StrataForge concrete strength · Case study

Problem

Concrete strength decides whether a slab or foundation can carry its load, so engineers tend to over-specify cement to be safe. Cement is the most expensive part of the mix and the biggest source of its carbon.

StrataForge is a stand-in name for a real construction materials client that I can’t name under contract. The firm had two years of mix trials but still designed mixes by rule of thumb. The task was to show which ingredients drive strength, test whether cheaper fly ash weakens concrete, and give the team a formula they could use.

Data

Because the client’s own records are confidential, I worked on the public UCI Concrete Compressive Strength dataset. It holds 1,030 mix trials. Each one has the amounts of seven ingredients in kg per cubic metre, the curing age in days and the measured strength in megapascals. Strength runs from 2.3 to 82.6 MPa and nothing is missing.

Cement, age and superplasticizer correlate most with strength, and water works against it. Age and superplasticizer are heavily skewed, which matters for a linear model.

How each ingredient correlates with strength, all 1,030 mixesPearson r, from −1 to 1

Show as table

Feature	r
Cement	0.498
Slag	0.135
Fly ash	-0.106
Water	-0.290
Superplasticizer	0.366
Coarse aggregate	-0.165
Fine aggregate	-0.167
Age	0.329

Superplasticizer per cubic metre

kg per cubic metre

379 mixes contain no superplasticizer at all. That spike is why taking its log later forced those rows out.

Show as table

kg per cubic metre	Value
none	379
0 to 4	32
4 to 8	192
8 to 12	314
12 to 16	59
16 to 20	30
20 to 24	14
24 to 32	10

Curing age at test

days

749 of 1,030 mixes were tested at 28 days or sooner, 425 of them at exactly 28 days, the industry standard. A thin tail runs out to a year, a strong right skew.

Show as table

days	Value
1 to 6	136
7 to 13	126
14 to 27	62
28	425
29 to 90	145
91 to 180	103
181 to 365	33

Approach

I built the model in R by forward stepwise selection. I fitted twelve candidate models, adding predictors in order of correlation and checking that every term stayed significant.

The skew in age and superplasticizer limited the plain models to about 61%. Taking logs of both lifted the fit sharply.

Strength against curing age, on a log scale

On a log scale of age the trend is close to a straight line, which is what a linear model needs. Strength rises fast in the first month and slowly after that.

Show as table (first 25 points)

age in days (log scale)	strength (MPa)
28	80
28	61.9
270	40.3
365	41.1
360	44.3
90	47
365	43.7
28	36.4
28	45.9
28	39.3
90	38.1
28	28
270	43
90	42.3
28	47.8
90	52.9
90	39.4
365	56.1
90	40.6
180	42.6
180	41.8
28	28.2
3	8.1
180	44.2
365	52.5

Superplasticizer is zero in 379 mixes, and the log of zero is undefined, so those rows dropped out and the final model uses 651 mixes.

Adjusted R² as the stepwise model grewhigher is better

Every bar uses adjusted R². The last bar is fitted on 651 mixes and the others on all 1,030, so part of that final jump comes from the smaller sample.

Show as table

Item	Value
Cement only	24.7%
+ superplasticizer	35.0%
+ age	48.0%
+ water, fine and coarse aggregate	56.8%
+ slag	59.4%
+ fly ash	61.3%
Logged age and superplasticizer, five predictors	81.3%

model_9 <- lm(concrete_strength ~ cement + superPlasticizer + water + age + slag,
              data = concrete_clean)   # superPlasticizer and age are logged
summary(model_9)   # adjusted R-squared 0.8135, every term p < 0.001
vif(model_9)       # 1.01 to 1.49, no multicollinearity

The final formula reads directly as engineering advice.

strength = 23.914 + 0.0974 × cement − 2.545 × ln(superplasticizer)
           − 0.2374 × water + 9.759 × ln(age) + 0.0683 × slag

More cement, more slag and longer curing raise strength. More water lowers it, which is why the water to cement ratio matters so much.

Hypothesis tests

Shapiro-Wilk showed strength is not normally distributed within the groups, so I used Kruskal-Wallis tests, plus a chi-square test for the two categories.

Question	Test	p-value	Result
Does coarse or fine concrete differ in strength?	Kruskal-Wallis	0.336	No significant difference
Do mixes with fly ash differ in strength?	Kruskal-Wallis	0.232	No significant difference
Is fly ash use linked to concrete category?	Chi-square	0.981	No association

A logistic regression also predicts whether a mix contains fly ash from its other ingredients, with an AIC of 145.12 and every predictor significant.

Evaluation

All five classical assumptions for linear regression were checked with residual plots, a Q-Q plot, a scale-location plot and variance inflation factors.

The original project only measured fit on the same rows it was trained on. In October 2026 I rebuilt the model from the raw spreadsheet, reproduced every coefficient, and ran 10-fold cross-validation repeated 20 times. The cross-validated R² is 0.80 ± 0.04 with an average error of 7.2 MPa, so the model holds up on mixes it has not seen.

What didn’t work

The report’s worked example predicted 100.4 MPa for a high-cement mix. The strongest mix ever tested was 82.6 MPa, so that prediction sits outside the data and a linear model should not be trusted there.

The fly ash tests found no significant difference, which is not the same as proving fly ash has no effect. Mixes with and without fly ash also differ in other ingredients, so a fair comparison would hold those fixed.

Limitations

The final model only applies to mixes that contain superplasticizer.
The cost example cuts cement by 90 kg per cubic metre on a 2,000 cubic metre job. At £120 per tonne that saves £21,600, but it does not count the 100 kg per cubic metre of slag added in its place.
The model was built on the public UCI dataset rather than the client’s own records, so its numbers describe that data.

Stack

R
dplyr
ggplot2
car
caret
corrplot
readxl
RVAideMemoire
React
Vite

Links

All work