Calculator
Civil & Structural Engineering FoS = R / D
Slope Stability Calculator
28 September 2026 11 min read
Overall slope factor from cohesive, frictional and driving resultants.
- Slope FoS
- 1.23
- Review urgently
Live statistics
Slope FoS
1.23
Verdict
Review urgently
Unit system
Cohesion c(kPa) Help
kPa
Along the slip.
Slip length(m) Help
m
Arc or plane length.
Normal resultant(kN) Help
kN
On the slip.
Friction angle(deg) Help
deg
Effective.
Driving force(kN) Help
kN
Downslope resultant.
Results update live as you type
How you compare — standard sections
| Section | Depth (mm) | Mass (kg/m) | A (cm²) | Iₓ (×10⁶ mm⁴) | Sₓ (×10³ mm³) |
|---|---|---|---|---|---|
| UB 305×165×40 · Steel I-beam | 304 | 40.3 | 51.3 | 85 | 560 |
| UB 457×191×67 · Steel I-beam | 454 | 67.1 | 85.5 | 294 | 1290 |
| UC 254×254×73 · Steel column | 254 | 73.1 | 93.1 | 114 | 895 |
| RC 300×500 (C30) · Concrete beam | 500 | 375 | 1500 | 3125 | 12500 |
| Glulam 140×450 · Timber beam | 450 | 29 | 630 | 1063 | 4725 |
| SHS 200×200×8 · Hollow section | 200 | 47.4 | 60.4 | 36.9 | 369 |
Indicative values for scheme design only. Always verify against the manufacturer / code tables (AISC Steel Manual, SCI Blue Book, ACI detailing) before ordering steel or pouring concrete.
Plans & details
Section
Slip section sketch
Trial circles, planes and water table on the critical cross-section.
Drain
Drainage detail
Crest cutoffs, face drains and toe blankets that keep FoS up in rain.
Check
Site verification
Confirm geology, tension cracks and seepage against the design section.
Typical details for scheme design. Replace placeholders with project drawings (1200×630 JPG or SVG line work in /public/plans/civil/). Do not scale from web images — use figured dimensions.
Design verification checklist
Work through before issuing results — 1/6 verified (17%).
Results
Critical
Slope factor
1.23-
Resisting over driving.
Resisting force
1106kN
Cohesion plus friction.
Verdict
Review urgentlycheck
Against 1.3 and 1.5.
Visualizations
Results Overview
1.2-Slope factor
1,106-Resisting force
Formula
Every slice method collapses to this ratio: what holds versus what pulls.
FoS = (c x L + N x tan phi) / T
Alternative Form
Want 1.3 temporary, 1.5 permanent
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| R | Resisting | kN | Cohesion plus friction. |
| T | Driving | kN | Downslope pull. |
| FoS | Factor | - | Margin on the slip. |
Explanation
Limit-equilibrium safety ratio.
Derivation
Limit-equilibrium ratio on the trial surface.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Step 1
Enter slip strength
Result: Hold fixed
Step 2
Enter driving pull
Result: Pull fixed
Step 3
Read FoS
Result: Stability issued
Final Answer
1.23
Worked Examples
See how the calculator works with different scenarios.
Scenario
c 20, L 25, N 1500, phi 22, T 900.
Inputs
c:20
La:25
N:1500
phi:22
T:900
Calculation
R is 500 + 606 = 1106 kN, FoS 1.23, review.
Result
FoS = 1.23
Interpretation
Below 1.3: flatten, drain or reinforce.
Understanding Your Result
Your Result: Slope factor = 1.23 - · Resisting force = 1106 kN · Verdict = Review urgently check
Typical range: 1.3 temporary, 1.5 permanent.
What This Means
Margin of the trial slip against sliding.
Why It Matters
The acceptance number for cuts and fills.
Professional Interpretation
Search circles, planes and wedges for the minimum.
Recommendations
- •Run dry and storm water cases side by side
- •Search several surfaces for the minimum FoS
- •Verify one line by hand per slope section
Learn — structural behaviour, deeply
Hold versus pull.
1 · The two equations that run every site
ΣF_y = 0
Vertical balance
Upward reactions must equal total downward load. Missing 1 kN here means a 1 kN shear error everywhere downstream.
