Obviously these two values of the shear are equations of two parallel
straight lines, that represented by equation (_f_) passing through
_A_, and that represented by equation (_h_) passing through _B_, the
constant vertical distance between them being _g_. Hence let _BF_ be
laid off negatively downward and _AG_ positively upward, each being
equal to _g_ by any convenient scale. The ordinates drawn from the
various positions 1, 2, 3 ... 6 of _g_ on _AB_ to _AD_ and _BC_ will
be the shears at _X_ produced by the load _g_ at any point of the
span, and determined by equations (_f_) and (_h_). The influence line,
therefore, for the section _X_ will be the broken line _ADCB_. When
_g_ is at _X_ the sign of the shear changes, since the latter passes
through a zero value.
If a train of weights _W₁_, _W₂_, _W₃_, etc., passes across the span,
the total shear at _X_ will be found by taking the sum of the vertical
intercepts between _AB_ and _ADCB_, drawn at the positions occupied by
the various single weights of the train. If those single weights are
expressed in terms of the unit load _g_, the shear _S_ will have the
value
1
_S_ = ---- ∑ _Wy_;
_g_
_y_ being the general value of the intercept between _AB_ and the
influence line. The latter shows that the greatest negative shear at
_X_ will exist when the greatest possible amount of loading is placed
on _AX_ only, while the greatest positive shear at the same section
will exist when _BX_ only is loaded. If _BX_ is the smaller segment of
span, the latter shear is called the “counter-shear,” and the former
the “main shear.”
If the loads are applied at panel-points of the span only, the
treatment is the same in general character as that employed for bending
moments. In Fig. 25_a_ let 4 and 5 be the panel-points between which
the load _g_ is found, and let the panel length be _p_. Also, let _z′_
be the distance of the weight _g_ from panel-point 4. The reactions at
_A_ and 4 will then be
_l - z p - z′_
_R_ = ------_g_ and _R₄_ = ----------- _g_.
_l p_
The shear at the section _X_ for any position of the weight _g_ will
then be
(_z′ z_)
_S_ = _R - R₄_ = _g_(--- - ---). (_k_)
(_p l_)
As this is the equation of a straight line, with _S_ and _z_ or _z′_
for the coordinates, the influence line for the panel in which the
section _X_ is located will be the straight line represented by _KL_ in
Fig. 25_a_.
Public-domain text, read in full here on John Shaqi.
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