A Unified Equation for Prediction of Concrete Strength at Various Ages Using the Ultrasonic Pulse Velocity

Abstract

Using ultrasonic pulse velocity (UPV) testing has long been regarded as an effective approach for evaluating concrete strength; however, because of the influence of numerous factors (e.g., mixture proportion, type of coarse aggregate, pozzolanic admixture, and age of the concrete), the relationship between UPV and concrete strength cannot be conclusively determined, thus hindering the application of this approach. The objective of this paper is to develop a unified equation for prediction of concrete strength at various ages using the ultrasonic pulse velocity. Firstly, this study investigated the relationship between the UPV evolution index and the strength evolution index of normal concrete with various mixture proportions, coarse aggregate types, and ages. Subsequently, the influence of adding pozzolanic materials on this relationship was considered. The experimental results showed that excellent correlation between the UPV evolution index and the strength evolution index was found and a unified equation was established in this study through regression analysis to represent the relationship between the UPV and the strength evolution index. Finally, the applicability of the unified equation was verified by data obtained from different studies. The results indicated that for all concrete specimens more than 3 days old, the error of the strength estimation decreased to within ±15%, demonstrating that this unified equation has substantial value for application in decision-making processes for major construction procedures (e.g., applying post tensioning or removing formwork supports).

1. Introduction

The use of ultrasonic pulse velocity (UPV) to evaluate the strength of concrete has always been a goal for scholars of nondestructive testing. Various long-term studies [1,2,3,4,5,6,7,8,9,10,11] have indicated that the volume distribution proportion of cementitious materials (i.e., cement and pozzolanic material) and filler materials (coarse and fine aggregates) influences the relationship between the UPV and strength of concrete. Furthermore, various types of coarse aggregates also affect the relationship between UPV and concrete strength [8,12]. Many researchers proposed various equations to describe the UPV-strength relationship [13,14,15,16], but the uncertainty of the concrete strength estimated by the proposed equations is still high. Therefore, in order to have a relatively small variation in estimating concrete strength using UPV, a relationship between UPV and strength must be established for the concrete having the same mixture proportion and aggregate type; it can then serve as the basis for quantitative assessment of concrete strength. Until now, because the relationship between UPV and strength of concrete remains unclear and cannot be systematically inferable, bottlenecks remain in the application of UPV for estimating concrete strength.

In their research on the relationship between UPV and strength of concrete, Lin et al. [8] indicated that, when concrete is viewed as a two-phase material including mortar and coarse aggregate, then, for a specific cement paste volume and single-type coarse aggregate (i.e., with a similar UPV), the relationship between the UPV and strength of hardened concrete can be simplified as subject to only the influence of coarse aggregate content. However, this relationship does not apply to early-age concrete, primarily because considerable differences can be observed in concrete with various water-to-cementitious-material ratios in terms of UPV and strength growth rate. Coarse aggregate type is another unneglectable factor because different aggregate types can substantially affect the UPV of concrete but exert a nonsignificant influence on the strength of normal concrete. Comprehensively considering the influences of concrete mixture proportion, age, and pozzolanic material content can render the relationship between concrete UPV and strength rather complicated, which is why the application of UPV nondestructive testing methods in estimating concrete strength has remained limited. In particular, concrete at an early age exhibits a high level of hydration, further complicating the investigation of the relationship between UPV and concrete strength [17,18]. A multivariate model treating UPV and curing age as variables was proposed and used to estimate the strength of concrete at various ages and the results showed that the corresponding 95% prediction limits were determined as ±11.8 MPa [19].

In this study, the testing results of the UPV and strength of concrete that had set for various ages and had different water-to-cementitious material ratios (w/cem), paste content, pozzolanic material content, aggregate content, and coarse aggregate type were compiled and analyzed to obtain a simplified and functional relationship between the UPV and strength development of concrete at different ages. Based on the idea that differences in mixture proportion and material type are eventually reflected in the final properties of hardened concrete, this study proposes a method for characterizing the relationship between the evolution index of UPV and strength in an attempt to clarify the complex relationship between UPV and strength in concrete. This study aims to establish a unified equation to represent the relationship between the UPV and the strength evolution index. The testing results of concrete with various mixture proportions were collected for strength estimation to verify the feasibility of using the unified equation to perform quantified evaluation of concrete strength at different ages.

