Civil Engineering Reference
In-Depth Information
cHAPtER 3
n
umericAL
i
ntegrAtion
And
d
ifferentiAtion
The
integration
of a continuous function is used to find the area under the
function and to evaluate integral relationships of functions.
Differentia-
tion
evaluates the rate of change of one variable with respect to another.
Examples of structural engineering problems involving integration and
differentiation include geometrical properties of centroids of areas and
volumes; moment of inertia; relationships between load, shear, moment,
rotation, and deflection of beams using the equation of the elastic curve;
and other strain energy relationships of structures involving shear, torsion,
and axial forces. Many methods exist to solve such types of problems with
varying levels of exactness. These and other problems will be covered in
the following chapters.
3.1
tRAPEZOiDAL RuLE
Consider a function
f
(
x
) graphed between points
a
and
b
along the x-axis
as shown in Figure 3.1. One approximation of the area under the curve is
to apply the trapezoidal rule by dividing the area into
n
strips of width
∆
x
.
Then, approximate the area of each strip as a trapezoid.
Calling the ordinates
f
(
x
i
)
= y
i
(
i
= 1, 2, 3,…,
n
,
n
+1), the areas of each
strip are as follows:
Ax
yy
Ax
yy
Ax
yy
A
n
+
+
+
…
=
∆
1
2
,
=
∆
2
3
,
=
∆
3
4
,
1
2
3
2
2
2
x
yy
+
=
∆
n
n
+
1
2
