3 background information the technical theory of the oblique transvers
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3. Background Information
The technical theory of the oblique transverse bending of the beam is based on two
suppositions: the hypothesis of flat sections and the theory of pure direct bending.
Pure oblique bending of a beam is one in which the internal forces in the cross section of
the beam are reduced only to the bending moment, the plane of action of which does not contain
any of the main central axes of inertia of the cross section of the rod during bending.
Unlike direct bending, the curved axis of the rod does not lie in the force plane, i.e.
deformation of the rod during oblique bending does not occur in the plane of the bending
moment, but in the plane rotated relative to the force by a certain angle towards the plane of least
stiffness of the rod during bending.
Since the hypothesis of flat sections is valid, the cross section rotates about the neutral
axis, remaining flat. The oblique bending line is not perpendicular to the force plane.
Oblique bending is considered simultaneous bending in two planes zx and zy, where the x
and y axes are the main central axes of inertia of the cross section of the rod. For this, the
bending moment Mis resolved into components with respect to the x and y axes. The normal
stress at the cross-sectional point is calculated as the algebraic sum of the stresses due to the
moments Mr and My, i. e.
σ =
Mxy + Myx
Ix
y+ -x,
Ly
(6.1)
where Mr. My-bending moments relative to the x, y axes; Ix. Iy-axial moments of inertia of the
cross-sectional area of the rod relative to the x, y axes; x, y- the coordinates of the point.
The highest stresses occur at the points of the section farthest from the neutral line.
In oblique bending, the full displacement of a point is defined as/n8 = √√u² + v²
(6.2)
where u― the displacement of the point in the x direction; - the displacement of the point in the
y direction.
Consider the oblique bending of the beam.
Bending moment Min section I
M = F.1
where F - the force on the beam; - the length of the beam.
Components of the moment relative to the main central axes of inertia x and y
MM cos = F · 1 · cos
(6.3)
(6.4)
My = M. sin = F · 1 · sin &
(6.5)
We define the stress at point A. Since each of the moments Mr and My causes the greatest
tensile stresses at this point, and since h=2b
σ =
y +
Ix
My
Ly
F.l.cos
F.l.sin
x =
+
63
¹b3
(6.6)
(6.7)
F-l
OA = 3. F¹ (cos + 2 sino)
ΤΑ
2 ba
To calculate the displacement of the beam, consider a cantilever beam of length 7.
stiffness E, moment of inertia Ix, loaded at the free end by force F, under conditions of direct
bending in the zy plane. The differential equation of the elastic line, with the origin at the fixed
end, is given by the following equation:
8" = Mx(z)
Elx
(6.8)
After double integration, using the beam apparatus constraints, it can be shown that the
deflection of end of the beam in the x and y direction are:
1 Fl³ sin
u =
3
Ely
(6.9)/n5.1 Procedure Part A: Deformation Meter Calibration
1. Measure beam dimensions including length, L, the beam height, H, the beam width, B. and
the Distance to the calculated section, L1 (found in INFO) & record in Table 6.1.
2. Determine Beam and Apparatus Properties including Modulus of Elasticity, E and the
Calibration Coefficients for the Dial (aka watch) Type Deflection Indicators, Kf (found in
INFO) & record in Table 6.1.
3. To calibrate the strain gauge, it is necessary to set the beam in a straight bending position.
For the highest accuracy, position the beam so that the deflections will be largest. This
occurs when the bending is in the plane of least rigidity when o = 90°.
4. Use the beam dimensions to calculate the Moments of Inertia about the X and Y axes, I, & I,
& record in Table 6.1
5. Load the beam in 10 N increments (by adding 1 kg masses) from 0 to 40 N, take digital
tensometer deformation readings of the strain gauge located on the upper surface, N3
(CHANNEL 3), and lower surfaces, N4 (CHANNEL 4) of the beam at each load stage &
record in Table 6.2.
6. Calculate the experiment results as follows & record in Table 6.2:
a. The change in Tensometer readings 4Ni, for each load step.
b. The average change in the readings for the Tensometer, AN, for each gauge.
c. The average change for all gages AN, (note: absolute value bars).
7. Calculate the theoretical values as follows & record in Table 6.2:
a. The normal stress per loading stage, Ao, using the change in force between loadings AF =
10N, the distance to the calculated section, 7= 650 mm and where maximum, at the top
and bottom surfaces, y = h/2./n8. To determine the calibration coefficient for deformation meter, compare the theoretical value
of the linear strain per loading stage, 4s, to the average change in meter readings for all gages
AN, & record in Table 6.2.
5.2Procedure Part B: Oblique Bending Experimental Stress and Displacement
1. Rotate the beam to the working position for oblique bending, o = 60°.
2. Load the beam in 10 N increments (by adding 1 kg masses) from 0 to 40 N. At each load,
record in Table 6.3:
a. The Tensometer readings for all four strain gages.
b. The dial-type indicator reading horizontal deflection of the end of the beam (Note:
because the deflection is negative, use the red number scale and smaller dial is for
the 100 scale for readings beyond a full rotation (i.e., greater than 100)).
c. The dial-type indicator (on the back side of the beam) reading vertical deflection of
the end of the beam. (Note: to rotate simulator view, hold down right mouse button,
because the deflection is negative, use the red number scale and smaller dial is for
the 100 scale for readings beyond a full rotation (i.e., greater than 100)).
