4.2.1. Determinacy and Stability

Trusses are statically determinate when all the bar forces can be determined from the equations of statics alone. Otherwise, the truss is statically indeterminate.

A statically determinate truss is one where the forces in the bars (members) can be found using just the basic equations of equilibrium. These equations are derived from Newton's laws, and they state that a system in static equilibrium must have all forces balanced and no net moments. If a truss cannot be analyzed this way, it’s classified as statically indeterminate, meaning that additional methods are needed to determine the forces in the members.

Imagine trying to find the forces acting on a group of three people holding a heavy beam. If enough information is provided about how they are positioned and the beam's weight, you can calculate how much force each person needs to exert simply using basic physics. This setup is 'statically determinate'. However, if their positions or connections add complexity and hinder clear calculations, you would need more detailed analysis, which is similar to dealing with a statically indeterminate truss.

A truss may be statically/externally determinate or indeterminate with respect to the reactions (more than 3 or 6 reactions in 2D or 3D problems respectively).

In a structural context, external determinacy refers to the ability to ascertain the support reactions of a truss using statics alone. In two-dimensional analysis, a truss can be external determinate if it has three or fewer external reactions. In three-dimensional cases, that number increases to six. If it has more reactions than these limits, the truss is considered externally indeterminate.

Consider a see-saw on two supports. If you were to add a third support point, it may introduce redundancy in how the weight is supported, complicating how you may analyze the forces acting on the see-saw. Similarly, if a truss has too many supports, it can make finding the forces at those points more complex and indeterminate.

A truss may be internally determinate or indeterminate. If we refer to j as the number of joints, R the number of reactions, and m the number of members, then we would have a total of m+R unknowns and 2j (or 3j) equations of statics (2D or 3D at each joint).

Internally, a truss is determined by its connectivity and member arrangement. Internal determinacy involves ensuring sufficient equations to solve for member forces. In a 2D truss, the relationship between the number of members (m), reactions (R), and joints (j) is essential: if the total unknowns (m + R) exceed the available equations (2j), then it is internally indeterminate. Conversely, if there are fewer unknowns than equations, the structure may be unstable.

Think of a family where each member must share chores. If there are more chores than members who can do them, they might need to cooperate to determine who does what—similar to a determinate system. If there are more members than chores, there could be confusion about who does what, leading to instability—akin to an indeterminate system.

If m < 2j (in 2D), the truss is not internally stable, and it will not remain a rigid body when it is detached from its supports.

For a 2D truss to be internally stable, there must be enough members to ensure that it can maintain its shape without external stiffness. Specifically, if the number of members (m) is less than twice the number of joints (2j), the truss will not hold its shape when the support is removed, indicating that it’s not rigid. This underscores the importance of member arrangement in structural designs.

Imagine a spider web; if there aren't enough lines connecting the points, the web collapses when a few points are removed (like supports being taken away). A well-constructed web (truss) needs enough connections to stay upright and not fall apart.

Since each joint is pin-connected, we can apply (ΣM = 0) at each one of them. Furthermore, the summation of forces applied on a joint must be equal to zero.

Each joint in a truss operates under the principle of equilibrium. At every pin-connected joint, the sum of the forces acting on it must equal zero, which reflects that the joint isn't moving (is in static equilibrium). This principle can be expressed using equilibrium equations such as ΣF_x = 0, ΣF_y = 0, and in 3D, the sum of moments ΣM also equals zero to ensure stability and balance.

Picture a playground swing. If a child is sitting perfectly still, the forces from the swing's connection to the frame and their weight balance each other out. Just as the swing rests motionlessly, a truss joint balances forces to maintain structural integrity.

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