2. Research Design and Testing Plan

2.1. Research Design

Concrete is composed of water, cementitious materials, and aggregates. At an early age, concrete exhibits high levels of hydration, and its UPV and strength positively correlate with the hydration of cementitious materials. By comparison, during the entire hydration process, the physical and chemical characteristics of aggregates do not change by age. Overall, variations in concrete UPV and strength during the hydration process are mainly affected by the water-to-cementitious-material ratio and the addition ratio of the pozzolanic material. Therefore, this research originally attributed increases in UPV and strength to the hydration level of cementitious materials. Moreover, the aggregate type and the mixture proportion primarily served as the reference for determining the final UPV and strength of the concrete. On the basis of these ideas, the concept of a relationship curve between the UPV and the strength evolution index was proposed [20]. By dividing the UPV (V_n) and strength (S_n) of concrete at various ages by the terminal UPV (V_t) and strength (S_t) after hardening, respectively, the evolution indexes V_g (Equation (1)) and S_g (Equation (2)) could be obtained.

$$V_g=\frac{V_n}{V_t}·100\%$$

(1)

$$S_g=\frac{S_n}{S_t}·100\%$$

(2)

The terminal UPV and strength of concrete may vary with variations in mixture proportions and they are used as normalized baselines in this study.

2.2. Testing Plan

Portland cement Type I served as the primary cementitious material, river sand with a unit weight of 2.62 ± 0.1 kg/cm³ was adopted as the fine aggregate, and crushed stone with a unit weight of 2.60 ± 0.1 kg/cm³ was used as the coarse aggregate. All aggregates were washed before being mixed. Test parameters included mixture proportion and pozzolanic material admixture; an appropriate amount of superplasticizer was also added to sustain workability. Cylindrical concrete specimens as shown in Figure 1 with dimensions of 10 cm (D) × 20 cm (H) were used in this study. They were all placed in water at 23 °C for curing and taken out for measurement. Excess moisture on the surface was wiped away using a dry cloth to ensure that the specimens were dry outside but saturated inside (ASTM C128 [21]). ASTM C597 [22] and ASTM C39 [23] methods were then employed to obtain the UPV and compressive strength of the concrete specimens, respectively.

Through a direct transmission mode, UPV was measured by a commercially available pulse meter with an associated transducer pair. The transducer pair had a nominal frequency of 54 kHz, which was relatively low compared to other ultrasonic applications and suitable for concrete. The principle of ultrasonic pulse velocity measurement involves sending a wave pulse into concrete and measuring the travel time for the pulse to propagate through the concrete. Knowing the path length, the measured travel time can be used to calculate the pulse velocity.

In this study, concrete that had a volume fraction of cement paste of 36% (V_paste) and had a sand-to-aggregate ratio (S/A) of 45% was used as the main research subject. The S/A refers to the percentage of fine aggregate in relation to the total aggregate weight. Variations were then introduced to this mixture proportion to investigate the effect of various factors on the relationship between UPV and the strength evolution index.

Table 1. Mixture proportions adopted to establish the reference relationship curve between UPV and the strength evolution index.

No	w/cem	S/A (%)	Mixture Proportion, kg/m3
			Vpaste (%)	Water	Cement	Super-Plasticizer	Fine Aggregate	Coarse Aggregate (Type 1)
NC1	0.3	45	36	164	583	10.90	737	908
NC2	0.4			193	502	7.25
NC3	0.5			218	440	2.66
NC4	0.6			234	392	1.00
NC5	0.7			248	354	0.00

presents the mixture proportions of concrete specimens that comprised normal Portland cement (V_paste = 36%, S/A = 45%) and had water-to-cementitious-material ratios (w/cem) ranging from 0.3–0.7. The specimens were numbered NC1 to NC5 to serve as a reference for establishing the relationship between UPV and the strength evolution index. Subsequently, to clarify the effect of various coarse-aggregate-to-fine-aggregate content proportions under a fixed paste volume ratio, the concrete mixture proportions listed in

Table 2. Mixture proportions of various levels of coarse aggregate content in normal concrete.

