Understanding Static Equilibrium Equations in Structural Analysis

Static equilibrium equations are fundamental principles used in engineering mechanics and structural analysis to determine unknown forces and moments acting on a structure or a body that is at rest (in a state of static equilibrium). These equations state that for a body to be in equilibrium, the net force and the net moment acting on it must be zero in all directions.

The three primary static equilibrium equations in a 2D plane are:

• $\sum F_x = 0$: The sum of all horizontal forces acting on the body is zero.
• $\sum F_y = 0$: The sum of all vertical forces acting on the body is zero.
• $\sum M = 0$: The sum of all moments about any point is zero.

In 3D analysis, there are six equilibrium equations: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$, $\sum M_x = 0$, $\sum M_y = 0$, and $\sum M_z = 0$. The question focuses on the general application, implying the core 2D equations or their 3D equivalents.

Determining Supporting Reactions Using Static Equations

One of the primary uses of static equilibrium equations is to determine the unknown forces and moments exerted by the supports on a structure. These are called supporting reactions. By considering the entire structure as a free body and applying the equilibrium equations, we can solve for the reaction components, provided the structure is statically determinate. A statically determinate structure is one where all support reactions and internal forces can be determined using only the equations of static equilibrium.

Determining Internal Forces: Shear Forces and Bending Moments

Once the external forces, including the supporting reactions, are known, the static equilibrium equations can also be used to determine the internal forces within the structure. Internal forces include axial forces, shear forces, and bending moments at any point along the structural member.

To find internal forces like shear force and bending moment at a specific point, one conceptually cuts the structure at that point and considers either segment as a free body. By applying the static equilibrium equations ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M = 0$) to this segment, the unknown internal forces and moments acting at the cut section can be determined.

• Shear force ($\sum F_y$ or $\sum F_x$ for vertical/horizontal cuts) is the internal transverse force component.
• Bending moment ($\sum M$) is the internal moment component.

Conclusion on Determinable Quantities

Based on the application of the static equilibrium equations:

• We can determine supporting reactions by analyzing the free body diagram of the entire structure.
• We can determine shear forces at different points by analyzing the free body diagram of segments of the structure created by imaginary cuts, applying force equilibrium ($\sum F_y = 0$ or $\sum F_x = 0$).
• We can determine bending moments at different points by analyzing the free body diagram of segments of the structure created by imaginary cuts, applying moment equilibrium ($\sum M = 0$).

Therefore, applying the static equations allows us to determine supporting reactions, shear forces, and bending moments in a statically determinate structure.

Revision Table: Static Equations Summary

Additional Information: Statically Determinate vs. Indeterminate Structures

It's important to note that the direct determination of all reactions and internal forces solely using static equilibrium equations is only possible for statically determinate structures. For statically indeterminate structures, there are more unknown reactions than available equilibrium equations. In such cases, additional methods are required, such as compatibility equations based on material properties and structural deformations (e.g., using the flexibility method, stiffness method, or slope-deflection method).

However, even in indeterminate structures, the static equilibrium equations are still valid and are used in conjunction with these other methods. The question specifically asks what "we may determine" by applying the static equations, and for determinate structures, all three quantities (reactions, shear, moment) are directly determinable.