ABSTRACT Ultra-high-performance concrete (UHPC) is known for its exceptional strength, durability, ductility, and toughness due to its dense cementitious matrix. However, its compact

structure presents challenges when exposed to high temperatures, making it prone to strength deterioration and spalling. This study aims to develop reliable relationships to predict UHPC’s

mechanical properties at high temperatures, including compressive, tensile, and flexural strengths, modulus of elasticity, peak strain, and compressive stress–strain curves. The UHPCs

considered in this study include UHPC without fibers (NF), UHPC with steel fibers (SF) only, UHPC with polypropylene fibers (PPF) only, and UHPC with a hybrid combination of SF and PPF. The

and trilinear equations. The modulus of elasticity and peak strain are described using linear and exponential combinations. The proposed compressive stress–strain relationships capture most

data but may not accurately represent all cases. However, significant discrepancies exist in the proposed equations for certain UHPC specimens, particularly for compressive, tensile, and

flexural strength in PPF specimens, compared to existing equations. The study suggests further testing to refine the constitutive relationships, including investigating different specimen

sizes and shapes, varying heating durations, and exploring alternative curing methods. SIMILAR CONTENT BEING VIEWED BY OTHERS TRIAXIAL COMPRESSION AND SHEAR STRENGTH CHARACTERISTICS OF

TWO-STAGE CONCRETE: AN EXPERIMENTAL STUDY Article Open access 26 November 2024 FLEXURAL BEHAVIOUR AND EVALUATION OF ULTRA-HIGH-PERFORMANCE FIBRE REINFORCED CONCRETE BEAMS CURED AT ROOM

TEMPERATURE Article Open access 24 September 2021 EXPERIMENTAL STUDY ON DYNAMIC MECHANICAL PROPERTIES OF HYBRID FIBER REINFORCED CONCRETE AT DIFFERENT TEMPERATURES Article Open access 09 May

2025 INTRODUCTION Ultra-high-performance concrete (UHPC), also referred to as reactive powder concrete (RPC), emerged in the 1990s as a revolutionary material in the construction industry.

It is distinguished by its superior mechanical and durability properties, stemming from its unique composition. UHPC features a low water-to-cement (w/c) ratio, high binder content, and

often incorporates fibre reinforcements, which together contribute to its exceptional compressive strengths ranging from 120 to 810 MPa, as well as its remarkable toughness, impermeability,

and resistance to environmental degradation. Its dense microstructure minimises permeability and enhances resistance to chloride ingress and chemical attacks, making it particularly suitable

for harsh environments1. These attributes position UHPC as a promising material for advanced structural applications, including protective defence infrastructure, high-performance bridge

decks, and long-term nuclear waste containment2. Despite these advantages, the dense cementitious matrix of UHPC presents a significant challenge during high-temperature exposure. The

inability of its compact structure to facilitate the release of water vapor results in increased internal pore pressure. This phenomenon often leads to thermal cracking, microstructural

degradation, and in extreme cases, explosive spalling under fire conditions3. Such vulnerabilities hold back UHPC’s widespread use, as its performance under elevated temperatures does not

align with the superior mechanical properties it exhibits at ambient conditions. Additionally, the lack of comprehensive predictive models for its behaviour under high-temperature scenarios

further limits its adoption4. The need to understand and accurately predict the behaviour of UHPC under elevated temperatures has become critical for its broader implementation. Exposure to

high temperatures alters key mechanical properties, including compressive and tensile strength, modulus of elasticity, and thermal expansion, while also inducing phenomena like creep and

thermal cracking. These changes can compromise the structural integrity and safety of UHPC-based structures in fire scenarios. To address these challenges, constitutive models that define

the stress–strain relationships and thermal responses of UHPC are essential. Such models serve as foundational tools for engineers to design fire-resistant structures, evaluate post-fire

damage, and formulate effective repair and rehabilitation strategies. Beyond safety considerations, accurate constitutive models enable performance-based engineering, allowing for optimised

structural designs that account for temperature-dependent changes in mechanical and thermal properties. These models also facilitate reliable numerical simulations, which are indispensable

for predicting the performance of UHPC under fire conditions. They consider critical factors such as temperature-induced creep, microstructural alterations, and stress redistribution within

UHPC elements. By addressing these aspects, such models not only enhance structural reliability but also promote the material’s potential for cost-effective and sustainable construction.

Current research on UHPC’s high-temperature performance is limited, with experimental data often focused on specific properties and lacking generalisability across different formulations and

fibre types. UHPC mixes vary widely depending on the combination of fibres, mineral admixtures, and curing methods, further complicating the prediction of its behaviour under fire exposure.

For instance, the addition of polypropylene fibres (PPF) is known to mitigate explosive spalling by forming vapor-release channels, but its effectiveness depends on the mix design and

curing process2. Similarly, hybrid fibres and alternative binders may offer enhanced performance but require more robust experimental validation. To address these gaps, this study focuses on

developing constitutive relationships that capture the mechanical behaviour of UHPC, include UHPC without fibers (NF), UHPC with steel fibers (SF) only, UHPC with polypropylene fibers (PPF)

only, and UHPC with a hybrid combination of SF and PPF,  under elevated temperatures. These relationships aim to provide a comprehensive understanding of key parameters such as compressive

and tensile strength, modulus of elasticity, peak strain, and stress–strain behaviour after high-temperature exposure. By leveraging and synthesising existing experimental data, this

research proposes predictive models that can guide the design of fire-resistant UHPC structures, advance the material’s broader adoption, and support the development of performance-based

design standards. LITERATURE REVIEW UHPC is known for its outstanding strength, durability, ductility, and toughness5,6. According to ASTM C18567 and ACI 2398, UHPC must achieve compressive

strengths of at least 120 MPa and 150 MPa, respectively. These properties are achieved through a combination of low water-to-binder (w/b) ratio, selected aggregates, cementitious materials,

chemical admixtures, and steel fibres. This section reviews the theoretical concepts of UHPC, its materials, mix designs, and curing methods. THEORETICAL CONCEPT UHPC’s exceptional

performance arises from its dense matrix of hydrated cement, fine reactive mineral admixtures, superplasticizers, and tightly bonded particles. A low w/c ratio reduces porosity but may

affect workability, which is compensated for with plasticizers9,10. This results in a homogeneous microstructure with a strong interfacial transition zone (ITZ), enhancing durability

