--- title: "Flexible vs rigid design; Westergaard temperature stresses" subject: "Transportation Engineering II" chapter: 2 key: "semester_7/transportation_engineering_ii:flexible-vs-rigid-design-westergaard-temperature" source: https://www.bctengineer.com/q/semester_7/transportation_engineering_ii/flexible-vs-rigid-design-westergaard-temperature site: BCT Engineering --- # Flexible vs rigid design; Westergaard temperature stresses Transportation Engineering II · Chapter 2: Highway Pavement - Asked in: 2072 Chaitra, 8 marks - Video: [Temperature stresses in rigid pavement](https://www.youtube.com/watch?v=ZcagKyhc4-Y) — Online Study - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/transportation_engineering_ii/flexible-vs-rigid-design-westergaard-temperature - Whole chapter: https://www.bctengineer.com/md/semester_7/transportation_engineering_ii/chapter-2 ## Question Differentiate between flexible pavement design and rigid pavement design. Describe Westergaard's concept for temperature stresses. ## Short answer Flexible pavement uses layered elastic theory, CBR/SN, and protects subgrade; rigid design treats slab as plate on elastic foundation using modulus k and flexural stress. Westergaard’s temperature concept models curling from top-bottom temperature difference, with reduced edge stress σ = CEαΔT/2 depending on slab length/radius ratio. ## Answer #### Flexible pavement design vs rigid pavement design | Basis | Flexible pavement design | Rigid pavement design | |---|---|---| | Structural action | Layered system; load spread through layers | Slab acting in flexure (plate on elastic foundation) | | Theory used | Burmister layered elastic theory, empirical CBR or AASHO methods | Westergaard, Bradbury, Pickett-Ray analysis | | Main design variable | Thickness of each layer (SN or total thickness) | Slab thickness and joint details | | Subgrade parameter | CBR or resilient modulus $M_R$ | Modulus of subgrade reaction $k$ | | Traffic input | Cumulative standard axles (msa, ESAL) | Wheel load, load repetition (fatigue), CVPD class | | Critical stress/strain | Tensile strain at bottom of bituminous layer; vertical strain on subgrade | Flexural tensile stress at slab edge/corner (load + temperature) | | Temperature | Affects stiffness of bituminous mix | Causes warping and frictional stresses, so joints are needed | | Codes | IRC:37, AASHTO-93, Asphalt Institute, Road Note 31 | IRC:58, AASHTO rigid, PCA | #### Westergaard's concept of temperature stresses Westergaard (1927) treated the slab as a plate on a dense-liquid foundation and analysed curling due to a temperature difference between top and bottom. - A temperature gradient $\Delta T$ across the thickness $h$ makes the top and bottom expand or contract differently, so the slab tends to curve. In the daytime the top is warmer and the slab edges curl down; at night the edges curl up. The slab weight and the subgrade reaction resist the curvature, creating stresses. - If the slab were completely restrained from curving, the stress would be $\sigma=\dfrac{E\alpha\Delta T}{2(1-\mu)}$. - For a finite slab free to curl, the stress is reduced by coefficients that depend on the ratio of slab length to radius of relative stiffness ($L/l$). Bradbury's version of Westergaard's result for the edge is $$ \sigma_{te}=\frac{C\,E\,\alpha\,\Delta T}{2} $$ and for the interior $\sigma_{ti}=\dfrac{E\alpha\Delta T}{2}\left(\dfrac{C_x+\mu C_y}{1-\mu^2}\right)$. - For short slabs ($L/l$ below about 4) $C$ is small, so closely spaced joints reduce warping stress; for slabs about 8-9 $l$ long $C$ reaches a maximum (about 1.08), and then decreases slightly. - The stress is tension at the bottom in the daytime and tension at the top at night; it is added to the load stress for the critical combination. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which subgrade parameters are used in flexible and rigid pavement design respectively? A. Modulus of subgrade reaction k for flexible; CBR for rigid B. CBR or resilient modulus for flexible; modulus of subgrade reaction k for rigid C. CBR for both D. Resilient modulus for rigid; k for flexible Answer: B. Flexible design uses CBR or M_R, while rigid design treats the slab as a plate on a foundation described by k. (Key point: Subgrade parameter) 2. What is the critical stress or strain for a rigid pavement? A. Tensile strain at the bottom of the bituminous layer B. Vertical strain on the subgrade C. Flexural tensile stress at the slab edge or corner from load plus temperature D. Rutting depth at the surface Answer: C. A rigid slab acts in flexure, so critical flexural tensile stress at edge/corner governs; the strains listed belong to flexible design. (Key point: Critical stress) 3. In Westergaard's temperature analysis, what happens to the slab in the daytime? A. The top is warmer, the edges curl down, and tension develops at the bottom B. The top is cooler, the edges curl up, and tension is at the top C. The slab does not curl because its weight prevents it D. The slab expands uniformly with no stress Answer: A. By day the top is warmer so the edges curl down; slab weight and subgrade reaction resist this and cause bottom tension. At night it reverses. (Key point: Curling direction) 4. How does the length of the slab (L/l ratio) affect the warping stress coefficient C? A. C is largest for very short slabs B. C is independent of slab length C. C is zero for slabs 8-9 l long D. For short slabs (L/l below about 4) C is small, so closely spaced joints cut warping stress; C peaks around 8-9 l Answer: D. A finite slab free to curl has reduced stress; the coefficient C rises with L/l to a maximum (about 1.08) near 8-9 l, so short slabs have low warping stress. (Key point: Effect of slab length on warping stress) ## Related in Transportation Engineering II - [Temperature stresses in rigid pavement; dowel bar design](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Atemperature-stress-rigid-dowel-bar-procedure) — Transportation Engineering II, chapter 2 - [Flexible vs rigid; radius of relative stiffness, k](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aflexible-rigid-differences-relative-stiffness-k) — Transportation Engineering II, chapter 2 - [Flexible pavement design with 8 CBR values](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aflexible-design-eight-cbr-340-cvd) — Transportation Engineering II, chapter 2 - [Flexible pavement failures; deflection overlay design](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aflexible-failures-deflection-overlay) — Transportation Engineering II, chapter 4 - [Repair vs rehabilitation; rigid pavement failures](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Arepair-vs-rehabilitation-rigid-failure-types) — Transportation Engineering II, chapter 4 - [Pavement design factors; IRC rigid design steps](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Apavement-factors-irc-rigid-steps) — Transportation Engineering II, chapter 2, asked 2× ## Related in other subjects - [Flexible versus rigid retaining structures](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Aflexible-vs-rigid-retaining) — Foundation Engineering, chapter 5, asked 3× - [Concept of rigid and deformable bodies](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aconcept-rigid-deformable-bodies) — Applied Mechanics, chapter 1 - [Concept of particle, rigid and deformable body](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aconcept-particle-rigid-deformable) — Applied Mechanics, chapter 1 - [Steel-aluminium bar with temperature rise: stresses](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asteel-aluminium-bar-temperature-supports) — Strength of Materials, chapter 3