CELE Structural Theory & Analysis — Analysis of Determinate StructuresSummary

A structure is statically determinate when the number of unknown reactions and internal forces equals the number of available equilibrium equations. For planar structures, the degree of static indeterminacy (DI) is calculated as DI = (3m + r) − (3n + e_c) for general frames, where m is the number of members, r is the number of reaction components, n is the number of joints, and e_c is the number of equations of condition (internal releases). Alternatively, for beams: DI = r − (3 + e_c). When DI = 0, the structure is determinate; DI > 0 indicates indeterminacy to that degree; DI < 0 signals instability.

Stability and determinacy are independent concepts. A structure can be statically determinate yet unstable if reaction forces are concurrent (meeting at a point) or parallel (no resistance to a particular direction of movement). Conversely, a structure can be statically indeterminate but stable. Stability requires that the reaction forces form a non-concurrent, non-parallel system capable of resisting forces in all directions. Always verify both determinacy (counting unknowns vs. equations) and stability (checking reaction arrangement).

An equation of condition is an additional equilibrium equation available due to internal releases (hinges, pins, or roller connections within the structure). Each internal hinge provides one equation of condition: the sum of moments on either side of the hinge must equal zero (ΣM = 0). For a hinge connecting multiple members, the number of equations of condition is (number of members at hinge − 1). These equations significantly reduce the degree of indeterminacy and enable analysis of compound (Gerber) beams.

Reactions are the forces and moments exerted by supports on a structure. They are found by applying the three fundamental equilibrium equations: ΣF_x = 0 (sum of horizontal forces), ΣF_y = 0 (sum of vertical forces), and ΣM = 0 (sum of moments about any point). For structures with internal releases, equations of condition (ΣM on each side of a hinge) are applied as additional relationships. The process involves selecting appropriate points for moment equilibrium (often supports or hinges) to eliminate unknowns strategically.

Internal forces represent the result of cutting a structure at a section and considering equilibrium of one free body side. Axial force N is the sum of forces parallel to the member axis; shear force V is the sum of transverse (perpendicular) forces; bending moment M is the sum of moments about the cut section. The internal forces on the left side of the cut equal and oppose those on the right side (action-reaction). Sign conventions are critical: typically, tension is positive for axial, upward shear on the left face is positive, and counterclockwise moment is positive. These internal forces are plotted as axial-force (N), shear-force (V), and bending-moment (M) diagrams.

A three-hinged arch is a curved member with two supports (usually at the same level) and an internal hinge at the crown or at a specified location. It has four reaction unknowns: horizontal and vertical components at each support (A_x, A_y, B_x, B_y). Despite having four unknowns, it is statically determinate because it has four equilibrium equations: global ΣF_x = 0, ΣF_y = 0, ΣM = 0, plus the hinge condition ΣM_C = 0 (sum of moments on one side of the hinge about the hinge point). The defining feature is the horizontal thrust H, which reduces the maximum bending moment compared with a simply-supported beam of the same span and loading. The arch shape and thrust work together to resist loads more efficiently.

A compound beam is an assembly of simple and cantilever beams connected by internal hinges or pins. The most efficient method for solving compound beams is to identify the suspended span (a simple or cantilever span that rests on adjacent spans via a hinge) and analyze it first, then carry its reaction onto the supporting span. This sequential analysis avoids setting up large systems of equations and recognizes the structural hierarchy. Each internal hinge provides an equation of condition (moment on one side of the hinge is zero or equals an applied moment at the hinge).

For beams (as opposed to general frames), a simplified determinacy formula applies: DI = r − (3 + e_c), where r is the number of reaction components and e_c is the number of equations of condition. This formula reflects that a single-member structure has three basic equilibrium equations. For example, a simply-supported beam with two supports (pin at one end, roller at the other) has r = 3, so DI = 3 − 3 = 0 (determinate). A continuous beam with three supports has r = 4, so DI = 4 − 3 = 1 (indeterminate to the first degree). An internal hinge adds one equation of condition, reducing DI by one.