compared to conventional concrete, which has larger pores and weaker interfacial bonding11. The inclusion of fibres, especially steel and carbon fibres, significantly improves toughness by

controlling crack initiation and propagation, see Fig. 112. RAW MATERIALS The raw materials used in UHPC include cement, fine aggregates, superplasticizers, and fibres, each playing a vital

role in the material’s performance. CEMENTITIOUS MATERIAL _Cement:_ Low-alkali Portland cements such as CEM I 52.5R and CEM I 42.5R are preferred due to their high strength and minimal

shrinkage13. Blast furnace slag cements (CEM III/A) are used in environments subject to temperature changes or chemicals. _Silica Fume (SFu):_ SFu fills gaps between cement particles,

enhances rheology, and generates secondary hydration products1. However, it increases viscosity, and alternatives like granulated blast furnace slag and limestone powder are sometimes

used14. Granulated Blast Furnace Slag (GBFS): Substituting cement with GBFS reduces water and superplasticizer requirements14. _Pulverized Fly Ash (PFA):_ When combined with SF or GBFS, PFA

improves performance, reducing hydration heat and shrinkage while lowering superplasticizer needs15,16. Steel Slag Powder (SS): SS improves fluidity and enhances strength when cured at high

temperatures17. _Limestone Powder (LP):_ LP improves fluidity but may lower compressive strength if overused18,19. INERT MINERAL ADMIXTURES Quartz powder (QP), with varying particle sizes,

enhances packing density and mechanical properties in UHPC20. AGGREGATES Quartz sand (QS) provides strength, but natural sands and recycled glass cullet can be cost-effective alternatives,

though they may slightly reduce mechanical properties21,22. SUPERPLASTICIZERS (SP) Polycarboxylates (PCE) are commonly used for their dispersion properties. Effective combinations of

methacrylate- and allyl ether-based PCE improve mix consistency13,23. FIBERS _Steel Fibers (SF):_ Steel fibres improve ductility, impact resistance, and crack control. The size,

distribution, and orientation of fibres significantly influence UHPC’s mechanical properties24. _Polypropylene Fibers (PPF):_ PPF prevents explosive spalling by creating micro-channels that

release vapor pressure at high temperatures25,26. MIX COMPOSITION The mix design for UHPC requires careful selection of raw materials to achieve optimal packing density. Numerical methods

like the Puntke method are used to optimise the water/binder ratio, which should remain between 0.2 and 0.24 to minimise capillary pore formation and drying shrinkage13,27,28. CURING AND

HEAT TREATMENT Curing methods include room temperature curing, heat curing, and autoclave curing. Standard room temperature curing is simple and cost-effective but may lead to weaker C-S-H

chains29. Heat curing at 90 °C for up to 12 days enhances compressive and flexural strength30. Autoclave curing produces the highest strength but is more complex and costly31. HEATING METHOD

Furnaces are used to expose UHPC specimens to elevated temperatures at a rate of 1–5 °C/min, maintaining the target temperature for 30–240 min before cooling25,32,33,34,35,36. SPECIMEN

SHAPE AND SIZE The influence of specimen shape and size on UHPC strength is debated, with some studies showing strength decreases with larger specimens, while others find negligible

effects37,38,39,40,41. UHPC PERFORMANCE AT AMBIENT AND ELEVATED TEMPERATURES At ambient temperature, UHPC has a compressive strength of 150–250 MPa, with a tensile strength of 8 MPa and a

modulus of elasticity of 50–70 GPa3. Exposure to high temperatures increases internal vapor pressure, leading to cracking once pressure exceeds tensile strength2. Studies show a three-stage

behaviour: stabilization up to 400 °C, strength reduction from 400 to 800 °C, and total strength loss above 800 °C42. The stress–strain curve demonstrates a parabolic decline after reaching

peak stress43. Recent research on UHPC at elevated temperatures has explored fibres like basalt and the effects of coarse aggregates and dynamic impact loading, which could enhance

understanding of UHPC’s performance under extreme conditions63,64,65,66,67,68,69. METHODS This study is a analytical modelling of UHPC mechanical properties to predict its behaviour at

elevated temperatures. The methodology of this study is shown below. DATA COLLECTION Existing UHPC experimental mixtures and results are collected online from over 60 accessible scholarly

books/journals/article websites. The number of specimens retrieved for each mechanical property is shown in Table 1. DATA CATEGORISATION, CONVERSION, AND NORMALISATION The gathered data is

categorised based on the mixture composition of each UHPC specimen and associated mechanical properties tested after exposure to elevated temperatures. In general, four specimen categories

are analysed: * a. UHPC without fibres (NF) * b. UHPC with steel fibres (SF) only * c. UHPC with polypropylene fibres (PPF) only * d. UHPC with steel fibres (SF) and polypropylene fibres

(PPF) – hybrid fibre specimens The availability of relevant existing data is the main reason for choosing these four categories. In this research, some mechanical properties have more than

four categories. However, the additional categories are still generated within the scope of the four categories listed above. Those additional categories are chosen from specimens with

different rates of particular fibres. The information on categories for each mechanical property is discussed in Sect. "Results and discussion". As various specimen shapes are used

by researchers, shape and size conversion factors are employed to get equivalent mechanical properties. The mechanical properties applied with shape conversion factors are compressive,

tensile, and flexural strength. According to Chen et al.44, the shape and size of specimens do not influence the modulus of elasticity significantly, therefore, the conversion factor is

negligible. Moreover, the shape and size conversion factors are not used for the stress–strain relationship and peak strain due to unavailability of data found. The standard shape and size

used for each mechanical property are chosen based on their prevalence in the existing research. The shape conversion factors are generally retrieved from various relevant existing research.

