Structural Analysis: Determinate Structures

Statically determinate structures are the workhorses of everyday engineering. They include many common beams, trusses, and simple frames whose support reactions and internal forces can be found using equilibrium alone. That feature is not just a mathematical convenience. It makes determinate systems transparent: you can trace how loads flow through members, understand where maximum forces occur, and check results with straightforward logic.

This article focuses on three core elements of determinate structural analysis: support reactions, internal forces, and influence lines.

What “statically determinate” really means

A structure is statically determinate when all unknown reactions and internal forces can be determined using only the equations of static equilibrium, without invoking material stiffness, deformation compatibility, or constitutive laws.

For a planar (2D) structure in static equilibrium, the available global equilibrium equations are:

• $\sum F_x=0$
• $\sum F_y=0$
• $\sum M=0$

If the number of unknown reaction components equals the number of independent equilibrium equations, the structure is externally determinate. If, in addition, the internal force system can be found solely from equilibrium of joints or cut sections, it is internally determinate.

Determinate does not mean “simple” in practice. It means the structure’s force response is governed by equilibrium. That makes results independent of member stiffness, a key point when comparing determinate and indeterminate systems.

Support reactions: the first step in analysis

Before internal forces can be computed, external reactions must be found. That starts with correctly identifying support types and the reaction components each can supply.

Common 2D supports and their reactions

• Roller support: one reaction force, typically vertical ($R_y$), no moment resistance.
• Pin support: two reaction forces ($R_x$, $R_y$), no moment resistance.
• Fixed support: two forces and a moment ($R_x$, $R_y$, $M$). In 2D, this introduces three unknowns at one support and often leads to indeterminacy unless balanced by releases elsewhere.

Reaction calculation for determinate beams and frames

For a simply supported beam (pin and roller), there are typically three unknown reaction components. With three global equilibrium equations, reactions follow directly. The same principle extends to determinate frames, provided the total number of external reaction unknowns does not exceed three in 2D, or the structure is arranged so additional reactions can be solved by isolating parts of the structure (for example, with internal hinges).

A practical approach:

1. Draw a clear free-body diagram (FBD) showing all applied loads, support reactions, and key dimensions.
2. Replace distributed loads with equivalent resultants where appropriate.
3. Take moments about a support to eliminate unknowns efficiently.
4. Use force balance in $x$ and $y$ to finish.

Errors at this stage propagate, so reaction checking matters. One fast check is to verify that the reaction resultants balance the total applied load and that moments balance about a convenient point.

Internal forces in determinate beams: shear and bending moment

Once reactions are known, internal actions in beams are typically expressed as:

• Shear force $V(x)$
• Bending moment $M(x)$
• (sometimes) axial force $N(x)$

Using the method of sections for beams

Conceptually, you “cut” the beam at a location $x$, expose internal forces on the cut, and enforce equilibrium on one side of the cut segment.

Key relationships guide interpretation:

• Concentrated loads cause jumps in shear.
• Applied couples cause jumps in bending moment.
• Distributed loads cause shear to vary continuously and moment to vary smoothly with changing slope.

For many common loading cases, shear and moment diagrams can be built piecewise by moving from left to right and applying equilibrium over short segments.

Why internal forces matter

Design checks are typically based on maxima:

• Maximum $|M|$ influences flexural stress and required section modulus.
• Maximum $|V|$ influences shear capacity and web design in steel or shear reinforcement in concrete.
• Axial force $N$ becomes critical in beam-columns or frame members under combined actions.

In determinate beams, these maxima are controlled by load placement and span conditions, which is exactly what influence lines help quantify.

Determinate trusses: axial force in members

Ideal trusses are assemblies of straight members connected at joints, with loads applied at joints. Under this idealization, members carry only axial force: tension or compression.