No.	w/cem	S/A (%)	Vpaste (%)	Mixture Proportion, kg/m3
				Water	Cement	Super-Plasticizer	Fine Aggregate	Coarse Aggregate (Type 2)
NC6	0.3	30	36	169	578	8.68	501	1165
NC7	0.4			197	500	4.99
NC8	0.5			219	440	2.20
NC9	0.6			235	392	0.98
NC10	0.7			248	354	0.00
NC11	0.3	45		169	578	8.68	752	915
NC12	0.4			197	500	4.99
NC13	0.5			219	440	2.20
NC14	0.6			235	392	0.98
NC15	0.7			248	354	0.00
NC16	0.3	60		169	578	8.68	1002	666
NC17	0.4			197	500	4.99
NC18	0.5			219	440	2.20
NC19	0.6			235	392	0.98
NC20	0.7			248	354	0.00

were adopted. Specimens numbered NC6–NC20 had w/cem ranging from 0.3–0.7, a fixed V_paste of 36%, and S/A of 30, 45 and 60%.

Table 3. Mixture proportions of various levels of fly ash concrete paste content.

NO.	w/cem	S/A (%)	Mixture Proportion, kg/m3
			Vpaste (%)	Water	Cement	Fly-Ash	Super-Plasticizer	Fine Aggregate	Coarse Aggregate (Type 1)
FC1	0.3	45	36	164	496	87	10.9	737	908
FC2	0.4			193	427	75	7.25
FC3	0.5			218	374	66	2.66
FC4	0.6			234	334	59	1.00
FC5	0.7			248	301	53	0.00
FC6	0.3		42	191	578	102	12.72	668	823
FC7	0.4			228	497	88	6.47
FC8	0.5			254	437	77	3.06
FC9	0.6			274	389	69	1.17
FC10	0.7			289	351	62	0.00

and

Table 4. Mixture proportions of various levels of slag concrete paste content.

NO.	w/cem	S/A (%)	Mixture Proportion, kg/m3
			Vpaste (%)	Water	Cement	Slag	Super-Plasticizer	Fine Aggregate	Coarse Aggregate (Type 2)
SC1	0.3	45	36	165	408	175	8.98	740	912
SC2	0.4			196	351	151	4.57
SC3	0.5			218	308	132	2.20
SC4	0.6			234	275	118	0.98
SC5	0.7			248	248	106	0.00
SC6	0.3		42	179	476	204	10.5	676	826
SC7	0.4			213	410	176	5.33
SC8	0.5			239	360	154	2.56
SC9	0.6			258	320	137	1.41
SC10	0.7			273	289	124	0.00

present the concrete mixture proportions containing pozzolanic materials. As indicated in Table 3, fly ash replaces cement at a ratio of 15% and two V_paste, 36 and 42%, were adopted; the specimens were labeled FC1–FC5. As seen in Table 4, slag replaces cement at a ratio of 30% and the two V_paste of 36 and 42% were adopted; the specimens were labeled SC1–SC10. The mixture proportions in Table 3 and Table 4 were used to investigate the effects of differences in the addition of pozzolanic materials and variation in the paste volume ratio on the relationship between UPV and the strength evolution index.

Two types of coarse aggregate (i.e., Type 1 and Type 2) were used in this study. Specimens listed in Table 1 (NC1–NC5) and Table 3 (FC1–FC10) adopted Type 1 coarse aggregate, whereas specimens in Table 2 (NC6–NC20) and Table 4 (SC1–SC10) adopted Type 2 coarse aggregate. Type 1 coarse aggregate had a specific weight of 2.60 and a UPV of 5175 m/s, whereas Type 2 coarse aggregate had a specific weight of 2.59 and a UPV of 5361 m/s.

The UPV and strength of three concrete cylinders were measured at all ages. The measurement results of the three specimens were then averaged to obtain the UPV and strength data of concrete at specific ages.

Table 5. The measured UPV and strength data for normal concrete.

