Graphical method for BM, shear, thrust in arch

The graphical method finds the reactions, then the resultant force on any section, from a force polygon and a linkage (funicular) polygon drawn to scale.

Steps

1. Draw the arch to a scale (span, rise, crown hinge $C$, supports $A$ and $B$) and mark the loads $P_1, P_2, \dots$ with their lines of action.
2. Force polygon: draw the load line $1\text{-}2\text{-}3\ldots$ to a force scale. Choose a pole $O$ and draw the rays $O1, O2, \dots$.
3. Reactions (using the hinge condition):
   • Combine the loads on the left of $C$ into their resultant $R_1$, and those on the right into $R_2$.
   • For $R_1$ alone, the right part $CB$ is unloaded, so the reaction at $B$ acts along $BC$. The line of $R_1$ meets line $BC$ at $E_1$; join $AE_1$, which is the direction of the reaction at $A$ for $R_1$. Close the triangle of forces.
   • Do the same for $R_2$ (the reaction at $A$ acts along $AC$; join its meeting point with $R_2$ to $B$).
   • Add the two sets of reactions vectorially. This gives $R_A$ and $R_B$ in magnitude and direction.
4. Linkage (thrust line) polygon: start from $A$ with a link parallel to the ray found from $R_A$ and continue the links parallel to the rays across each load. The polygon passes through $A$, $C$ and $B$ (this fixes the pole).
5. Bending moment at a section $D$: measure the vertical intercept $\delta$ between the arch axis and the linkage polygon at $D$. Then

$$M_D = H\times\delta$$

where $H$ is the pole distance (horizontal thrust) scaled from the force polygon. Since $M_D = M_{beam} - H\,y = H\,(y_{thrust} - y)$, the moment is positive (sagging) where the linkage polygon lies above the arch axis and negative (hogging) where it lies below. 6. Normal thrust and radial shear: read from the force polygon the resultant force $R_D$ on the section (the ray, or the sum of the reaction and the loads left of $D$). Draw the tangent to the arch at $D$. Resolve $R_D$:

$$N = R_D\cos\phi \ (\text{along tangent}), \qquad V = R_D\sin\phi \ (\text{perpendicular to tangent})$$

where $\phi$ is the angle between $R_D$ and the tangent.

   Space diagram                Force polygon
      P1   P2                     1 *----.
       |    |        C               |     '. O (pole)
    .--+----+--.   (hinge)         2 *------* 
   A     link polygon  B             |
   (through A, C, B)               3 *
   M = H x (vertical intercept)    H = pole distance

7. Check that the linkage polygon passes through all three points $A$, $C$, $B$ (the hinge condition) and that the force polygon closes.