Learn: Limit-state utilization Ed/Rd
A resistance check compares what the structure must carry with what it can resist in a particular failure mode. The utilization ratio places demand and resistance on a common scale: 1 is the boundary for this one comparison.
What the formula is saying
Divide design effect Ed by design resistance Rd. A value such as 0.80 means demand uses 80% of the supplied resistance; a value above 1 means this specific inequality is not satisfied.
Read the symbols in plain language
- Ed
- Design action effect
The already-factored effect being checked, such as a force or moment; use the same quantity and unit as the resistance.
consistent unitUse consistent unit as the base unit shown here. Ed and Rd must describe the same quantity and use exactly the same unit, such as kN with kN or kN·m with kN·m. The units cancel. Enter nonnegative demand magnitude and positive resistance.
- Rd
- Design resistance
Resistance for the same failure mode, quantity and unit as the action effect, after the required design factors.
consistent unitUse consistent unit as the base unit shown here. Ed and Rd must describe the same quantity and use exactly the same unit, such as kN with kN or kN·m with kN·m. The units cancel. Enter nonnegative demand magnitude and positive resistance.
- Ed/Rd
- Result to find
Limit-state utilization Ed/Rd. Dividing matching quantities gives a dimensionless utilization ratio, not a global safety certificate.
ratio / no unit
Sort out the units first
Ed and Rd must describe the same quantity and use exactly the same unit, such as kN with kN or kN·m with kN·m. The units cancel. Enter nonnegative demand magnitude and positive resistance.
Assumptions before calculating
Both values have already been derived for the same design situation, limit state and failure mode. Signs have been converted to compatible demand and resistance magnitudes before this scalar comparison.
This is a first-generation Eurocode teaching relationship or an explicitly simplified coefficient calculation. The numbers supplied here are exercise data, not a recommendation for any country. Check the adopted edition, relevant clause, National Annex, applicability conditions and all other limit states before any real design.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find Ed/Rd and explain the result in the stated output unit.
- Ed · Design action effect
- 80 consistent unit
- Rd · Design resistance
- 100 consistent unit
Identify the design demand
Use the demand magnitude for the exact failure mode and design situation being checked.
(80) = 80 same as EdCompare with the supplied resistance
Dividing matching quantities gives a dimensionless utilization ratio, not a global safety certificate.
(80) ÷ (100) = 0.8
Does this worked answer make sense?
Equal demand and resistance give 1. Doubling demand doubles utilization, while doubling resistance halves it. Report the governing check alongside the number.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- Ed · Design action effect
- 120 consistent unit
- Rd · Design resistance
- 100 consistent unit
Identify the design demand
Use the demand magnitude for the exact failure mode and design situation being checked.
(120) = 120 same as EdCompare with the supplied resistance
Dividing matching quantities gives a dimensionless utilization ratio, not a global safety certificate.
(120) ÷ (100) = 1.2
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- Ed · Design action effect
- 90 consistent unit
- Rd · Design resistance
- 120 consistent unit
Find: Learn: Limit-state utilization Ed/Rd
A hint, not the answer
Divide design effect Ed by design resistance Rd. A value such as 0.80 means demand uses 80% of the supplied resistance; a value above 1 means this specific inequality is not satisfied.
Ed and Rd must describe the same quantity and use exactly the same unit, such as kN with kN or kN·m with kN·m. The units cancel. Enter nonnegative demand magnitude and positive resistance.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Identify the design demand
Use the demand magnitude for the exact failure mode and design situation being checked.
(90) = 90 same as EdCompare with the supplied resistance
Dividing matching quantities gives a dimensionless utilization ratio, not a global safety certificate.
(90) ÷ (120) = 0.75
Avoid the common trap
Do not reverse Ed and Rd. Do not use a negative demand to obtain an apparently favourable ratio, or mix a bending moment with an axial force.
When this method applies — and when it does not
A ratio at or below 1 is not a declaration that the structure is safe. Interaction, stability, fatigue, deformation, detailing and other modes still require their own checks. Do not compare an unfactored demand with a factored resistance without justification.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
One idea understood. Keep going.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Limit-state utilization Ed/Rd. Basis-of-design reading: partial factors, design values, and combinations of actions. The displayed coefficients are supplied exercise data.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.