However, a few shape conversion factors were retrieved through estimation due to the unavailability of the data. The conversion factors were estimated by considering the shape conversion

factor of the closest size of the specimen. The conversion factor lists are shown in Tables 2, 3, and 4. Then, the converted strength at each temperature is normalised to that at room

temperature. DATA PLOTTING AND ANALYSIS The normalised strength of each specimen for compressive, tensile, flexural strength, modulus of elasticity, and peak strain are grouped according to

their categories and plotted into charts as nodes. These charts consist of mechanical property strength on the Y-axis and temperature on the X-axis. Subsequently, for the stress–strain

relationship, the data retrieved from digitiser software, are plotted to charts as line graphs with stress magnitude in the Y-axis and strain magnitude in the X-axis. Ultimately, the

proposed constitutive relationships are developed on the distribution of these nodes and line graphs. RESULT COMPARISON The proposed equations are compared to the available existing

equations from existing experiments to check their accuracy and congruity to existing data. Ultimately, the result and conclusion of the analytical model are discussed, followed by the

limitations and recommendations for future study. RESULTS AND DISCUSSION This section presents and discusses the categories of each mechanical property and their associated constitutive

relationships. The developed relationships’ applicability beyond the tested conditions, such as in large-scale projects or under dynamic loading, can be extended to numerical analysis and

finite element modelling of UHPC structural elements, where the developed relationships can be used as material property inputs in these analyses and models. COMPRESSIVE STRENGTH The

specimen’s mixtures for compressive strength are categorised into 7 groups. Each category represents the mixtures with different rates of fibres. While elevated temperatures can accelerate

strength gain initially, excessive temperatures can lead to reduced strength and durability due to a combination of incomplete hydration, increased shrinkage, and potential degradation of

the cement paste. These 7 categories are proposed to increase the uniformity of specimen group analysed. More importantly, the available existing relationships for specific mixtures are also

considered in the categorisation. Table 5 shows the categories and their associated number of existing equations. The equation proposed for compressive strength consists of multiple linear

equations for both the strength increase and decrease to reflect the concrete behaviour. The chart and proposed equations for each category are shown and discussed in the following

subsection. 0%NF The normalised strengths at each temperature are plotted in the chart as nodes. Based on the distribution of the nodes, the constitutive relationship is developed.

Initially, linear, polynomial, and exponential regression analyses were conducted to determine the coefficients of the constitutive relationship. However, the R2 values generated are very

low (< 0.4). Therefore, the reliability of these equations is insufficient. Thus, using the trial-and-error method, the coefficient is modified accordingly until the equation’s results

align with the nodes’ distribution. The constitutive relationship for this category is shown in Eq. 1.

$$\frac{{f}_{c}^{T}}{{f{\prime}}_{c}}=\left\{\begin{array}{c}0.99+\left(\frac{T}{1000}\right), 20\,^\circ C\le T\le 200\,^\circ C\\ 1.32-\left(\frac{T}{1000}\right), 200\,^\circ C<T\le

1000\,^\circ C\end{array}\right.$$ (1) The constitutive relationship above is plotted in the chart as a line graph. The nodes’ distributions and the line graph are shown in Fig. 2. Based on

Fig. 2, the proposed constitutive relationship appears to run along the middle point of the normalised strength range at each temperature. The strength increases up to 9% when the

temperature is between room temperature and 200 °C. Then, it decreases to 17% of its original strength as the temperature rises to the maximum tested temperature. No existing equation is

found to compare the proposed constitutive relationship with. >0–1%SF Using the same method as the previous category, the constitutive relationship for this category is shown in Eq. 2.

$$\frac{{f}_{c}^{T}}{{f^{\prime}}_{c}}=\left\{\begin{array}{c}0.99+0.5 x\left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 300\,^\circ C\\ 1.68 -1.8 x\left(\frac{T}{1000}\right), 300\,^\circ

C<T\le 1000\,^\circ C\end{array}\right.$$ (2) The nodes and constitutive relationship for this category are plotted in the chart shown in Fig. 3. In this category, the concrete

compressive strength increases up to 1.14 of its original strength as the temperature rises to 300 °C, followed by a strength slump to less than 0.25 of the initial strength at 800 °C.

Bilinear equations are used to reflect the strength fluctuation due to temperature change. There are four existing relationships created by other researchers to be compared with the proposed

equations of this category. These four existing and proposed relationships are plotted in the same chart and shown in Fig. 4. The proposed equations fit between both extreme values

generated by other researchers (see Fig. 4). The strength fluctuation by the proposed equations differs from most existing equations. However, the proposed equations for this category have a

very similar trend to those proposed by49. >1–2% SF The constitutive relationship of this category is shown in Eq. 3.

$$\frac{{f}_{c}^{T}}{{f^{\prime}}_{c}}=\left\{\begin{array}{c}0.99+0.33 x \left(\frac{T}{1000}\right), 20\,^\circ C\le T\le 300^\circ C\\ 1.6-1.7 x\left(\frac{T}{1000}\right), 300\,^\circ

C<T\le 900\,^\circ C\end{array}\right.$$ (3) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 5. The existing specimens typically show an

increase in compressive strength as the temperature increases to 300 °C (see Fig. 5). Then, the strength drops as the temperature keeps rising. Based on Eq. 3, the specimens’ strength is

estimated to increase to 1.09, before dropping to 0.24 at 900 °C. The comparison of existing and proposed equations is shown in Fig. 6. The compressive strength at each temperature generated

by Eq. 3 is between the extreme strengths proposed by other equations. Equation 3 also generated a very similar result to equations by Gong et al.49. 2–3% SF The constitutive relationship

of this category is shown in Eq. 4. $$\frac{{f}_{c}^{T}}{{f{\prime}}_{c}}=\left\{\begin{array}{c}0.99+0.31 x\left(\frac{T}{1000}\right), 20\,^\circ C\le T\le 300\,^\circ C\\ 1.76-2.25

x\left(\frac{T}{1000}\right), 300\,^\circ C<T\le 600\,^\circ C\\ 0.991-0.97 x\left(\frac{T}{1000}\right), 600\,^\circ C<T\le 900\,^\circ C\end{array}\right.$$ (4) The nodes and

constitutive relationship for this category are plotted in the chart in Fig. 7. From Fig. 7, the proposed equations show an increase in compressive strength to 1.08 as the temperature

increases to 300 °C. Then, the strength drops with two gradients. First, when the temperature rises from 300 to 600 °C with a steeper gradient. The strength drops down to 0.41 initial

strength. The second drop is from 600 to 900 °C with a flatter steepness. The comparison of existing and proposed equations is shown in Fig. 8. Equation 5 shows a similar rate of strength

increase as proposed by Gong et al.49 when the temperature goes up to 300 °C. Moreover, as the temperature increases from 600 to 800 °C, the Eq. 5 shows a more similar trend to the existing

relationship proposed by Zheng et al.50. >0–0.3%PPF The constitutive relationship of this category is shown in Eq. 5. The nodes and constitutive relationship for this category are plotted

in the chart in Fig. 9. The proposed relationship implies that the UHPC specimen’s strength generally increases slightly to 1.04 when the temperature rises to 200 °C. Subsequently, the

strength decreases linearly as the temperature continues to rise. However, the strength estimated by the proposed equations does not align with the equation presented by Zheng et al.43.

