This module explores the characteristics of forces, various types of force systems, and methods for resolving and composing forces. In architectural structures, forces rarely act in isolation; they form complex systems that must be simplified to be analyzed.

When multiple forces act together on a body, they form a "force system." We classify them based on their lines of action to determine the correct mathematical method of analysis. The primary classifications are based on whether forces share a plane (Coplanar vs. Spatial) and whether their lines of action intersect (Concurrent vs. Parallel vs. Non-Concurrent).

In a right-handed 3D coordinate system, forces are expressed using Cartesian unit vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, corresponding to the $x$, $y$, and $z$ axes, respectively.

A force vector $\mathbf{F}$ is written as:

The magnitude of the force vector is determined using the 3D Pythagorean theorem:

The direction of $\mathbf{F}$ is defined by coordinate direction angles $\alpha$, $\beta$, and $\gamma$ measured from the positive $x$, $y$, and $z$ axes. The components are related by direction cosines: $F_x = F \cos \alpha$, $F_y = F \cos \beta$, and $F_z = F \cos \gamma$.

To analyze a structure, we often need to simplify the force system. We do this through two fundamental processes: Resolution (breaking a force apart) and Composition (adding forces together).

1. Force Resolution (Components): If a diagonal cable pulls on a wall with a force at an angle from the horizontal, it is difficult to add it directly to other vertical or horizontal forces. We resolve it into orthogonal (perpendicular) components using trigonometry. By finding the X (horizontal) and Y (vertical) components, we know exactly how much force is trying to slide the wall sideways versus lifting it upwards.

2. Composition (Resultant): Composition is the reverse of resolution. If multiple forces act on a structural joint, we want to mathematically combine them to find a single Resultant Force. The resultant is the single theoretical force that produces the exact same external effect on the rigid body as the entire original system of forces combined.

Varignon's Theorem mathematically proves that the total moment produced by a complex resultant force about any specific point is perfectly equal to the sum of the individual moments produced by all of its component forces about that exact same point.

While 2D analysis only considers the X and Y axes, real buildings exist in 3D space. To analyze a true spatial force system, such as wind hitting the asymmetrical corner of a skyscraper, we must introduce the Z-axis (depth).

When calculating the resultant of a 3D concurrent force system, you expand the 2D procedure to include the third dimension. You resolve every force into three distinct components ($F_x, F_y, F_z$), sum them independently along each axis to find $R_x, R_y, R_z$, and then calculate the 3D Resultant Magnitude using the expanded Pythagorean theorem.

Finding the exact direction of a 3D resultant is more complex than a single 2D angle. It requires calculating "Direction Cosines"—the angles the resultant vector makes with the X, Y, and Z axes ($\alpha, \beta, \gamma$). Modern architectural engineering relies heavily on 3D Cartesian vector mathematics and matrix algebra to solve these complex spatial systems computationally.

In modern structural engineering software (like SAP2000 or ETABS), spatial (3D) force systems are not solved using simple trigonometry. Instead, forces and moments are represented mathematically as vectors and tensors within a matrix formulation.

A force vector in 3D space is formally defined using Cartesian unit vectors ($\mathbf{i}, \mathbf{j}, \mathbf{k}$):

The resultant of a complex 3D force system is computed via matrix addition, which allows computers to solve millions of simultaneous equilibrium equations for large-scale architectural frames efficiently.

In advanced 3D spatial systems, a complex force and moment combination can be simplified into a Wrench Resultant: a single resultant force and a collinear couple moment vector, useful for complex architectural structural nodes.

The Principle of Transmissibility states you can slide a force anywhere along its line of action without changing its external effect on the rigid body (e.g., the support reactions remain identical).

However, this principle fails completely when analyzing the internal forces and stresses of a building.

Before you can resolve or compose complex force systems, you must mathematically isolate the body. While covered deeply in Equilibrium, the first step of resolving forces is ensuring your system boundary is strictly defined. You cannot mathematically combine a wind force pushing on a wall with the tension in a completely unrelated interior cable. All forces in your system must be acting upon the exact same isolated free body.