Presentation on theme: "Stress in any direction"— Presentation transcript:
1Stress in any direction Use arrow keys on your keypad or click the mouse to navigate through the presentationStress in any direction
2Two dimensional state of stress, and the stress element XYF1F2F3F4FnMStress ElementtxysxsyPYsyPsxXtxyccwtxycwWhen a set of co-planer external forces and moments act on a body,the stress developed at any point ‘P’ inside the bodycan be completely defined by the two dimensional state of stress:sx = normal stress in X direction,sy = normal stress in Y direction, andtxy = shear stress which would be equal but opposite inX (cw) and Y (ccw) directions, respectively.The 2D stress at point P is described by a box drawn with its faces perpendicular to X & Y directions, and showing all normal and shear stress vectors (both magnitude and direction) on each face of the box. This is called the stress element of point P.
3The stress formulae that we have learnt thus far, can determine the 2D stresses developed inside a part, ONLY ALONG A RECTANGULAR AXIS SYSTEM X -Y, that is defined by the shape of the part.For example, X axis for a cantilever beam is parallel to its length,and Y axis is perpendicular to X.YFor a combined bending and axial loading (F1, F2 etc.) of this cantilever beam:the normal and shear stress at a point P, can be determined using the formulae, such as,sx= Mv/I+P/A,txy=VQ/(Ib).VtxysxsyPXfYF1UPF2XNote that, these formulae can only determine stresses parallel to X and Y axis, and the stress element is aligned with X-Y axis.The question is, what would be the values of normal and shear stresses at the same point P, if the stresses are measured along another rectangular axis system U-V, rotated at an angle f with the X-Y axis system ?
4The Problem is: given sx, sy, txy and f, Knowing the 2D stresses at point P along XY coordinate system,we want to determine the 2D stresses for the same point P, when measured along a new coordinate system UV,which is rotated by an angle f with respect to the XY system.txyXYsxsysuvtuvusvXfFvuXF1YF2PfXYufF2PThe Problem is: given sx, sy, txy and f,can we determine su, sv, tuv ?
5THIS IS HOW WE CAN ACHIEVE THAT 2. To maintain static equilibrium, let the internal normal and shear stresses su & tuv, respectively are developed on the cut plane1. We cut the stress element by an arbitrary plane at an angle f. This plane is normal to u-axistxysxsyfsxtxyXYsysutuvffLLsinfLcosfUtxy(LBsinf)sx(LBcosf)txy(LBcosf)sy(LBsinf)su(LB)tuv(LB)f4. If the thickness of the element is B, then the force acting on each face of the element will be equal to the stress multiplied by the area of the face.3. Let, L be the length of the cut side. Then the other two sides are Lsinf & Lcosf
6CONTINUING 5. Forces acting on the faces = force x area txyLBsinfcosfftxyLBsin2ftxyLBcos2fsu(LB)tuv(LB)syLBsinfcosfsyLBsin2fsxLBcos2fsxLBcosfsinfCONTINUING5. Forces acting on the faces = force x areatxy(LBsinf)sx(LBcosf)txy(LBcosf)sy(LBsinf)su(LB)tuv(LB)f6. Resolving each force in u & v directionsEquating forces in u-direction:suLB = sxLBcos2f + syLBsin2f + 2txyLBsinfcosfOr, su = sxcos2f + sysin2f + 2txysinfcosf ………..(1)Equating forces in v-direction:tuvLB = txyLBcos2f - txyLBsin2f - sxLBsinfcosf+ syLBsinfcosfOr, tuv = txy(cos2f - sin2f) – (sx-sy) sinfcosf ……. (2)
7Replacing the square terms of trigonometric functions by double angle terms and rearranging : Equations 3, 4 & 5 gives us the 2D stress values, if measured along U-V axis which is at an angle f from X-Y axis.Since both sets of stresses refer to the stress of the same point, the two sets of stresses are also equivalent.txyXYsxsyfsutuvvsvXfu
9implements these three equations Mohr’s circleimplements these three equationsby a graphical aid, which simplifies computation and visualization of the changes in stress values (su, sv & tuv) with the rotation angle f of the measurement axis.s-st-tMohr circle is plotted on a rectangular coordinate system in which the positive horizontal axis represents positive (tensile) normal stress (s), and the positive vertical axis represents the positive (clockwise) shear stress (t).Thus the plane of the Mohr circle is denoted as s-t plane.In this s-t plane, the stresses acting on two faces of the stress element are plotted.txysxXcwtxyYsxsyXx faces have stress:(sx & txy)txyYsyccwFor a stress elementY faces have stress: (sy,-txy)
