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tau_axis X &Y >kP v_label_1 &\ >l v_label_2 &_ >lɀv_label_3 &b >mv_label_4 &e >n` v_label_5 &h >o# v_label_6 &k >o v_label_7 &n >` v_label_8 &q >avv_label_9 &t >bE v_label_10 &w >c v_label_11 &z >cրv_label_12 &} >dv_label_13 & >ex v_label_14 & >f@ v_label_15 & >g v_label_16 & >g v_label_17 & 2& v_label_18 & 2X v_label_19 & 2{v_label_20 & 2v_label_21 & 2v_label_22 & 2`v_label_23 & 3 v_label_24 & 3R v_label_25 ś45 & ]= txtCenter & 7DtxtRadius & $ y scale_note . % 0 i @ right_marker . % : i @ left_marker @ @ @ @ ?$ u radius Math abs _parent scale_of_grid GridScaling dScale ScaleTheGrid dXOrigin center ShiftTheGrid grid20pixels DeltaPixels MakeSureGridShows thetaPrincipal txtCenter _width iDigits sString C= formatDecimals gotoAndPlay localX circle_mc xy_diameter radius_target _x localY _y rotatedX Rotation_X_Component rotatedY Rotation_Y_Component txtRadius R= show_grid_labels mask_width mask_height leftmost_line_stress rightmost_line_stress leftmost_line ceil rightmost_line floor no_of_lines topmost_line_stress botmost_line_stress topmost_line botmost_line no_of_vlines tau_axis _visible g1 log g1int idigits i h_label_ sTick h_label_1 h_label_2 h_label_3 h_label_4 h_label_5 h_label_6 h_label_7 h_label_8 h_label_9 h_label_10 h_label_11 h_label_12 h_label_13 h_label_14 h_label_15 h_label_16 h_label_17 h_label_18 h_label_19 h_label_20 h_label_21 h_label_22 h_label_23 h_label_24 h_label_25 v_label_ v_label_1 v_label_2 v_label_3 v_label_4 v_label_5 v_label_6 v_label_7 v_label_8 v_label_9 v_label_10 v_label_11 v_label_12 v_label_13 v_label_14 v_label_15 v_label_16 v_label_17 v_label_18 v_label_19 v_label_20 v_label_21 v_label_22 v_label_23 v_label_24 v_label_25 scale_note right_marker left_marker gridscale 1 grid sq
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y? tau_axis circle_mc _visible InitializeStrings dimension_sigma dimension_tau construction_details filter_0 filter_1 filter_2 filter_3 strip gotoAndPlay filter_0.strip thetaPrincipal filter_0.txtOne filter_0.dot txtOne sDim hSigma x s MakeHTMLgreek _parent txtExplain sText First, we will plot the state of stress on the x face of the stress element. The normal stress on the x face is = sSigmaX . filter_1.strip filter_1.txtOne filter_1.dot hTau xy t The shear stress on the x face of the stress element is sTauX tauXY A negative shear stress on the x face rotates the stress element clockwise; therefore, we plot this stress above the axis. A positive shear stress on the x face rotates the stress element counterclockwise; therefore, we plot this stress below the x ( , ) Plot this point and label it x (to denote that it represents the state of stress on the x face). filter_2.strip filter_2.txtOne filter_2.dot y Next, we will plot the state of stress on the y face of the stress element. The normal stress on the y face is sSigmaY filter_3.strip filter_3.txtOne filter_3.dot The shear stress on the y face of the stress element is A negative shear stress on the y face rotates the stress element counterclockwise; therefore, we plot this stress below the A positive shear stress on the y face rotates the stress element clockwise; therefore, we plot this stress above the y ( sTauY Plot this point and label it y (to denote that it represents the state of stress on the y face). show_the_circle @2 # O = O O I O O O
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Z6OJ 6 [ ? filter_1 F Z6OJ 6[ ? filter_0 F Z6OJ 6a ? dimension_sigma F Z6OJ% 6!a ? dimension_tau @ ? @ ? U tau_axis circle_mc _visible InitializeStrings twoTheta thetaPrincipal eqn_center eqn_radius dimension_sigma dimension_tau filter_0 filter_1 filter_2 filter_3 plain_circle gotoAndPlay txtOne sDim x ( sSigmaX , sTauX ) filter_1.strip filter_1.txtOne filter_1.dot y ( sSigmaY sTauY filter_3.strip filter_3.txtOne filter_3.dot construction_details draw_the_line draw_diameter draw_diameter.txtCenter _parent txtExplain sText Draw a line connecting points x and y. This line is the diameter of Mohr's circle. draw_the_center_dot _rotation txtCenter C hSigma s MakeHTMLgreek The point where the diameter crosses the axis is the center of the circle. Note: the center of the circle will always lie somewhere on the axis. Label this point C. Next, we need to compute the coordinates of C and the value of the circle radius R. hSigmaX x hSigmaY y The location of C is found from the average of the two normal stresses: and . sigmaX sigmaY center strip filter_0.strip filter_0.txtOne filter_0.dot filter_2.strip filter_2.txtOne filter_2.dot The radius R is found from the Pythagorean theorem: tauXY radius dTemp thetaPrincipalRadians Math cos abs dimension_sigma.txtOne dimension_sigma.pos_mask dimension_sigma.dim_lines formatDecimals sin dimension_tau.txtOne dimension_tau.dim_lines Use the center and the radius to draw the circle. This circle represents every possible combination of normal stress and shear stress as the orientation of our reference axes is changed. show_the_circle @2 # O = O O O O
