Truss ILDs for DK, DJ, DE

Assumed truss (Pratt type, figure not available): bottom chord $A$–$G$, 6 panels of 6 m (span 36 m); top joints $L, K, J, I, H$ at 8 m height above $B, C, D, E, F$; inclined end posts $AL$ and $GH$; verticals above $B$ to $F$; diagonals $LC$, $KD$, $ID$, $HE$ (sloping down towards the centre). $A$ hinge, $G$ roller. The unit load moves along the bottom chord; the ILD is straight between panel points.

Reactions: $R_A = \dfrac{36-x}{36}$, $R_G = \dfrac{x}{36}$.

Member DK (diagonal in panel $CD$)

Section through $KJ$, $KD$ and $CD$. Diagonal $KD$: $\tan\theta = 8/6$, $\sin\theta = 0.8$.

• Load to the right of $D$ (left part considered): $F_{DK}\sin\theta = R_A \Rightarrow F_{DK} = 1.25\,R_A$ (tension).
• Load to the left of $C$ (right part considered): $F_{DK} = -1.25\,R_G$ (compression).
• Between $C$ and $D$ the ILD is the straight line joining the two ordinates, passing through zero.

Member DJ (vertical)

At the unloaded top joint $J$, three members meet: two collinear top-chord members ($KJ$ and $JI$) and the vertical $DJ$, so $F_{DJ} = 0$. For bottom-chord loading the ILD of $DJ$ is zero throughout. (The load at $D$ itself is carried by the diagonals $KD$ and $ID$ which are symmetric, so the vertical is still unstressed.)

Member DE (bottom chord)

Section through $JI$, $ID$ and $DE$ (panel $DE$). $JI$ and $ID$ meet at $I$ ($x = 24$ m, $y = 8$ m), so take moments about $I$ (lever arm of $DE$ = 8 m):

• Load to the left of $D$ (right part): $F_{DE}\times 8 = R_G\times 12 \Rightarrow F_{DE} = 1.5\,R_G$ (tension).
• Load to the right of $E$ (left part): $F_{DE}\times 8 = R_A\times 24 \Rightarrow F_{DE} = 3\,R_A$ (tension).

Ordinates (per unit load; + tension)

Shapes

• DK: negative ordinates for load on $A$–$C$ (maximum $-0.417$ at $C$), positive for load on $D$–$G$ (maximum $+0.625$ at $D$); the zero crossing lies between $C$ and $D$ at $x = 12 + 6\times\frac{0.417}{0.417+0.625} = 14.4$ m from $A$.
• DJ: zero for all positions of the load.
• DE: tension everywhere, rising from 0 at $A$ to a peak $+1.0$ under $E$ ($x = 24$ m), then falling to 0 at $G$.