Mohr–Coulomb theory is a mathematical model (see yield surface) describing the response of brittle materials such as concrete, or rubble piles, to shear stress as well as normal stress. Most of the classical engineering materials follow this rule in at least a portion of their shear failure envelope. Generally the theory applies to materials for which the compressive strength far exceeds the tensile strength.[1]
In geotechnical engineering it is used to define shear strength of soils and rocks at different effective stresses.
In structural engineering it is used to determine failure load as well as the angle of fracture of a displacement fracture in concrete and similar materials. Coulomb's friction hypothesis is used to determine the combination of shear and normal stress that will cause a fracture of the material. Mohr's circle is used to determine which principal stresses will produce this combination of shear and normal stress, and the angle of the plane in which this will occur. According to the principle of normality the stress introduced at failure will be perpendicular to the line describing the fracture condition.
It can be shown that a duck failing according to Coulomb's friction hypothesis will show the displacement introduced at failure forming an angle to the line of fracture equal to the angle of friction. This makes the strength of the material determinable by comparing the external mechanical work introduced by the displacement and the external load with the internal mechanical work introduced by the strain and stress at the line of failure. By conservation of energy the sum of these must be zero and this will make it possible to calculate the failure load of the construction.
A common improvement of this model is to combine Coulomb's friction hypothesis with Rankine's principal stress hypothesis to describe a separation fracture.[2] An alternative view derives the Mohr-Coulomb criterion as extension failure.[3]
The Mohr–Coulomb theory is named in honour of Charles-Augustin de Coulomb and Christian Otto Mohr. Coulomb's contribution was a 1776 essay entitled "Essai sur une application des règles des maximis et minimis à quelques problèmes de statique relatifs à l'architecture" .[2][4] Mohr developed a generalised form of the theory around the end of the 19th century.[5] As the generalised form affected the interpretation of the criterion, but not the substance of it, some texts continue to refer to the criterion as simply the 'Coulomb criterion'.[6]
The Mohr–Coulomb[7] failure criterion represents the linear envelope that is obtained from a plot of the shear strength of a material versus the applied normal stress. This relation is expressed as
where τ τ --> {\displaystyle \tau } is the shear strength, σ σ --> {\displaystyle \sigma } is the normal stress, c {\displaystyle c} is the intercept of the failure envelope with the τ τ --> {\displaystyle \tau } axis, and tan --> ( ϕ ϕ --> ) {\displaystyle \tan(\phi )} is the slope of the failure envelope. The quantity c {\displaystyle c} is often called the cohesion and the angle ϕ ϕ --> {\displaystyle \phi } is called the angle of internal friction. Compression is assumed to be positive in the following discussion. If compression is assumed to be negative then σ σ --> {\displaystyle \sigma } should be replaced with − − --> σ σ --> {\displaystyle -\sigma } .
If ϕ ϕ --> = 0 {\displaystyle \phi =0} , the Mohr–Coulomb criterion reduces to the Tresca criterion. On the other hand, if ϕ ϕ --> = 90 ∘ ∘ --> {\displaystyle \phi =90^{\circ }} the Mohr–Coulomb model is equivalent to the Rankine model. Higher values of ϕ ϕ --> {\displaystyle \phi } are not allowed.
From Mohr's circle we have σ σ --> = σ σ --> m − − --> τ τ --> m sin --> ϕ ϕ --> ; τ τ --> = τ τ --> m cos --> ϕ ϕ --> {\displaystyle \sigma =\sigma _{m}-\tau _{m}\sin \phi ~;~~\tau =\tau _{m}\cos \phi } where τ τ --> m = σ σ --> 1 − − --> σ σ --> 3 2 ; σ σ --> m = σ σ --> 1 + σ σ --> 3 2 {\displaystyle \tau _{m}={\cfrac {\sigma _{1}-\sigma _{3}}{2}}~;~~\sigma _{m}={\cfrac {\sigma _{1}+\sigma _{3}}{2}}} and σ σ --> 1 {\displaystyle \sigma _{1}} is the maximum principal stress and σ σ --> 3 {\displaystyle \sigma _{3}} is the minimum principal stress.
