Beam Check:
formulas and assumptions
This article explains the method behind Beam Check. Every formula is shown and every assumption stated: first the inputs and section data, then the stiffness analysis, then the checks, and finally a worked example.
Part A: Inputs and section data
1. Notation and sign convention
L: total beam length (entered in mm, analysed in m)
EI: bending stiffness of one beam (kNm²)
GA: shear stiffness of one beam, optional (kN)
w_sw: self-weight of one beam (kN/m)
N: number of beams acting together (1 to 10)
q: uniform load over the full length (kN/m)
q1, q2: trapezoidal load intensity at its start and end (kN/m)
P: point load (kN)
k: spring support stiffness (kN/mm)
M, V, δ, R: bending moment (kNm), shear force (kN), deflection (mm), support reaction (kN)
Ma, Va, Ra: allowable (safe working) moment, shear and reaction for one beam
n: deflection limit divisor (span ÷ n)
Sign convention:
x is measured from the left end of the beam.
Loads are positive acting downward.
Bending moment is positive when sagging (tension at the bottom).
Deflection is drawn negative downward; the checks use its size only.
Reactions are positive acting upward. A negative reaction at a tied support means the support is holding the beam down.
2. Section library
Choosing a library section fills in EI, self-weight and the safe working moment and shear. You can overwrite any value, which switches the section to Custom.
Values per beam (Ma kNm / Va kN / weight kg/m / I cm⁴ / EI kNm²):
Ladder Beam: 13.5 / 18.0 / 11.9 / 2,600 / 5,271
Unit Beam: 27.7 / 18.5 / 14.5 / 10,572 / 21,320
Haki 750: 41.3 / 30.6 / 7.5 / 17,182 / 11,856
Haki 450: 15.7 / 12.7 / 4.2 / 4,481 / 3,087
Apollo X 750: 42.9 / 45.4 / 8.0 / 17,097 / 11,796
ASP 750: 36.5 / 30.7 / 6.5 / 16,563 / 11,594
Layher 450: 13.94 / 12.32 / 5.0 / 4,481 / 3,137
Layher 750: 33.5 / 25.6 / 6.0 / 15,195 / 10,637
Dessa D78: 38.87 / 31.66 / 5.5 / 16,520 / 11,399*
Triangle Rail: 11.2 / 21.6 / 10.7* / 281 / 576*
Steel tube 48.3 Ø: 1.12 / 29.10 / 4.37 / 13.77 / 28.2*
Aluminium tube 48.3 Ø: 1.33 / 25.52 / 1.65 / – / 10.45
Scaffold board 38 × 225: 0.475 / 4.60 / 6.0 / – / 29.0
Values marked * are derived, not taken from a datasheet:
Dessa D78 (aluminium): EI = 69 GPa × 16,520 cm⁴ = 11,399 kNm²
Triangle Rail (steel assumed): EI = 205 GPa × 281 cm⁴ = 576 kNm²
Triangle Rail weight: 13.6 cm² × 7,850 kg/m³ ≈ 10.7 kg/m
Steel tube 48.3 × 4.0: EI = 205 GPa × 13.77 cm⁴ = 28.2 kNm²
Assumptions:
Safe working values and stiffness come from manufacturers' datasheets and must be checked against current manufacturer information.
Galvanised tubes in good condition are taken as new.
Scaffold board self-weight is 25 kg/m², which is 6 kg/m for one 225 mm board.
No library section includes GA, so library sections are analysed for bending deformation only unless you enter GA.
3. Number of beams
N beams placed side by side are analysed as one combined beam. Section values are entered for one beam and multiplied by N:
Bending stiffness used: EI total = N × EI
Shear stiffness used: GA total = N × GA
Self-weight used: w_sw total = N × w_sw
Applied loads (q, q1, q2, P) are totals for all N beams and are not multiplied.
Forces in each beam: M ÷ N, V ÷ N, R ÷ N
Assumptions:
The beams share the load equally and deflect together. They must be the same section and length, and supported at the same points.
Load sharing is assumed perfect. Uneven loading, uneven support or one beam taking more than its share are not considered.
Deflection is not divided by N, because all beams deflect by the same amount.
N = 1 is a single beam. The quick buttons set N = 1 to 4; up to 10 can be typed.
4. Loads
Self-weight from mass: w_sw = kg/m × 9.81 ÷ 1000 (kN/m)
Uniform load used: w = q + N × w_sw (self-weight only if "Add self-weight" is ticked)
Trapezoidal load at position x: q(x) = q1 + (q2 − q1) × (x − a) ÷ (b − a), for a ≤ x ≤ b
Total applied load: W = w × L + Σ (q1 + q2) ÷ 2 × (b − a) + Σ P
Assumptions:
All loads are static and vertical, and act through the beam's shear centre. Eccentricity, torsion, dynamic and impact effects are not included.
Loads are working (unfactored) loads, compared with safe working values. No partial safety factors are applied.
