EquiBeam – frequently asked questions

EquiBeam determines the bending moment M(x), based on equilibrium equations that include internal forces for a selected part of the beam, and calculates values at a chosen cross-section using the loads and reactions. This allows you to read the moment value at any point along the beam.

• Cross-section at point x: the program “cuts” the beam within a given interval and analyzes one side of the beam (left/right).
• Equilibrium equations of the cut part: it formulates equations for V(x), M(x), and N(x) based on the equilibrium conditions of the system.
• Distributed and concentrated loads: includes concentrated forces, moments, and distributed loads (resultants over intervals), so M(x) is obtained as a function within a given interval.
• Value at any point: after assembling the relationships, the program allows calculation and reporting of M(x) for any x, and highlights values at characteristic points (under forces, at supports, at the boundaries of distributed loads).

In addition, the shear force diagram T(x) is generated in parallel, which helps verify the moment diagram, since local extrema of M(x) occur where T(x)=0.

Bending moments analysis in EquiBEam app

Yes. EquiBeam calculates the shear force V(x)at any point of the beam, based on the internal force equations for the beam segment at the selected position x. At points where a concentrated force is applied, the program distinguishes the value “just before” and “just after” that point.

• Cross-section at point x: the program “cuts” the beam at the selected location and considers one side of the system.
• Vertical force equilibrium: from the equation ∑Fy=0, it determines T(x) by summing the reactions and loads acting on that side of the beam.
• Distributed loads q(x): the contribution of q(x) is accumulated over the segment length, so V(x) changes as a function (for q=const, it is linear).
• Diagram jumps: at a concentrated force, the V(x) diagram has a jump, so the program can provide V(x⁻) and V(x⁺).
• Value at any point: after assembling the piecewise relationships, EquiBeam provides V(x) for any and also at characteristic points.

Additionally, the course of V(x) is consistent with the distributed load: within a given interval, the relationship dV/dx=q(x) applies, which helps verify the correctness of the scheme and loading.

Shear force analysis in EquiBeam app

EquiBeam calculates beam deflection using the virtual force method by allowing the user to select hyperstatic reactions as unknowns and determining them from canonical equations based on the real-load state P and unit-load states. Once these are found, the program calculates beam deflections using Mohr’s method (virtual work).

• Selection of hyperstatic unknowns: the user can indicate which reactions/interactions are treated as hyperstatic (degree of static indeterminacy).
• Primary system: the program “releases” the selected restraints, creating a statically determinate system for the basic calculations.
• Real-load state P: it calculates diagrams/internal actions due to the real load in the primary system.
• Unit-load states: it runs unit-load cases for each hyperstatic unknown (influence functions / flexibility coefficients).
• Analytical or graphical integration: coefficients in the canonical equations can be calculated analytically or graphically (e.g. from diagrams).
• Canonical equations: it solves the classical system

and determines the values of the hyperstatic reactions .

• Deflection of the real system: after determining , the program calculates the remaining reactions and internal forces by superposition or directly from the fully restrained system.
• Mohr’s method: the final deflections or slope angle at a point are determined using Mohr’s method (virtual work), based on moments from the real and unit states.

This gives you both the final deflection values and the full calculation path of the virtual force method (P+unit states →canonical equations →deflections), which can be easily compared with a classroom solution.

Virtual force method calculations in EquiBeam

EquiBeam analyzes the effect of a distributed load q(x) on the bending moment M(x) by introducing it directly into the internal force equations and determining the course of M(x) along the entire beam. EquiBeam supports constant and linearly varying loads, both in the classical approach and in formulations such as the Clebsch method.

• q(x) written directly into the equations: distributed loads are included directly in the equations for V(x) and M(x), without manually replacing them with several equivalent forces.
• Load types: constant and linearly varying loads (triangular/trapezoidal) over any beam segment.
• Diagram relationships: within a segment, the relationships dV/dx=q(x) and dM/dx=V(x) hold, so q(x) determines the shape of M(x).
• Interval form: the program builds piecewise functions M(x) between load change points (start/end of q, concentrated forces, supports).
• Influence reading: it allows quick reading of M(x) at any point and indicates extrema resulting from the course of T(x).

This makes it possible to directly compare how changing the value or type of distributed load affects the moment diagram — both numerically and graphically.

Distributed load (EquiBeam app)

Yes – EquiBeam determines support reactions for a multi-span beam, taking into account intermediate supports and connections between spans. It can do this in several ways: by splitting the system into rigid parts with interaction forces, or by cleverly selecting equations (moments at hinges, forces relative to “skates”).

