Truss ILDs for LC, LK, LD, lower chord

Assumed truss (Pratt type): bottom joints $A$–$G$, top joints $N, M, L, K, J, I, H$ directly above them, 6 panels of 6 m (span 36 m), height 8 m; verticals at every joint; diagonals $NB$, $MC$, $LD$, $JD$, $IE$, $HF$ (sloping down towards the centre). $A$ hinge, $G$ roller. The load moves on the lower chord. Reactions: $R_A = \dfrac{36-x}{36}$, $R_G = \dfrac{x}{36}$. The diagonal $LD$ has length $10$ m, $\sin\theta = 0.8$.

Member LK (top chord, panel $CD$)

Section through $LK$, $LD$ and $CD$ (panel $CD$); $LD$ and $CD$ meet at $D$ ($x = 18$ m), so take moments about $D$ (lever arm 8 m): $F_{LK} = -\dfrac{M_D}{8}$.

• Load right of $D$: $F_{LK} = -\dfrac{18R_A}{8} = -2.25R_A$; load left of $C$: $F_{LK} = -\dfrac{18R_G}{8} = -2.25R_G$ (compression).

Member LD (diagonal)

Same section: the panel shear is carried by the diagonal, $F_{LD} = \pm\dfrac{V}{0.8}$.

• Load right of $D$: $F_{LD} = +\dfrac{R_A}{0.8} = 1.25R_A$ (tension).
• Load left of $C$: $F_{LD} = -1.25R_G$ (compression).

Member LC (vertical)

At the unloaded top joint $L$ the vertical balances the vertical component of the diagonal: $F_{LC} = -0.8\,F_{LD}$.

Ordinates (per unit load; + tension, − compression)

Shapes

• LK: compression only, triangle with peak $-1.125$ under $D$ (centre).
• LD: compression (peak $-0.417$ at $C$) for load left of the panel, tension (peak $+0.625$ at $D$) for load to the right; the neutral point is at $x = 12+6\times\frac{0.417}{0.417+0.625} = 14.4$ m from $A$.
• LC: tension for load on the left (peak $+0.333$ at $C$), compression to the right (peak $-0.5$ at $D$); the sign change lies between $C$ and $D$ at $x = 12 + 6\times\frac{0.333}{0.333+0.5} = 14.4$ m.