Shear center

The point where the torsional moments becomes nullified or The point where the shear loads should act to prevent the torsion of beam

Find out C/S center of gravity

Calculate Moment of Inertia and relate it to steiners theorem

Determine shear flow and shear stress by using static moments

Calculate the resultant forces acting on the beam by which bending moment is calculated and through which you can find the shear force acting on and shear center wrt the coordinate axis you are taking lets say y axis

The shear center M is the point in which transverse forces Vz and Vy must act so that a torsional effect does not occur in addition to bending of the beam under consideration. This can be expressed in such a way that it is required that the circumferential unit shear flows according to (6.28) do not cause a moment about the x− axis

ring integral of Tyrt M ds = 0, ring integral Tzrt M ds = 0.

1) take an arbiatary point as D and calculate the distance using it

rt M = rt D − (yM − yD) cos α − (zM − zD)sin α

and then use above formulae  Tyrt M ds =  Tyrt Dds − (yM − yD)  Ty dz ds ds + (zM − zD)  Ty dy ds ds = 0.to conclude Ym and Zm.

there is a formulae for it

General rules: - If a cross section has one axis of symmetry, the shear center M is always located on this axis of symmetry. - If the cross-section under consideration is point-symmetric or double-symmetric, then the shear center M is always located at the center of gravity S. - If the cross section under consideration consists of a number of thin-walled straight segments that all intersect at one point, then the shear center M lies exactly at the intersection of the segments

They dont consider Shear strains and are neglected

The Timoshenko beam theory:

Similar to the Euler-Bernoulli beam theory, we assume that cross-sections remain plane and no warping occurs during deformation. However, the normal hypothesis is discarded, meaning that cross-sections still do not warp when the beam deflects, but that the cross-sections in the deformed state do not necessarily have to be orthogonal to the longitudinal axis of the beam (Fig. 16.1). Shear strains in the beam are thus explicitly allowed, but we assume that they are constant over the cross-section thickness. • We assume linear-elastic material behavior and assume that Hooke’s law is valid. • It is assumed that the cross-sectional shape does not change during deformation as a result of a given load. • We assume geometric linearity and assume that the deformations are small with respect to the length of the beam, but also with respect to the dimensions of its cross-section. • In this chapter, straight beam structures are considered exclusively

Thimoshenko beam thory in contrast with EB BT allows for shear strains which will be constant over the height due to anglw siy.but however shear flow resulting from transverse shear force is parabolic over those segments of c/s that run in direction of action of Vz

The theory assumes that the shear strain (and hence the shear stress) is uniformly distributed across the cross-section of the beam. However, this assumption does not hold true for most practical cases. In reality, the shear stress distribution is not uniform:

• For a rectangular cross-section, for example, the shear stress is zero at the top and bottom surfaces and reaches a maximum at the neutral axis.

A common way to determine the shear correction factor K is to calculate the strain energy introduced into the system due to shear for both the actual parabolic case (16.30) and the constant case (16.17) and equate them from which the shear correction K can be obtained

for arbitary case then is A more general way to represent the influence of transverse shear deformations is based on the calculation of the so-called reference or effective cross section Aef f . The reference cross-section or the effective area Aef f represents the area of the cross-section which is effectively involved in the shear transfer. Again, we obtain statements about Aef f via energetic considerations and first consider the related shear strain energy provided by the shear force Vz along the shear strain γs(s)