Copyright ©2014 Pearson Education, All Rights Reserved Chapter Objectives Navigate between rectilinear co-ordinate systems for stress components Determine principal stresses and maximum in-plane shear stress Determine the absolute maximum shear stress in 2D and 3D cases Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved In-class Activities Reading Quiz Applications General equations of plane-stress transformation Principal stresses and maximum in-plane shear stress Mohr’s circle for plane stress Absolute maximum shear stress Concept Quiz Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved READING QUIZ 1) Which of the following statement is incorrect? The principal stresses represent the maximum and minimum normal stress at the point When the state of stress is represented by the principal stresses, no shear stress will act on the element When the state of stress is represented in terms of the maximum in-plane shear stress, no normal stress will act on the element For the state of stress at a point, the maximum in-plane shear stress usually associated with the average normal stress. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved APPLICATIONS Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved

GENERAL EQUATIONS OF PLANE-STRESS TRANSFORMATION The state of plane stress at a point is uniquely represented by three components acting on an element that has a specific orientation at the point. Sign Convention: Positive normal stress acts outward from all faces Positive shear stress acts upwards on the right-hand face of the element Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved

GENERAL EQUATIONS OF PLANE-STRESS TRANSFORMATION (cont) Sign convention (continued) Both the x-y and x’-y’ system follow the right-hand rule The orientation of an inclined plane (on which the normal and shear stress components are to be determined) will be defined using the angle θ. The angle θ is measured from the positive x to the positive x’-axis. It is positive if it follows the curl of the right-hand fingers. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved

GENERAL EQUATIONS OF PLANE-STRESS TRANSFORMATION (cont) Normal and shear stress components: Consider the free-body diagram of the segment Copyright ©2014 Pearson Education, All Rights Reserved

GENERAL EQUATIONS OF PLANE-STRESS TRANSFORMATION (cont) +ΣFx’ = 0; σx’ ∆A – (τxy ∆A sin θ) cos θ – (σy ∆A sin θ) sin θ – ( τxy ∆A cos θ) sin θ – (σx ∆A cos θ) cos θ = 0 σx’ = σx cos2 θ + σy sin2 θ + τxy (2 sin θ cos θ) +ΣFy’ = 0; τx’y’ ∆A + (τxy ∆A sin θ) sin θ – (σy ∆A sin θ) cos θ – ( τxy ∆A cos θ) cos θ + (σx ∆A cos θ) sin θ = 0 τx’y’ = (σy – σx) sin θ cos θ + τxy (cos2 θ – sin2 θ) σx’ = σx + σy 2 σx – σy cos 2θ + τxy sin 2 θ + τx’y’ = – σx + σy 2 sin 2θ + τxy cos 2 θ σy’ = σx + σy 2 σx – σy cos 2θ – τxy sin 2 θ – Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved VARIABLE SOLUTIONS Please click the appropriate icon for your computer to access the variable solutions Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 1 The state of plane stress at a point on the surface of the airplane fuselage is represented on the element oriented as shown in Fig. 9–4a. Represent the state of stress at the point on an element that is oriented 30° clockwise from the position shown. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 1 (cont) Solutions The element is sectioned by the line a-a. The free-body diagram of the segment is as shown. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 1 (cont) Solutions Applying the equations of force equilibrium in the x’ and y’ direction, Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 1 (cont) Solutions Repeat the procedure to obtain the stress on the perpendicular plane b–b. The state of stress at the point can be represented by choosing an element oriented. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 2 The state of plane stress at a point is represented by the element shown in Fig. 9–7a. Determine the state of stress at the point on another element oriented 30° clockwise from the position shown. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 2 (cont) Solutions From the sign convention we have, To obtain the stress components on plane CD, Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 2 (cont) Solutions To obtain the stress components on plane BC, The results are shown on the element as shown. Copyright ©2014 Pearson Education, All Rights Reserved

IN-PLANE PRINCIPAL STRESS The principal stresses represent the maximum and minimum normal stress at the point. When the state of stress is represented by the principal stresses, no shear stress will act on the element. Solving this equation leads to θ = θp Copyright ©2014 Pearson Education, All Rights Reserved

IN-PLANE PRINCIPAL STRESS (cont) Copyright ©2014 Pearson Education, All Rights Reserved

IN-PLANE PRINCIPAL STRESS (cont) Solving this equation leads to θ = θp; i.e Copyright ©2014 Pearson Education, All Rights Reserved

