified by the product of the axial load times the lateral deformation, P-Delta effects)?
11. Time dependent effects (such as creep, which is extremely important in prestressed concrete, or cable-stayed concrete bridges).

When preparing for structural analysis, engineers face numerous questions that guide their modeling approach. For instance, deciding whether to model a structure in 2D or 3D will significantly affect the complexity and accuracy of the analysis. The decisions about whether to include elements like secondary braces or to account for material properties can change the predicted behavior of the structure under loads, impacting safety and performance. Each question leads to a different simplification or assumption that should reflect the actual behavior of the structure as closely as possible.

Think of a wedding cake. If you only focus on the top tier, you might ignore how the weight from the middle tier affects it. Each layer has its own importance, just like the elements of a structure. When engineers analyze buildings, they must consider how small factors (like a missing tier) can influence the entire design.

We should differentiate between two coordinate systems:

1. Global: to describe the structure nodal coordinates. This system can be arbitrarily selected provided it is a Right Hand Side (RHS) one, and we will associate with it upper case axis labels, X; Y; Z.
2. Local: system is associated with each element and is used to describe the element internal forces. We will associate with it lower case axis labels, x; y; z.

In structural analysis, two coordinate systems are essential for accurate calculations. The global coordinate system is used to define the overall structure and locations of various nodes. It is noted by uppercase letters (X, Y, Z). The local coordinate system relates specifically to individual elements, such as beams or trusses, and utilizes lowercase letters (x, y, z). This distinction offers clarity, allowing engineers to analyze forces acting on specific components relative to both the entire structure and individual parts.

Think of a concert hall. The global coordinate system is like the overall layout of the entire building, showing exits and entrances. The local system is similar to focusing on a particular musical instrument or section of musicians on stage, analyzing how they perform in relation to the entire concert. This dual approach ensures every performer contributes effectively to the overall performance.

The sign convention in structural analysis is completely different than the one previously adopted in structural analysis/design. In matrix structural analysis, the sign convention adopted is consistent with the prevailing coordinate system.

In structural analysis, understanding the sign convention is crucial because it establishes how forces and moments are represented mathematically. Unlike previous methods that might define positive moments in a particular way, the sign convention in matrix analysis aligns with the coordinate system, where counter-clockwise moments are considered positive. This consistency aids in avoiding confusion during analysis and ensures correct interpretations of results.

Consider a map where you decide that north will always be at the top; this consistency helps everyone navigate the map without confusion. Similarly, having a standard sign convention in engineering means that all calculations and interpretations will follow the same rules, preventing misunderstandings.

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