Beyond Surface Modulus: Shape-Aware Crack-Index Assessment of Mass Concrete Foundations Under Early-Age Thermal Stress

Tien-Toi Pham; Ngoc-Tuyen Tran; Chi-Cong Vu; Hong-Hai Tran; Hong-Ha Le

doi:10.2478/cee-2027-0013

Beyond Surface Modulus: Shape-Aware Crack-Index Assessment of Mass Concrete Foundations Under Early-Age Thermal Stress

Civil and Environmental Engineering

AHEAD OF PRINT

By: Tien-Toi Pham, Ngoc-Tuyen Tran, Chi-Cong Vu, Hong-Hai Tran and Hong-Ha Le

Open Access

|Aug 2026

Figures & Tables

Time-dependent baseline relationships for the hydration-heat and mechanical-property models (K = 51 °C, α = 1.516 day⁻¹, a = 4.0, b = 0.85, *fc′*(28) = 35 MPa). Values in (b) are normalised to the 28-day reference

Table 1:

Thermal crack-index acceptance bands (Korea Concrete Institute, 2009)

Criterion	Thermal crack index I_cr
To prevent cracks	I_cr ≥ 1.5
To limit cracks	1.2 ≤ I_cr < 1.5
To allow cracks	0.7 ≤ I_cr < 1.2

Table 2:

Baseline material and thermal properties

Property	Concrete	Sub-soil	Unit
Specific heat c	0.25	0.20	kcal·kgf⁽⁻¹⁾ °C⁽⁻¹⁾
Density ρ	2400	1800	kgf·m⁽⁻³⁾
Thermal conductivity k	2.3	1.7	kcal·m⁽⁻¹⁾·hr⁽⁻¹⁾·°C⁽⁻¹⁾
Convection (atm / steel form)	4.5 / 4.5 (vary)	12	kcal·m⁽⁻²⁾·hr⁽⁻¹⁾·°C⁽⁻¹⁾
Ambient T_amb	27.1 (vary)	–	°C
Casting T₀	15 (vary)	–	°C
28-day compressive strength f′_c(28)	350	–	kgf·cm⁽⁻²⁾
Strength gain (a, b)	4.0, 0.85	–	–
28-day modulus E_c	2.99×10⁵	1.0×10⁻⁵	kgf·cm⁽⁻²⁾
Thermal expansion α_th	1.0×10⁻⁵	1.0×10⁻⁵	°C⁽⁻¹⁾
Poisson’s ratio	0.18	0.20	–
Cement content	350	–	kg·m⁽⁻³⁾
Heat source (K; α)	51.0; 1.516	–	°C; day⁽⁻¹⁾

Table 3:

Surface-modulus parametric matrix (G-series)

ID	Geometry	Dimensions [m]	A_cool [m²]	V [m³]	M [m⁻¹]
G1	Small cube	2 × 2 × 2	20	8	2.50
G2	Reference cube	3 × 3 × 3	45	27	1.67
G3	Large cube	4 × 4 × 4	80	64	1.25
G4	Very large cube	5 × 5 × 5	125	125	1.00
G5	Thin slab	6 × 6 × 1.5	72	54	1.33
G6	Thick slab	6 × 6 × 3	108	108	1.00
G7	Column	2 × 2 × 6	52	24	2.17
G8	Pile cap	4 × 4 × 2	48	32	1.50

Temperature–time histories at the seven monitoring nodes for the baseline case

Temperature contour of the quarter model at the instant of peak core temperature

Table 4:

Effect of time step Δt

Case	Δt [h]	Steps	T_max,core [°C]	ΔT_cs,max [°C]	I_cr,min	CPU [min]
A1	0.5	360	63.93	25.22	1.688	10
A2	1.0	180	63.84	24.21	1.735	07
A3	2.0 (ref)	90	65.16	26.17	1.793	03
A4	4.0	45	67.73	27.97	1.519	02
A5	8.0	23	72.69	31.41	1.284	01

Table 5:

Effect of element size

Case	Edge [mm]	Grid cells	T_max,core [°C]	ΔT_cs,max [°C]	I_cr,min	CPU [min]
B1	500	3	64.77	17.50	1.464	01
B2	375	4	64.98	21.89	1.456	01
B3	300	5	65.10	24.49	1.543	02
B4	250 (ref)	6	65.16	26.17	1.793	03
B5	187.5	8	65.23	28.23	1.470	08
B6	150	10	65.27	29.38	1.396	17

Table 6:

Effect of convection coefficient h_c.