If a trapezoidal load is entered with From greater than To, the ends are swapped so each intensity stays with its position.
Up to 24 supports, 12 point loads and 12 trapezoidal loads can be entered.
5. Supports and end conditions
Knife (rigid) support: vertical displacement v = 0; free to rotate.
Spring support: vertical force R = −k × v, with k converted from kN/mm to kN/m (× 1000); free to rotate.
Tied: the support can push up or hold down.
Lift-off: the support can only push up. It is removed if it would go into tension (see section 9).
Fixed end: v = 0 and rotation θ = 0 at x = 0 or x = L.
Free end: no restraint.
Assumptions:
Supports act at a single point (knife edge). Bearing length and local effects at the support are ignored.
A rigid support placed at a fixed end is absorbed into the fixed end, and its reaction is reported with the fixed end.
The beam must be stable: at least two supports at different positions, or at least one fixed end.
Part B: Stiffness analysis
6. Dividing the beam into elements
The beam is solved with the direct stiffness method. It is split into finite beam elements, each with two nodes, and each node has two unknowns: vertical displacement v and rotation θ.
Base mesh: 240 equal elements over the length L
Extra nodes at every support, every point load and the start and end of every trapezoidal load
Number of unknowns: 2 × number of nodes
Nodes closer together than L × 10⁻⁷ are merged. Because supports and point loads always sit on nodes, their effects are exact for the model.
7. Element stiffness
Each element of length l uses the Timoshenko beam stiffness matrix. When shear stiffness is not included, φ = 0 and it becomes the standard Euler–Bernoulli matrix.
Shear parameter: φ = 12 EI ÷ (GA × l²), or φ = 0 if GA is not used
Stiffness factor: c = EI ÷ (l³ × (1 + φ))
Element stiffness matrix k = c × the matrix below, acting on [v1, θ1, v2, θ2]:
Row 1: 12, 6l, −12, 6l
Row 2: 6l, (4 + φ)l², −6l, (2 − φ)l²
Row 3: −12, −6l, 12, −6l
Row 4: 6l, (2 − φ)l², −6l, (4 + φ)l²
Assumptions:
Linear elastic material and small deflections. EI and GA are constant along the beam.
GA is used only when "Include shear stiffness" is ticked and GA is greater than zero.
The beam is assumed to be laterally restrained. Lateral-torsional buckling, local buckling of chords or lacing, and slip at beam splices are not modelled; the safe working values are relied on for these.
8. Load vector
The distributed load on each element is taken to vary in a straight line from p1 at its left node to p2 at its right node, with p = −w (upward positive). The equivalent nodal loads are:
Force at node 1: f1 = l × (7p1 + 3p2) ÷ 20
Moment at node 1: m1 = l² × (3p1 + 2p2) ÷ 60
Force at node 2: f2 = l × (3p1 + 7p2) ÷ 20
Moment at node 2: m2 = −l² × (2p1 + 3p2) ÷ 60
Point loads are added directly as −P at their node.
A trapezoidal load is applied to an element when the element's midpoint lies within the load. Since the load's ends are nodes, this is exact.
9. Solution and lift-off
The system solved is K × u = F.
Spring support: k × 1000 is added to the stiffness at the support node.
Rigid restraint: that unknown is fixed at zero.
Reaction at a rigid support: R = (K × u − F) at that unknown.
Reaction at a spring support: R = −k × 1000 × v.
K is solved by banded LDLᵀ factorisation. For lift-off supports the solution is repeated:
Solve with all active supports.
Remove any active lift-off support with R < 0 (it would be pulling down).
Put back any removed support where the beam has moved down through it (v < 0).
Repeat until nothing changes, up to 40 times. If it does not settle, the last trial is shown with a warning.
Tied supports are never removed. If a tied support ends up in tension, a warning asks you to check the tie-down or coupler. If removing supports leaves the beam unstable, no result is given.
10. Internal forces and equilibrium
The end forces of each element come from f = k × u − f_load, which gives the shear Vl and moment Ml at its left end. At distance s from the left node, with w1 and w2 the downward load at each end of the element:
Shear force: V(s) = Vl − w1 × s − (w2 − w1) × s² ÷ (2l)
Bending moment: M(s) = Ml + Vl × s − w1 × s² ÷ 2 − (w2 − w1) × s³ ÷ (6l)
Deflection: straight-line interpolation between nodal values, × 1000 for mm
Equilibrium check: Σ R = W, shown on the reaction tile
Assumptions:
Maximum values are taken at element ends (nodes). With 240 or more elements this captures peaks to within a fraction of a percent.
Vertical reactions at fixed ends are included in Σ R; their fixing moments are shown in the reactions table.
The schematic and deflected shape are drawn to an exaggerated scale.
Part C: Checks
11. Spans and cantilevers
The restraint points are all active supports plus any fixed ends, sorted along the beam. The beam is then split into:
Span: the distance between two neighbouring restraint points.
