Statics: Internal Hinge Analysis

Internal hinges are not just small components in a structure's diagram; they are powerful design features that control how forces and moments flow through a system. Mastering their analysis is essential for determining support reactions in complex, determinate structures like multi-span beams and three-hinged arches. This skill transforms an seemingly unstable assembly into a solvable puzzle by providing the extra equations you need.

The Internal Hinge as a Moment Release

An internal hinge is a specialized connection inserted within a structural member, such as a beam or frame. Its defining characteristic is that it cannot transmit internal bending moment. In other words, the bending moment at the hinge is always zero. However, it can and does transmit shear force and axial force.

Think of a door hinge: it allows the door to rotate freely (carrying no moment) but still holds the door up and transfers its weight to the frame (carrying shear). An internal hinge in a beam functions identically. This moment release turns a single, continuous member into two connected segments that can rotate independently relative to each other. From an analytical perspective, this release reduces the structure's internal rigidity and provides an additional equilibrium condition: $M_{hinge}=0$.

Generating Additional Equilibrium Equations

For a rigid body in 2D statics, you have three standard equations of equilibrium: $\sum F_x=0$, $\sum F_y=0$, and $\sum M=0$. A single, fully rigid structure only provides these three equations to solve for unknowns. An internal hinge introduces a fourth independent equation.

This is because the "sum of moments about the hinge must equal zero" can be applied to either free-body diagram (FBD) created by cutting at the hinge. Since the hinge carries no moment, if you take the FBD of the segment to the left of the hinge, the sum of moments about that hinge point must be zero. The same applies to the segment on the right. You can use one of these moment equations as your additional condition. This is crucial for solving statically determinate structures that would otherwise be unsolvable with only the three global equations.

Splitting the Structure at the Hinge

The most reliable analytical procedure is to physically disassemble the structure at the internal hinge. You draw separate free-body diagrams for each segment created by this split.

At the cut location, you must represent the internal forces the hinge transmits. Since the moment is zero, you only draw two unknown force components: typically a horizontal ($H$) and a vertical ($V$) force. Crucially, these forces are equal and opposite on the FBDs of the two connecting segments, following Newton's Third Law. This step is non-negotiable; treating the hinge forces correctly is the key to the entire solution.

For example, consider a beam A-C-B with a hinge at C. You would draw one FBD for segment A-C and a second FBD for segment C-B. On the left end of segment C-B, you show forces $H_C$ and $V_C$. On the right end of segment A-C, you show the same forces but in opposite directions.

Determining Support Reactions

With the structure split into multiple FBDs, you now have more equations than unknowns on a global scale. The standard solution sequence is:

1. Start with the segment that has the fewest unknowns. Often, this is a segment with only one support and the hinge forces. For instance, if segment A-C is a simply supported beam with a pin at A and the hinge at C, its FBD has three unknowns ($A_x$, $A_y$, and the hinge forces at C). You can solve for all three using its three equilibrium equations.
2. Transfer the solved hinge forces. Once you calculate $H_C$ and $V_C$ from the first segment, apply them (with their now-known magnitudes and correct directions) to the adjacent segment's FBD.
3. Solve the remaining segment. The second segment's FBD now has known hinge forces acting on it, leaving only its support reactions as unknowns. Use its three equilibrium equations to solve for these final reactions.

This methodical, segment-by-segment approach breaks a complex problem into a series of simpler ones.

Applications to Multi-Span Beams and Arches

The principles of internal hinge analysis find direct application in two classic determinate structures.

Multi-Span (Gerber) Beams: These are continuous beams over multiple supports made statically determinate by the insertion of internal hinges at specific locations. A common rule is that for a beam with $n$ supports, you need $n-2$ internal hinges to make it determinate. When analyzing a three-span beam with hinges, you split it into three simply supported beams (each with their own FBDs) connected by hinge forces. You always analyze the end spans first, as they typically have fewer unknowns.

Three-Hinged Arches: A parabolic or circular arch with a pin support at each base and an internal hinge at the crown is a classic determinate structure. The internal hinge prevents the arch from developing indeterminate internal moments due to temperature change or support settlement. To solve for the four support reactions ($A_x$, $A_y$, $B_x$, $B_y$), you use:

• Global equilibrium: $\sum F_x=0$, $\sum F_y=0$, $\sum M_A=0$ (3 equations).
• The hinge condition: $\sum M_C=0$ on either the left or right segment (1 equation).

This gives you the four equations needed. The hinge ensures the arch's thrust forces are purely axial under symmetric loading.

Common Pitfalls

1. Forgetting to Split the FBD: The most common error is trying to analyze the entire structure as one piece. You must draw separate FBDs by cutting at the internal hinge to access the additional equilibrium equation. The single global FBD only ever yields three useful equations.
2. Incorrect Hinge Force Direction: When drawing the two separate FBDs, the hinge forces must be shown as an action-reaction pair. If the vertical hinge force $V_C$ is drawn upward on the right end of the left segment, it must be drawn downward on the left end of the right segment. Assuming the same direction on both will lead to incorrect answers.
3. Applying the Hinge Condition Incorrectly: The condition $M_{hinge}=0$ must be applied to the FBD of a segment about the hinge point. It is not a global equation. You cannot sum moments about the hinge for the entire structure unless you have already found all forces acting across the cut, which you haven't at the start.
4. Solving Segments in an Illogical Order: Attempting to solve the segment with more unknowns first often leads to dead ends. Always identify which segment (often an end span) has only three unknowns—a support reaction and the two hinge forces—and begin your analysis there.

Summary

• An internal hinge is a moment-release connection that transmits shear and axial force but cannot carry internal bending moment ($M_{hinge}=0$).
• This property provides an additional independent equilibrium equation, allowing the solution of statically determinate structures like multi-span beams and three-hinged arches.
• The fundamental analysis technique is to split the structure at the hinge, drawing separate free-body diagrams for each segment and representing the transmitted hinge forces as equal and opposite action-reaction pairs.
• Reactions are found by solving the segments sequentially, starting with the one containing the fewest unknowns, calculating the hinge forces, and transferring them to solve the adjacent segment.
• This methodology is directly applied to analyze Gerber beams and three-hinged arches, which rely on internal hinges to achieve static determinacy and predictable force paths.