The document discusses various types of combined stresses: 1) Biaxial bending stresses are the sum of stresses from bending in two perpendicular directions. 2) Combined bending and axial stresses add the axial stress to the bending stress. 3) Eccentrically loaded axial members have extra bending stress due to eccentricity. 4) Normal and shear stresses on a stress element are defined. 5) Principal stresses are the maximum normal stresses on perpendicular planes. 6) Maximum shear stress and its orientation are also defined. 7) Mohr's circle is introduced as a graphical method to represent the stresses on a plane.

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

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

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

4. 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

5. 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 

6. 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 

7. 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 

10. G.)Mohr’s Circle

11. G.)Mohr’s Circle