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# CES Mechanics I Lecture 7

RWTH Aachen University, Bachelor program in Computational Engineering Science

Topics:

• Equilibrium analysis. 5.1.2

Analysis of equilibrium problems usually proceeds in three steps:

1. Draw the free-body diagram.
2. Write equilibrium equations in terms of the unknowns.

The unknowns are typically reaction forces, but sometimes also applied loads, angles or dimensions necessary for equilibrium.

A great deal of practice is needed to write the most convenient equations for a given problem, which can be solved one at a time, thus avoiding a solution of a coupled system of equations. Both the selection of the kinds of equilibrium equations, and their sequence, can be varied.

3. Solve the equation system for the unknowns.

In most cases, the number of the equations available will be the same as the number of unknowns. Such system is called statically determinate or statisch bestimmt.

In some cases, the number of the equations will be smaller than the number of unknowns. A rigid bar attached on both ends by a pin support is an example. Such system is called statically indeterminate or statisch unbestimmt and a full solution requires the knowledge of internal forces in the rigid body; a field called deformable body mechanics, which will be discussed in the third semester of CES. In some special cases, some of the unknowns can be obtained from the equilibrium equations alone.

In some cases, the number of the equations will be larger than the number of unknowns. A rigid bar attached on one end by a pin support is an example. Such system cannot remain in equilibrium for most loads applied to it.

• Equilibrium analysis of composite bodies. 5.3

More detailed description will be added soon

• Two- and three-force bodies.

If a rigid body is subjected to forces, either from applied loads or support reactions, at exactly two points, and no couples or moments are acting, it can be assumed that the overall force vectors at those two points will be colinear, equal and opposite, with only their magnitude to be determined. No further equilibrium equations can be written for this rigid body, and the unknown magnitude will usually come from the equilibrium of another component of the same multi-body system.

If a rigid body is subjected to forces at exactly three points, the overall force vectors at those three points must be concurrent, i.e., they must intersect at a single point.

Corresponding Gross et al. Statik chapters are shown in red.

Key techniques:

• Separating a FBD into two or more complementary FBDs.
• Accounting for internal reaction forces and couple at a section.
• Accounting for internal reaction forces at a pin joint.
• Applying equilibrium equations to all or some of the individual FBDs.
• Simplifying FBDs by identifying two- and three-force bodies.

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