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Unformatted text preview: Lecture 9 1 Lecture 9 1 Outline: Stress transformations Maximum and minimum stresses Mohr’s circle 1. Stresses on inclined planes When a body is in Equilibrium under acting forces, any piece cut from the body is also in equilibrium. As shown in Fig.1. a cube cut from the body is subjected to normal and shear stresses acting on its surfaces. In general the stresses act on all six surfaces of the cube but the stresses are shown only on four surfaces in Fig.1, because we will not be considering 3 dimensional stress states and limit our selves to plane stress. As also shown in Fig. 1 if an inclined surface is cut from the cube the remaining triangular prism is also in equilibrium. Therefore there should also be stresses acting on the inclined surface which can be decomposed into a normal stress (that is normal to the inclined plane) and a shear stress that is in tangential direction to the plane. Cube in static equilibrium Triangular prism in static equilibrium Fig. 1 Pieces cut from the body are in Equilibrium under the stresses acting on their surfaces 1 Mechanics of Solids Dr Emre Erkmen Lecture 9 2 So far, we have built the relations between the stresses acting on the crosssection and the internal forces, since the internal forces were also acting on the crosssection of the beam as shown in Fig. 2. N My A I σ = − VQ T Ib J ρ τ = + Fig. 2 Relations between the stresses acting on the crosssectio and the internal forces A force vector can be described in terms of its components. The directions of the components are selected according to the orientation of the axes. Because the axes can be in various directions the force can be described in terms of its components in various different ways. Similarly, the stresses can be described in different ways. As components of the same force in different directions have different values, stresses acting on different surfaces (dotted line in Fig.2) will, in general, have different values. Uniaxial stress As discussed in Lecture 3, normal (axial) force acting on a bar will cause only normal (axial) stresses x σ on the crosssection (Fig.3), hence a uniaxial stress state. However, on the inclined surface shown in Fig.3 below in addition to the normal stress, there will also be shear stresses. Fig. 3. Normal (axial) stresses on the crosssection due to axial force Lecture 9 3 Therefore, in general, the stresses acting on the inclined surface can be expressed in terms of a normal stress component θ σ and a shear stress component θ τ . To find the stresses on the inclined surface that makes an angle of θ with the crosssectional surface, we can write the equilibrium equations on the triangular prism. Fig. 4. Triangular prism in equilibrium Fig. 5. Relations between the areas of the surfaces The force components in the normal and tangential direction to the inclined surface due to stress x σ acting on the crosssectional area A are shown in Fig. 6 Fig. 6. Force components due to x σ A The equilibrium of forces in the direction that is normal to the inclined surface due to stresses acting...
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