$$\frac{{f}_{c}^{T}}{{f{\prime}}_{c}}=\left\{\begin{array}{l}0.99+0.25 x\left(\frac{T}{1000}\right), 20\,^\circ C\le T\le 200\,^\circ C\\ 1.235-0.97 x\left(\frac{T}{1000}\right), 200\,^\circ

C<T\le 900\,^\circ C\end{array}\right.$$ (5) >0.3–2%PPF The constitutive relationship of this category is shown in Eq. 6

$$\frac{{f}_{c}^{T}}{{f{\prime}}_{c}}=\left\{\begin{array}{l}0.98+1 x \left(\frac{T}{1000}\right), 20\,^\circ C\le T\le 300\,^\circ C\\ 1.776-1.65 x \left(\frac{T}{1000}\right), 300\,^\circ

C<T\le 1000\,^\circ C\end{array}\right.$$ (6) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 10. Equation 6 estimates an increase in

compressive strength up to 1.28 at 300 °C before slumping down at the maximum temperature tested. There is no existing equation found for this category. 1–2% SF AND 0.1–0.2%PPF The

constitutive relationship of this category is shown in Eq. 7 $$\frac{{f}_{c}^{T}}{{f{\prime}}_{c}}=\left\{\begin{array}{l}0.99+0.2 x\left(\frac{T}{1000}\right), 20\,^\circ C\le T\le

400\,^\circ C\\ 1.75-1.7x\left(\frac{T}{1000}\right), 400\,^\circ C<T\le 1000\,^\circ C\end{array}\right.$$ (7) The nodes and constitutive relationship for this category are plotted in

the chart in Fig. 11. Equation 7 indicates that the compressive strength of this mixture category will slightly increase to 1.07 at 400 °C. Then, the temperature decreases rapidly to around

20% of the original strength at 900 °C. The comparison of existing and proposed equations is shown in Fig. 12. The proposed relationship and existing relationship by Zheng et al.35 show a

linear strength increase with different gradients when the specimens are heated up to 400 °C. Equation 7 estimates the strength to decrease linearly, while Zheng et al.36 estimate that the

strength will decrease in a polynomial way. However, Huang et al.51 proposed the UHPC strength increases as the temperature rises to 200 °C, followed by two different rates of decrease until

the maximum tested temperature. 0.5–2%SF AND 0.2–2%PPF The constitutive relationship of this category is shown in Eq. 8

$$\frac{{f}_{c}^{T}}{{f{\prime}}_{c}}=\left\{\begin{array}{l}0.977+1.18 x\left(\frac{T}{1000}\right), 20\,^\circ C\le T\le 400\,^\circ C\\ 2.29 -2.1 x\left(\frac{T}{1000}\right), 400\,^\circ

C<T\le 1000\,^\circ C\end{array}\right.$$ (8) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 13. Equation 8 estimates the specimen compressive

strength to increase up to 1.45 of the room temperature strength at 400 °C, then slumps down to less than 0.2 at 1000 °C. There is no existing relationship found for this category. 2.5–3%SF

AND 0.1–1%PPF The constitutive relationship of this category is shown in Eq. 9 $$\frac{{f}_{c}^{T}}{{f{\prime}}_{c}}=\left\{\begin{array}{l}0.99+0.65 x\left(\frac{T}{1000}\right),

20\,^\circ C\le T\le 400\,^\circ C\\ 2.13-2.2 x\left(\frac{T}{1000}\right), 400\,^\circ C<T\le 800\,^\circ C\\ 0.53 -0.2 x\left(\frac{T}{1000}\right), 800\,^\circ C<T\le 1000\,^\circ

C\end{array}\right.$$ (9) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 14. Equation 9 demonstrates the behaviour of specimens in this category

into three stages. The first stage is a strength increase up to 1.25 at 400 °C. The second stage is a significant drop to 0.37 at 800 °C. Lastly the strength decreases with a relatively low

rate to 33% of the original strength at 1000 °C. There is no existing equation found for this category. TENSILE STRENGTH The specimen’s mixtures for tensile strength are grouped into 5

categories. These 5 categories represent different types of fibres used whilst considering the available existing relationships to be compared with. The specimen included in this research

are those tested using splitting tensile test. Table 6 shows the categories and their associated number of existing equations. The equation proposed for tensile strength consists of multiple

linear equations for both the strength increase and decrease to reflect the concrete behaviour. The chart and proposed equations for each category are shown and discussed in the following

subsection. 1.0S%NF The constitutive relationship of this category is shown in Eq. 10 $$\frac{{f}_{t}^{T}}{{f{\prime}}_{t}}=\left\{\begin{array}{l}0.99+0.5x\left(\frac{T}{1000}\right),

20\,^\circ C\le T\le 200\,^\circ C\\ 1.32 -1.15x\left(\frac{T}{1000}\right), 200\,^\circ C<T\le 800\,^\circ C\end{array}\right.$$ (10) The nodes and constitutive relationship for this

category are plotted in the chart in Fig. 15. From the behaviour of relatively low number of specimens for this category, Eq. 10 implies that the tensile strength increases up to 1.09 of the

room temperature’s strength when being heated up to 200 °C. Then the strength drops down to 0.4 at 800 °C. There is no existing equations found for this category. >0–3%SF The

constitutive relationship of this category is shown in Eq. 11 $$\frac{{f}_{t}^{T}}{{f{\prime}}_{t}}=\left\{\begin{array}{l}1+0.25x\left(\frac{T}{1000}\right), 20\,^\circ C\le T\le

400\,^\circ C\\ 1.92-2.05 x\left(\frac{T}{1000}\right), 400\,^\circ C<T\le 1000\,^\circ C\end{array}\right.$$ (11) The nodes and constitutive relationship for this category are plotted in

the chart in Fig. 16. For this category, the tensile strength is estimated to rise 1.1 after heating up to 400 °C. Then, the strength plummets to 0.075 when the temperature increases to 900

