3.3. Equations of Conditions

If a structure has an internal hinge (which may connect two or more substructures), then this will provide an additional equation ((cid:6)M = 0 at the hinge) which can be exploited to determine the reactions.

An internal hinge is a flexible connection in a structure that allows rotation without translating. This means that in a structure with such hinges, we can apply an additional equation for equilibrium, stating that the moment (M) at that hinge equals zero. This is useful because it helps us find the reactions at supports in more complex structures by adding more information to our equations of equilibrium.

Think of an internal hinge like a door hinge. When you open or close a door, it swings around its hinge. Similarly, a structure can rotate around an internal hinge without moving away from its connected parts, and this allows engineers to solve for forces in the structure more easily.

Those equations are often exploited in trusses where each connection is a hinge to determine reactions.

In a truss, which is a framework composed of beams connected at joints, each connection behaves like a hinge. This allows for the use of the equations of conditions, particularly because such connections enable easy rotation without additional forces. By applying the hinge equations along with static equilibrium equations, engineers can effectively determine the reaction forces that help stabilize the structure.

Imagine a simplified truss like a bike's frame. The joints, where the tubes meet, can pivot slightly just like hinges. This order of flexibility is crucial when calculating how much weight the bike can support without collapsing.

In an inclined roller support with S and S horizontal and vertical projection, then the reaction R would have, Fig. 3.2. R S x = y (3.3) R S y x.

In this context, an inclined roller support is a type of support allowing a structure to roll along one axis while restricting it in the other. The reactions at this support have both horizontal (Sx) and vertical (Sy) components. The equation R (which is the resultant reaction) indicates that the horizontal projection (Sx) is equal to the vertical projection (Sy) at a specific angle. This relationship helps in calculating the resultant forces acting at the support based on its orientation.

Picture a skateboard ramp. When you put a skateboard on a sloped ramp (the inclined roller support), the downward force due to gravity has to be resolved into two components: one that pushes sideways along the ramp and another that pushes straight down. Understanding these forces helps in designing safe and effective ramps.

In statically determinate structures, reactions depend only on the geometry, boundary conditions and loads.

A structure is considered statically determinate if the reactions can be calculated solely from the static equilibrium equations. This means you can determine the force reactions based solely on the applied loads and the shape of the structure without needing additional information. This simple relationship makes these structures straightforward for analysis and design, as they do not require complex calculations or additional considerations for internal force distributions.

Think of a classic triangle made of sticks. If you push down on the top corner, you can use simple geometry to find out how much force each stick at the bottom experiences. In contrast, if you have a more complicated shape, like a human-made sculpture with joints that flex, you can't determine the forces just by looking at it—you might need to consider more complex factors.

If the reactions cannot be determined simply from the equations of static equilibrium (and equations of conditions if present), then the reactions of the structure are said to be statically indeterminate.

Statically indeterminate structures are those where you cannot find the support reactions using just the equations of equilibrium and conditions. This situation arises when there are more unknown forces than the equations available to solve them. As a result, to analyze these structures, additional information or methods must be used, often involving material properties or deformations.

Imagine a complicated bridge that has extra cables and supports. If you try to calculate how much each cable is pulling based only on the weights hanging from it, you might end up with more questions than answers because of the added complexity. You'd need more detailed data about the materials or how much bend they're allowing to accurately assess the forces.

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