ΣM = 0
Moment balance
Moments about ANY point must cancel. Take moments about one support to eliminate it and solve the other directly.
ΣF_x = 0
Horizontal balance
Zero for vertical-only loading — but the pin still needs checking once wind, soil or inclined loads appear.
Worked derivation (simply supported, point load): ΣMleft = 0 ⇒ Rb·L − P·a = 0 ⇒ Rb = P·a/L, then ΣFy = 0 ⇒ Ra = P − Rb = P·(L−a)/L. UDL adds w·L/2 per side acting at midspan. That is the entire calculation — everything else is bookkeeping.
2 · Support types — what each one can (and cannot) do
| Support | Reactions | Allows | Watch out |
|---|---|---|---|
| Roller | 1 (vertical) | Horizontal slide + rotation | Provides NO lateral restraint — pair with a pin or bracing. |
| Pin / hinge | 2 (vertical + horizontal) | Rotation only | Moment released here — detail the hinge, never weld it rigid by accident. |
| Fixed / clamped | 3 (+ moment) | Nothing (fully restrained) | Attracts large moments; settlement or temperature causes locked-in stress. |
| Propped cantilever | 3 (indeterminate ×1) | Rotation at prop | Needs compatibility (deflection = 0 at prop) — equilibrium alone is not enough. |
| Continuous (3+ supports) | 4+ (indeterminate) | Rotation over interior piers | Solve by superposition / moment distribution / software; reactions shift with stiffness. |
3 · Factor-impact table — what moves your reactions most
| Factor | Typical impact | Why |
|---|---|---|
| Load position (a/L) | Linear, dominant | Moving P toward a support shifts ~100% of P onto it. Ra = P·(L−a)/L — position matters more than span. |
| UDL intensity (w) | ∝ w·L total | Each extra 1 kN/m on a 6 m beam adds 3 kN per reaction. Self-weight alone is often 1–5 kN/m on concrete. |
| Span length (L) | ∝ L for UDL | Doubling L doubles UDL reactions and quadruples UDL moment (wL²/8). Keep spans tight where possible. |
| Overhang length (c) | ∝ c/L uplift | Uplift at the far support = P·c/L. A 2 m overhang with 10 kN on a 5 m span pulls down 4 kN — needs hold-down. |
| Fixity (pinned → fixed) | Redistributes ~10–30% | Fixed ends attract moment and shed midspan moment. Never assume fixity you have not detailed. |
| Support settlement (Δ) | 0 (det.) / large (indet.) | Determinate beams just rotate; continuous/fixed beams develop locked-in moments ≈ 3EIΔ/L². |
| Dynamic / impact | +10–100% | Static reactions must be amplified for cranes, machinery, footfall (see ASCE 7 Ch. 4, AISC DG11). |
| Load combination | × 1.2–1.6 (LRFD) | 1.2D + 1.6L + 0.5(Lr/S/R) governs most beams. Reactions for strength ≠ reactions for service checks. |
4 · Code provisions you must cite on drawings
- ASCE 7 load combinations (LRFD): 1.4D · 1.2D + 1.6L + 0.5(Lr/S/R) · 1.2D + 1.0W + 1.0L … Strength reactions use factored loads; bearing/deflection checks use service (unfactored) loads.
- ACI 318 (concrete): strength-reduction ϕ = 0.65–0.90; development/lap lengths Ch. 25; one-way shear + punching Ch. 22. Reactions feed directly into Vu ≤ ϕVn.
- AISC 360 (steel): Ch. F/G/H for moment/shear/combined; bearing at supports per J7–J8; bracing per App. 6. Connection must develop the computed reaction — a beam is only as strong as its seat.
- Eurocode basis (EN 1990): ULS 1.35G + 1.5Q; SLS G + Q (characteristic). National annexes adjust factors — confirm the project annex.
- Serviceability (all codes): live-load deflection ≈ L/360 floors, L/240 roofs; total ≈ L/240. Vibration: AISC DG11 / SCI P354 for rhythmic or office floors.
5 · Pro workflow (5 minutes per beam)
- Sketch the free-body diagram: every load incl. self-weight, every support, dimensions.
- Solve ΣM then ΣF — write both lines on the calc sheet so a checker can follow.
- Verify: re-take moments about the OTHER support; residual must be ≈ 0.