 °C. The comparison of the proposed equation to existing ones is shown in Fig. 17. The proposed equations appear to generate a different behaviour compared to the existing equations

proposed. Equation 11 estimates a slight strength increase followed by a strength drops, whereas Zheng et al.43 show a relatively lower rate of increase. However, two other research estimate

no strength increase at all. Moreover, the strength deterioration rate estimated by the proposed equation also does not align with the strength reduced by Zheng et al.43,60. >0–1.2%PPF

The constitutive relationship of this category is shown in Eq. 12 $$\frac{{f}_{t}^{T}}{{f{\prime}}_{t}}=\left\{\begin{array}{l}1.02+0.55x\left(\frac{T}{1000}\right), 20\,^\circ C\le T\le

300\,^\circ C\\ 1.134-1.12 x\left(\frac{T}{1000}\right), 300\,^\circ C<T\le 900\,^\circ C\end{array}\right.$$ (12) The nodes and constitutive relationship for this category are plotted in

the chart in Fig. 18. The proposed relationship implies that the UHPC specimen’s tensile strength generally increases slightly to 1.13 when the temperature rises to 300 °C. Then, the

strength decreases to 0.04 linearly as the temperature reaches 900 °C. However, the strength fluctuation estimated by the proposed equations is not in agreement with the equation presented

by Ju et al.33. >1–2%SF AND 0.05–0.2%PPF The constitutive relationship of this category is shown in Eq. 13

$$\frac{{f}_{t}^{T}}{{f{\prime}}_{t}}=\left\{\begin{array}{l}1.022-0.7x\left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 200\,^\circ C\\ 1.132-1.25 x \left(\frac{T}{1000}\right),

200\,^\circ C<T\le 900\,^\circ C\end{array}\right.$$ (13) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 19. Based on the existing data, Eq. 13

estimates that the tensile strength of this UHPC’s category will drop at 2 different rates. The first drop from the room temperature to 200 °C is slightly lighter, with the strength reduced

to 0.88 × the original strength. Subsequently, the strength drops with a steeper gradient to 0.07 at 900 °C. This proposed relationship does not completely align with the equation by

Banerji and Kodur52, but both generate similar strength estimation at 600–750 °C. 0.5–3%SF AND 0.03–0.75%PPF The constitutive relationship of this category is shown in Eq. 14

$$\frac{{f}_{t}^{T}}{{f{\prime}}_{t}}=\left\{\begin{array}{l}1.002-0.1x\left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 200\,^\circ C\\ 1.188 -1.1 x\left(\frac{T}{1000}\right), 200\,^\circ

C<T\le 1000\,^\circ C\end{array}\right.$$ (14) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 20. The proposed relationship implies that the

UHPC specimen’s strength for this category will drop at 2 different rates. It starts with a slight drop from room temperature to 200 °C, reducing the strength to 0.98 the original strength.

Subsequently, the strength will slump to 0.14 at 1000 °C. There are no existing equations found for this category. FLEXURAL STRENGTH The specimen’s mixtures for flexural strength are grouped

into 6 categories. Table 7 shows the categories and their associated number of existing equations. The equation proposed for UHPC’s flexural strength consists of multiple linear equations

for both the strength increase and decrease to reflect the concrete behaviour. The chart and proposed equations for each category are shown and discussed in the following. 0%NF The

constitutive relationship of this category is shown in Eq. 15 $$\frac{{f}_{f}^{T}}{{f{\prime}}_{f}}=\left\{\begin{array}{l}1.005-0.22x \left(\frac{T}{1000}\right), 20\,^\circ C \le T \le

200\,^\circ C\\ 1.225 -1.3 x \left(\frac{T}{1000}\right), 200\,^\circ C<T \le 900\,^\circ C\end{array}\right.$$ (15) The nodes and constitutive relationship for this category are plotted

in the chart in Fig. 21. Equation 15 estimates that the flexural strength of this UHPC category drops in 2 different gradients as the temperature rises. It starts with a slight drop from

room temperature to 200 °C, reducing the strength to 0.97 the original. Then, the strength will slump to 0.06 at 1000 °C. There are no existing equations found for this category. >0–3%SF

The constitutive relationship of this category is shown in Eq. 16. $$\frac{{f}_{f}^{T}}{{f{\prime}}_{f}}=\left\{\begin{array}{l}1+0.15x \left(\frac{T}{1000}\right), 20\,^\circ C \le T \le

200\,^\circ C\\ 1.3 -1.35 x \left(\frac{T}{1000}\right), 200\,^\circ C<T \le 900\,^\circ C\end{array}\right.$$ (16) The nodes and constitutive relationship for this category are plotted

in the chart in Fig. 22. The proposed relationships estimate that the UHPC specimen’s flexural strength will slightly increase to 1.03 when the temperature rises to 200 °C. Then, the

strength plummets to 0.09 linearly as the temperature reaches 900 °C. The strength fluctuation generated by the proposed equations shows a similar trend as the strength as the equation by

Zheng et al.43. The strength initially increases up to 200 °C before decreasing until failure. The rate of strength predicted at 300–600 °C is also similar. >0–1.2%PPF The constitutive

relationship of this category is shown in Eq. 17. $$\frac{{f}_{f}^{T}}{{f{\prime}}_{f}}=\left\{\begin{array}{l}0.992+0.45x \left(\frac{T}{1000}\right), 20\,^\circ C \le T \le 300\,^\circ C\\

1.682 -1.85 x \left(\frac{T}{1000}\right), 300\,^\circ C<T \le 900\,^\circ C\end{array}\right.$$ (17) The nodes and constitutive relationship for this category are plotted in the chart

in Fig. 23. The proposed relationships estimate that the UHPC specimen’s flexural strength will increase to 1.13 when the temperature rises to 300 °C. Then, the strength plummets to 0.02 as

the temperature reaches 900 °C. The Eq. 17 does not appear to be aligned with the equations proposed by Zheng et al.35. The only similarity between Eq. 17 to the existing equation is a

strength increase from room temperature to 300 °C. 1–2%SF AND 0.1–0.2%PPF The constitutive relationship of this category is shown in Eq. 18

$$\frac{{f}_{f}^{T}}{{f{\prime}}_{f}}=\left\{\begin{array}{l}1.02-0.55x \left(\frac{T}{1000}\right), 20\,^\circ C \le T \le 200\,^\circ C\\ 1.134 -1.12 x \left(\frac{T}{1000}\right),