Case	h_c [kcal m⁻² h⁻¹ °C⁻¹]	T_max,core [°C]	ΔT_cs,max [°C]	I_cr,min
C1	1.5	65.64	16.55	2.471
C2	4.5 (ref)	65.16	26.17	1.793
C3	8.0	64.87	29.49	1.612
C4	12.0	64.68	31.03	1.537

Table 7:

Effect of placement temperature T₀

Case	T₀ [°C]	T_max,core [°C]	ΔT_cs,max [°C]	I_cr,min
D1	15 (ref)	65.16	26.17	1.793
D2	20	69.92	29.57	1.647
D3	25	74.69	32.97	1.518
D4	30	79.48	36.39	1.404
D5	35	84.28	39.80	1.302

Table 8:

Effect of ambient temperature T_amb

Case	T_amb [°C]	T_max,core [°C]	ΔT_cs,max [°C]	I_cr,min
E1	20	64.82	31.02	1.614
E2	25	65.06	27.60	1.737
E3	27.1 (ref)	65.16	26.17	1.793
E4	30	65.31	24.20	1.876
E5	35	65.59	20.84	2.033

Table 9:

Effect of cement content (F-series)

Case	C [kg m⁻³]	K [°C]	T_max,core [°C]	ΔT_cs,max [°C]	I_cr,min
F1	300	43.7	58.07	21.26	2.16
F2	350 (ref)	51.0	65.16	26.17	1.79
F3	400	58.3	72.25	31.10	1.53
F4	450	65.6	79.35	36.02	1.33

Table 10:

Effect of surface modulus M on the thermal response (G-series)

Case	Geometry	M [m⁻¹]	T_max,core [°C]	ΔT_cs,max [°C]	I_cr,min	t(T_max,core) [h]	P_cr,JCI [%]
G1	2 × 2 × 2 cube	2.50	61.54	21.15	2.103	44	2.8
G2	3 × 3 × 3 cube (ref)	1.67	65.16	26.17	1.79	62	5.6
G3	4 × 4 × 4 cube	1.25	66.52	30.92	1.295	78	20.6
G4	5 × 5 × 5 cube	1.00	67.01	33.67	1.095	94	37.7
G5	6 × 6 × 1.5 slab	1.33	58.82	13.56	4.523	45	0.1
G6	6 × 6 × 3 slab	1.00	64.40	23.00	2.553	71	1.2
G7	2 × 2 × 6 column	2.17	61.82	24.69	1.743	50	6.2
G8	4 × 4 × 2 pile cap	1.50	63.23	23.41	1.778	54	5.7

Diagnostic M-only plots for (a) T_max,core, (b) ΔT_cs,max, and (c) I_cr,min

Table 11:

Regression-model performance for geometry descriptors and crack-index response

Response	Candidate model	Predictors	R²	RMSE	MAE	LOOCV RMSE	Interpretation
T_max,core	M-only	M	0.227	2.276 °C	1.623 °C	2.742 °C	Weak mixed-geometry descriptor
T_max,core	Quadratic heat-flow model	d_min, d_min²	0.940	0.633 °C	0.449 °C	1.108 °C	Strong screening relation
ΔT_cs,max	M-only	M	0.080	5.494 °C	4.195 °C	6.824 °C	Not adequate
ΔT_cs,max	Two-variable model	M, d_min	0.869	2.073 °C	1.710 °C	3.788 °C	Acceptable for screening
I_cr,min	M-only	M	0.003	1.002	0.717	1.220	Invalid for crack-index prediction
I_cr,min	Inverse gradient model	ΔT_cs,max	0.951	0.223	0.177	0.302	Best performing and physically interpretable

Relationship between I_cr,min and ΔT_cs,max for the updated G-series

Table 12:

Comparative sensitivity of peak core temperature T_max,core and minimum crack index I_cr,min

Parameter	Range studied	ΔT_max,core [°C]	ΔI_cr,min	Dominant response
Time step Δt	0.5–8 h	8.79	0.51	Numerical stability
Element size	150–500 mm	0.50	0.39	Numerical resolution
Convection coefficient h_c	1.5–12 kcal m⁻² h⁻¹ °C⁻¹	0.96	0.93	I_cr,min
Placement temperature T₀	15–35 °C	19.14	0.49	T_max,core
Ambient temperature T_amb	20–35 °C	0.77	0.42	I_cr,min
Cement content C	300–450 kg m⁻³	21.28	0.83	T_max,core
Surface modulus M (cubic subset G1–G4)	1.00–2.50 m⁻¹	5.47	1.008	I_cr,min
Geometry/surface-modulus portfolio (full G-series)	1.00–2.50 m⁻¹	8.19	3.428	Shape-dependent I_cr,min response

References

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DOI: https://doi.org/10.2478/cee-2027-0013 | Journal eISSN: 2199-6512 (formerly 1336-5835) | Journal ISSN: 1336-5835

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Language: English

Submitted on: May 25, 2026

Accepted on: Jun 19, 2026

Published on: Aug 18, 2026

Published by: University of Žilina

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

Mass concrete,

Thermal crack index,

Surface modulus,

Multi-geometry analysis,

Hydration heat,

MIDAS Civil

Related subjects:

Business and economics,

Business management,

Industries,

Environmental management

© 2026 Tien-Toi Pham, Ngoc-Tuyen Tran, Chi-Cong Vu, Hong-Hai Tran, Hong-Ha Le, published by University of Žilina
This work is licensed under the Creative Commons Attribution 4.0 License.

AHEAD OF PRINT

Beyond Surface Modulus: Shape-Aware Crack-Index Assessment of Mass Concrete Foundations Under Early-Age Thermal Stress

Figures & Tables

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