- Draw shear then moment (reactions are the SFD starting values).
- Size the support: bearing stress, padstone, bolt/weld group, foundation pressure.
- File: keep the input → reaction → SFD/BMD → support chain on one page.
6 · Field traps (where reactions go wrong)
- • Forgetting self-weight: a 300×600 concrete beam weighs 4.5 kN/m — on 6 m that is 13.5 kN per support.
- • Assuming fixity: a “fixed” sketch with two bolts is a pin. Model what you detail.
- • Ignoring uplift: overhang + wind can reverse a bearing — light roofs lift off.
- • Mixing factored/service: never compare a factored reaction against an allowable soil pressure.
- • Point load at the wrong station: 0.5 m error on a 10 kN load swings reactions by P·Δa/L.
7 · The 3-check rule (sign-off standard)
- Math check: ΣFy and ΣM both close to zero with project units printed.
- Sanity check: reactions sum to total load; heavier side sits under heavier load; UDL symmetry gives Ra ≈ Rb.
- Load-path check: every reaction lands on something sized for it — beam → seat → column → footing → soil, no floating numbers.
Practical Tips
- •Split dead vs live early — factoring depends on it
- •Include self-weight; concrete often dominates
- •Report service values to 1 decimal for bearings and footings
Common Mistakes
Service load checked against factored capacity
Keep a service column (settlement, soil, deflection) and a factored column (strength) side by side.
Self-weight omitted
Add member weight into the total first.
Wrong units in one term
Convert everything to the page unit system before combining terms.
Assumptions
This calculator makes the following assumptions to provide estimates:
Static gravity loading: No impact, vibration or seismic amplification unless the page applies one.
Nominal material properties: Code-grade materials per project data; confirm lightweight or high-strength cases.
Scheme-level accuracy: Closed-form code clauses for preliminary and check calculations.
Limitations
Be aware of these limitations when using this calculator:
Verify supports and pattern: Pattern loading, continuity and support settlement need their own runs.
Confirm local code values: Loads, factors and minimums vary by jurisdiction.
Frequently Asked Questions
Answers to common questions about this calculator.
Which surface?
The minimum over all trials governs.
Drained or undrained?
Run both: end of construction and long term.
Is this final?
No, it is a single-surface ratio. A licensed engineer must confirm final design.
How do I find reactions for a simply supported beam?
Take moments about the left support: Rb = (P·a + w·L·L/2) / L, then Ra = P + w·L − Rb. Always re-check by taking moments about the right support — the residual must be ≈ 0.
What does a negative (downward) reaction mean?
The support must pull DOWN instead of pushing up — net uplift from overhangs, pattern loading or wind. Provide a hold-down strap, anchor or ballast; a gravity-only bearing will lift off.
Do I include the beam self-weight?
Yes, always. A 300×600 mm concrete beam weighs ~4.5 kN/m — on a 6 m span that adds ~13.5 kN per support. Bundle it into the UDL before solving.
Service vs factored loads — which do I report?
Report BOTH in separate columns: service (unfactored D+L) for deflection, bearing, settlement and soil checks; factored LRFD (1.2D + 1.6L) for member strength. Never compare a factored reaction with an allowable soil pressure.
What is the difference between a UDL and a UVL?
A UDL (uniformly distributed load) is constant intensity over a length (kN/m). A UVL (uniformly varying / triangular) runs zero-to-max — e.g. soil or hydrostatic pressure. They integrate differently, so use the matching calculator.
How do I turn a floor area load into a beam line load?
Multiply the area load (kN/m²) by the tributary width (m) draining to the beam: line load = area load × tributary width. Add the beam self-weight, partitions and finishes first.
What is tributary area?
The floor/roof area whose load flows to one member — bounded by halfway lines to adjacent members. Tributary area × area load = the point/line load that member must carry.
Which load combination usually governs a beam?
For strength: 1.2D + 1.6L (+ 0.5 for roof live/snow/rain) per ASCE 7 LRFD. Service D+L governs deflection and bearings; 0.9D + 1.0W governs uplift and overturning.
LRFD vs ASD — does it change the reactions?
No — equilibrium is the same. The design PHILOSOPHY differs: LRFD factors loads up (1.2D+1.6L) against factored resistance (ϕRn); ASD keeps service loads against allowable stresses (Rn/Ω). Keep the two systems in separate columns.