200\,^\circ C<T\le 900\,^\circ C\end{array}\right.$$ (18) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 24. Equation 18 estimates that the

flexural strength of this UHPC category drops in 2 different gradients as the temperature rises. It starts with a slight drop from room temperature to 200 °C, reducing the strength to 0.91

the original. Then, the strength drops at a relatively steeper rate to 0.13 at 900 °C. The proposed strength drop rate shows a different strength shift trend to the equation proposed by Li

and Liu53. However, both equations appear to have a similar strength when the temperature is around 500 °C. 0._5–2%SF AND 0.25–0.8%PPF_ The constitutive relationship of this category is

shown in Eq. 19 $$\frac{{f}_{f}^{T}}{{f{\prime}}_{f}}=\left\{\begin{array}{l}1.011-0.53x \left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 700\,^\circ C\\ 1.86-1.75 x

\left(\frac{T}{1000}\right), 700\,^\circ C<T\le 1000\,^\circ C\end{array}\right.$$ (19) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 25.

Equation 19 estimates that the flexural strength of this UHPC category drops in 2 different gradients as the temperature rises. It starts with a strength drop from room temperature to 500 

°C, reducing the strength to 0.75 the original. Then, the strength slumps to 0.12 at 1000 °C. There are no existing equations found for this category. 3% SF AND 0.1–1%PPF The constitutive

relationship of this category is shown in Eq. 20 $$\frac{{f}_{f}^{T}}{{f{\prime}}_{f}}=\left\{\begin{array}{l}0.99+0.31 x \left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 400\,^\circ C\\

1.76-2.25 x \left(\frac{T}{1000}\right), 400\,^\circ C< T\le 800\,^\circ C\\ 0.991-0.97 x \left(\frac{T}{1000}\right), 800\,^\circ C< T\le 1000\,^\circ C\end{array}\right.$$ (20) The

nodes and constitutive relationship for this category are plotted in the chart in Fig. 26. Equation 20 estimates that the flexural strength of this UHPC category drops at 3 different rates

as the temperature rises. The first drop rate occurs from room temperature to 500 °C, reducing the strength to 0.83 the original. Then, the strength slumps to 0.54 at 600 °C with the

steepest rate compared to the other rates of this category. Ultimately, the strength drops with a flatter gradient to 0.22 at 1000 °C. There are no existing equations found for this

category. MODULUS OF ELASTICITY The specimen’s mixtures for modulus of elasticity are grouped into 4 categories. Table 8 shows the categories and their associated number of existing

equations. The equations proposed for UHPC’s modulus of elasticity consist of a combination of linear and exponential equations. The chart and proposed equations for each category are shown

and discussed in the following. 0%NF The constitutive relationship of this category is shown in Eq. 21. $$\frac{{E}_{c}^{T}}{{E}_{o}}=\left\{\begin{array}{l}1.023-1.13x

\left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 200\,^\circ C\\ 1.07-1.5 \left(\frac{T}{1000}\right)+0.7 x{\left(\frac{T}{1000}\right)}^{2}, 200\,^\circ C<T\le 1000\,^\circ

C\end{array}\right.$$ (21) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 27. Equation 21 estimates that the MOE of this UHPC category drops in 2

different gradients as the temperature rises. At first, the MOE decreases linearly from room temperature to 200 °C, reducing its value to 0.8 of the room temperature value. Then, the

strength slumps exponentially to 0.27 at 1000 °C. There are no existing equations found for this category. >0–3%SF The constitutive relationship of this category is shown in Eq. 22

$$\frac{{E}_{c}^{T}}{{E}_{o}}=\left\{\begin{array}{l}1.02-0.96x \left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 300\,^\circ C\\ 2.26-6.58 \left(\frac{T}{1000}\right)5 x

{\left(\frac{T}{1000}\right)}^{2}, 300\,^\circ C<T\le 500\,^\circ C\\ 0.62-1.15 \left(\frac{T}{1000}\right)+0.6 x {\left(\frac{T}{1000}\right)}^{2}, 500\,^\circ C<T\le 1000\,^\circ

C\end{array}\right.$$ (22) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 28. Equation 21 estimates that the modulus of elasticity of this UHPC

category drops at 3 different gradients as the temperature rises. The first gradient is a linear drop from room temperature to 300 °C, reducing its value to 0.73 of the room temperature

value. Next, the value plummets exponentially to 0.22 at 500 °C. Finally, the MOE decreases exponentially with a relatively flat gradient to 0.07 at 1000 °C. The comparison of the proposed

equations to the existing ones is shown in Fig. 29. Equation 21 shows a relatively similar result to the equations proposed by Tai et al.54 from room temperature to around 300 °C compared to

the other existing equations. However, when the temperature is about 500–700 °C, equations by Zheng et al.50 also appear to align with both Eq. 21 and Tai et al.54. Then, when the

temperature is around 800 °C, the existing equation by Way and Wille55 shows a similar result to Eq. 21. >0–0.3%PPF The constitutive relationship of this category is shown in Eq. 23

$$\frac{{E}_{c}^{T}}{{E}_{o}}=\left\{\begin{array}{l}1.023-1.13x \left(\frac{T}{1000}\right), 20\,^\circ C\le T\le 200\,^\circ C\\ 1.07-1.5 \left(\frac{T}{1000}\right)+0.7 x

{\left(\frac{T}{1000}\right)}^{2}, 200\,^\circ C<T\le 1000\,^\circ C\end{array}\right.$$ (23) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 

30. Equation 23 estimates that the modulus of elasticity of this UHPC category drops at 2 different gradients as the temperature rises. At first, the modulus of elasticity decreases linearly

from room temperature to 200 °C, reducing its value to 0.91 of the room temperature value. Then, the strength slumps exponentially to 0.28 at 1000 °C. There are no existing equations found

for this category. _>1–2%SF_ > _0.1–0.2%PPF_ The constitutive relationship of this category is shown in Eq. 24 $$\frac{{E}_{c}^{T}}{{E}_{o}}=\left\{\begin{array}{l}1.05-2.4x

\left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 200\,^\circ C\\ 0.875-1.7 \left(\frac{T}{1000}\right)+0.85 x {\left(\frac{T}{1000}\right)}^{2}, 200\,^\circ C<T\le 1000\,^\circ

C\end{array}\right.$$ (24) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 31. Equation 24 estimates that the modulus of elasticity of this

category drops at 2 different rates as the temperature rises. At first, the modulus of elasticity decreases significantly from room temperature to 200 °C, reducing its value to 0.57 of the

room temperature value. Then, the strength slumps exponentially to 0.04 at 1000 °C. The comparison of Eq. 24 to the existing ones is shown in Fig. 32. The Eq. 24 shows a relatively high

similarity to the equation proposed by Banerji and Kodur52. Both show a MOE value reduction during exposure to increasing temperature. This behaviour is different to the estimation proposed

by Li56, as they predict that the MOE value will increase slightly when the temperature rises to 200 °C. Ultimately, all equations appear to have high similarity in MOE value estimation at

500–750 °C. PEAK STRAIN The specimen mixtures for peak strain are grouped into four categories. Table 9 shows these categories along with the number of existing equations associated with

each. The equations proposed for the peak strain of UHPC consist of both linear and exponential forms. The chart and proposed equations for each category are presented and discussed below.

