A means of simulating loads in a truss is to represent each truss as a matrix of vectors (a tensor).

If external forces are exerted only at junctions of linear struts, which are assumed not to flex or stretch in any fashion, then the struts can be treated as simply the means of transferring force from one junction to the next. Forces within the truss can be approximated as being exerted by one junction on another. Thus, information on all forces within the truss can be contained within some matrix of vectors F, where each element vector F_ij of the matrix corresponds to the vector force exerted by junction i on junction j.

i.e., Vector F₁₂ would be the force exerted by junction 1 on junction 2, Vector F₂₁ would be the force exerted by junction 2 on junction 1, Vector F₃₅ would be the force exerted by junction 3 on junction 5, Vector F₁₁ would be the force exerted by junction 1 on itself (an external force), etc.

Since each element of the matrix is a vector, itself composed of three orthogonal elements (the x, y, and z components of each vector) the matrix of vectors F can be represented as a set of three independent matrices of scalars: F_x, the matrix of x-components for the force vectors; F_y, the matrix of y-components; and F_z, the matrix of z components. F_x, F_y and F_z are components of F.

F = F_x i + F_y j + F_z k

is true for the matrix F just like

F_ij = F_ij,x i + F_ij,y j + F_ij,z k

is true for each individual force vector.

If the truss is assumed to be static (not accelerating, all forces in equilibrium), then the matrices F_x, F_y and F_z will all have certain properties in common.

Property A: SF_ij,e = 0, for all rows i and all bases e

(i.e., the sum of each row of each component matrix will be zero)

Property B: SSF_ij,e = 0, for all bases e

(i.e., the sum of all elements of the component matrix will be zero)

Property C: SF_ii,e = 0, for all bases e

(i.e., the sum of elements corresponding to external forces – the diagonal sum – will be zero)

Property D: Det(F_e) = 0, for all bases e

(i.e., the determinant of each component matrix is zero)

Property E: Fij,e = -Fji,e for all bases e

(i.e., the force from junction i to junction j is always the exact opposite of the force from j to i.)

The matrix of vectors, F, with the exception that it does not have a determinant, will share the summation properties of its components because it is the sum of its components.

Now, how do you use that?

Well, consider a simple bridge truss, consisting of five junctions.

The forces between junctions in the truss can be written in a 5x5 matrix, F.

Because opposite forces within the same strut cancel each other out (Property E), we can write F as:

The values where the force is zero correspond to places in the truss where there is no strut connecting the corresponding points, and thus, according to our initial assumptions, no force.

The sum of elements in each row is zero, giving us these force equations (Property A):

F₁₁ + F₁₂ + F₁₅ = 0

-F₁₂ + F₂₂ + F₂₃ + F₂₄ + F₂₅ = 0

-F₂₃ + F₃₃ + F₃₄ = 0

-F₂₄ - F₃₄ + F₄₄ + F₄₅ = 0

-F₁₅ - F₂₅ - F₄₅ + F₅₅ = 0

The sum of the diagonal elements is also zero (Property C).

F₁₁ + F₂₂ + F₃₃ + F₄₄ + F₅₅ = 0

Given initial values for the external forces (diagonal elements), these six equations can be used to determine the value of each element of the matrix using simple algebra. However, the results are not general for all trusses. However, if this 5x5 matrix of three-dimensional vector elements is instead written as a 5x5x3 array, it can be entered into a computer program for bootstrap (incremental) calculation of a solution using the methods of linear algebra. This will work for any truss, of any size.