Truss analysis by the method of joints

Member forces of a simple triangular truss under a load at the apex: the equilibrium of the joint is written as a matrix equation and solved in one line, then the bottom chord follows from the next joint.

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Truss member forces found by solving the joint equilibrium equations.

The calculation

P := 30 kN
s := 6 m
h_t := 2 m
alpha := atan(h_t/(s/2)) =
alpha*180/pi =
K_m := [cos(alpha), -cos(alpha); sin(alpha), sin(alpha)]
F := lusolve(K_m, [0; -P/(1 kN)]) =
F_AC := F[1,1]*1 kN = kN
F_BC := F[2,1]*1 kN = kN
F_AB := -F_AC*cos(alpha) = kN
R_A := P/2 = kN

Results for these inputs

Roof angle 33.7°.
Both rafters carry 27.0 kN in compression (negative).
The tie carries 22.5 kN in tension.
Each support reaction is 15 kN.

How it works

At the apex the horizontal and vertical components of the two member forces must balance the load: two equations, two unknowns. lusolve solves the system; with more joints, the same matrix simply grows. Tension is positive, compression negative.

These calculations use classical textbook mechanics, not a design code. For design, use the method, factors and checks that apply to your work and have the result checked by a competent person.

Bigger trusses

For larger trusses, write the equilibrium of every joint into one matrix, or use the stiffness matrix method. See also the matrix calculator.

Questions

How does the method of joints work?

Each joint is in equilibrium, so the forces in the members meeting there, resolved horizontally and vertically, must balance the loads and reactions. Writing these equations for the joints gives a linear system for the member forces.

Can Equation Pad solve large systems?

Yes. lusolve solves systems of any size; matrices can be typed, built from blocks or taken from a spreadsheet.

Also used for

Equation Pad runs in your browser, with nothing to install, and we never see or store your calculations. Results must be checked by a competent person before they are relied on; see the disclaimer.

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