2D Rigid Plane Frame Elements

CHAPTER 7Element Formulations

Physical Intuition: Mechanics of Rigid Plane Frames

In structural and computational mechanics, a plane frame is a skeletal structure composed of members that lie in a single two-dimensional plane and are connected by rigid joints. Unlike truss members, which only carry axial loads, or beams, which are typically analyzed for transverse loads and bending alone, frame members are subjected to a complex combination of axial forces, transverse shear forces, and bending moments simultaneously.

Structural Context and Applications

Real-world engineering structures—such as portal frames in industrial warehouses, multi-story building frames, offshore platform jackets, machine frames, and aerospace crane structures—rely on rigid joints to transmit moments. The joints are constructed using welds, heavy gusset plates with high-strength pretensioned bolts, or monolithic concrete pours. These connections are designed to be rotationally stiff, meaning they do not permit relative rotation between the members meeting at the joint.

To analyze these systems accurately, we must model them using 2D Rigid Plane Frame Elements. This element represents the structural synthesis of two simpler elements:

The Axial Bar Element: Resists axial elongation and contraction (tension and compression).

The Euler-Bernoulli Beam Element: Resists transverse deflection, shear, and flexural bending.

The Coupling of Axial and Flexural Deformations

A crucial concept in frame analysis is how axial and bending deformations couple:

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Local Uncoupling: Under the assumptions of linear elastic, small-deformation theory, axial and flexural behaviors remain completely independent (uncoupled) within the member's *local* coordinate system. The axial force depends only on axial displacement, and the bending moment depends only on transverse displacement and rotation.

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Global Coupling: In the *global* coordinate system, members are oriented at arbitrary angles. When the structure deforms, a global horizontal displacement at a joint will project onto both the local longitudinal axis and the local transverse axis of a slanted member. Thus, a single global displacement activates both axial stretching and bending deflection. This coordinate-based mapping is the source of geometric coupling, and it is what allows a frame to distribute loads through both axial and bending mechanisms across the entire structure.

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Geometric Non-Linearity (P-Delta Effects): While our primary formulation is linear, in advanced analysis (large deformation theory), a high compressive axial force reduces a member's bending stiffness (compression softening / buckling), while a tensile axial force increases it (tension stiffening). This is known as *physical-geometric coupling*. In this chapter, we focus on the linear elastic formulation where coupling occurs through joint equilibrium and coordinate transformation.

Rigid Joints vs. Pinned Joints

The behavior of a rigid joint is fundamentally different from a pinned joint:

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Pinned Joints (Trusses): The members are connected by frictionless pins. The joints cannot resist moments, and the members are free to rotate independently. The relative angle between members changes under load. Consequently, members act as pure two-force members, carrying only axial load.

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Rigid Joints (Frames): The connection is monolithic. The angles between the member ends at the joint remain constant during deformation. If the joint rotates by an angle $\theta$, the end of every member connected to that joint must rotate by the exact same angle $\theta$. The joint rotates as a single rigid body, enabling the transmission of bending moments from one member to another. This makes frames statically highly indeterminate (hyperstatic), but also significantly stiffer and more redundant than trusses.

Degrees of Freedom (DOFs)

To fully describe the deformation of a 2D frame element, each node must have three degrees of freedom in both the local and global coordinate frames:

$u$ (or $u'$ in local): Translation along the horizontal axis (axial displacement).

$v$ (or $v'$ in local): Translation along the vertical axis (shear/transverse displacement).

$\theta$ (or $\theta'$ in local): Rotational displacement in the plane of the structure (about the out-of-plane $Z$-axis).

Since the element has two nodes, a 2D plane frame element possesses a total of six degrees of freedom, resulting in a $6 \times 6$ element stiffness matrix.

Coordinate Systems: Local vs. Global Frames

To establish the finite element equations, we must work with two distinct coordinate systems:

Local Coordinate System ($x'-y'$): Unique to each individual member. The local $x'$-axis is aligned with the longitudinal axis of the member, pointing from the start node (Node $i$) to the end node (Node $j$). The local $y'$-axis is perpendicular, oriented $90^\circ$ counterclockwise. In this frame, the element stiffness is easy to define because the axial and bending behaviors are decoupled.

Global Coordinate System ($x-y$): A fixed Cartesian coordinate system used for the entire frame structure. All node coordinates, boundary conditions, and external loads are defined in this system.

Orientation and Direction Cosines

The orientation of a frame member is defined by the angle $\beta$ that the local $x'$-axis makes with the positive global $x$-axis, measured counterclockwise.

Let the global coordinates of Node $i$ be $(x_i, y_i)$ and Node $j$ be $(x_j, y_j)$. The length $L$ of the member is:

$$L = \sqrt{(x_j - x_i)^2 + (y_j - y_i)^2}$$

The orientation is defined by the direction cosines:

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Horizontal Direction Cosine ($l$):

$$l = \cos\beta = \frac{x_j - x_i}{L}$$

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Vertical Direction Cosine ($m$):

$$m = \sin\beta = \frac{y_j - y_i}{L}$$

These values satisfy $l^2 + m^2 = 1$. They act as projection coefficients to transform vector quantities (displacements, forces) between the local and global coordinate systems.

Extended chapter reference

Coupled axial-bending members, rigid joints, releases, and load paths

Plane-frame elements combine bar and beam behavior in one transformed matrix. Their three nodal DOFs reproduce axial extension, transverse bending, and in-plane rotation across rigidly connected members.

Assemble the six-DOF local frame stiffness matrix.

Transform member equations and distributed loads into global axes.

Model hinges, offsets, and end releases without creating mechanisms.

Recover axial force, shear, moment, and support reactions.

Visual map

From formulation to verified result

Each stage produces evidence for the next. A reliable result preserves traceability from the physical idealization to the final engineering check.

Equation sheet

Governing relationships

Local response

$$\{f_l\}=[k_l]\{d_l\}+\{f_l^0\}$$

Combines deformation forces with fixed-end forces.

Transformation

$$[k_g]=[T]^T[k_l][T]$$

Rotates a member matrix into structural axes.

Energy check

$$U=\frac{1}{2}\{d\}^T[K]\{d\}$$

Global strain energy must remain nonnegative.

Engineering comparison

Selection and interpretation table

Joint model	Transferred actions	Matrix treatment	Example
Rigid	Axial, shear, moment	Shared translations and rotation	Welded steel joint
Pinned	Axial and shear	Release end moment DOF	Ideal truss connection
Semi-rigid	All with finite rotation	Add rotational spring	Bolted end plate
Offset	Actions through eccentric arm	Rigid-link transform	Beam-column face connection

Verification and application

Checks before accepting the result

No unintended hinge mechanism
Member and global axes are visible
Fixed-end forces are included
Story shear equals applied lateral load

Industry application

A portal frame under wind can be modeled quickly with frame elements, but realistic base and beam-column rotational stiffness may change drift and moment distribution substantially compared with ideal fixed or pinned joints.