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center_radius 6 b plain_circle 6 g draw_diameter 6&s m6 eqn_center 68 m6p eqn_radius @ ? @ &8?: , tau_axis PassVariablesToCircleMC eqn_principal _visible dimension_tau dimension_sigma construction_details hSigma s MakeHTMLgreek _parent txtExplain sText The principal stresses are the extreme values of normal stress that occur at any orientation for a specific state of stress. circle_mc gotoAndPlay nt_diameter hSigma1 1 The most positive normal stress is termed . Inspection of Mohr's circle reveals that the most positive value of is found by simply adding the radius R to the center C of the circle. Note: Although is the most positive normal stress, may be a negative number. tween_1 We calculate from... eqn_sigma1 center radius hSigma2 2 The most negative normal stress . From Mohr's circle, we see that the most negative value of is found by subtracting the radius R from the center C coordinates. Note: Although is the most negative normal stress, may be a positive number. tween_2 Calculate from.... eqn_sigma2 show_the_circle @2 # = O O O I g =
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eqn_principal @ ? @ &? M tau_axis PassVariablesToCircleMC circle_mc xy_diameter arc_sweeper_negative gotoAndStop arc_sweeper_positive gotoAndPlay eqn_thetaP sigmaX sigmaY tauXY thetaPrincipal twoTheta _visible dimension_sigma dimension_tau filter_0 filter_1 filter_2 filter_3 plain_circle h2thetaP 2 p q MakeHTMLgreek theta_target theta_text sString construction_details _parent txtExplain sText We must determine the orientation of the principal stresses relative to our x-y coordinate system. To do this, we first compute the angle between the radial line to point x and the radial line to principal stress 1. We use the geometry of Mohr's circle to compute this angle, which we will call . dTemp thetaPrincipalRadians Math cos abs dimension_sigma.txtOne dimension_sigma.pos_mask dimension_sigma.dim_lines txtOne sDim formatDecimals sin dimension_tau.txtOne dimension_tau.dim_lines How do we decide which value for the angle (either sAnglePos or sAngleNeg ) is the correct one?
The geometry of the circle will tell us. We are trying to find the angle between point x and point 1. The correct angle is = . The magnitude of is sAngleMagnitude . hSigma1 1 s Therefore; the principal stress acts in the x direction. sweeper_flag acts in the y direction. To get from point x to point 1 on the circle, we must rotate in a clockwise direction. sweeping static in a counterclockwise direction. show_the_circle @2 # = N N R N N R R O
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theta_principal 6& m; + eqn_thetaP @ ? @ &? / tau_axis PassVariablesToCircleMC dimension_sigma _visible dimension_tau eqn_tau_max hTau max t MakeHTMLgreek hS1 1 MakeHTMLs1s2 hS2 2 construction_details _parent txtExplain sText Looking at Mohr's circle, we observe that the maximum shear stress corresponds to the top and bottom points on the circle (i.e., and ). The magnitude of is equal to the circle radius R. circle_mc gotoAndPlay nt_diameter hSigma s Unlike the principal stresses, however, the normal stress corresponding to is generally not equal to zero. The normal stress at hs2 is equal to the center coordinate C of the circle. Two points on the circle have the same magnitude of shear stress . We'll label the point that has positive shear stress ... tween_s1 We'll label the point that has negative shear stress tween_s2 avg We calculate and the magnitude of from C and R... center radius show_the_circle @2 # = O O O
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tau_max 6& m6x eqn_tau_max @ ? @ &? S tau_axis PassVariablesToCircleMC circle_mc xy_diameter arc_sweeper_negative gotoAndStop arc_sweeper_positive gotoAndPlay eqn_thetaS sigmaX sigmaY tauXY thetaMaxShear twoTheta thetaPrincipal _visible dimension_sigma dimension_tau filter_0 filter_1 filter_2 filter_3 plain_circle h2thetaS 2 s q MakeHTMLgreek theta_target theta_text sString hS1 1 MakeHTMLs1s2 hS2 construction_details _parent txtExplain sText We must determine the orientation of the maximum shear stress relative to our x-y coordinate system. To do this, we first compute the angle between the radial line to point x and the radial line to either or . We use the geometry of Mohr's circle to compute this angle, which we will call . dTemp thetaPrincipalRadians Math cos abs dimension_sigma.txtOne dimension_sigma.pos_mask dimension_sigma.dim_lines txtOne sDim formatDecimals sin dimension_tau.txtOne dimension_tau.dim_lines How do we decide which value for the angle (either sAnglePos sAngleNeg ) is the correct one?
