In statics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem


where $F_1$, $F_2$ and $F_3$ are the magnitudes of the three coplanar, concurrent and non-collinear vectors which keep the object in static equilibrium, and $\alpha$, $\beta$ and $\gamma$ are the angles directly opposite to the vectors (see Figure 1).

Figure 1

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy. Its proof is essentially based on the law of sines.

On the Internet there are hundreds of static equilibrium problems where they apply Lami's theorem to a three-force system, see for instance Dubey - Engineering Mechanics: Statics and Dynamics, section 3.10. Although Dubey's book is recent (2013), there is not a single equilibrium problem based on a four-force system. Coincidentally, the author of this note has come across questions on the Internet questioning the possibility of applying Lami's theorem for more than three forces. In this note we give a generalization of Lami's theorem for four forces.

Theorem 1 (Generalization). If four coplanar, concurrent and non-collinear forces act upon an object, and the object remains in static equilibrium, then

Figure 2


where $A$, $B$, $C$ and $D$ are the magnitudes of the four vectors and $\alpha'$, $\beta'$, $\gamma'$ and $\delta'$ are the angles between them (see Figure 2).

Proof. Consider the quadrilateral formed by the four vectors in such a manner that the head of one touches the tail of another (see Figure 3) and denote $\Delta$ its area. If $\alpha$, $\beta$, $\gamma$ and $\delta$ are the interior angles of the quadrilateral, then its area can be written as Figure 3 $$\Delta=\frac12AD\sin{\alpha}+\frac12BC\sin{\gamma}=\frac12AB\sin{\beta}+\frac12CD\sin{\delta}\tag{3}$$ and as $\sin{\alpha'}=\sin{(\pi-\alpha)}=\sin{\alpha}$, and similarly for $\beta'$, $\gamma'$ and $\delta'$, the relation in $(2)$ follows. $\square$

Theorem 1 is a generalization in the sense that if one of the vectors vanishes, the relation we obtain is that of Lami's theorem. Indeed, for instance suppose $C=0$, then the relation $(2)$ reduces to $$D\sin{\alpha'}=B\sin{\beta'},\tag{4}$$ which is Lami's theorem.

Questions: Is Lami's theorem used in the design of cranes as the image below suggest? If so, could the generalization presented be used in the design of more efficient cranes given that we have 4 vectors instead of 3?

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  • $\begingroup$ I am working on a method that replaces the use of vector components (as Lami's theorem does for three-force systems) for four-force based systems, both in 2D and 3D. So any approach based on vector components falls outside my purposes. $\endgroup$ Commented Jul 14, 2022 at 19:34

1 Answer 1


Sure. I don't think I've ever directly used the formula, but when you derive the equations you obviously end up with the same calculation in cases where your members have has 3 vectors. So yes i have used the same derived formula for optimizations. This thing comes up quite often in calculating member force in brackets. The general form is useful for trusses, but honestly i would use matrix solvers for that.

Though, this formula is nice for the crane operators so that they can check their maximum loads and load conditions. Though usually they look it form a graph.

  • $\begingroup$ I have edited my question to include a generalization for a general quad where I remove the angles restriction. Are cranes designed using quadrilateral of forces? Because all cranes I have seen use a triangle of forces. $\endgroup$ Commented Jul 12, 2022 at 16:19
  • 1
    $\begingroup$ No trusses are designed out of triangles (because a quad has a degree of freedom) which obviously due to adjacency have 3, 4, 5 or 6 forces on a node. Anyway statics for trusses is usually done by matrix notation as its easy to input and let the computer solve, global minima is relatively easy to search with a computer. Especially since the same notation translates quite easily to a finite element method input this accounts for the beams bending which yout equations dont help with. $\endgroup$
    – joojaa
    Commented Jul 12, 2022 at 16:33
  • $\begingroup$ However, I have never seen Lami's theorem for 4 vectors, do you have a reference? $\endgroup$ Commented Jul 12, 2022 at 21:29

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