311 Ch17

IX.) Combined Stresses A.) Biaxial Bending  b = M x + M y S x S y

B.) Combined Bending & Axial Stresses  = + P + M A S

C.) Eccentrically Loaded Axial Members M = Pe , where “e” = “eccentricity”  = + P + Mc , A I Replace M with Pe:  = + P + Mc A I

D.) Stress Element  x  = Normal Stress in x direction  y  = Normal Stress in y direction  xy  = Shear Stress in x and y direction  y  xy  x  xy x  xy  xy  y  x

D.) Combined Normal & Shear Stresses  n  =  x  cos 2  y  sin 2  xy  sin  cos   v  = (  x  y  sin  cos  xy  cos 2  sin 2   y  n  v  xy  x  xy y x 

E.) Maximum Normal Stresses (Principal Stresses)  1,2  = 0.5(  x  y    (  x  y   +  xy   The maximum normal stresses occur on a plane whose normal is at the angle  p with the x-axis, determined from: tan(2  p ) =  xy   (  x  y 

F.) Maximum Shear Stress  v,max  =   (  x  y   +  xy  The maximum shear stress occurs on a plane whose normal is at the angle  v with the x-axis, determined from: tan(2  v ) = (  x  y   xy 

G.)Mohr’s Circle

Otto Mohr developed a useful graphical method to visualize the principal stress equations.

The coordinates of a point on Mohr’s Circle give the normal and shear stresses on a given plane……

G.)Mohr’s Circle

and the orientation of the line from the center of the circle to the given point gives the orientation of the plane on which those normal and shear stress are acting.