Learn: Concrete secant modulus — study relation
Concrete stiffness controls elastic deformation and is not numerically equal to its compressive strength. This empirical expression estimates a mean secant modulus from mean cylinder strength for the stated concrete model.
What the formula is saying
Divide fcm by the reference strength 10 MPa, raise the dimensionless ratio to 0.3, then multiply by 22 GPa. The exponent makes stiffness grow more slowly than strength.
Read the symbols in plain language
- fcm
- Mean compressive strength
Mean compressive strength. Divide by the reference strength expressed in the same MPa unit.
MPaOne megapascal equals one N/mm² and 1000 kPa.
- Ecm
- Result to find
Concrete secant modulus — study relation. The coefficient is calibrated to return gigapascals when the strength input uses megapascals.
GPa
Sort out the units first
Enter fcm in MPa; the formula returns Ecm in GPa. These are deliberately different units: 1 GPa = 1000 MPa. The 10 and 22 are tied to this calibrated unit convention.
Assumptions before calculating
Use the first-generation normal-weight concrete relationship with the reference aggregate assumptions. The teaching strength range is fcm from 20 to 98 MPa, corresponding to fck from 12 to 90 MPa through fcm = fck + 8.
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 Ecm and explain the result in the stated output unit.
- fcm · Mean compressive strength
- 38 MPa
Normalize the mean strength
Divide by the reference strength expressed in the same MPa unit.
(38) ÷ 10 = 3.8Apply the empirical exponent
The exponent models the slower increase of elastic stiffness with concrete strength.
(3.8)^0.3 ≈ 1.492571274Find the estimated secant modulus
The coefficient is calibrated to return gigapascals when the strength input uses megapascals.
22 × (1.492571274) ≈ 32.83656803 GPa
Does this worked answer make sense?
Doubling fcm increases Ecm by 2^0.3, not by 2. A modulus of about 30 GPa is about 30,000 MPa, far above a concrete strength measured in tens of MPa.
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.
- fcm · Mean compressive strength
- 48 MPa
Normalize the mean strength
Divide by the reference strength expressed in the same MPa unit.
(48) ÷ 10 = 4.8Apply the empirical exponent
The exponent models the slower increase of elastic stiffness with concrete strength.
(4.8)^0.3 ≈ 1.600930104Find the estimated secant modulus
The coefficient is calibrated to return gigapascals when the strength input uses megapascals.
22 × (1.600930104) ≈ 35.22046229 GPa
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.
- fcm · Mean compressive strength
- 33 MPa
Find: Learn: Concrete secant modulus — study relation
A hint, not the answer
Divide fcm by the reference strength 10 MPa, raise the dimensionless ratio to 0.3, then multiply by 22 GPa. The exponent makes stiffness grow more slowly than strength.
Enter fcm in MPa; the formula returns Ecm in GPa. These are deliberately different units: 1 GPa = 1000 MPa. The 10 and 22 are tied to this calibrated unit convention.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Normalize the mean strength
Divide by the reference strength expressed in the same MPa unit.
(33) ÷ 10 = 3.3Apply the empirical exponent
The exponent models the slower increase of elastic stiffness with concrete strength.
(3.3)^0.3 ≈ 1.430718464Find the estimated secant modulus
The coefficient is calibrated to return gigapascals when the strength input uses megapascals.
22 × (1.430718464) = 31.47580621 GPa
Avoid the common trap
Do not substitute fck where fcm is required. Do not report the answer as MPa without converting, and do not turn the power 0.3 into multiplication by 0.3.
When this method applies — and when it does not
Aggregate type can change stiffness; measured values may be needed. Cracked-section stiffness, creep-adjusted modulus, early-age development and nonlinear tangent stiffness are not provided here.
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.
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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: Concrete secant modulus — study relation. First-generation EN 1992 teaching: material properties and the relevant bending, shear, serviceability, detailing or prestress relationship. Read the applicability conditions as well as the expression.
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.