38
3 Investigation of Euler–Bernoulli Beams in the Elastic Range
Fig. 3.1 General configuration for Euler–Bernoulli beam problems: a example of boundary conditions and external loads; b cross-sectional area (bending occurs in the X -Z plane)
for constant distributed load (q Z = q 0 = const.) to obtain the general analytical solution of the problem:
u Z (X ) =
1
E I Y
q 0 X
4
24
+
c 1 X
3
6
+
c 2 X
2
2
+ c 3 X + c 4
,
(3.1)
where the four constants of integration c i (i = 1, . . . , 4) must be determined based
on the boundary conditions (see Table 3.3). The following equations for the shear
force Q Z (X ), the bending moment M Y (X ), and the rotation ϕ Y (X ) were obtained
based on one-, two- and three-times integration and might be useful to determine
some of the constants of integration:
Q Z (X ) = −q 0 X − c 1 ,
(3.2)
M Y (X ) = −
q 0 X
2
2
− c 1 X − c 2 ,
(3.3)
ϕ Y (X ) = −
du Z (X )
dX
= −
1
E I Y
q 0 X
3
6
+
c 1 X
2
2
+ c 2 X + c 3
.
(3.4)
The internal reactions in a beam become visible if one cuts—at an arbitrary location X —the member in two parts. As a result, two opposite oriented shear forces Q Z
and bending moments M Y can be indicated. Summing up the internal reactions from
both parts must result in zero. Their positive direction is connected with the positive
coordinate directions at the positive face (outward surface normal vector parallel to
the positive X -axis). This means that at a positive face the positive reactions have
the same direction as the positive coordinate axes, see Fig. 3.2.
Once the internal bending moment M Y is known, the normal stress σ X can be
calculated:
σ X (X, Z ) =
M Y (X )
I Y
Z (X ) ,
(3.5)