No.	Day 1		Day 3		Day 7		Day 14		Day 21		Day 28
	S *	V **	S	V	S	V	S	V	S	V	S	V
NC1	44	4342	66	4549	69	4614	71	4651	71	4706	74	4726
NC2	25	4091	45	4483	48	4573	63	4627	62	4662	63	4691
NC3	12	3869	25	4345	38	4499	43	4585	46	4622	46	4615
NC4	7	3586	19	4196	24	4362	32	4451	35	4529	36	4528
NC5	4	3248	12	4033	18	4224	20	4344	22	4413	23	4422
NC6	38	4303	55	4477	58	4546	65	4609	-		67	4706
NC7	23	4050	34	4307	43	4467	49	4517			54	4593
NC8	12	3794	23	4093	28	4336	34	4376			36	4450
NC9	6	3441	13	3888	18	4155	22	4240			24	4300
NC10	4	3315	7	3705	11	4010	14	4084			16	4134
NC11	42	4222	56	4452	62	4513	65	4593			71	4612
NC12	25	4021	39	4290	46	4404	50	4481			54	4563
NC13	14	3803	27	4133	37	4295	43	4388			46	4451
NC14	8	3538	17	3847	23	4158	28	4219			30	4245
NC15	5	3396	10	3772	14	4001	18	4104			22	4164
NC16	45	4260	59	4414	63	4494	68	4583			74	4629
NC17	25	3984	42	4240	45	4322	51	4453			55	4498
NC18	13	3719	25	4033	34	4221	41	4291			46	4369
NC19	9	3577	18	3855	25	4142	27	4213			35	4279
NC20	7	3382	14	3771	21	4037	26	4084			29	4141

* S: Strength (MPa), ** V: velocity (m/s).

and

Table 6. The measured UPV and strength data for pozzolanic concrete.

No.	Day 3		Day 7		Day 14		Day 21		Day 28		Day 56		Day 70
	S	V	S	V	S	V	S	V	S	V	S	V	S	V
FC1	58	4514	66	4594	78	4622	80	4675	84	4716	91	4760	92	4795
FC2	39	4418	47	4556	56	4593	57	4621	61	4682	67	4720	70	4760
FC3	24	4211	31	4426	38	4486	38	4512	41	4532	47	4620	53	4633
FC4	16	4028	22	4216	27	4324	30	4370	32	4373	36	4479	38	4510
FC5	10	3810	14	4100	17	4240	19	4273	21	4325	25	4393	25	4425
FC6	60	4449	69	4581	74	4631	81	4675	83	4670	91	4714	95	4717
FC7	37	4319	44	4460	49	4558	52	4565	59	4602	65	4626	65	4612
FC8	24	4192	29	4318	32	4412	38	4422	38	4492	44	4554	46	4586
FC9	13	3886	16	4137	19	4198	22	4239	25	4341	27	4389	29	4412
FC10	8	3722	11	3928	11	4055	13	4086	14	4132	17	4201	18	4280
SC1	68	4530	78	4609	84	4679	82	4697	95	4682	99	4760	102	4799
SC2	46	4461	60	4546	66	4624	68	4696	69	4699	72	4701	74	4786
SC3	27	4275	34	4402	44	4499	44	4553	49	4597	50	4636	54	4670
SC4	18	4082	28	4314	34	4396	37	4489	39	4500	40	4519	42	4621
SC5	10	3855	18	4141	23	4297	26	4351	26	4380	29	4458	31	4468
SC6	58	4415	76	4566	87	4606	86	4643	92	4685	98	4737	99	4748
SC7	29	4284	40	4432	46	4538	49	4562	53	4578	53	4666	53	4699
SC8	15	4080	23	4195	27	4336	30	4396	33	4452	35	4480	37	4557
SC9	9	3783	14	4034	21	4206	21	4261	25	4278	26	4413	28	4410
SC10	6	3647	12	3863	15	4336	17	4195	18	4243	20	4275	22	4384

summarize all the measured UPV and strength data pairs for normal concrete and pozzolanic concrete, respectively, at various ages.

3. Relationship between UPV and Strength Evolution Index of Concrete

3.1. Relationship between UPV and Strength in Normal Concrete

Figure 2a shows the testing results of NC1–NC5, which adopted the mixture proportions in Table 1. The UPV–strength curve of concrete with a w/cem of 0.3 is located on the upper end of the curve, followed sequentially by the curves of concrete with w/cem of 0.4, 0.5, 0.6, and 0.7. The measurements were conducted at specimen ages of 1, 3, 7, 14, 21, and 28 days. For the concrete with a w/cem of 0.3, the testing results revealed a relatively high UPV and strength evolution index; nearly all points on the relationship curve became adjacent to each other after 3 days. As the w/cem increased, the various points that comprised the relationship curve gradually became scattered, and the relationship curves moved downwards. In other words, concrete strength decreased in accordance with increases in the w/cem. Figure 2a clearly shows that the relationship curves of concrete with various water-to-cementitious-material ratios were mutually independent and dispersed.