Pin, roller or fixed — how do I choose the support model?
Model what you DETAIL. Masonry/steel seats and two-bolt end plates are pins (moment released); moment frames, welded rigid joints and well-anchored cantilevers earn fixity. When in doubt, the pinned assumption is conservative for midspan moment but NOT for uplift.
What deflection limits apply to beams?
Typical code limits: live-load deflection L/360 for floors (L/240 roofs), total-load L/240, cantilevers 2L/360-ish per IBC Table 1604.3. Always check under SERVICE loads with the correct (cracked, long-term) stiffness.
How do I convert kN ↔ kips and m ↔ ft?
1 kN = 0.2248 kips (1 kip = 4.448 kN); 1 m = 3.2808 ft. Moments: 1 kN·m = 0.7376 kip·ft. Every calculator on this site has an SI/Imperial toggle that converts all displayed values.
Shear vs bending moment — which sizes what?
Shear sizes stirrups/webs and connections (V ≤ ϕVn); bending moment sizes flexural steel and the section (M ≤ ϕMn via section modulus S: stress = M/S). Deflection is governed by stiffness EI, a third independent check.
Where does the maximum moment occur?
Where shear crosses zero: under a midspan point load, or at x = R/w under UDL (span moment = R·x − w·x²/2). On continuous beams also check the hogging peaks OVER interior supports — they often govern.
What is section modulus and how do I use it?
S = I/c (mm³): the geometric bending efficiency of a section. Bending stress = M/S; required S = factored moment / allowable (or design) stress. Compare against the standard-section table on each page.
How do I quickly check a steel beam?
Three checks per AISC 360: bending (Ch. F incl. lateral-torsional buckling over the unbraced length), shear (Ch. G), and deflection (L/360). Then verify the connection develops the end reaction.
How do I quickly check a concrete beam?
Flexure per ACI 318: ϕMn = ϕ·As·fy·(d − a/2) with ϕ = 0.9 (tension-controlled); one-way shear Vc + Vs (Ch. 22); deflection via minimum thickness or direct computation with cracked stiffness and long-term multiplier.
What is development length and why does it matter?
The embedment needed for a bar to reach yield (ld, ACI 318 Ch. 25). A perfectly computed moment capacity is worthless if the bar pulls out — extend bars past inflection points and hook or develop them at supports.
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Glossary
Technical terms used in mortgage calculations explained in plain language.
- Reaction
- Force (or moment) a support exerts on a member to maintain static equilibrium (ΣF = 0, ΣM = 0).
- Simply supported
- Beam on one pin + one roller. Resists vertical load; allows rotation and (at roller) horizontal movement.
- Fixed support
- Clamped end resisting force AND moment; prevents rotation. Used in cantilevers and fixed frames.
- UDL / UVL
- Uniformly distributed load (constant kN/m) vs. uniformly varying load (triangular, zero-to-max).
- Tributary area
- Floor/roof area whose load drains to one member. Line load = area load × tributary width.
- Factored load
- Service load × LRFD load factor (e.g. 1.2D + 1.6L) for strength design per ASCE 7.
- Serviceability
- Deflection, cracking and vibration limits under day-to-day (unfactored) loads.
- Uplift
- Net downward-hold requirement: a negative (downward-pulling) reaction, common on overhangs.
Authoritative References
Trusted sources and standards referenced by this calculator.
ACI 318 Code & Commentary (American Concrete Institute) AISC Steel Construction Manual ASCE 7 Hazard Tool — design loads Eurocodes (EN 1990–1999) — Basis of Structural Design Hibbeler — Structural Analysis (textbook companion) NIST — Wind & Seismic design resources
Applicable Standards
ACI 318 — Building Code Requirements for Structural ConcreteAISC 360 — Specification for Structural Steel BuildingsASCE 7 — Minimum Design Loads and Associated CriteriaEurocode 2 (EN 1992) — Design of Concrete StructuresEurocode 3 (EN 1993) — Design of Steel StructuresIBC — International Building Code (structural chapters 16–23)
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Page Information
Last Updated:
2026-09-28
Version:
2.0.0
References:
License:
Free to use for educational and professional purposes.