Strain Energy Method

[Zouatine]

Dear All,

Respected All hope you are doing well,

Kindly , I'm trying to understand how to derive the shape functions of Timoshenko beam using the concept of Virtual Work method. The theory is done on 3D beam element (Fig 1) .

Fig 1

Considering all the deformation in all the directions, the deformations ( disps , rotations ) are derived as well as the strain (Fig 2) :

Fig 2

Coming to the formulas of the strain energy in case of the 3D beam is defined as follows (Fig 3):

Fig3

This is the strain energy expression for general cross-section shape.

My understanding problem appears here, according to the slide , in case we do have a symmetrical cross-section , the two terms in the next Fig 4 are zero :

Fig 4

Would you please clear the point , why these two elements are zero in case of symmetrical cross-section ?


Thank You ,
Waiting for you kind reply,

 

[WARose]

why these two elements are zero in case of symmetrical cross-section

IIRC, the cross-section is in equilibrium. [Net] Strain energy develops along the length.

 

[Zouatine]

Thank you Sir for your reply,

Would you please elaborate more in your answer because I'm new in this topic. your help is much appreciated .

Thank you looking forward for your reply.

 

[WARose]

Would you please elaborate more in your answer because I'm new in this topic.

Each cross section is in equilibrium. (Considering it by itself.) Ergo the net strain energy must be zero. (Maybe the better way to think of it is the distortion/displacement one way is equal to what it is in the other direction (in-plane).) But considering the full length of the beam, the net isn't zero. So that's why Eq.6 isn't equal to zero.