Building Design and Construction Handbook - Merritt - Ventech!

5.42 SECTION FIVE
not appreciably greater than the average for the web section alone, if it is assumed
that the flanges take no shear.
5.5.14 Combined Shear and Bending Stress
For deep beams on short spans and beams made of low-strength materials, it is
sometimes necessary to determine the maximum stress ƒ� on an inclined plane
caused by a combination of shear and bending stress—v and ƒ, respectively. This
stress ƒ�, which may be either tension or compression, is greater than the normal
stress. Its value may be obtained by application of Mohr’s circle (Art. 5.3.6), as
indicated in Fig. 5.10, but with ƒ y � 0, and is
5.5.15 Beam Deflections
�
��
2
ƒ ƒ 2
ƒ� � � v � (5.60)
2 2
When a beam is loaded, it deflects. The new position of its longitudinal centroidal
axis is called the elastic curve.
At any point of the elastic curve, the radius of curvature is given by
EI
R � (5.61)
M
where M � bending moment at the point
E � modulus of elasticity
I � moment of inertia of the cross section about the neutral axis
Since the slope dy/dx of the curve is small, its square may be neglected, so that,
for all practical purposes, 1/R may be taken equal to d 2 y/dx 2 , where y is the
deflection of a point on the curve at a distance x from the origin of coordinates.
Hence, Eq. (5.61) may be rewritten
2 dy
M � EI (5.62)
2 dx
To obtain the slope and deflection of a beam, this equation may be integrated, with
M expressed as a function of x. Constants introduced during the integration must
be evaluated in terms of known points and slopes of the elastic curve.
Equation (5.62), in turn, may be rewritten after one integration as
B M
� � � � � dx (5.63)
B A
A EI
in which � A and � B are the slopes of the elastic curve at any two points A and B.
If the slope is zero at one of the points, the integral in Eq. (5.63) gives the slope
of the elastic curve at the other. It should be noted that the integral represents the
area of the bending-moment diagram between A and B with each ordinate divided
by EI.