The plane truss shown in the figure is hinge-supported at E and F. The truss is subjected to vertical downward force at R and horizontal force at G. (Figure not to scale) The force (in kN) along with its nature in member JF is
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Find the support reactions first (F_y=25 kN, from moments about E), then use a vertical section through the last panel and vertical equilibrium to isolate member JF.
Step 1: Set coordinates and find the support reactions.
Place E at the origin. Using the 3 m panel widths and the two 3 m storey heights shown in the figure: G=(0,3), R=(9,6), and F=(12,0). The 40 kN load acts rightward (+x) at G, and the 20 kN load acts downward (-y) at R.
Taking moments about E (E and F sit at the same level, y=0, so a horizontal reaction at F has zero moment arm about E and drops out of this equation regardless of the exact support type at F):
\[
\Sigma M_E = 0:\quad -(3)(40) - (9)(20) + 12F_y = 0 \;\Rightarrow\; -120-180+12F_y=0 \;\Rightarrow\; F_y = 25\text{ kN (upward)}
\]
From \(\Sigma F_y=0\): \(E_y + F_y = 20 \Rightarrow E_y = -5\) kN. From \(\Sigma F_x=0\): \(E_x + F_x = -40\) kN (the horizontal components split between the two hinges, but this split does not change the vertical reaction found above).
Step 2: Trace the load path along the top chord using zero-force joints.
At joint L, only two non-collinear members meet (L-M and L-G) with no load, so both must carry zero force. That zero cascades: at joint M (members L-M, M-N, and diagonal G-M, no load) the same argument forces M-N and G-M to zero; the same pattern repeats at joint N, giving zero in N-R as well. So the entire top chord segment from L up to R carries no force before the load point, and the 40 kN entering at G is carried across by the diagonals and the middle chord instead.
Step 3: Bring the 40 kN load down through the middle chord.
At joint G, with the zero-force members above already known, horizontal equilibrium under the 40 kN load gives the middle-chord member G-H a compressive force of 40 kN, and this same value is carried through H-I and I-J by the same style of joint-by-joint equilibrium along the middle chord, since each of these joints has no external load of its own.
Step 4: Use a vertical section through the last panel.
Cut the truss with a vertical section just to the right of the R-J-V line, isolating the corner piece containing S, K and F. Because the top-chord member entering this piece (R-S) traces back to the same zero-force chain identified in Step 2, it carries no force, and the only members left standing in the vertical (y) direction across this cut are the reaction \(F_y\) and the diagonal member J-F (inclined at 45 degrees, since it spans 3 m horizontally and 3 m vertically). Vertical equilibrium of this isolated piece gives:
\[
\Sigma F_y = 0:\quad F_y + T_{JF}\sin 45^\circ = 0 \;\Rightarrow\; T_{JF} = -\sqrt{2}\,F_y = -25\sqrt{2}\text{ kN}
\]
The negative sign means the member is in compression, so this line of reasoning gives \(25\sqrt{2}\) kN compression.
Step 5: Be upfront about where this does not match the given key.
The reaction calculation in Step 1 (\(F_y=25\) kN) only needs the load positions and support levels, so it is solid. However, the exact bracing pattern inside each panel (which members are verticals versus diagonals in the crowded region around R, J and V) could not be pinned down with full certainty from the printed figure, and a small change there shifts which members share the load at the J-F cut. Following the reasoning above consistently gives \(25\sqrt{2}\) kN compression (option C), which does not match the answer key's option A (\(10\sqrt{2}\) kN compression). The compression sense agrees with the key; the working above is the most defensible reading of the figure, and the numeric mismatch is flagged here for a check against a clearer copy of the truss diagram rather than silently forced to agree with the key.
\[ \boxed{T_{JF}\text{ works out to }25\sqrt{2}\text{ kN (compression) by this method; flagged against the stated key of }10\sqrt{2}\text{ kN compression.}} \]
Note: As per the official GATE 2026 answer key, the correct option is (A).
