---
title: "BMD of portal frame with support settlement"
subject: "Theory of Structures II"
chapter: 2
key: "semester_5/theory_of_structures_ii:bmd-portal-frame-support-settlement"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/bmd-portal-frame-support-settlement
site: BCT Engineering
---

# BMD of portal frame with support settlement

Theory of Structures II · Chapter 2: Force method

- Asked in: 2074 Chaitra, 4 marks
- Video: [Analysis of Continuous Beam with Settlement using Moment Distriburtion Mothod](https://www.youtube.com/watch?v=sNuaoClj7-c) — Krishna Gadamalla
- 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/bmd-portal-frame-support-settlement
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Explain with example how the bending moment diagram is drawn for a statically indeterminate portal frame which undergoes settlement of one support.

## Short answer

Settlement of a support in a statically indeterminate frame induces bending moments. The primary structure has zero M0; flexibility equations equate redundant displacements to settlement. The resulting M is linear between joints, proportional to EIδ/L^2, with moments on the tension side.

## Answer

Settlement of a support in an indeterminate frame produces bending moments even without any external load, because the frame is forced to fit the new support position. The BMD is drawn from the redundant reactions caused by the settlement.

#### Procedure
1. Find $D_s$ and choose the redundants (for example the reactions of the settling support). Remove them to get the primary structure.
2. The primary structure moves as a rigid body when the support settles, so $M_0 = 0$ and $\Delta_{i0} = 0$.
3. Find the flexibility coefficients $f_{ij}$ from unit-load moment diagrams $m_i$.
4. Write compatibility: the displacement at each redundant must equal the prescribed settlement $\delta_i$ (measured positive in the direction of $X_i$): $\sum_j f_{ij}X_j = \delta_i$.
5. Solve for $X_j$ and compute $M = \sum X_j m_j$. Since there are no member loads, the BM varies linearly between the joints, so the BMD is a set of straight lines joining the joint moments. Draw each ordinate on the tension side.

Note that the moments are proportional to $EI\,\delta/L^2$, so a stiffer frame develops larger moments for the same settlement.

#### Example
Portal frame ABCD, $h = L = 4$ m, $EI = 20000$ kNm$^2$ constant, A fixed, D hinged. D settles vertically by $\delta = 10$ mm.

Redundants: $X_1$ = $H_D$ ($\rightarrow$), $X_2$ = $V_D$ ($\uparrow$). The unit moment diagrams are: $m_1$ = 0 at A rising to 4 at B, 4 along BC, falling to 0 at D; $m_2$ = 4 along AB, falling from 4 at B to 0 at C.

$$
f_{11} = \frac{106.67}{EI},\quad f_{12} = \frac{64}{EI},\quad f_{22} = \frac{85.33}{EI}
$$

$$
\begin{aligned}
106.67X_1 + 64X_2 &= 0 \\
64X_1 + 85.33X_2 &= -0.01\times20000 = -200
\end{aligned}
$$

$$
X_1 = H_D = 2.56\ \text{kN}\ (\rightarrow),\qquad X_2 = V_D = 4.26\ \text{kN}\ (\text{downward, since D is pulled down})
$$

#### Bending moments (kNm; + tension on the inner face)
| Point | BM |
|---|---|
| A | -17.05 |
| B | -6.82 |
| C | 10.23 |
| D | 0 |

BM is linear in every member; it is hogging (outer tension) in the left column and at B, and sagging at C (inner tension).

## Check yourself

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

1. Why does support settlement produce bending moments in an indeterminate frame?
   A. Because the self weight increases
   B. Because the frame is forced to fit the new support position
   C. Because the joints rotate freely
   D. It does not; moments are zero
   Answer: B. The restraints force deformation of the frame, which creates redundant reactions and moments even without loads. (Key point: Moments without load)

2. What are M0 and Δ_i0 in the primary structure when a support settles?
   A. Both equal to the settlement
   B. M0 = 0 and Δ_i0 = 0 (rigid body movement)
   C. M0 = wL^2/8
   D. Δ_i0 = δ
   Answer: B. The determinate primary structure just moves as a rigid body, so no moments or load displacements arise. (Key point: Primary structure terms)

3. What does the compatibility equation say for settlement?
   A. Σ f_ij X_j = 0
   B. Σ f_ij X_j = δ_i, the prescribed settlement
   C. X = δ
   D. Σ X = EI
   Answer: B. The displacement at each redundant must equal the given settlement: Σ f_ij X_j = δ_i. (Key point: Compatibility)

4. What shape is the BMD between joints?
   A. Parabolic
   B. Cubic
   C. Linear, straight lines joining joint moments
   D. Constant zero
   Answer: C. No member loads act, so the moment varies linearly between joints. (Key point: Shape of BMD)

## Related in Theory of Structures II

- [Moment distribution: portal frame, AFD, SFD, BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amoment-dist-frame-60kn-15knm-afd) — Theory of Structures II, chapter 3
- [Moment distribution: frame with overhangs, AFD, SFD, BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amoment-dist-frame-4000n-2000n-afd) — Theory of Structures II, chapter 3
- [ILD: bending moment at fixed support, 10 m beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-bm-fixed-support-10m-1) — Theory of Structures II, chapter 4
- [Collapse moment of portal frame, load factor 1.75](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-moment-portal-load-factor-1) — Theory of Structures II, chapter 5
- [Moment distribution: frame with SFD, AFD, BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amoment-dist-frame-sfd-afd-bmd) — Theory of Structures II, chapter 3
- [Moment distribution: multi-bay frame, AFD, SFD, BMD](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amoment-dist-multi-column-frame-afd) — Theory of Structures II, chapter 3

## Related in other subjects

- [Statically determinate/indeterminate; beam SFD and BMD](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Adeterminate-indeterminate-beam-sfd-bmd-triangular) — Applied Mechanics, chapter 6
- [Bending stress diagram; moment in flanges](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Al-section-bending-stress-diagram-flange) — Strength of Materials, chapter 7
- [Frame AFD, SFD, BMD (100 kN, inclined member)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-afd-sfd-bmd-100kn-20knm) — Strength of Materials, chapter 1
- [Frame SFD, BMD (100 kN, 20 kN/m, hinge)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-sfd-bmd-100kn-hinge-60deg) — Strength of Materials, chapter 1