SEC

Returns the secant of a number.

Syntax:

SEC(number)

returns the secant of number.

Example:

SEC(0): returns 1, the secant of 0.

Application:

Imagine a vertical column in a building that is subject to a compressive load, P, applied at its top. The column has a certain length, L, and a bending stiffness, EI, which is a measure of its resistance to bending. Due to the load, the column will deflect laterally (bend) by a small amount.

The deflection curve of the column can be described by a differential equation. The maximum lateral deflection, δmax, at the top of the column is given by the following equation:

$δ_{ma x} = δ_{0} sec (\frac{L}{2} \frac{P}{E I})$

where:

δmax is the maximum lateral deflection at the top of the column.
δ0 is the initial eccentricity of the load (the distance the load is offset from the column's central axis, which causes the initial bending).
L is the length of the column.
P is the compressive load applied to the column.
EI is the bending stiffness of the column.
sec is the secant function, sec(x)=1/cos(x).

An engineer needs to determine the maximum deflection of a steel column. The column has the following properties:

Length (L): 10 meters
Bending Stiffness (EI): 3×107N⋅m2
Initial Eccentricity (δ0): 0.05 meters (5 cm)

The engineer wants to see how the maximum deflection changes as the applied load (P) increases. This is crucial for ensuring the column remains stable and does not buckle.

The engineer can create a table to calculate δmax for different values of P.

First, let's calculate the term inside the secant function: $k = \frac{L}{2} \frac{P}{E I}$ .

$k = \frac{10}{2} \frac{P}{3 \times 1 0 ^{7}} = 5 \frac{P}{3 \times 1 0 ^{7}}$

Now, we can calculate the maximum deflection for various loads.

	Load, P (Newtons)	$k = 5 \frac{P}{3 \times 1 0 ^{7}}$	SEC(k)	Maximum Deflection, δmax=δ0SEC(k) (meters)
	A	B	C	D
1	1×105	0.288	1.042	0.05×1.042=0.0521
2	5×105	0.645	1.181	0.05×1.181=0.0591
3	1×106	0.913	1.503	0.05×1.503=0.0752
4	1.5×106	1.118	2.112	0.05×2.112=0.1056
5	2×106	1.291	3.593	0.05×3.593=0.1797
6	2.5×106	1.443	9.940	0.05×9.940=0.4970
7	2.96×106	1.570*	≈∞	≈∞

* Note: k=1.570 radians is very close to π/2≈1.5708, where cos(π/2)=0. Since sec(x)=1/cos(x), the value of the secant function approaches infinity as x approaches π/2.

Interpretation of the Results

The table clearly shows how the maximum deflection of the column increases dramatically as the applied load approaches a critical value. This critical load is known as the Euler buckling load, Pcr. When k approaches π/2, the cosine of the angle approaches zero, and the secant function, and thus the deflection, approaches infinity. This point represents the load at which the column becomes unstable and buckles, leading to catastrophic failure.

The secant function in this example is a critical tool for structural engineers to:

Understand the non-linear behavior of columns under compression.
Predict the maximum deflection and ensure it stays within safe limits.
Determine the buckling load for a given column, which is essential for safe design.