7.2. Combined axial tension (compression) and bending
The beam is represented in Fig. 7.3 which is subject to the combined action of bending and axial tension. The transverse load causing bending can also be more complicated.
To determine the sum stresses the principle of the superposition is applied. The tension stresses of the force F1 at all points of the cross section, as it is well known, are equal, they are determined by the formula
, (7.15)
Fig. 7.3.
or in a general case (under any axial load) by the formula
where
is the normal force at the section considered. The bending stresses
according to the formula
Hence, the sum stresses at any point are
(7.16)
In the
given case the section is dangerous at the rigidly clamped end where
the maximum bending moment acts and is equal
At this section the more loaded points will be the ones which are
located along the line AB
because at them the stresses in tension and the most stresses in
bending are added up (
).
At the point along the line DC
stresses will be lesser:
(7.17)
For the bars equally working in tension and compression the strength conditions are as follows
(7.18)
If the transverse load is complicated to determine the danger section and the maximum bending moment it is necessary to draw first the bending moment diagram.
The received relations are also correct under the action of the compression force but the stress will be negative and the most (by absolute value) stresses will be at the points along the line DC. It is necessary to note that under the action of the compression force the above given formulas are correct only for the bars of the large rigidity, i.e. those whose axial compression force influence upon the deformation of bending is insignificant and may not be considered.
In the case of tension and unsymmetrical bending the stresses are determined by the formula
(7.19)
For bars from the materials equally working in tension and compression with the cross section having the angle points equidistant from the principal x and y-axis (the rectangular type, the double - T profile and the like) the strength condition has the form
(7.20)
For the bars manufactured from materials differently working in tension and compression the strength checking must be performed both by tension and compression stresses.
7.3. Eceentrical tension (compression)
Very often the longitudinal load can be applied not to the centroid of the cross section of the bar but with some displacement (an accentricity) about the section axis (Fig. 7.4 a).
a)
Zero
line
b)
Fig. 7.4.
By using
the section method we would display at any bar cross section the
normal force
and the bending moments which are equal about the x-axis
and about the y-axis
Therefore the stress at any point of the cross section with the coordinates x and y are determined like that in axial tension and bending into two planes, i.e. by the formula which is analogous to the formula (7.18):
(7.21)
For the sections having the overhanging angle points the stresses are determined by the formula
(7.22)
where
and
are the section modulus about x and y-axis.
At the section shown in Fig. 7.4 b the maximum stresses will be at the point E as here the tension stresses caused by both the axial tension and bending are summed up in two planes:
(7.23)
The minimum (in algebraic sense) stresses will be at the point D:
(7.24)
Thereby they can appear both tension and compression ones.
The strength condition with the tension stresses has the form
(7.25)
If the point of the force application is on one of the principal axes of the section, for example, on the y-axis the previous formula is simplified:
(7.26)
To determine the location of the danger points for the arbitrary cross section it is necessary to find the zero line location.
By equating to zero the stress we get the zero line equation:
(7.27)
where x0 and y0 are the variable coordinates of the zero line points.