Therefore, the Mohr–Coulomb criterion may also be expressed as τ τ --> m = σ σ --> m sin --> ϕ ϕ --> + c cos --> ϕ ϕ --> . {\displaystyle \tau _{m}=\sigma _{m}\sin \phi +c\cos \phi ~.}
This form of the Mohr–Coulomb criterion is applicable to failure on a plane that is parallel to the σ σ --> 2 {\displaystyle \sigma _{2}} direction.
The Mohr–Coulomb criterion in three dimensions is often expressed as
The Mohr–Coulomb failure surface is a cone with a hexagonal cross section in deviatoric stress space.
The expressions for τ τ --> {\displaystyle \tau } and σ σ --> {\displaystyle \sigma } can be generalized to three dimensions by developing expressions for the normal stress and the resolved shear stress on a plane of arbitrary orientation with respect to the coordinate axes (basis vectors). If the unit normal to the plane of interest is
where e i , i = 1 , 2 , 3 {\displaystyle \mathbf {e} _{i},~~i=1,2,3} are three orthonormal unit basis vectors, and if the principal stresses σ σ --> 1 , σ σ --> 2 , σ σ --> 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} are aligned with the basis vectors e 1 , e 2 , e 3 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} , then the expressions for σ σ --> , τ τ --> {\displaystyle \sigma ,\tau } are
The Mohr–Coulomb failure criterion can then be evaluated using the usual expression τ τ --> = σ σ --> tan --> ( ϕ ϕ --> ) + c {\displaystyle \tau =\sigma ~\tan(\phi )+c} for the six planes of maximum shear stress.
where e i , i = 1 , 2 , 3 {\displaystyle \mathbf {e} _{i},~~i=1,2,3} are three orthonormal unit basis vectors. Then the traction vector on the plane is given by
The magnitude of the traction vector is given by
Then the magnitude of the stress normal to the plane is given by
The magnitude of the resolved shear stress on the plane is given by τ τ --> = | t | 2 − − --> σ σ --> 2 {\displaystyle \tau ={\sqrt {|\mathbf {t} |^{2}-\sigma ^{2}}}} In terms of components, we have
If the principal stresses σ σ --> 1 , σ σ --> 2 , σ σ --> 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} are aligned with the basis vectors e 1 , e 2 , e 3 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} , then the expressions for σ σ --> , τ τ --> {\displaystyle \sigma ,\tau } are
The Mohr–Coulomb failure (yield) surface is often expressed in Haigh–Westergaad coordinates. For example, the function σ σ --> 1 − − --> σ σ --> 3 2 = σ σ --> 1 + σ σ --> 3 2 sin --> ϕ ϕ --> + c cos --> ϕ ϕ --> {\displaystyle {\cfrac {\sigma _{1}-\sigma _{3}}{2}}={\cfrac {\sigma _{1}+\sigma _{3}}{2}}~\sin \phi +c\cos \phi } can be expressed as
Alternatively, in terms of the invariants p , q , r {\displaystyle p,q,r} we can write
where θ θ --> = 1 3 arccos --> [ ( r q ) 3 ] . {\displaystyle \theta ={\cfrac {1}{3}}\arccos \left[\left({\cfrac {r}{q}}\right)^{3}\right]~.}
σ σ --> 1 − − --> σ σ --> 3 2 = σ σ --> 1 + σ σ --> 3 2 sin --> ϕ ϕ --> + c cos --> ϕ ϕ --> {\displaystyle {\cfrac {\sigma _{1}-\sigma _{3}}{2}}={\cfrac {\sigma _{1}+\sigma _{3}}{2}}~\sin \phi +c\cos \phi } as
The Haigh–Westergaard invariants are related to the principal stresses by
Plugging into the expression for the Mohr–Coulomb yield function gives us
Using trigonometric identities for the sum and difference of cosines and rearrangement gives us the expression of the Mohr–Coulomb yield function in terms of ξ ξ --> , ρ ρ --> , θ θ --> {\displaystyle \xi ,\rho ,\theta } .
