--- title: "Strain energy and real work deflection, overhang beam" subject: "Theory of Structures I" chapter: 3 key: "semester_4/theory_of_structures_i:real-work-strain-energy-overhang-5m" source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/real-work-strain-energy-overhang-5m site: BCT Engineering --- # Strain energy and real work deflection, overhang beam Theory of Structures I · Chapter 3: Analysis by the Virtual Work Method - Asked in: 2075 Baisakh, 12 marks - Video: [Deflection of overhanging beam by strain energy method](https://www.youtube.com/watch?v=2nu-K8fVLpw) — GANGADHAR HUGAR - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/real-work-strain-energy-overhang-5m - Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3 ## Question Determine strain energies due to bending and shear in the overhanging beam shown in the figure below and also determine the deflection at C by using the real work method. $E = 200\ \text{kN/mm}^2$, $G = 80\ \text{kN/mm}^2$. [Figure: beam ABC, A hinge, B roller support 5 m from A, free end C 2 m beyond B carrying a 20 kN downward load; rectangular cross-section 100 mm × 120 mm.] ## Short answer Strain energy: bending U_b = 0.648 kNm (648 J), shear U_s = 0.0007 kNm (0.7 J). Real work method gives deflection at C from ½PΔ = U_b+U_s, so Δ_C = 64.9 mm downward. Shear energy is negligible. ## Answer #### Given data Overhanging beam: span $AB = 5$ m (A hinge, B roller), overhang $BC = 2$ m, load $P = 20$ kN at C. Section 100 mm $\times$ 120 mm ($b = 0.1$ m, $d = 0.12$ m). $E = 200$ kN/mm$^2 = 2\times10^8$ kN/m$^2$, $G = 80$ kN/mm$^2 = 8\times10^7$ kN/m$^2$. ```text A B C ^------------^------+ 20 kN | 5 m | 2 m v ``` $$ I = \frac{0.1\times0.12^3}{12} = 1.44\times10^{-5}\ \text{m}^4,\quad EI = 2880\ \text{kNm}^2,\quad A = 0.012\ \text{m}^2 $$ #### Reactions $$ R_A\times5 = 20\times2 \Rightarrow R_A = 8\ \text{kN (down)},\qquad R_B = 28\ \text{kN (up)} $$ #### Bending moments and shear forces (magnitudes) | Portion | $M$ | $V$ | |---|---|---| | $AB$ ($x$ from A, 0 to 5) | $8x$ (hogging, 40 kNm at B) | 8 kN | | $CB$ ($s$ from C, 0 to 2) | $20s$ (hogging) | 20 kN | #### Strain energy due to bending $$ U_b = \int_0^5\frac{(8x)^2}{2EI}dx + \int_0^2\frac{(20s)^2}{2EI}ds = \frac{1}{2EI}\left[\frac{64\times125}{3} + \frac{400\times8}{3}\right] = \frac{3733.33}{2\times2880} $$ $$ U_b = 0.6481\ \text{kNm} = 648.1\ \text{J} $$ #### Strain energy due to shear For a rectangle, the form factor is $\alpha = 1.2$: $$ U_s = \frac{\alpha}{2GA}\left[8^2\times5 + 20^2\times2\right] = \frac{1.2\times1120}{2\times8\times10^7\times0.012} = 7.0\times10^{-4}\ \text{kNm} = 0.70\ \text{J} $$ #### Deflection at C (real work method) External work = total strain energy: $$ \tfrac12 P\Delta_C = U_b + U_s = 0.6481 + 0.0007 = 0.6488\ \text{kNm} $$ $$ \Delta_C = \frac{2\times0.6488}{20} = 0.06488\ \text{m} $$ **Answer:** $U_b = 0.648$ kNm (648 J), $U_s = 0.0007$ kNm (0.70 J), and $\Delta_C = 64.9$ mm downward (bending alone gives 64.81 mm; shear adds only 0.07 mm). The shear energy is about 0.1% of the bending energy. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For the 5 m span with 2 m overhang loaded by 20 kN at C, the reaction at A is: A. 8 kN downward B. 8 kN upward C. 40 kN upward D. 20 kN downward Answer: A. Moments about B give R_A x 5 = 20 x 2, so R_A = 8 kN downward; R_B = 28 kN upward. (Key point: Reactions) 2. The bending moments in the beam are: A. 8x in AB and 20s in CB (s from C) B. 28x in AB and 20s in CB C. 8x in AB and 20 in CB D. 20x in AB and 8s in CB Answer: A. In AB, M = R_A x = 8x (40 kNm at B); in the overhang, M = 20s measured from C. (Key point: Bending moment expressions) 3. For the rectangular section, the shear strain energy uses form factor alpha equal to: A. 1.0 B. 1.2 C. 2.0 D. 0.5 Answer: B. Rectangle: alpha = 1.2, in U_s = (alpha / 2GA) integral of V^2 dx. (Key point: Shear form factor) 4. How is deflection at C obtained? A. (1/2) P Delta_C = U_b + U_s B. P Delta_C = U_b C. Delta_C = U_b / P^2 D. Delta_C = U_s x 2 Answer: A. External work (1/2)P Delta_C equals total strain energy 0.6488 kNm, giving Delta_C = 64.9 mm. (Key point: Real work for deflection) 5. Compared with bending, the shear strain energy is: A. About 50% of bending B. About 10% of bending C. About 0.1% of bending D. Equal Answer: C. U_s = 0.70 J against U_b = 648 J, so shear is about 0.1% and is usually neglected. (Key point: Shear contribution) ## Related in Theory of Structures I - [Overhang beam free end deflection by virtual work](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Abeam-vw-free-end-d-2065) — Theory of Structures I, chapter 3 - [Moment area: overhang beam free-end deflection](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-overhang-10kn-m-2078) — Theory of Structures I, chapter 4 - [Conjugate beam: overhanging beam free end](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aconjugate-overhang-300-150) — Theory of Structures I, chapter 4 - [Strain energy deflection at A, spring support](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Astrain-energy-beam-spring-deflection-a) — Theory of Structures I, chapter 2 - [Conjugate beam: slope and deflection at free end](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aconjugate-slope-defl-free-end-2073) — Theory of Structures I, chapter 4 - [Conjugate beam: slope and deflection at free end C](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aconjugate-udl-2knm-10kn-5kn-2068) — Theory of Structures I, chapter 4, asked 2× ## Related in other subjects - [T-beam with overhang: max bending stress](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3At-beam-4kn-m-10kn-overhang) — Strength of Materials, chapter 7 - [Beam ABC with wire support: BMD and SFD](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Abeam-wire-support-sfd-bmd) — Strength of Materials, chapter 1 - [Rotation and deflection at free end, BMD](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Abeam-fem-rotation-deflection-free-end) — Computational Techniques in Civil Engineering, chapter 4 - [Cantilever: slope and deflection at free end](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acantilever-slope-deflection-50kn-40kn) — Strength of Materials, chapter 7