0%NF The constitutive relationship of this category is shown in Eq. 25 $$\frac{{\varepsilon }_{c}^{T}}{{\varepsilon }_{o}}=\left\{\begin{array}{l}0.993+0.34 x \left(\frac{T}{1000}\right),

20\,^\circ C \le T\le 400\,^\circ C\\ 0.65+1.795-1.5 x {\left(\frac{T}{1000}\right)}^{2}, 400\,^\circ C<T\le 1000\,^\circ C\end{array}\right.$$ (25) The nodes and constitutive

relationship for this category are plotted in the chart in Fig. 33. Using Eq. 25, the peak strain of this UHPC category changes at two different rates as the temperature rises. Initially,

the peak strain increases linearly from room temperature to 400 °C, reaching 1.13 times the room temperature value. Then, it increases exponentially to a maximum of 1.19 at 600 °C before

diminishing to 0.95 at 1000 °C. No existing equations for this category have been found. >0–3%SF The constitutive relationship of this category is shown in Eq. 26 $$\frac{{\varepsilon

}_{c}^{T}}{{\varepsilon }_{o}}=\left\{\begin{array}{l}0.99+0.9x \left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 300\,^\circ C\\ -0.55+6 \left(\frac{T}{1000}\right), 300\,^\circ C<T\le

600\,^\circ C\\ 3.25+1.56 \left(\frac{T}{1000}\right)-3.05 x {\left(\frac{T}{1000}\right)}^{2}, 600\,^\circ C<T\le 1000\,^\circ C\end{array}\right.$$ (26) The nodes and constitutive

relationship for this category are plotted in the chart in Fig. 34. Equation 26 estimates that the peak strain of this UHPC category change at 3 different rates as the temperature rises. The

first shift is when the peak strain increases linearly to 1.26 of the value in room temperature at 300 °C. Then, the peak strain linearly with a steeper gradient to 3.05 at 600 °C. As the

temperature gets hotter than 600 °C, the value drops exponentially to 2.18 at 900 °C. The proposed equations of this category show a generally similar peak strain change trend to the

equations proposed by Tai et al.54. The values generated by these equations are very close particularly at around 200–400 °C and 700–800 °C. However, as the temperature is within 400–700 °C,

the proposed equations estimate a relatively higher increase compared to the values estimate by Zheng et al.50 and Tai et al.54. The comparison of Eq. 26 to the existing ones is shown in

Fig. 35. >0–0.3%PPF The constitutive relationship of this category is shown in Eq. 27 $$\frac{{\varepsilon }_{c}^{T}}{{\varepsilon }_{o}}=0.925+3.85 x \left(\frac{T}{1000}\right)-1.7 x

{\left(\frac{T}{1000}\right)}^{2}, 20\,^\circ C \le T\le 1000\,^\circ C$$ (27) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 36. The proposed

equations of this category show a generally similar peak strain change trend to the equations proposed by Tai et al.54. The values generated by these equations are very close particularly at

around 200–400 °C and 700–800 °C. However, as the temperature is within 400–700 °C, the proposed equations estimate a relatively higher increase compared to the values estimate by Zheng et

al.50 and Tai et al.54. _>1–2% SF AND_ > _0.1–0.2% PPF_ The constitutive relationship of this category is shown in Eq. 28 $$\frac{{\varepsilon }_{c}^{T}}{{\varepsilon

}_{o}}=\left\{\begin{array}{l}0.95+2.3x \left(\frac{T}{1000}\right), 20\,^\circ C \le T\le 300\,^\circ C\\ -1.35+11.1 x \left(\frac{T}{1000}\right)-4 x {\left(\frac{T}{1000}\right)}^{2},

200\,^\circ C<T\le 1000\,^\circ C\end{array}\right.$$ (28) The nodes and constitutive relationship for this category are plotted in the chart in Fig. 37. Equation 28 estimates that the

peak strain of this category increases at 2 different rates as the temperature rises. At first, the peak strain value increases linearly from room temperature to 300 °C, rising its value to

1.64 of the room temperature value. Then, the strength escalates exponentially to 5.4 at 1000 °C. The comparison of Eq. 28 to the existing ones is shown in Fig. 38. The Eq. 28 estimates peak

strain value with a high similarity at room temperature to 300 °C to the value Abid et al. (2019) estimated. Moreover, the value generated by Eq. 28 appears to align with the equation

proposed by Banerji and Kodur52 at 400–550 °C. However, as the temperature increases over 500 °C, the proposed equation estimates a value within both existing equations. STRESS–STRAIN

RELATIONSHIP (SVS) The specimen’s mixtures for stress–strain relationship are grouped into 4 categories. Table 10 shows the categories and their associated number of existing equations. The

equations proposed for the stress–strain relationship of the four categories above are adopted from the main equations proposed by Zheng et al.35 shown in Eq. 29. This equation proposed by

Zheng et al.36 was originally for UHPC specimens with > 0–3%SF. Table 11 shows the value of α and β used in the original equation. In this research, the variable α and β in Eq. 29 are

modified accordingly for each category and temperature to reasonably match the existing svs data. The chart and variables α and β for each category are discussed in the following.

$$y=\left\{\begin{array}{c}\alpha x+\left(5-4\alpha \right){x}^{4}+\left(3\alpha -4\right){x}^{5}, 0 \le x\le 1\\ \frac{x}{\beta {(x-1)}^{2}+x} , x\ge 1\end{array}\right.$$ (29) \(x=