Compiling the relationship curves of concrete at different ages yielded Figure 2b, which exhibits that the distribution of the relationship curve between UPV and strength is highly related to the concrete age. Along with increases in the age of the concrete, the UPV and strength relationship curve of the concrete moved to the upper right of the figure. In terms of the shape of relationship curve, a substantial radian change was observed at an age of 1 day; however, the degree of change decreased over time. The curve shapes of all concrete specimens became fairly similar after 14 days. Figure 2b reveals that the increase in UPV due to relatively low w/cem is more apparent when the concrete is at early ages, but the difference in cementitious material content exerts less influence on increases in UPV when the concrete has a longer period of curing time. Figure 2b shows that the relationship curves of concrete specimens at different ages are independent and scattered. In Figure 2a,b, the relationship between concrete UPV and strength is weak; thus, a simple yet general regression curve could not be obtained.

Dividing the UPV (V_n) and strength (S_n) of specimens at various ages in Figure 2a by their corresponding terminal UPV (V_t) and strength (S_t) of concrete, one can obtain the relationship between the UPV evolution index (V_g) and the strength evolution index (S_g) of concrete with different mixture proportions, as depicted in Figure 3. Figure 3 indicates that, as the w/cem decreased, the UPV and strength evolution index became closer to the top of the figure. In other words, smaller w/cem corresponded to a higher UPV and strength evolution index when the concrete was at an early age, and vice versa. In Figure 3, the distribution of the UPV and strength evolution index data points of specimens with different mixture proportions and at different ages is highly concentrated. By inputting the data into an exponential function for regression analysis, an optimal regression curve, $S_g=0.0685e^{\left(0.0728·V_g\right)}$, was obtained, where R² = 0.968. The regression results for Figure 3 reveal that the UPV and strength evolution index of concrete specimens with different water-to-cementitious-material ratios at different ages exhibited satisfactory correlation. Therefore, the UPV and strength evolution index of normal concrete with different mixture proportions should be capable of being simplified into a single regression curve.

3.2. Influence of the Proportion of Coarse-to-Fine Aggregates

This section explores the influence of differences in the content of coarse and fine aggregates on the relationship between concrete UPV and strength under the condition of a fixed volume of paste. The specimens were prepared according to the mixture proportions listed in Table 2; specifically, the V_paste was 36%, the S/A were 30, 45 and 60%, and the specimens were tested at 1, 3, 7, 14, and 28 days. Figure 4a presents the relationship curves of UPV and strength for concrete specimens with different sand–aggregate ratios. Noticeable variation was apparent among these relationship curves; specifically, increases in the S/A caused the relationship curves to move to the right. The results indicated that the relationship curves of specimens with different sand–aggregate ratios were fairly scattered and that no simplified single relationship curve could be obtained.

When the UPV and strength data of specimens with different mixture proportions and at different ages in Figure 4a were converted into UPV and strength evolution index, a relationship curve could be replotted (Figure 4b). This figure reveals that the distribution of data points for the UPV and strength evolution index is highly concentrated. An exponential function could then be employed for regression analysis. The optimal curve was determined to be $S_g=0.0612{\mathrm{e}}^{\left(0.0740·V_g\right)},$ where R² = 0.966. The regression results of Figure 4b again indicate that the UPV and strength evolution index of the concrete specimens with various S/A and at different ages, which were obtained with reference to the terminal UPV and strength evolution index, exhibited strong correlation. Thus, the relationship between concrete UPV and strength evolution index could be simplified into a single regression curve.

Regarding the possible influence of different sources of coarse aggregate on the relationship between concrete UPV and strength, the testing results of two concrete groups, namely NC1–NC5 listed in Table 1 and NC11–NC1 listed in Table 2, were adopted for comparison and analysis. Even though these two groups of concrete had identical mixture proportions, they exhibited dissimilar UPV values because of their different coarse aggregate sources. Figure 5a presents the relationship curves of UPV and strength for concrete with different sources of coarse aggregate. A significant difference can be observed in the relationship curves of the two groups; the increase in the UPV of coarse aggregate (Type 2) leads the relationship curve to move to the right.