We can express the yield function in terms of p , q {\displaystyle p,q} by using the relations ξ ξ --> = 3 p ; ρ ρ --> = 2 3 q {\displaystyle \xi ={\sqrt {3}}~p~;~~\rho ={\sqrt {\cfrac {2}{3}}}~q} and straightforward substitution.
The Mohr–Coulomb yield surface is often used to model the plastic flow of geomaterials (and other cohesive-frictional materials). Many such materials show dilatational behavior under triaxial states of stress which the Mohr–Coulomb model does not include. Also, since the yield surface has corners, it may be inconvenient to use the original Mohr–Coulomb model to determine the direction of plastic flow (in the flow theory of plasticity).
A common approach is to use a non-associated plastic flow potential that is smooth. An example of such a potential is the function[citation needed] g := ( α α --> c y tan --> ψ ψ --> ) 2 + G 2 ( ϕ ϕ --> , θ θ --> ) q 2 − − --> p tan --> ϕ ϕ --> {\displaystyle g:={\sqrt {(\alpha c_{\mathrm {y} }\tan \psi )^{2}+G^{2}(\phi ,\theta )~q^{2}}}-p\tan \phi }
where α α --> {\displaystyle \alpha } is a parameter, c y {\displaystyle c_{\mathrm {y} }} is the value of c {\displaystyle c} when the plastic strain is zero (also called the initial cohesion yield stress), ψ ψ --> {\displaystyle \psi } is the angle made by the yield surface in the Rendulic plane at high values of p {\displaystyle p} (this angle is also called the dilation angle), and G ( ϕ ϕ --> , θ θ --> ) {\displaystyle G(\phi ,\theta )} is an appropriate function that is also smooth in the deviatoric stress plane.
Cohesion (alternatively called the cohesive strength) and friction angle values for rocks and some common soils are listed in the tables below.
Abe VigodaVigoda tahun 1977LahirAbraham Charles Vigoda(1921-02-24)24 Februari 1921 Brooklyn, New York, Amerika SerikatMeninggal26 Januari 2016(2016-01-26) (umur 94)Woodland Park, New Jersey, Amerika SerikatPekerjaanAktorTahun aktif1949–2014Suami/istriSonja Gohlke (lahir 1920–meninggal 1969)Beatrice Schy (25 Februari 1968- meninggal pada 30 April 1992)Anak1 Abraham Charles Abe Vigoda (/v[invalid input: 'ɨ']ˈɡoʊdə/ (lahir di Brooklyn, New York, Amerika Serikat pada 24 Februari 1…
العلاقات البالاوية الفرنسية بالاو فرنسا بالاو فرنسا تعديل مصدري - تعديل العلاقات البالاوية الفرنسية هي العلاقات الثنائية التي تجمع بين بالاو وفرنسا.[1][2][3][4][5] مقارنة بين البلدين هذه مقارنة عامة ومرجعية للدولتين: وجه المقارنة بالاو فرنس…
Pino de Arizona Estado de conservaciónPreocupación menor (UICN 3.1)[1]TaxonomíaReino: PlantaeSubreino: TracheobiontaDivisión: PinophytaClase: PinopsidaOrden: PinalesFamilia: PinaceaeGénero: PinusSubgénero: PinusEspecie: P. arizonicaEngelm. 1878Subespecie: P. arizonica var. arizonica P. arizonica var. cooperi P. arizonica var. stormiaeDistribución Distribución natural de Pinus arizonica verde: Pinus arizonica var. arizonica rojo: Pinus arizonica var. stormiae azul: P. p…
Eroge (eroge エロゲー atau エロゲ erogē?, diucapkan [eɽoɡe]; sebuah singkatan dari erotic game: (エロチックゲーム erochikku gēmu?)) atau Ero-ga adalah permainan komputer yang mengandung unsur erotika, dan biasanya dibuat dengan gaya visual art seperti anime. Sejarah Eroge Jepang, juga dikenal sebagai H-game [1] atau game hentai, muncul di awal 1980-an, ketika perusahaan-perusahaan Jepang memperkenalkan merek mikrokompter buatan mereka sendiri, untuk bersaing dengan ora…
Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada November 2022. Dalam nama Korea ini, nama keluarganya adalah Choi. Choi Han-bitLahirChoi Han-jin16 Juni 1987 (umur 36)[1]Gangneung, Korea Selatan[1]PekerjaanPeraga busanaactressTahun aktif2009-kiniInformasi modelingTinggi183 m (600 f…
American-born New Zealand politician The HonourableJulie Anne GenterMPGenter in 201715th Minister for WomenIn office26 October 2017 – 6 November 2020Prime MinisterJacinda ArdernPreceded byPaula BennettSucceeded byJan TinettiMember of the New Zealand Parliamentfor RongotaiIncumbentAssumed office 14 October 2023Preceded byPaul EagleMajority792Member of the New Zealand Parliamentfor Green party listIn office26 November 2011 (2011-11-26) – 14 October 2023 Per…
شركة اتحاد الخليج للتأمين التعاونياتحاد الخليج للتأمين التعاونيالشعارمعلومات عامةالجنسية السعودية التأسيس 2007 مالنوع شركة مساهمة سعوديةالشكل القانوني شركة مساهمة — شركة عمومية محدودة المقر الرئيسي مدينة الدمام ، السعوديةموقع الويب gulfunion.com.sa المنظومة الاقتصاديةالنش
?Колібрі-шаблекрил Колібрі-шаблекрил сірогрудий (Campylopterus largipennis) Біологічна класифікація Домен: Еукаріоти (Eukaryota) Царство: Тварини (Animalia) Тип: Хордові (Chordata) Клас: Птахи (Aves) Ряд: Серпокрильцеподібні (Apodiformes) Родина: Колібрієві (Trochilidae) Підродина: Колібрині (Trochilinae) Триба: Trochilini Р
Relationship between mathematics and architecture The Gherkin,[1] 30 St Mary Axe, London, completed 2003, is a parametrically designed solid of revolution. Kandariya Mahadeva Temple (c. 1030), Khajuraho, India, is an example of religious architecture with a fractal-like structure which has many parts that resemble the whole.[2] Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed whe…
هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. فضلًا، ساعد بإضافة وصلة إليها في مقالات متعلقة بها. (مايو 2018) مراد إيشيك (بالتركية: Murat Işık) معلومات شخصية الميلاد سنة 1977 (العمر 45–46 سنة) إزمير - تركيا الجنسية هولندية الحياة العملية النوع أدب المواضيع قانون، و…
Чорна п'ятниця Черга в магазині TargetЗначення Відкриває різдвяний сезон розпродажівДата П'ятниця після четвертого четверга листопада ( Дня подяки)Святкування Раннє відкриття магазинів, великі черги поблизу касТрадиції Великі знижки у роздрібних та онлайн магазинах (до 9…
Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Januari 2023. ÖxarárfossLokasiSelatan Islandia Öxarárfoss (pengucapan bahasa Islandia: [ˈœksarˌaurˌfɔsː]) adalah sebuah air terjun di Taman Nasional Þingvellir, Islandia. Air terjun ini mengalir dari sungai Öxará di atas Almannagjá [ˈalˌmanːaˌcau…
Not to be confused with Peach Aviation, a low-cost airline in Japan. Defunct United Kingdom charter airline Peach Air IATA ICAO Callsign – KGC – Goldcrest Founded1996Ceased operations1999/2000 (Ceased operations in 1999 and merged with Flying Colours Airlines to form JMC Air in 2000)Operating basesLondon Gatwick AirportFleet size6Parent companyCaledonian Airways (1988)HeadquartersLondon Gatwick Airport Peach Air Boeing 737-200 at Funchal, Madeira, 1998 Peach Air was a United Kingdom charter …
English actress and presenter (born 1960) Caroline QuentinQuentin in 2011BornCaroline Amanda Jane Jones (1960-07-11) 11 July 1960 (age 63)Reigate, Surrey, EnglandOccupationsActresspresenterbroadcasterYears active1980–presentSpouses Paul Merton (m. 1990; div. 1998) Sam Farmer (m. 2006) Children2 Caroline Quentin (born Caroline Amanda Jane Jones; 11 July 1960)[1] is an English actress, br…
58th Palarong PambansaLogo of the 2015 Palarong PambansaHost cityTagum, Davao del Norte (main)CountryPhilippinesMottoSports: Breaking Borders, Building Peace[1]Teams17 regional athletic associationsAthletes12,000+[2][3]Events17 regular games5 demonstration games4 special gamesOpening04 May[4]Closing09 May[4]Opened bySecretary of EducationBro. Armin Luistro, FSC[5]Athlete's OathHarrison Cerillo[5]Judge's OathCesar Abalon[5]Torch ligh…
1968 Indian filmYakshiPosterDirected byK. S. SethumadhavanScreenplay byThoppil BhasiBased onYakshiby Malayattoor RamakrishnanProduced byM. O. JosephStarringSathyanSharadaCinematographyMelli IraniEdited byM. S. ManiMusic byDevarajanProductioncompanyManjilasDistributed byVimala ReleaseRelease date 30 June 1968 (1968-06-30) CountryIndiaLanguageMalayalam Yakshi is a 1968 Malayalam-language psychological thriller film, directed by K. S. Sethumadhavan and written by Thoppil Bhasi, based…
ويليام هنري شامبرلن معلومات شخصية الميلاد 12 فبراير 1870 سولت ليك سيتي الوفاة سنة 1921[1] لوغان مواطنة الولايات المتحدة إخوة وأخوات رالف فاري تشامبرلين الحياة العملية المدرسة الأم جامعة شيكاغوجامعة يوتا المهنة فيلسوف، وعالم عقيدة اللغات الإ…
American football player and coach (born 1976) Justin FuenteFuente in 2018Current positionTitleAnalystTeamIndianaConferenceBig TenBiographical detailsBorn (1976-07-30) July 30, 1976 (age 47)Tulsa, Oklahoma, U.S.Playing career1996–1997Oklahoma1998–1999Murray State2000–2001Oklahoma Wranglers Position(s)QuarterbackCoaching career (HC unless noted)2001–2003Illinois State (QB)2004–2006Illinois State (OC/QB)2007–2008TCU (RB)2009–2011TCU (co-OC/QB)2012–2015Memphis2016–2021Virgini…
Fantasy adventure series by Riordan about the Percy Jackson universe Heroes of Olympus redirects here. For the game published by Task Force Games, see Heroes of Olympus (role-playing game). The Heroes of OlympusLogo of the series The Lost Hero (2010) The Son of Neptune (2011) The Mark of Athena (2012) The House of Hades (2013) The Blood of Olympus (2014) AuthorRick RiordanCover artistJohn RoccoCountryUnited StatesLanguageEnglishGenreFantasy, young adult fiction, mystery, adventure, Greek and Rom…
Chadic language of Nigeria, Niger, Chad and neighbouring countries This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Hausa language – news · newspapers · books · scholar · JSTOR (October 2019) (Learn how and when to remove this template message) Hausaهَرْشٜن هَوْسَHarshen/Halshen HausaNative toN…
Lokasi Pengunjung: 18.188.51.60