---
title: "ILD: shear at section x-x, two-span beam"
subject: "Theory of Structures II"
chapter: 4
key: "semester_5/theory_of_structures_ii:ild-shear-section-xx-two-span"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/ild-shear-section-xx-two-span
site: BCT Engineering
---

# ILD: shear at section x-x, two-span beam

Theory of Structures II · Chapter 4: Influence line (IL) for continuous beams

- Asked in: 2072 Kartik, 6 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-shear-section-xx-two-span
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-4

## Question

Draw influence line diagram for shear at the section x-x for the two-span continuous beam shown in the figure below. Draw the ordinate at 2 m interval. [Figure: two-span beam, hinged at the left end, rollers at the middle and right end; each span 8 m; section x-x at 4 m from the middle support, in the right span.]

## Short answer

Use V = simple-span shear minus M_B/L_2 for the section in the right span. There is a unit jump at the section: -0.406 to +0.594 at x = 12 m; ordinates are zero at A, B and C, positive in AB up to about 0.096, and negative between B and the section.

## Answer

Sign convention: sagging moment (+), hogging moment (-); shear positive when the left part tends to go up; reaction positive upward. Ordinates are for a unit load moving along the beam.

#### Reading of the figure
Two equal spans $AB = BC = 8$ m, hinged at A, rollers at B and C. Section x-x is in BC, 4 m from B (12 m from A).

#### Method
For a section in span BC the shear is that of the simple span BC corrected for the support moment $M_B$ (which is hogging, so it changes the end reaction at C by $M_B/L_2$):
$$
V_{x} = V_{\text{simple}} - \frac{M_B}{L_2}
$$
where $V_{\text{simple}} = 0$ for a load in AB, $+(L_2 - z)/L_2$ for a load right of the section ($z$ measured from B, z > 4), and $-z/L_2$ for a load left of the section in BC. With $L_1 = L_2 = 8$: $M_B = -\dfrac{x(64-x^2)}{256}$ for a load in AB, $M_B = -\dfrac{y(64-y^2)}{256}$ for a load in BC.

Sample: load at x = 4 (in AB): $M_B = -0.75$, $V = -(-0.75)/8 = +0.094$.
Sample: load at the section from the left ($z = 4$): $V_{simple} = -0.5$, $M_B = -0.75$, $V = -0.5 + 0.094 = -0.406$; from the right: $+0.5 + 0.094 = +0.594$.

#### Ordinates (2 m interval)
| x from A (m) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 |
|---|---|---|---|---|---|---|---|---|---|
| $V_{x-x}$ | 0 | 0.0586 | 0.0938 | 0.082 | 0 | -0.168 | -0.4062 / 0.5937 | 0.3086 | 0 |

#### ILD shape
```text
                                    |*
                                    | ***
                                    |    ***
         **********                 |       ***
*********----------*********--------|----------**
                            ****    |
                                ****|
                                    |

^                       ^                       ^
```
Jump of 1.0 at the section ($-0.406$ to $+0.594$). The ILD is positive but small in AB (maximum about 0.096) and negative in BC between B and the section.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For a section in BC, V_x =
   A. V_simple + M_B x L2
   B. V_simple only
   C. V_simple - M_B/L2
   D. M_B/L2 - 1
   Answer: C. M_B changes the end reaction at C by M_B/L2. (Key point: Formula)

2. Jump at the section is
   A. 1.0, from -0.406 to +0.594
   B. 0.5
   C. 0.094
   D. 0.406
   Answer: A. Shear ILD jumps by unity at the section. (Key point: Jump)

3. For a load at x = 4 in AB, V is
   A. -0.094
   B. +0.094
   C. 0
   D. +0.5
   Answer: B. V = -(-0.75)/8 = +0.094. (Key point: Load in AB)

## Related in Theory of Structures II

- [ILD: moment at middle support, two-span beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-moment-mid-support-two-span) — Theory of Structures II, chapter 4
- [ILD: moment at section x-x, continuous beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-moment-section-xx-two-span) — Theory of Structures II, chapter 4
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## Related in other subjects

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