MAXIMUM IN-PLANE PRINCIPAL STRESS The state of stress can also be represented in terms of the maximum in-plane shear stress. In this case, an average stress will also act on the element. Solving this equation leads to θ = θs; i.e And there is a normal stress on the plane of maximum in-plane shear stress Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 3 When the torsional loading T is applied to the bar in Fig. 9–13a, it produces a state of pure shear stress in the material. Determine (a) the maximum in-plane shear stress and the associated average normal stress, and (b) the principal stress. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 3 (cont) Solutions From the sign convention we have, Maximum in-plane shear stress is For principal stress, Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 3 (cont) Solutions If we use Thus, acts at as shown in Fig. 9–13b, and acts on the other face Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 4 When the axial loading P is applied to the bar in Fig. 9–14a, it produces a tensile stress in the material. Determine (a) the principal stress and (b) the maximum in-plane shear stress and associated average normal stress. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 4 (cont) Solutions From the established sign convention, Principal Stress Since no shear stress acts on this element, Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 4 (cont) Solutions Maximum In-Plane Shear Stress To determine the proper orientation of the element, Copyright ©2014 Pearson Education, All Rights Reserved

MOHR’S CIRCLE OF PLANE STRESS A geometrical representation of equations 9.1 and 9.2; i.e. Sign Convention: σ is positive to the right, and τ is positive downward. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 5 Due to the applied loading, the element at point A on the solid shaft in Fig. 9–18a is subjected to the state of stress shown. Determine the principal stresses acting at this point. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 5 (cont) Solutions Construction of the Circle From Fig. 9–18a, The center of the circle is at The reference point A(-12,-6) and the center C(-6, 0) are plotted in Fig. 9–18b.The circle is constructed having a radius of Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 5 (cont) Solutions Principal Stress The principal stresses are indicated by the coordinates of points B and D. The orientation of the element can be determined by calculating the angle Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 6 The state of plane stress at a point is shown on the element in Fig. 9–19a. Determine the maximum in-plane shear stress at this point. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 6 (cont) Solutions Construction of the Circle We first construct of the circle, The center of the circle C is on the axis at From point C and the A(-20, 60) are plotted, we have Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 6 (cont) Solutions Maximum In-Plane Shear Stress. Max in-plane shear stress and average normal stress are The counter-clockwise angle is Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 7 The state of plane stress at a point is shown on the element in Fig. 9–20a. Represent this state of stress on an element oriented 30°counterclockwise from the position shown. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 7 (cont) Solutions Construction of the Circle We first construct of the circle, The center of the circle C is on the axis at From point C and the A(-8, -6) are plotted, we have Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 7 (cont) Solutions Stresses on 30° Element From the geometry of the circle, The stress components acting on the adjacent face DE of the element, which is 60° clockwise from the positive x axis, Fig. 9–20c, are represented by the coordinates of point Q on the circle. Copyright ©2014 Pearson Education, All Rights Reserved

ABSOLUTE MAXIMUM SHEAR STRESS State of stress in 3-dimensional space: Copyright ©2014 Pearson Education, All Rights Reserved

ABSOLUTE MAXIMUM SHEAR STRESS (cont) State of stress in 3-dimensional space: Copyright ©2014 Pearson Education, All Rights Reserved

ABSOLUTE MAXIMUM SHEAR STRESS (cont) State of stress in 3-dimensional space: Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 8 The point on the surface of the cylindrical pressure vessel in Fig. 9–24a is subjected to the state of plane stress. Determine the absolute maximum shear stress at this point. Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 8 (cont) Solutions An orientation of an element 45° within this plane yields the state of absolute maximum shear stress and the associated average normal stress, namely, Same result for can be obtained from direct application of Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved EXAMPLE 8 (cont) Solutions By comparison, the maximum in-plane shear stress can be determined from the Mohr’s circle, Copyright ©2014 Pearson Education, All Rights Reserved

Copyright ©2014 Pearson Education, All Rights Reserved CONCEPT QUIZ 1) Which of the following statement is untrue? In 2-D state of stress, the orientation of the element representing the maximum in-plane shear stress can be obtained by rotating the element 45° from the element representing the principle stresses. In 3-D state of stress, the orientation of the element representing the absolute maximum shear stress can be obtained by rotating the element 45° about the axis defining the direction of σint. If the in-plane principal stresses are of opposite sign, then the absolute maximum shear stress equals the maximum in-plane stress, that is, τabs max = (σmax – σmin)/2 Same as (c) but the principal stresses are of the same sign. Copyright ©2014 Pearson Education, All Rights Reserved