代做MAE 157 - Lightweight structures代做Prolog

MAE 157 - Lightweight structures

Definition of the coordinate system

When defining the coordinate system, we will place the origin of the system at one end of the beam, as shown in Figure 1, and we will designate the axial axis as z. Next, we will define the x and y axes in such a way that they are perpendicular to each other and perpendicular to the z-axis, forming a right-handed coordinate system.

Definition of the positive forces and moments

After defining the coordinate system, it is important to have a convention about the forces sign. Keep in mind that forces and moments are vector quantities. This means they have magnitude, direction, and orientation. And the sign only represents their orientation.

Different authors may use various sign conventions. In this course, we will employ the sign convention depicted in Figure 2.

To understand the sign convention it is important to keep in mind that Newton’s third law applies. Meaning that if we cut the structure at a certain section, we will have two pieces. If we look at one face the forces will have a certain direction, but if we look at the other face, the direction of the forces will be the opposite.

In this course, we will follow the convention depicted in figure 2. It is important to note that if we cut the structure at a certain value of z we will get two faces (FACE A and FACE B as shown in figure 2). In this course, we will consider shear and moment (Mx, My, Mz, Sx and Sy) as positive when they go in the same direction as  the x, y and z axes in FACE B. Finally, we will consider axial force (N) as positive when the force is in tension.

Different views of the beam

Sometimes, instead of working with the 3-dimensional view, it is more convenient to work with 2d representations. You can find below some 2D representations of the same beam with the positive forces and moments.

Relationship between moment and shear

After defining the positive orientation of the moments and forces, it may be useful to know the relationship between the moment and shear. To find them, we can cut the beam and observe the forces and moments that act on one piece of beam as shown in figure 6 and 7.

It is important to note that moment and shear may vary with z. We can see that if we follow this sign convention, the following equations hold:

Derivation of the previous relationshiops

To derive the previous equation, we only need to do the balance of moments. If we do the summation of moment

from figure 6 (ignoring the higher order term  dzdz), we get:

Simplifying terms, we get:

If we do the same procedure on figure 7, we get:

Simplifying terms, we get: