Physical Intuition and Continuum Mechanics Context

In finite element analysis, transitioning from one-dimensional (1D) elements—such as spring, bar, and beam elements—to two-dimensional (2D) and three-dimensional (3D) elements marks a crucial conceptual leap. In 1D structures, the physical behavior is constrained along a single line, and stress states are uniaxial. However, real-world structural components, such as pressure vessel walls, shear panels in aerospace structures, gusset plates in civil infrastructure, and turbine blades, experience complex multi-axial states of stress and strain.

To model these continuous systems, we must discretize a 2D domain. The simplest and most historically significant 2D continuum element is the Constant Strain Triangle (CST), introduced in the seminal 1956 paper by Turner, Clough, Martin, and Topp. The CST element is a three-node triangular element with two translational degrees of freedom (DOFs) per node, resulting in a total of six DOFs per element.

The CST element is categorized as a sub-parametric or isoparametric linear triangular element. Its primary characteristic is that the displacement field within the element is assumed to vary linearly with the spatial coordinates $x$ and $y$. Because the strain field is defined by the spatial derivatives (gradients) of the displacement field, taking the derivative of a linear function yields a constant. Consequently, the strain (and by extension, the stress via Hooke's Law) is uniform throughout the entire domain of the element. This feature gives the element its name: the Constant Strain Triangle.

While geometrically versatile and easy to implement, the CST element exhibits significant physical limitations, such as shear locking and poor convergence in bending-dominated fields. Understanding the theory, mathematical formulation, and limitations of the CST element is essential for selecting the correct element types in engineering simulations.

2D Elasticity and Governing Equations

Before formulating the CST element, we must review the equations of 2D elasticity. In a general 3D continuum, the state of stress at a point is described by a symmetric $3 \times 3$ tensor containing six independent components. In 2D elasticity, we reduce the problem to two dimensions by asserting specific physical assumptions. This yields two distinct states: Plane Stress and Plane Strain.

1. The Plane Stress Assumption

The Plane Stress condition is applicable to thin, flat bodies loaded exclusively in their middle plane. Examples include thin plates, brackets, shear webs, and diaphragms.

The shear strains associated with the $z$-axis remain zero: $\gamma_{xz} = \gamma_{yz} = 0$.

where $E$ is Young's Modulus and $\nu$ is Poisson's ratio.

2. The Plane Strain Assumption

The Plane Strain condition is applicable to long, thick bodies where the geometry and loading do not vary along the longitudinal direction ($z$-axis). Examples include concrete gravity dams, tunnels, retaining walls, long pressurized pipes, and rolling mills.

The transverse shear stresses remain zero: $\tau_{xz} = \tau_{yz} = 0$.

> [!NOTE] > The shear modulus $G$ is identical in both formulations: > $$G = \frac{E}{2(1 + \nu)}$$ > The term in the bottom-right corner of both $[D]$ matrices, when multiplied by the scaling factor, simplifies directly to $G$. Specifically: > * Plane Stress: $\frac{E}{1-\nu^2} \cdot \frac{1-\nu}{2} = \frac{E}{2(1+\nu)} = G$ > * Plane Strain: $\frac{E(1-\nu)}{(1+\nu)(1-2\nu)} \cdot \frac{1-2\nu}{2(1-\nu)} = \frac{E}{2(1+\nu)} = G$

Shape Functions in Cartesian and Area Coordinates

To interpolate the displacement field within the triangular element, we define shape functions. These shape functions can be represented using either Cartesian coordinates $(x, y)$ or Area (Natural) coordinates $(L_1, L_2, L_3)$.

1. Area Coordinates (Natural Coordinates)

Area coordinates are a system of natural coordinates specifically designed for triangular domains. Consider a triangular element with vertices 1, 2, and 3, and an internal point $P(x, y)$. If we draw lines from $P$ to each of the three vertices, we divide the total area $A$ into three sub-triangles: $A_1$, $A_2$, and $A_3$, where $A_i$ is the area of the triangle opposite to node $i$.

The area coordinates $L_1, L_2, L_3$ are defined as the ratios of these sub-areas to the total area $A$:

By definition, the sum of these sub-areas must equal the total area:

This constraint shows that only two of the three area coordinates are independent.

2. Coordinate Transformation

The Cartesian coordinates $(x, y)$ of any point $P$ inside the triangle can be expressed as a linear combination of the nodal coordinates $(x_i, y_i)$ weighted by the area coordinates:

Along with the normalization constraint, we can write this coordinate transformation in matrix form:

3. Linear Shape Functions ($N_i$)

In the Finite Element Method, the shape functions $N_i$ interpolate nodal displacements to represent the displacement at any point within the element:

For a Constant Strain Triangle, the shape functions are equal to the area coordinates:

This formulation satisfies key finite element requirements:

For instance, at Node 1, $L_1 = 1$ and $L_2 = L_3 = 0$.

This property enables the element to undergo rigid-body translation without generating artificial strains.

Characteristics and Limitations of CST Elements

The Constant Strain Triangle is easy to formulate and computationally efficient, but its physical assumptions introduce structural behaviors that must be managed:

1. Step-Like Stress/Strain Fields

Because the strains are constant within each element, the strain and stress fields across a mesh of CST elements are discontinuous at element boundaries. In regions with high stress gradients (e.g., near holes, notches, or crack tips), the physical stresses vary rapidly. A CST mesh approximates this variation as a series of constant steps. To obtain accurate stress fields in these areas, a highly refined mesh is required.

2. Shear Locking (Bending-Dominated Problems)

The most critical limitation of the CST element is shear locking (also known as parasitic shear). When a CST element is subjected to pure bending, the linear displacement field cannot represent the true quadratic curvature.

Under pure bending, the actual normal strains vary linearly through the depth, and the shear strains are zero. However, when a single CST element is bent, its straight edges are forced to remain straight. This displacement constraint induces artificial shear strains within the element.

Since the element cannot bend without shearing, it behaves as if it is much stiffer than it actually is. This artificial stiffness is called shear locking. In bending-dominated applications (such as beams or thin shells modeled with CST elements), the predicted displacements are significantly smaller than the analytical solutions unless an extremely fine mesh is used.

3. Remedy

To avoid shear locking and accelerate convergence, higher-order elements are typically preferred: