Mohr circle mohr circle anaysisand application
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Mohr's circle is a graphical representation used to analyze the state of stress at a point. It relates the normal and shear stresses acting on planes of all orientations passing through that point. The document discusses how to construct Mohr's circle through an example and how it can be used to analyze principal stresses, maximum shear stress, and normal and tangential stresses on any given plane. Applications of Mohr's circle include studying brittle deformation features like faults and fractures, pore pressure effects, and stability of structures like bridges, dams and tunnels.
Mohr circle mohr circle anaysisand application
- 1. Under guidance of: DR. HARISINGH GOUR VISHWAVIDYALAYA SAGAR, M.P ( A CENTRAL UNIVERSITY) Bibhukalyan mohapatra Presented by: Regd-Y18251010 DEPARTMENT OF APPLIED GEOLOGY MOHR CIRCLE: ANALYSIS AND APPLICATIONS Dr A.K.SHANDILYA
- 3. CONTENTS INTRODUCTION CONSTRUCTION ANALYSIS APPLICATIONS BEGINNING NOTE(STRESS AND STRAIN)
- 4. Stress is the force applied to an object. In geology, stress is the force per unit area that is placed on a rock. Types of Stress • Compressive stress applied from opposite sides towards the center. • Resulting strain – rock becomes shorter and thicker. Shortening strain: Block becomes shorter parallel to stress and thicker perpendicular to stress
- 5. • Tensional stress(extensional stress) is applied from opposite sides away the center. • Resulting strain – rock becomes longer and thinner. Shear stress is applied in opposite directions parallel to the center. Resulting strain– warping, but no shortening or elongation. Elongation strain: Block becomes longer parallel to stress and thinner perpendicular to stress Shear strain:Block neither shortens nor elongates parallel to stress
- 6. • NORMAL STRESS(n): acts perpendicular to the surface to which a force has been applied and can be either compressional(positive values) or tensional(negative values) in character. • DIRECTED STRESS: varies with direction, hence a stone slab creates directed stress because there is a force in one direction only.
- 7. Unlike the stress that can’t be observed in the geological bodies, strain is observable as a consequence of stress. If the geological bodies are transported in bulk from one place to another due to which position of a particles in the body change, they are called to have undergone rigid body translation or displacement. STRAIN The strain can be defined as a change in shape, location or size of the body, geological or otherwise due to application of force of stress.
- 8. INTRODUCTION • Mohr’s circle is a graphical representation of a general state of stress at a point. It is a graphical method used for evaluation of principal stresses, maximum shear stress; normal and tangential stresses on any given plane. • The Mohr circle describes the normal and shear stress acting on planes of all possible orientations through a point in the rock. MOHR CIRCLE OF STRESS • Mohr’s circle is named after German Civil Engineer Otto Mohr .He developed the graphical method in 1882.
- 9. IN THE GIVEN SLIDE WE CAN STUDY STEP BY STEP PROCESS OF CONSTRUCTION OF MOHR’S CIRCLE THROUGH AN EXAMPLE (1) (2) 0 (3) (4)
- 10. In Mohr's circle , the two principal stress planes that are actually separated by 90º are represented 180º apart i.e., to achieve a rotation of 90º you need to rotate by 180º. Therefore for every 1º rotation you need to rotate by 2º on the Mohr's circle. Note the use of double angles IN 3D AXIS THE ANGLE BETWEEN PRINCIPAL PLANES IS 90° BUT AS WE ARE REPRESENTING IT ON A SEMI-CIRCLE SO WE ARE TAKING ANGLE AS 2Θ SO ACTUAL ANGLE CAN BE FOUND OUT BY DIVIDING IT BY 2.
- 11. ANALYSIS : MOHR CIRCLES UNDER VARYING VALUES OF STRESSES (1) (2) (3) (4) (5) (6) Characteristic states of stress are illustrated in the Mohr diagram for 3D stress
- 12. CONTINUES............ •Brittle Deformation – Faults, fractures and Joints •Ductile deformation – Folds, foliations and lineations •Brittle materials have a small or large region of elastic behavior but only a small region of ductile behavior before they fracture. •Ductile materials have a small region of elastic behavior and a large region of ductile behavior before they fracture. MOHR CIRCLE IS MOSTLY APPLIED IN BRITTLE DEFORMATION CASES AS SUDDEN RUPTURE DO OCCUR WHILE IT IS RARELY APPLIED IN DUCTILE DEFORMATION i.e.IN FINITE STRAIN CASE. TYPE OF DEFORMATION DECIDES MOHR CIRCLE
- 13. CONTINUES……. Role of pore pressure AS PORE PRESSURE ACTS ,INCREASE IN TOTAL STRESS WILL BE SLIGHT NOT ALL PORE PRESSURE BECAUSE OF ABSORPTION OF STRESS BY ELASTIC DEFORMATION IN THE GRAINS.