\frac{\varepsilon }{{\varepsilon }_{c,T}}\) and \(y= \frac{\sigma }{{f}_{c,T}}\) Where, ε = specimen strain (μm) εc,T = specimen peak strain at associated temperature (μm) σ = concrete

stress (MPa) fc,T = concrete maximum compressive strength (MPa) α = independent variables β = independent variables 0%NF The proposed α and β values at various temperatures for this category

are shown in Table 12. There is no value suggested for temperature 300, 500, 700, and 900 °C due to unavailability of existing data to be analysed. The typical charts showing existing svs

curves and proposed relationship curves for each temperature are shown by Figs. 39, 40 and 41. Based on the plots of the proposed relationships shown in the figures above, the proposed

relationships appear to be reasonably accurate in representing the existing experimental data. However, there are a few relatively large discrepancies between the proposed equation and the

results produced by Zdeb et al.57 at 200 °C and by Hager et al.58 at 800 °C. >0–0.3%PPF The proposed α and β values at various temperatures for this category are shown in Table 13. The

typical charts showing existing svs curves and proposed relationship curves for each temperature are shown by Figs. 42, 43, 44 and 45. Based on the plots of the proposed relationship shown

in figures above, the proposed relationships appear to be reasonably accurate in representing the existing experimental data. However, there are relatively large discrepancies shown by the

proposed equation to the experimental result by59. >0–3%SF The proposed α and β values at various temperatures for this category are shown in Table 14. The typical charts showing existing

svs curves and proposed relationship curves for each temperature are shown by Figs. 46, 47, 48, 49, 50 and 51. Based on the plots of the proposed relationship shown in figures above, the

proposed relationships appear to be reasonably accurate in representing the majority of the existing experimental data. However, due to the large range of data in this category, the proposed

relationships also generate a relatively significant discrepancy to the data at every temperature analysed. _>1–2%SF AND_ > _0.1–0.2%PPF_ The proposed α and β values at various

temperatures for this category are shown in Table 15. There is no value suggested for temperature of 800 °C due to unavailability of existing data. Figures 52, 53, 54, 55, 56 and 57 display

the charts showing existing svs curves and the proposed relationship curves for each temperature. The proposed relationships in this category demonstrate lower accuracy compared to the other

categories. The proposed relationship exhibits relatively higher similarity to the results by Sarwar61 and the least similarity to the results by Abid et al.59. CONCLUSION The following

conclusions can be drawn from this research: DETERIORATION OF MECHANICAL PROPERTIES UNDER HIGH TEMPERATURES * The UHPC’s mechanical properties degrade with increasing temperature. * Residual

compressive strength exhibits a bilinear or trilinear trend, characterised by an initial linear increase in strength: * o 20 °C to 200 °C for NF (non-fibrous) and PPF (polypropylene fibre)

specimens. * o 20 °C to 300 °C for SF (steel fibre) specimens. * o 20 °C to 400 °C for hybrid fibre specimens. * This is followed by one or two linear decreases as the temperature continues

to rise, eventually leading to near-complete loss of strength. * Proposed models for SF and hybrid fibre specimens align well with existing equations, demonstrating consistency in residual

compressive strength trends. CONSTITUTIVE RELATIONSHIPS FOR TENSILE AND FLEXURAL STRENGTH * Tensile strength is represented by bilinear relationships, while flexural strength follows

bilinear or trilinear trends depending on specimen type. * For the modulus of elasticity and peak strain, the experimental results exhibit trends that combine linear and exponential

behaviours, which are effectively captured in the proposed relationships. COMPRESSIVE STRESS–STRAIN RELATIONSHIP * The proposed compressive stress–strain model, adapted from the Zheng et

al.35,36 equations, incorporates modified α and β parameters to reflect observed behaviour. * The modified model demonstrates good agreement with most experimental datasets across varying

temperatures but does not fully capture all experimental variations, highlighting areas for refinement. OVERALL ACCURACY OF PROPOSED RELATIONSHIPS * Proposed equations for compressive

strength, tensile strength, flexural strength, peak strain, and modulus of elasticity show strong alignment with available experimental data and existing models. * However, discrepancies are

more pronounced for PPF specimens, particularly in tensile and flexural strength predictions, compared to SF and hybrid fibre specimens. * Compressive strength equations are the most

reliable, while those for tensile strength, flexural strength, modulus of elasticity, and peak strain require further validation due to limited experimental datasets. RECOMMENDATIONS FOR

FUTURE RESEARCH * Expanded experimental studies on UHPC exposed to high temperatures are critical, focusing on its initial compressive, tensile, and flexural strength effects on

post-exposure behaviour. * Additional testing and refinement of constitutive models are needed to improve the accuracy of tensile, flexural, modulus of elasticity, and stress–strain

relationships for PPF and hybrid fibre-based UHPC. * These efforts will enhance understanding of UHPC’s mechanical performance and enable the development of more robust and universally

applicable predictive models. DATA AVAILABILITY The datasets used and/or analysed during the current study available from the corresponding author on reasonable request. REFERENCES *

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references AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Materials and Structures Innovation Group, School of Engineering, The University of Western Australia, Perth, WA, Australia Rodrick

Passion Simanjuntak & Farhad Aslani Authors * Rodrick Passion Simanjuntak View author publications You can also search for this author inPubMed Google Scholar * Farhad Aslani View author

publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS Rodrick Passion Simanjuntak: Formal analysis; Investigation; Methodology; Software; Validation;

Visualization; Roles/Writing—original draft. Farhad Aslani: Funding acquisition; Supervision; Project administration; Resources; Conceptualization; Methodology; Data curation; Investigation;

Visualization; Writing—review & editing. CORRESPONDING AUTHOR Correspondence to Farhad Aslani. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests.

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copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/. Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Simanjuntak, R.P., Aslani, F. Constitutive

relationships for ultra-high-performance concrete at elevated temperatures. _Sci Rep_ 15, 4957 (2025). https://doi.org/10.1038/s41598-025-88788-6 Download citation * Received: 17 July 2024 *

Accepted: 30 January 2025 * Published: 10 February 2025 * DOI: https://doi.org/10.1038/s41598-025-88788-6 SHARE THIS ARTICLE Anyone you share the following link with will be able to read

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KEYWORDS * Ultra-high-performance concrete * Elevated temperatures * Mechanical properties * Constitutive relationships