--- title: "Series and parallel phase model; solids in hcp" subject: "Concrete Technology and Masonry Structure" chapter: 2 key: "semester_5/concrete_technology_and_masonry_structure:series-parallel-phase-model" source: https://www.bctengineer.com/q/semester_5/concrete_technology_and_masonry_structure/series-parallel-phase-model site: BCT Engineering --- # Series and parallel phase model; solids in hcp Concrete Technology and Masonry Structure · Chapter 2: Structure of concrete - Asked in: 2080 Baisakh, 3+3 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/concrete_technology_and_masonry_structure/series-parallel-phase-model - Whole chapter: https://www.bctengineer.com/md/semester_5/concrete_technology_and_masonry_structure/chapter-2 ## Question Explain the series and parallel phase model of mortar, interface and coarse aggregate of concrete during load transfer. Write in short about the solids present in hydrated cement phase. ## Short answer The parallel model assumes phases share strain, giving upper bound modulus E_c = E_m V_m + E_a V_a. The series model assumes phases share stress, giving lower bound 1/E_c = V_m/E_m + V_a/E_a. Actual modulus lies between; the weak interface reduces stress transfer. Hydrated cement solids are C-S-H gel, calcium hydroxide, calcium sulphoaluminates, and unhydrated grains. ## Answer #### Series and parallel models Concrete is idealised as a composite of **mortar (hcp + sand)**, the **interface (ITZ)** and **coarse aggregate**. Their arrangement under load decides how the stress is transferred. ```text Parallel model Series model load ↓ ↓ ↓ load ↓ +----+---+----+ +---------+ | M | A | M | | mortar | | | | | +---------+ +----+---+----+ |aggregate| same strain +---------+ each phase same stress, shares load strains add ``` - **Parallel (Voigt) model**: the phases act side by side; each has the same strain and the load is shared in proportion to stiffness. Upper bound of modulus: $$ E_c = E_m V_m + E_a V_a $$ - **Series (Reuss) model**: the phases are in layers one after the other; each carries the same stress and strains add. Lower bound: $$ \frac{1}{E_c} = \frac{V_m}{E_m} + \frac{V_a}{E_a} $$ where $V$ is the volume fraction (subscripts $m$ = mortar, $a$ = aggregate). The actual modulus of concrete lies between the two bounds (a combination of both models, as in Hirsch's model); including the weak ITZ in series reduces it. Load transfer is by bond/shear at the interface, so the weak interface causes microcracks and nonlinearity. #### Solids in hydrated cement paste - **C-S-H gel** (50-60%): main binding product. - **Calcium hydroxide** (20-25%): hexagonal crystals. - **Calcium sulphoaluminates** (15-20%): ettringite, monosulphate. - **Unhydrated cement grains.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the parallel (Voigt) model, which holds? A. Same stress, strains add B. Same strain, load shared by stiffness; upper bound Ec = Em Vm + Ea Va C. Lower bound D. Aggregate carries no load Answer: B. Phases side by side share strain and give the upper bound. (Key point: Parallel model) 2. The series (Reuss) model gives... A. Upper bound B. Lower bound 1/Ec = Vm/Em + Va/Ea with equal stress C. Equal strain D. Exact value Answer: B. Layers carry same stress; strains add, giving the lower bound. (Key point: Series model) 3. Where does actual concrete modulus lie? A. Above both B. Between the two bounds C. Below both D. Equal to aggregate E Answer: B. Real concrete combines both arrangements (Hirsch). (Key point: Actual modulus) 4. Which is the largest solid in hcp? A. Calcium hydroxide B. C-S-H gel 50-60% C. Ettringite D. Unhydrated cement Answer: B. C-S-H is 50-60% of solids. 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