If a structure is statically determinate (DI = 0), it can be solved using equilibrium equations alone. If it is indeterminate (DI > 0), additional compatibility (deformation) equations are required, leading to methods like the slope-deflection equation, moment distribution, or virtual work. This relationship determines the entire analysis approach.

Internal hinges (or other releases like pins or partial moment connections) provide equations of condition. Each internal hinge adds one equation, reducing the degree of indeterminacy by one. Gerber beams leverage this principle: an internal hinge at a suspended span allows it to be analyzed first, then its reaction becomes a load on the supporting span.

Reactions are the boundary conditions resulting from external loads. Internal forces (N, V, M) are derived from reactions using equilibrium of cut sections. Internal forces directly produce stresses in the material (σ = N/A, τ = V/A_w, σ = M/I), which must be compared against allowable stresses or design capacities (per NSCP 2015, AISC 360, ACI 318) to verify safety.

The horizontal thrust H in a three-hinged arch depends on the shape (radius of curvature) and the loading. The hinge condition ΣM_C = 0 at the crown (or hinge location) relates the vertical reactions, span, and loading to the horizontal thrust. The arch's efficiency (lower maximum moment) increases with rise and horizontal thrust for a given span and loading.

A compound beam's internal hinges create a hierarchy: suspended spans are independent and can be analyzed first, their reactions then loading the supporting spans. This structure-function relationship enables the suspended-span-first method, avoiding simultaneous equations and recognizing load paths naturally.

Beams use the simplified formula DI = r − (3 + e_c) because they have a single load path. Frames (with multiple members and joints) use the general formula DI = (3m + r) − (3n + e_c). The difference reflects that frames have more degrees of freedom and require explicit accounting of all members and joints.

Simply-supported bridge spans are determinate; their reactions are found directly from equilibrium. Continuous bridges span multiple supports and are indeterminate; however, intermediate hinges (expansion joints) can convert spans to determinate subunits. Analyzing determinate spans first is essential before advancing to indeterminate continuous-bridge analysis. The horizontal thrust in arched or cable-stayed bridges carries significant design implications for abutments and piers (as per NSCP 2015 Bridge Design Code).

Individual stories of a building can sometimes be analyzed as determinate frames under lateral loads (e.g., wind) if the lateral-load-resisting system (moment frames or braced frames) is configured appropriately. Understanding determinacy helps identify where symmetry and geometry make a structure determinate, reducing analysis complexity. Reactions from determinate frame analysis guide foundation and column-base design (ACI 318 foundation requirements).

Simple trusses are typically determinate or just-determinate (m = 2n − 3 for a planar truss with n joints and m members). Reaction analysis for determinate trusses is straightforward; member forces are then found using method of joints or method of sections. Determinate trusses are common in roof and bridge applications because they avoid locked stresses from fabrication tolerances.

Cantilever beams and overhangs attached to longer beams can form compound (Gerber) systems. The suspended-span-first method efficiently analyzes such systems: solve the cantilever or overhang first, then treat its reaction as a load on the main beam. This approach is practical in architectural design where overhangs and cantilevered balconies are common.

In reinforced concrete, moment-release details (such as seats or pin supports) are designed to accommodate the internal forces expected from determinate behavior. In steel structures, pin connections, rocker supports, and hinged details are fabricated based on reactions from determinate analysis. Incorrect determinacy assessment leads to inadequate connection design (AISC 360, ACI 318).

Three-hinged arches appear in heritage structures, some modern roof systems, and pedestrian bridges. Their efficiency (lower maximum moment due to horizontal thrust) makes them attractive for long spans. The crown hinge accommodates thermal expansion and differential settlement, reducing locked stresses. Analyzing the horizontal thrust is essential for proportioning arch ribs, abutments, and tie rods (if any).

Understanding determinacy and equations of condition helps engineers trace load paths intuitively. A designer recognizing that a structure is determinate can quickly sketch its reaction distribution and internal force flow, informing conceptual design decisions about member sizes, support types, and load-sharing before detailed analysis. This skill is valued in professional practice and examined in the PRC exam's design-based questions.