Determinacy of planar trusses

A common determinacy check for a planar truss is:

$$m+r=2j$$

where:

• $m$ = number of members
• $r$ = number of reaction components
• $j$ = number of joints

If the equality holds and the geometry is stable, the truss is typically statically determinate. If $m+r>2j$, the truss is statically indeterminate. If $m+r<2j$, it is unstable or a mechanism.

This is a necessary check, not a guarantee. A truss can satisfy the count and still be unstable if it has poor geometry (for example, untriangulated panels).

Methods: joints and sections

• Method of joints: apply joint equilibrium ($\sum F_x=0$, $\sum F_y=0$) sequentially. Efficient when you need forces in many members.
• Method of sections: cut through up to three unknown member forces in a planar truss and use equilibrium of one cut part. Efficient when you need forces in a few specific members, such as a chord member near midspan.

Interpretation is essential: chord members often carry larger axial forces due to global bending behavior, while web members route shear through diagonals and verticals.

Determinate frames: combining axial force, shear, and moment

Frames differ from trusses because members can carry bending. Even so, many frames are made determinate by including internal hinges or by limiting reaction components.

Internal hinges as determinacy tools

An internal hinge transmits forces but not bending moment. It creates a convenient point where $M=0$, and it allows a frame to be separated into sub-bodies for reaction calculations.

For a frame with an internal hinge, you can:

1. Split the structure into two parts at the hinge.
2. Write equilibrium for each part separately.
3. Use the hinge force components as interface unknowns, solved through equilibrium.

This technique is common in three-hinged arches and hinged portal frames, both classic determinate forms.

Influence lines: how moving loads create maximum effects

Influence lines describe how a response quantity at a specific point varies as a unit load moves across the structure. For determinate structures, influence lines can be constructed using equilibrium and basic geometry, making them especially useful for bridges, crane girders, and any structure dominated by moving loads.

What an influence line represents

An influence line for a quantity $Q$ (reaction, shear at a section, or moment at a section) gives $Q$ as a function of the load position. If a real load $P$ moves across, the response is:

$$Q=P\cdot \text{(influence ordinate at load position)}$$

For multiple loads, responses add by superposition, which is valid under linear elastic behavior and small deflections, assumptions typically used in standard structural analysis.

Influence lines for determinate beams (typical cases)

• Support reaction influence line: generally linear between supports for a simply supported beam.
• Bending moment at a section: forms a peaked, piecewise linear shape with a maximum when the load is at the section.
• Shear at a section: has a jump at the section location, reflecting the sign change depending on whether the load is just left or just right of the cut.

Practical use: locating maximum moment and shear

For a single concentrated moving load on a simply supported span:

• Maximum positive moment at a given section occurs when the load is at that section.
• Maximum shear at a section occurs when the load is placed adjacent to the section, on the side that produces the larger magnitude.

For groups of moving loads (axle loads), influence lines let you position the load train to maximize the area under the load pattern times the influence ordinates. In practice, engineers often combine influence lines with standard truck or lane load models to identify critical positions.

A disciplined workflow for determinate analysis

Determinate analysis is most reliable when it follows a consistent sequence:

1. Model correctly: confirm supports, connections, and whether truss assumptions are justified.
2. Solve reactions: use clean FBDs and equilibrium, then check totals.
3. Compute internal forces: sections for beams and trusses, joint equilibrium for trusses, sub-body equilibrium for hinged frames.
4. Identify extremes: use diagrams and influence lines to locate maximum $M$, $V$, and $N$ under expected load patterns.
5. Sanity-check behavior: verify symmetry, boundary conditions (for example, $M=0$ at simple supports), and qualitative load paths.

Why determinate structures still matter

Even when modern projects use indeterminate systems for efficiency and redundancy, determinate structures remain foundational. They are often used directly in practice and are also the clearest way to learn load paths, equilibrium thinking, and moving-load effects. Mastering reactions, internal forces, and influence lines in determinate beams, trusses, and frames builds the intuition and technical discipline that carries into every advanced topic in