- 14. CONTINUES……… Pore pressure increases chances of failure
- 15. CONTINUES……. σ s Different rock behave differently to pore pressure In sandstone and shale σ1 is same but σ3 is different in both rocks, so differential stress is more for sandstone than for shale, so amount of pore pressure needed for sandstone deformation is less than for shale .
- 16. WHY SHEAR FRACTURES DO NOT FORM AT 45° TO THE LARGEST PRINCIPAL STRESS ? Navier and Coulomb both showed that Shear fractures do not simply form along the theoretical surfaces of maximum shear stress. Maximum resolved shear stress on a plane is obtained when the plane is oriented 45° to the maximum principal stress(Θ=45°).This fact is easily extracted from the Mohr diagram , where the value for shear stress is at its maximum when 2Θ=90°.However,in this situation the normal stress σn across the plane is fairly large. Both σs and σn decrease as Θ increases, but σn decreases faster than σs. The optimal balance between σn and σs depends on the angle of internal friction φ,and is predicted by the Coulomb criterion to be around 60° for many rock types .At this angle (Θ=60°) σs is still large ,while σn is considerably less. The angle depends also on the confining pressure (depth of deformation),temperature and pore fluid ,and experimental data indicate that there is a wide scatter even for the same rock type and conditions. P1 is the plane of maximum resolved shear stress (2Θ=90°) and forms at 45° to σ1.The plane P2 oriented at 30° to σ1 has a slightly lower shear stress (the difference is Δσs ),but a much lower normal stress (by Δσn).It is therefore easier for a shear fracture to form along P2 than along P1.
- 17. APPLICATIONS: The Coulomb fracture criterion occurs as two straight lines(red) in the Mohr diagram . The circles represent examples of critical states of stress . The blue line represents Griffith criterion for comparison . The combination of the two is sometimes used(GC in the tensile regime and CC in the compressional regime). CC, Coulomb criterion ; GC , Griffith criterion ;C , the cohesive strength of the rock ; T , the tensile strength of the rock Where…… σs =critical shear stress C=cohesive strength σn =normal stress Φ=angle of internal friction
- 18. (a)Stable state of stress (b) Critical situation ,where the circle touches the envelope . This is when the rock is at the verge of failure ,also called critically stressed (c)Unstable situation where the state of stress is higher than that required for failure Three different fracture criteria combined in Mohr space . Different styles of fracturing are related to confining pressure: (a)Tensile fracture (b)hybrid or mixed mode fracture (c) shear fracture (d)semi-ductile shear bands (e) plastic deformation
- 19. The effect of pumping up the pore fluid pressure Pt in a rock . The Mohr circle is “pushed” to the left (the mean stress is reduced) and a shear fracture will form if the fracture envelope is touched while σ3 is still positive . A tensile fracture forms if the envelope is reached in the tensile field , as shown in(b) (low differential stress) Coulomb envelope modified for tensile and ductile failure The effect of a pre-existing fracture(plane of weakness),illustrated in the Mohr diagram . The criterion for reactivation(frictional sliding) is different from that of an unfractured rock of the same kind , and the differential stress required to reactivate the fracture is considerably smaller than that required to generate a new fracture in the rock.
- 20. SOME APPLICATIONS…………………………. In detection of stability of • Bridges • Dams • Tunnels • Slope
- 21. MOHR’S STRAIN DIAGRAM • We had the following Equations for the strain ellipse: ’= (’1+’2/2)-((’2-’1/2) cos 2’) And ’ =/= (’2-’1 /2) sin2 • These equations are in same from as the equations for the circle; X=c – r cos y= rsin
- 22. • Where the center of the circle is located a (c,0) on the x-axis and the circle has a radius of “r”. Thus the above equations define a circle with a center at (c,0)=(’2+’1/2,0) andradiusr=(’2-’1/2) • These equations are the Mohr’s circle for finite strain. The strain can represented by Mohr’s circle graphically
- 23. BIBLIOGRAPHY: • Structural geology by HAAKON FOSSEN • Structural geology by AK JAIN • YouTube videos by PROFESSOR TK BISWAL STRUCTURAL GEOLOGY IIT BOMBAY • Class notes by AWADESH KANT SONI SIR
- 24. THANK YOU