10t s -s -t Y V X X sx txy su U tuv sy sv X (sx,txy) su U sx txy sy tuv DRAWING MOHR CIRCLEYsxtxysyXsuUVtuvsvXfStart by drawing the original stress element with its sides parallel to XY axis, and show the normal and the shear stress vectors on the element.Draw the s-t rectangular axis and label them.On the s-t plane, plot X with normal and shear stress values of sx and txy, and Y with values sy and –txy.Join X and Y points by a straight line, which intersects the horizontal s axis at C. C denotes the average normal stress savg=(sx+sy)/2 .The line CX denotes X axis, and line CY denotes Y axis in Mohr circle. Name them.Draw the Mohr circle using C as the center, and XY line as the diameter.To find stress along the new UV axis system, draw a line UV rotated at an angle 2f from the XY line. CU line denotes U axis, and CV denotes V axis.The normal and shear stress values of the points U and V on the s-t plane denote the stresses in U and V directions, respectively.This way we can find stresses for an element rotated at any desired angle f.tsuX (sx,txy)Shear stress axis (t)UsxtxysyX axis2ftuvs-sCtuvNormal stress axis (s)txyY axisVsvY(sy,-txy)savg=(sx+sy)/2-t
11PROOF -s X (sx,txy) Shear stress axis (t) U (su,tuv) sx txy X sy X axis2fShear stress axis (t)Y(sy,txy)sxsy2qNormal Stress axis (s)Y axissavg=(sx+sy)/2U (su,tuv)V (sv,txy)aX (sx,txy)XYsxtxysyXsuUVtuvsvXf
12Principal Normal Stresses s1 & s2, and Max Shear Stress tmax YXIn the Mohr circle, for a rotation of 2q angle, the XY axis line becomes horizontal. In the rotated axis s1-s2, the shear stress vanishes.The element will have only normal stresses s1 & s2, and s1 being the maximum normal stress. s1 & s2 are called the Principal normal stresses.s2qs1xYtmintmaxs2savgs1tmaxq’Y(sy,-txy)sxsysavg-s-totxyX axisY axistsX (sx,txy)tmax2q’(savg,tmax)(savg,-tmax)Similarly, if the XY axis line is rotated by an angle 2q ‘ to make it vertical, then the shear stress maximizes and the element will have normal stress = savg and Maximum shear stress = tmax2qs2s1
13t -s s o -t Formulea for Principal Normal Stresses & Max Shear Stress YsxtxysyXtmax(savg,tmax)X (sx,txy)X axistxysy2q’-s2qsxsos2s1savgtxyY axisY(sy,-txy)-t(savg,-tmax)s1s2YqXtmintmaxsavgxYfPrincipal normal stress elementMaximum shear stress element
14t s -s -t Determining su, sv & tuv Given sx, sy, txy & f Y sx txy sy X X (sx,txy)U (su,tuv)sxX axistxysy2f2qtuvs-sCtuvtxyY axisV (sv,txy)svY(sy,-txy)savg=(sx+sy)/2-t
16YXFor a stress element withsx=20,000 psi,sy= psi, andtxy= 5000 psi.4,000 psi5,000 psi20,000 psiDraw the Mohr Circle and, draw two stress elements properly oriented for (i) the principal normal stresses, and (ii) max shear stresses element.t-tDraw the stress element along XY axis.Draw the s-t axes for mohr circlePlot point X for sx=20K, txy=5kPlot point Y for sy= -4K, txy=-5kDraw line XY and show X & Y axes.Draw the circle with XY as the diameter(8k,13k)tmaxX (20k,5k)X axis5ks-so67.420k22.6s2= -5k-4k8kR=13Ks1=21k5kY axisY(-4k,-5k)(8k,-13k)This completes the Mohr circle. Next, the stress elements
17t -s o s -t Y X Y s2 X tmax s1 X (20k,5k) tmin tmax x 5,000 psi20,000 psi4,000 psiThe principal normal stress axis will be rotated CWDraw the principal stress axis 11.3o CW from XY axis.Show the principal stresses.PRINCIPAL NORMAL STRESS ELEMENTYXs2t-t5k(8k,13k)21ktmaxq=11.3s15kX (20k,5k)STRESS ELEMENT FOR tMAXX axis5ks-so67.4The tmax axis will be rotated CCWDraw the tmax stress axis 33.7o CCW from XY axis.Show the the stresses.20k22.6s2= -5k-4k8kR=13Ks1=21k5kYY axistminY(-4k,-5k)8ktmax13k8k33.7x(8k,-13k)8k13k8kThat completes the drawing of the two stress elements
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