The geometry of the circle will tell us. If we wish to find the angle between point x and point , the correct angle is = . The magnitude of is sAngleMagnitude . hTauMax max t Therefore; the maximum shear stress hSigma1 acts on the x and y faces. sweeper_flag To get from point x to point on the circle, we must rotate in a clockwise direction. in a counterclockwise direction. show_the_circle @2 # = N N R N N R R O
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theta_shear 6& {d6P eqn_thetaS @ ? @ &?! tau_axis construction_details PassVariablesToCircleMC circle_mc nt_diameter _visible xy_diameter arc_sweeper_negative gotoAndStop arc_sweeper_positive gotoAndPlay eqn_sigmaN center radius theta thetaPrincipal sigmaN tauNT twoTheta dimension_sigma dimension_tau htheta q MakeHTMLgreek hthetaequals = Math formatDecimals h2thetaequals 2 hsigma s htau t hsigmaN n htauNT nt _parent SetupTheRotatedElement rotated_element highlight_hole txtExplain sText We want to compute the normal stress and the shear stress acting on a surface specified at an angle of relative to the x face of the stress element (i.e., the face with the x axis as its normal). highlight _rotation The outward normal to the specified surface is called the n axis. Since a plane is defined by its normal, we term the specified surface the n face. The direction parallel to the specified surface is called the t axis. The face whose normal is the t axis is called the t face. The positive t direction is defined such that the n-t axes form a right-handed coordinate system. Mohr's circle is plotted in a - coordinate system. Since the angle is defined in an x-y coordinate system, we cannot directly apply it geometrically in Mohr's circle. We must relate our angular measure to the point on Mohr's circle that corresponds to the state of stress on the x face. This is the point on the circle labeled x. dTemp (counterclockwise in this case). (clockwise in this case). Since Mohr's circle is derived in terms of double-angle trigonometry functions, we must double our specified angle when working on the circle. Starting at point x, we rotate around the circumference of the circle in the same direction as in the x-y coordinate system Label this point n. Rotate 2(90) counterclockwise from n and label this point t. The coordinates of point n are the normal stress acting on the n face. Similarly, the coordinates of point t are the normal and shear stresses acting on the t face. PassVariablesToEqnSigmaN BetaArcs theta_target The coordinates of point n tell us and on the surface specified by the angle (i.e., the n face). Next, we use the geometry of the circle to compute the coordinates for point n.
We note that the angle from point x to point 1 is abs . As previously discussed, the magnitude of the angle from point x to point n is ... From the circle, we observe that the angle from point n to the axis is b = betaAngle . The circle center C is located at . The circle radius is R = . Using C, R, and the angle , we can now compute using trigonometry. sSig negative sSigTens compression positive tension sTauAbove above sClock clockwise below counterclockwise Since is , the normal stress on the n face is . Point n is the axis; therefore, the shear stress acting on the n face in the t direction rotates the element . cos dimension_sigma.txtOne dimension_sigma.pos_mask dimension_sigma.dim_lines sigmaX sigmaY txtOne sDim R cos( ) sin dimension_tau.txtOne dimension_tau.dim_lines tauXY R sin( We could follow a similar procedure to find on the t face. show_the_circle @2 # = HL @ H * = N O A I * = N O c B I * = N O $ = N O N N R N N R
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4 4 theta_shear mohr gotoAndPlay construction_details R NQ O O 3 NH 3 O $ J % D @'
4 @'
x 4 theta_shear mohr gotoAndPlay construction_details R NP O O updateAfterEvent = % J @ @ @ @ @ @ @ @ @ ? @ ? & = PassVariablesToMohrMC = sigmaN mohr
gotoAndPlay R select_stress_display @ # given @2 #' 2 SetupTheRotatedElement =
compute_sigmaN & D @'
@'
4 / sigmaN mohr gotoAndPlay construction_details R NQ O O = NH = O & T ' D @'
4 @'
s / sigmaN mohr gotoAndPlay construction_details R NP O O updateAfterEvent = ' T @ @ @ @ @ @ @ @ @ ? @