Group 4

DISTRIBUTEDFORCES

For all practical purposes these lines of action will be concurrent at a single point G, which is called the center of gravity of the body.

CENTERS OF MASS

1. Lines

CENTROIDS OF LINES, AREAS, AND VOLUMES

2. Areas

CENTROIDS OF LINES, AREAS, AND VOLUMES

3. Volumes

CENTROIDS OF LINES, AREAS, AND VOLUMES

AREA

When the density of ƥ is small but has constant thickness t, we can model it as surface area of A. so the mass of the element become

Again, if ƥ and t are constant over the entire area, the coordinates of the center of mass of the body also become the coordinates of the centroid C of the surface area. The coordinates may be written

EXAMPLE QUESTION

Determine the distance h from the base of triangle of altitude h to the centroid of its area!

So, the two sides of the triangle have the same result and considered a new base with corresponding new altitude. The centroid lies at the intersection of the median. Since, the distance of this point from any side is one-third, the altitude of the triangle with that side considered the base.

VOLUME

THEOREMS OF PAPPUS

BEAMS—EXTERNAL EFFECTS

• Beams are structural members which offer resistance to bendingdue to applied loads.

• Most beams are long prismatic bars, and the loadsare usually applied normal to the axes of the bars.

Types of Beam

DISTRIBUTED LOADS

The Formula

BEAMS—INTERNAL EFFECTSShear, Bending, and Torsion• The force V is called the shear

force

• the couple M is called thebending moment

• the couple T is called a torsional moment.

• These effects represent the vector components of the resultant of the forces acting on a transverse section of the beam as shown in the lower part of the figure.

CABLE

Flexible Cable Parabolic Cable

Catenary Cable