# 2.1.3 Deflections under the longitudinal force

In this section we determine the magnitude and direction of the deformation produced by the axial force
. Solution to this problem will give the middle column (2) of tensor
(see (2) in **chapter 2.1.1**).

(1)

(2)

(3)

The force
acting in the cantilever axis direction produces moment
that results in deformation called here the **vertical bending of y-type (Fig. 1)**.

**Fig. 1. Vertical deflection of the y-type.**

In spite of the formal resemblance to vertical bending of z-type (see **chapter 2.1.2**), the deformation profile in this case is quite different. The equation describing the y-type bending reads (compare with (7) in **chapter 2.1.2**):

(4)

Boundary conditions remain the same: and . For the solution we find:

(5)

Thus, the tip vertical deflection due to this type of deformation is as follows:

(6)

Comparing (6) and (3) and taking into account the expression for the common multiplier
(see (12) in **chapter 2.1.2**), we get:

(7)

The angle of the beam end deflection is given by the following formula:

(8)

From formula (8) and diagram for the beam vertical bending of y-type **(Fig. 1)** it is easy to derive the tip deflection
induced by the force
application:

(9)

From (2), (7) and (9) it is easy to obtain:

(10)

Taking into account that , we get:

(11)

Finally, we calculate the components of the matrix (3) from **chapter 2.1.1** second column. From expressions (6–8) it follows that

(12)

Because under the influence of the force the top cantilever surface does not bend in the direction, then

(13)

## Summary.

- The y-type deflection is a result of the axial bending force action.
- To find the components of the inverse stiffness tensor corresponding to the y-type deflection, one should solve the problem of the beam static deflection which is reduced to the ordinary differential equation of the second order.
- The axial force results in the tip deflection not only in the longitudinal but also in vertical direction and in the deflection angle appearance.