Structures, Models, and Assumptions

3D, 2D, or 1D Structures

Structures can be categorized based on their external geometry. If a structure's three dimensions are of similar size, it is referred to as a three-dimensional (3D) structure or a 3D solid (Figure 1).

On the other hand, if one dimension of the structure is much smaller than the other two dimensions, the structure is called a plate or shell, depending on whether the undeformed configuration is flat or curved, respectively. The two large dimensions are commonly referred to as the in-plane directions, while the smaller dimension is called the thickness, which can be constant or varying. For plates and shells, a reference surface can be defined using two in-plane coordinates. Thus, they can be called two-dimensional (2D) structures.

If one dimension of the structure is much larger than the other two dimensions, the structure is known as a beam. The large dimension is often referred to as the span or axis of the beam, and a reference line can be defined for the beam using the axial coordinate. The reference line can be as general as a spatial curve, as seen in the case of initially curved and twisted beams. Consequently, these structures are also referred to as one-dimensional (1D) structures. The two small dimensions are called the cross-section (constant or varying) of beam structures.

If a beam structure has a wall thickness much smaller than the cross-sectional dimension, it is termed a thin-walled beam. In such cases, all three dimensions of the structure are of different magnitudes.

Collectively, plates, shells, beams, and thin-walled beams are known as dimensionally reducible structures, a term coined by the author of to emphasize the fact that one or more small dimensions can be eliminated to construct a corresponding simplified structural model with reduced dimensions while still adequately predicting their behaviors.

As far as mathematical modeling is concerned, this classification of structures is based solely on their external geometry, while their internal construction can be arbitrary. For example, a porous material can be modeled as a 3D solid (Figure 2), a flat panel with grid stiffeners can be modeled as a plate (Figure 2), and a high-aspect-ratio wing without clearly defined cross-sections of an aircraft can be modeled as a beam. In this sense, all engineering structural systems, regardless of their complexity, can be considered as combination of structural components in terms of 3D structures, plates, shells, and beams with the possibility of having complex internal constructions.

It is essential to clarify that our classification of structures, based on their external geometric characteristics, does not restrict the materials that can be used to construct these structures. Therefore, a structure categorized as a beam, plate, shell, or 3D solid can be made from either isotropic (e.g. aluminum) or anisotropic materials (e.g., carbon fiber-reinforced composites). If a structure is made of a single material filling every point of the structural body, it is referred to as a homogeneous structure. Conversely, if the structure is made of different materials, it is termed a heterogeneous structure.

Furthermore, it is important to note that our classification of structure is independent of the loading conditions they may experience. While engineers often associate beam structures with solid bodies subject to bending, a beam structure, as classified here, can be subject to forces and moments in all three directions, including axial force, transverse shear forces, torque about the beam axis, and bending moments about the two transverse directions. In this manner, the classification provides a comprehensive framework for understanding and analyzing various structural behaviors, regardless of the loading conditions they may encounter.

Structural Mechanics Models and Associated Assumptions

The field of applied mechanics known as structural mechanics is concerned with predicting displacements, strains, stresses, failure, and other phenomena for various types of structures using mathematical models. The classification of structures as 3D structures, plates, shells, and beams is motivated by their geometric characteristics, which in turn lead to the construction of different models for analyzing these structures.

Beam models, also called 1D structural models, are constructed for beam-like structures. The field variables in these models are functions of a single coordinate representing the reference line along the beam.

On the other hand, plate and shell models, also known as 2D structural models, are designed for plate- and shell-like structures. The field variables in these models are functions of two in-plane coordinates representing the reference surface of the plate and shell.

It is important to note that the notation of 1D, 2D, or 3D refers to the number of coordinates needed to describe the analysis domain, not the dimensionality of the behavior. For instance, a 1D beam model can describe displacements and rotations of the structure in all three directions.

In structural mechanics, each structural model comprises three sets of equations: kinematics, kinetics, and energetics (constitutive relations). The constitutive relations, which describe the material behavior, are the only equations that change for different materials used to construct the structure. Predicting these constitutive relations is referred to as constitutive modeling. The constitutive relations are given in terms of material properties such as Young's modulus, Poisson's ratio, shear modulus, and strength constants that can be measured using physical testing or calculated using micromechanics.

For homogeneous structures, the 3D material properties can be directly input into a structural analysis using a 3D solid model. Additionally, for isotropic homogeneous structures, the geometric characteristics can be combined with the 3D properties to compute the required structural stiffness for plate, shell, and beam models (e.g., extension stiffness EA, bending stiffness EI, and torsion stiffness GJ for beam models). These stiffness parameters are determined based on certain a priori assumptions, such as the Euler-Bernoulli assumptions for beams.

However, when dealing with homogeneous structures made of anisotropic materials or heterogeneous structures, constitutive modeling becomes more complex.

For example, for composite laminates, many refined assumptions, such as higher-order assumptions, zigzag assumptions, and layerwise assumptions, have been introduced to better capture the kinematics along the smaller dimensions of the structure (e.g. the thickness of a composite plate).

Structural analyses are routinely carried out using finite element analysis (FEA) codes such as Abaqus, Ansys, and Nastran. These analyses use 3D solid elements, 2D plate or shell elements, or 1D beam elements (see Figure 3), each corresponding to a specific structural model that includes three sets of equations: kinematics, kinetics, and constitutive relations.

In engineering, different beam models are commonly employed, including the Euler-Bernoulli model, the Timoshenko model, and the Vlasov model. The Euler-Bernoulli model, also referred to as the classical beam model, neglects transverse shear effects. The Timoshenko model, also known as the first-order shear deformation theory (FSDT) for composite beams, accounts for transverse shear effects. The Vlasov model accounts for non-uniform torsion effects.

For plate and shell structures, commonly used models include the Kirchhoff-Love model and the Reissner-Mindlin model. The Kirchhoff-Love model, also called the classical plate and shell model, or the classical lamination theory (CLT) for composite laminated plates and shells, neglects the transverse shear effects. The Reissner-Mindlin model, also known as FSDT for composite plates and shells, accounts for the transverse shear effects.

For 3D structures, the commonly used model is the Cauchy continuum model, which is also known as the classical continuum model.

Sometimes, specific models and their associated assumptions are used to name a structure in literature and technical communications. For instance, terms like Euler-Bernoulli beam or Timoshenko beam imply that a structure must adhere to certain assumptions to be modeled using these beam theories. In reality, a beam structure is merely a solid with a much larger axial dimension compared to the other two dimensions. This structure can be analyzed using various models, such as the 3D Cauchy continuum model, the Euler-Bernoulli model, the Timoshenko model, or other higher-order models, depending on the behavior one aims to capture. In this context, it is better to dis-associate the model from the structure and the associated assumptions. Assumptions according to Euler-Bernoulli or Timoshenko are not essential, and it is possible for us to develop a Euler-Bernoulli beam model without using the corresponding Euler-Bernoulli assumptions.