--- title: "ILD as system-specific; focal point recurrence" subject: "Theory of Structures II" chapter: 4 key: "semester_5/theory_of_structures_ii:ild-system-specific-focal-point-recurrence" source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/ild-system-specific-focal-point-recurrence site: BCT Engineering --- # ILD as system-specific; focal point recurrence Theory of Structures II · Chapter 4: Influence line (IL) for continuous beams - Asked in: 2071 Shrawan, 5 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/ild-system-specific-focal-point-recurrence - Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-4 ## Question Explain influence line diagram as a system specific diagram. Derive the expression of the recurrent formula for focal point ratio considering two consecutive spans for loading on right spans. ## Short answer An influence line is system-specific: it depends only on spans, supports and stiffness, not on loading, and can be used for any load once drawn. For loading on the right, the left focal ratio recurrence is k_{n+1} = (L_{n+1}/I_{n+1})/[(2-k_n)L_n/I_n + 2L_{n+1}/I_{n+1}], with hinged end k = 0 and fixed end k = 1/2. ## Answer #### ILD as a system-specific diagram An influence line shows how one response (a reaction, shear or moment at one fixed location) varies as a unit load moves over the structure. It depends only on the **system**: the spans, support types, flexural rigidities and the chosen response, and not on the loads. So: - the ILD of $R_B$ for a propped cantilever (zero at A, 1 at B, curved between) is quite different from that of a simply supported beam of the same length, and a beam with spans 6 m + 8 m has different ILDs from one with 8 m + 8 m; - once the ILD for a system is drawn, it can be used for any loading on that same system: $Q = \sum P_i y_i$ for point loads, $Q = w\times(\text{area of ILD under the load})$ for a UDL. By contrast, a BMD or SFD is **load-specific**: it shows one response along the whole beam for one given loading. An ILD is system-specific and response-specific. #### Recurrence formula of the left focal ratio (loading on right spans) Consider a continuous beam with spans numbered from the left; span $n$ lies between supports $n-1$ and $n$, with length $L_n$ and moment of inertia $I_n$. Let the load be on a span to the right of span $n+1$ (all spans from the left end up to $n+1$ unloaded), and let moments be taken with one sign convention (hogging or sagging) at all supports. **Three-moment equation** at support $n$ for the unloaded spans $n$ and $n+1$ (the load terms are zero): $$ M_{n-1}\frac{L_n}{I_n} + 2M_n\left(\frac{L_n}{I_n}+\frac{L_{n+1}}{I_{n+1}}\right) + M_{n+1}\frac{L_{n+1}}{I_{n+1}} = 0 $$ Because the moment in an unloaded span is linear, the moments at its two ends have opposite signs and a fixed ratio. Define the **left focal ratio** of span $n$: $$ k_n = -\frac{M_{n-1}}{M_n} $$ and similarly for span $n+1$: $M_n = -k_{n+1}M_{n+1}$. Substitute $M_{n-1} = -k_nM_n$ in the equation: $$ M_n\left[(2-k_n)\frac{L_n}{I_n} + 2\frac{L_{n+1}}{I_{n+1}}\right] = -M_{n+1}\frac{L_{n+1}}{I_{n+1}} $$ so that $M_n/M_{n+1} = -k_{n+1}$ gives the **recurrence formula** $$ \boxed{\,k_{n+1} = \frac{L_{n+1}/I_{n+1}}{(2-k_n)\,L_n/I_n + 2\,L_{n+1}/I_{n+1}}\,} $$ For equal spans with constant $EI$ this becomes $k_{n+1} = \dfrac{1}{4-k_n}$. **Starting value:** if the left end of the beam is hinged, $M_0 = 0$ so $k_1 = 0$; if it is fixed, the slope there is zero, $2M_0 + M_1 = 0$, so $k_1 = \tfrac12$. **Focal point:** the straight BMD of unloaded span $n$ is $M(x) = M_n\left[1 - (1+k_n)\dfrac{x}{L_n}\right]$, with $x$ measured from support $n$, so it crosses the beam axis at $x = \dfrac{L_n}{1+k_n}$ from support $n$. This point is the left focal point, and it is the same for any loading on the spans to the right of span $n$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. An ILD depends on A. The loads applied B. Only the beam length C. The system (spans, supports, EI, response), not the loads D. The load position only Answer: C. Once drawn it serves any loading on the same system. (Key point: System-specific) 2. For a UDL, the response from an ILD is A. w x area of ILD under the load B. w x maximum ordinate C. Sum of ordinates D. w / area Answer: A. Point loads use P_i y_i; UDL uses w times area. (Key point: Use) 3. k_1 for a hinged left end and fixed left end are A. 1/2 and 0 B. 0 and 1/2 C. 0 and 1 D. 1 and 1/2 Answer: B. Hinged: M0 = 0; fixed: 2M0 + M1 = 0 gives k = 1/2. (Key point: Starting values) 4. The recurrence is derived by A. Applying virtual work to span n B. Equating deflections at the supports C. Substituting M_(n-1) = -k_n M_n into the three-moment equation with zero load terms D. Integrating the BMD Answer: C. The three-moment equation with M_n = -k_(n+1)M_(n+1) gives the formula. 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