• Division into rigid parts: it splits the beam into fragments/spans and writes equilibrium equations for each part separately, introducing unknown interaction forces at the connections.
• Connections –
• Hinges (rotation released): it uses moment equations about hinges to eliminate unknowns and quickly close the system.
• Sliders (translation released): it writes force equations in directions parallel to the release to accelerate the solution.
• Loads over multiple spans: includes concentrated forces, moments, and distributed loads on different segments, creating a consistent system of equations.
• Step-by-step solution: after formulating the equations, the program eliminates successive unknowns (substitution / linear system solution) until all reactions are obtained.
• When analyzing a statically indeterminate system: additional compatibility equations are created (e.g. force method / elastic curve equation).

As a result, a multi-span beam is solved “just like in class”: either by cutting it into parts and applying equilibrium, or by selecting equations at points that simplify the calculations.

Multi-span beam reaction forces in EquiBeam app

EquiBeam determines the maximum deflection by calculating the deflection curve w(x) and identifying its extreme value along the beam length. In methods based on the elastic curve equation or the Clebsch formulation, the maximum can be calculated directly; in other methods, deflections are provided at user-selected points, while the maximum is still visible on the diagram.

• Elastic curve equation: it builds w(x) from the classical approach EI·w” = M(x) and determines the deflection extremum on this basis.
• Clebsch method: it expresses deflection/derivatives as a single function with terms of the form ⟨x-a⟩, which makes it easier to determine w(x) and its maximum.
• Other methods: it calculates displacements at user-selected points (e.g. mid-span, under a load, at characteristic cross-sections).
• Deflection diagram: for each method, it generates the plot of w(x), where the maximum can be read visually.

In practice, you get the deflection both numerically (maximum / selected points) and in the form of a diagram, which makes it easier to quickly verify the problem and interpret where the beam deflects the most.

Displacement calculations – Clebsch_method in EquiBeam

Yes. EquiBeam analyzes statically indeterminate beams by solving them with analytical methods based on displacement compatibility and beam deflection. Several approaches are available, so you can choose the method depending on the type of structure and what you want to learn.

• Differential equation of the elastic curve: uses the relationship EI·w” = M(x) together with boundary/continuity conditions to determine unknowns.
• Clebsch method: represents displacements w(x) as a single function, greatly simplifying calculations for continuous beams (constant material, constant cross-section, no hinges).
• Force method: selection of hyperstatic reactions, real-load state P + unit-load states, solution of canonical equations.
• Menabrea–Castigliano method: energy-based approach (minimum energy / derivatives of strain energy) to determine hyperstatic unknowns.
• Extensions: numerical methods are also planned, including FEM (finite element method) and the displacement method.

Thanks to this, EquiBeam allows you to calculate hyperstatic reactions “in several ways,” which is excellent for verifying results and comparing methods in academic problems.

Statically indeterminate beam calculations in EquiBeam app

EquiBeam determines extreme points on the bending moment diagram M(x) automatically by searching for locations where the shear force V(x)=0.

• Extremum condition: the program checks where the T(x) diagram intersects the zero level (because dM/dx=V).
• Load intervals: it analyzes each interval separately (between supports, concentrated forces/moments, start/end of distributed load).
• Determination of point x*: it finds the exact location x where V(x*)=0 and calculates M(x*) there.
• Characteristic points: additionally, it checks moments at discontinuity locations (concentrated forces, boundaries of q, hinges), since these may also be candidates for extreme values.

In practice, you immediately get both the coordinate of the critical point and the value of M(x), without manually searching the entire diagram.

Bending moment extremum calculations in EquiBeam app

EquiBeam helps students solve beam problems by guiding them step by step through the analytical solution using the selected method, while also showing the complete set of results (reactions, diagrams, deflections). This makes it easy to compare your own calculations, find mistakes, and prepare for quizzes and exams.

• Choice of calculation method: the student can select the approach (e.g. equilibrium equations, virtual force method, elastic curve, Clebsch method, Menabrea–Castigliano) depending on the topic.
• Step-by-step guidance: the program formulates equations and solves them in stages until the final result is reached (reactions → internal forces → deflections).
• Calculation verification: easy comparison with a handwritten solution: reactions, values of V(x), M(x)at specific points, extrema.
• Clear diagrams: generates the courses of V(x), M(x), and w(x), helping users understand the effect of loads and supports.
• Point-based work: allows calculation/reading of values at any and at user-selected points (e.g. under a force, at mid-span).
• Learning from mistakes: quick correction of signs, lever arms, equivalent distributed loads, and boundary conditions — without wasting time recalculating everything from scratch.

It is one of the best tools for learning, because it combines classroom methods with instant result verification and visualization, making it much easier to understand the solution process before an assessment.

Beams calculations in EquiBeam app