Solid Elliptical Section Civil Engineering (CE) Notes | EduRev

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Civil Engineering (CE) : Solid Elliptical Section Civil Engineering (CE) Notes | EduRev

The document Solid Elliptical Section Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Advanced Solid Mechanics - Notes, Videos, MCQs & PPTs.
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Solid elliptical section

The first cross section shape that we consider is that of an ellipse. That is we study a bar with elliptical cross section, whose boundary is defined by the function,
                                                     Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                               (9.58)

where a and b are constants and we have assumed that the major and minor axis of the ellipse coincides with the coordinate basis, as shown in the figure 9.11.

Choosing the Prandtl stress function to be,

                                                            Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                                     (9.59)       

 where C is a constant, we find that it satisfies the boundary condition (9.50).

  

Solid Elliptical Section Civil Engineering (CE) Notes | EduRev

 Substituting the stress function, (9.59) in the equation (9.47) we obtain                                                                      

Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                  (9.60)  

Using (9.59) and (9.60) in (9.57), twisting moment is obtained as
 

  Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                 (9.61)

 Corresponding to the stress function (9.59) the shear stresses are computed to be
   Solid Elliptical Section Civil Engineering (CE) Notes | EduRev  (9.62)

where Ixx = πab3/4, Iyy = ba3/4 and we have substituted for the constant, C from equation (9.60). The variation of these shear stresses over the cross section is indicated in figure 9.12.

The resultant shear stress in the xy plane is given by,

Solid Elliptical Section Civil Engineering (CE) Notes | EduRev       (9.63)

It is clear from equation (9.63) that the extremum shear stress occurs at (0,0) or the boundary of the cross section. It can be seen that at (0,0) minimum

 

  Solid Elliptical Section Civil Engineering (CE) Notes | EduRev
Figure 9.12: Shear stresses in a bar with elliptical cross section subjected to end torsion 

shear stress occurs, as τ = 0 at this location. At the boundary the extremum shear stresses would occur at (0,±b) and (±a,0). Thus, the extremum shear stresses are τext = 2T/(πab2) at (0,±b) and τext = 2T/(πba2) at (±a,0). If a > b, then the maximum shear stress in the elliptical cross section is,
       Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                  (9.64)

Substituting (9.59) in equations (9.45) and (9.46) and rearranging we obtain 

  Solid Elliptical Section Civil Engineering (CE) Notes | EduRev           (9.65)

Solving the differential equations (9.65) we obtain

Solid Elliptical Section Civil Engineering (CE) Notes | EduRev             (9.66)


where D0 is a constant. Since, there is no rigid body translation, we require that ψ(0,0) = 0. Hence, D0 = 0. Thus, we find that the section warps into a diagonally symmetric surface as shown in figure 9.13.

Rearranging the equation (9.61) we can get Ω as a function of torque, T
as

Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                  (9.67)

 The entire displacement field, (9.33) for a bar with elliptical cross section is now known that we have found the warping function (9.66). The deformation of a bar with the elliptical cross section computed using this displacement field is shown in figure 9.14

 

 

  Solid Elliptical Section Civil Engineering (CE) Notes | EduRev

Figure 9.13: Warping deformation of a elliptical cross section due to end torsion

Before concluding this section let us see the error that would have been made if one analyzes the elliptical section as a closed section. The polar moment of inertia for the elliptical section could be computed and shown to be J = πab(a2 + b2)/4. Then, the angle of twist per unit length computed from (9.17) would be,
Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                   (9.68)

The ratio of Ωcl/Ω, computed from equations (9.68) and (9.67) is
Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                        (9.69)

 Thus, for a given torque open section twists more. Next, we examine the shear stresses. If one uses (9.17) to compute the shear stresses, they would erroneously conclude that the maximum shear stress occurs at (±a,0), by virtue of these points being farthest from the center of twist and the value of this maximum shear stress would be,
Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                   (9.70)

 

 

 

Solid Elliptical Section Civil Engineering (CE) Notes | EduRev  

Figure 9.14: A bar with elliptical cross section subjected to end twisting moment

 

Solid Elliptical Section Civil Engineering (CE) Notes | EduRev

Comparing equations (9.64) and (9.70) we find that
Solid Elliptical Section Civil Engineering (CE) Notes | EduRev                   (9.71) 

Thus, closed section underestimates the stresses in the bar.

Hence, from both strength and serviceability point of view assuming the elliptical cross section to be closed would result in an unsafe design.

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