Energy methods:

Strain energy under axial load:

     Consider a member of constant cross sectional area A, subjected to axial force Pas shown in Fig. 2.8. Let E be the Young’s modulus of the material. Let the member be under equilibrium under the action of this force, which is applied through the centroid of the cross section. Now, the applied force P is resisted by uniformly distributed internal stresses given by average stress σ =P/A as shown by the free body diagram (vide Fig. 2.8). Under the action of axial load P applied at one end gradually, the beam gets elongated by (say) . This may be calculated as follows. The incremental elongation of du small element of length  dx of beam is given by,

clip_image001clip_image002

clip_image003

Now the work done by external loads   W= 1/2XPu                                    (3.0) 

    In a conservative system, the external work is stored as the internal strain energy. Hence, the strain energy stored in the bar in axial deformation is,

clip_image001[4]

Substituting equation (2.0) in (4.0) we get,

clip_image002[4]

Strain Energy in Bending :

clip_image003

Fig .6

Consider a beam AB subjected to a given loading as shown in figure.

Let,

M = The value of bending Moment at a distance x from end A.

From the simple bending theory, the normal stress due to bending alone is expressed as.

clip_image002[6]

clip_image001[6]

Castigliano’s theorem:

clip_image002

clip_image004

clip_image002[4]clip_image004[4]

clip_image002[6]clip_image004[6]

Numerical:

clip_image002[8]

clip_image004[8]clip_image006

clip_image002[10]

clip_image004[10]

clip_image002[12]

Maxwell’s Theorem

The displacement of a point B on a structure due to a unit load acting at point A is equal to the displacement of point A when the unit load is acting at point B, that is, fBA = fAB.

clip_image003

   

image

image

The rotation of a point B on a structure due to a unit couple moment acting at point A is equal to the rotation of point A when the unit couple moment is acting at point B, that is, αBA = αAB.

These are useful for 2nd-degree-indeterminate and higher structures.

Unit load method:

clip_image004[12]

clip_image002[16]

clip_image004[14]

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s