Cantilever: the overhang from the end of the beam to the first or last restraint point.
The largest deflection within each span or cantilever is recorded.
12. Utilisation checks
Moment: uM = max |M| ÷ (N × Ma)
Shear: uV = max |V| ÷ (N × Va)
Reaction: uR = max |R| ÷ (N × Ra), over all supports and fixed ends
Deflection limit for a span: δ limit = span length ÷ n
Deflection limit for a cantilever: δ limit = 2 × overhang ÷ n
Deflection: uδ = the highest value of max |δ| ÷ δ limit over all spans and cantilevers
Reaction presets (Ra per beam):
Right-angle coupler, Class A: 6.1 kN
Right-angle coupler, Class B: 9.1 kN
2 × Class A couplers: 12.2 kN
2 × Class B couplers: 18.2 kN
Assumptions:
M, V and R are totals for all N beams, so dividing by N × allowable is the same as checking each beam's share against its own allowable. When N is more than 1, the summary shows the total allowable as "N × per-beam value" and lists the forces in each beam.
The reaction limit is per beam, assuming one coupler or support per beam at each support position.
The same allowable moment is used for sagging and hogging, and the same allowable shear in both directions. If a section's hogging capacity is lower, enter the lower value.
Moment and shear are checked separately; their combined effect is not checked.
A blank limit means that check is not carried out. A new job starts with a deflection limit of span ÷ 270.
Two couplers are assumed to share the load equally (2 × the single-coupler value).
Deflection includes support movement at springs, so it is total deflection, not deflection relative to the supports.
13. Result
ANALYSED: no limits have been entered.
FAILS: any checked utilisation is above 1.00. Each check over the limit is listed.
OK: all checked utilisations are 1.00 or less. The governing check (highest utilisation) is named.
Utilisation bars turn amber above 90% and red above 100%. The 90% band is a visual warning only. Supports that lift off are flagged and carry no load, and tied supports in tension are flagged.
Worked example
14. Ladder Beam bridge (default job)
Inputs: L = 6,400 mm. Tied knife supports at x = 200 mm and 6,200 mm, giving a 6.0 m span with 200 mm overhangs at each end. Four 3.0 kN point loads at x = 1,400, 2,600, 3,800 and 5,000 mm. Uniform load 0.3 kN/m plus self-weight. Limits: Ma = 13.5 kNm, Va = 12.0 kN, Ra = 18.2 kN (2 × Class B couplers), deflection span ÷ 200.
One beam (N = 1):
Self-weight = 11.9 × 9.81 ÷ 1000 = 0.1167 kN/m
w = 0.3 + 0.1167 = 0.4167 kN/m
Total load W = 0.4167 × 6.4 + 4 × 3.0 = 14.67 kN
Reactions (symmetrical) = 14.67 ÷ 2 = 7.33 kN each
Maximum shear = 7.33 − 0.4167 × 0.2 = 7.25 kN
Maximum moment at mid-span (x = 3.2 m) = 7.333 × 3.0 − 0.4167 × 3.2² ÷ 2 − 3 × 1.8 − 3 × 0.6 = 12.67 kNm
Maximum span deflection = 9.07 mm (from the stiffness analysis, EI = 5,271 kNm²)
Maximum cantilever deflection = 0.96 mm (upward)
Checks:
Moment: 12.67 ÷ 13.5 = 94%, OK, governs
Shear: 7.25 ÷ 12.0 = 60%, OK
Span deflection: 9.07 ÷ (6000 ÷ 200 = 30.0 mm) = 30%, OK
Cantilever deflection: 0.96 ÷ (2 × 200 ÷ 200 = 2.0 mm) = 48%, OK
Reaction: 7.33 ÷ 18.2 = 40%, OK
Same job with two beams (N = 2):
EI total = 2 × 5,271 = 10,542 kNm²
Self-weight = 2 × 0.1167 = 0.2334 kN/m, so w = 0.5334 kN/m
Total load W = 0.5334 × 6.4 + 12 = 15.41 kN
Checks:
Moment: 13.19 kNm total (6.59 kNm per beam) ÷ (2 × 13.5 = 27.0) = 49%, OK, governs
Shear: 7.60 kN total (3.80 kN per beam) ÷ (2 × 12.0 = 24.0) = 32%, OK
Span deflection: 4.72 ÷ 30.0 mm = 16%, OK
Cantilever deflection: 0.50 ÷ 2.0 mm = 25%, OK
Reaction: 7.71 kN total (3.85 kN per beam) ÷ (2 × 18.2 = 36.4) = 21%, OK
Linear-elastic stiffness analysis, with Timoshenko beam elements when GA is included. Check section data against current manufacturer information. Product names are trademarks of their respective owners and are used only to identify the product; this site is not affiliated with or endorsed by any manufacturer. Results are for preliminary estimation and educational purposes only. They do not replace professional engineering judgement, applicable codes and standards, or design verification by a qualified engineer. Use at your own risk.