Truss ILDs for L0U1, L0L1, U2L2

Assumed truss (figure not available): Pratt truss, 6 panels of 5 m (span 30 m), height 5 m; verticals $U_iL_i$ and diagonals $U_1L_2$, $U_2L_3$, $U_4L_3$, $U_5L_4$; end posts $L_0U_1$ and $L_6U_5$ at 45°. $L_0$ hinge, $L_6$ roller. Reaction for unit load at $x$ from $L_0$: $R_{L_0} = (30-x)/30$.

Method

• $L_0U_1$ (end post): joint $L_0$, vertical equilibrium: $F\sin45^\circ = -R_{L_0}$, so $F_{L_0U_1} = -1.414\,R_{L_0}$ (always compression).
• $L_0L_1$: horizontal equilibrium at $L_0$: $F_{L_0L_1} = -F_{L_0U_1}\cos45^\circ = +R_{L_0}$ (always tension).
• $U_2L_2$ (vertical): the section through $U_2U_3$, $U_2L_3$ and $L_2L_3$ gives the diagonal $F_{U_2L_3} = \sqrt2\,V$, where $V$ is the shear in panel $L_2L_3$ (left forces upward positive). At the unloaded joint $U_2$ the vertical balances the diagonal: $F_{U_2L_2} = -F_{U_2L_3}/\sqrt2 = -V$. For a unit load at $L_2$: $V = \tfrac23 - 1 = -\tfrac13$, so $F_{U_2L_2} = +0.333$. For a unit load on the top chord at $U_2$, the load itself is added at the joint, which gives $-0.667$.

Ordinates when the unit load moves on the bottom chord (+ tension)

Ordinates when the unit load moves on the top chord

Shapes (bottom-chord loading)

• $L_0U_1$: always compression. 0 at $L_0$ (load directly on the support), peak $-1.179$ at $L_1$, then a straight line decreasing to 0 at $L_6$. Beyond $L_1$ the ordinate is $-1.414\,R_{L_0}$.
• $L_0L_1$: always tension. 0 at $L_0$, peak $+0.833$ at $L_1$, then falling linearly to 0 at $L_6$ (ordinate $=R_{L_0}$).
• $U_2L_2$: $+0.167$ at $L_1$, $+0.333$ at $L_2$, $-0.5$ at $L_3$, then $-0.333$, $-0.167$ and 0. The sign changes between $L_2$ and $L_3$, at $x = 10 + 5\times\frac{0.333}{0.833} = 12$ m from $L_0$, so the vertical is stressed in both tension and compression (the part of the ILD with a negative area is larger).
• With top-chord loading, the ordinates of $L_0U_1$ and $L_0L_1$ are the same linear functions of $R_{L_0}$ (taken at the top joints), while $U_2L_2$ takes $-0.667$ at $U_2$, since the load acts directly on the vertical.