The UPV and strength data presented in Figure 5a of specimens with different mixture proportions and at different ages were again converted into UPV and strength evolution index to construct relationship curves, as depicted in Figure 5b. Figure 5b reveals that the distribution of data points for the UPV and strength evolution index of the specimens with different mixture proportions and at different ages is again highly concentrated. Inputting the data into an exponential function for regression analysis yielded an optimal curve, $S_g=0.0649{\mathrm{e}}^{\left(0.0734·V_g\right)}$, where R² = 0.967.

The results presented in this section with respect to differences in aggregate proportion and source of coarse aggregate revealed that the regression relationship between concrete UPV and the strength evolution index remained satisfactory. The following sections investigate whether the addition of pozzolanic materials and changes in paste volume ratio exert significant influence on the relationship between concrete UPV and the strength evolution index.

3.3. Influence of Pozzolanic Material and Paste Volume Ratio

The testing results in the previous section revealed that differences in the S/A and coarse aggregate type did not have a significant influence on the correlation between UPV and the strength evolution index. This section examines the influence of adding pozzolanic materials and altering the paste volume ratio. The specimens were prepared according to the mixture proportions in Table 3 for fly ash concrete and those in Table 4 for slag concrete. Because a lower evolution index in strength was observed for concrete samples in which pozzolanic materials such as fly ash and slag were added compared with normal concrete, the time points for testing were 3, 7, 14, 21, 28, 56, and 70 days. During analysis, the UPV and strength values of specimens at an age of 70 days were adopted as the terminal UPV and strength values for concrete with pozzolanic material.

Figure 6a displays the testing results of specimens produced according to the mixture proportions listed in Table 3 for fly ash concrete. As indicated in Figure 6a, the regression curves of the UPV and strength evolution index were nearly identical for fly ash concrete with different paste volume ratios (i.e., 36 and 42%). Figure 6b presents the testing results of specimens produced according to the mixture proportions listed in Table 4 for slag concrete. The regression curves of the UPV and strength evolution index for slag concrete with different paste volume ratios (i.e., 36 and 42%) again nearly overlap. The results of a regression analysis of Figure 6a,b reveal that changes in paste volume ratio did not affect the regression curves of the UPV and strength evolution index of concrete with pozzolanic material. However, the terminal UPV and strength values used for concrete with pozzolanic material were measurement results obtained at an age of 70 days.

To evaluate the difference between normal concrete and concrete with pozzolanic material, the data of normal concrete specimens (NC1–NC5) with identical paste volume ratios (V_p = 36%) and sand–aggregate ratios (S/A = 45%) were combined with the data of pozzolanic concrete for analysis. Figure 7 displays the relationship between the UPV and strength evolution index for specimens NC1–NC5, FC1–FC5, and SC1–SC5. The terminal UPV and strength data for the normal and pozzolanic concrete were the test results at 28 and 70 days, respectively. Figure 7 demonstrates that the data points for the UPV evolution index had a strong correlation with the data points for the strength evolution index. An exponential function was then adopted for a regression analysis. The optimal fitting curve was obtained as ${\mathrm{S}}_g=0.0599{\mathrm{e}}^{\left(0.0742·V_g\right)}$, where R² = 0.951. This result indicates that, as long as the UPV and strength values measured at 70 days were adopted as the terminal UPV and strength for the pozzolanic concrete, the relationship between the UPV and strength evolution index of normal and pozzolanic concrete should still have been able to be expressed using a single relationship curve.

3.4. A Unified Equation for the Relationship between UPV and the Strength Evolution Index

The aforementioned results indicated that the adoption of various mixture proportions in this study, namely with respect to the water-to-cementitious material ratio, paste volume ratio, sand–aggregate ratio, type of aggregate source, and amount of pozzolanic material, exhibited limited influence on the concrete UPV and strength evolution index. Therefore, the test data of all mixture proportions in Table 1, Table 2, Table 3 and Table 4 were compiled to conduct a regression analysis on the UPV and strength evolution index (Figure 8). As indicated in Figure 8, the data points for the UPV and strength evolution index of concrete specimens with various mixture proportions exhibited a concentrated distribution. Specifically, these data points were distributed evenly along both sides of the regression curve of the relationship, suggesting favorable correlation between the UPV and the strength evolution index of concrete specimens with different mixture proportions at different ages. For practical application, the relationship between the UPV and the strength evolution index should be as simplified as much as possible. The regression function as given in Equation (3), derived from Figure 8, was used as a unified equation for subsequent estimations of the concrete strength evolution index using the UPV evolution index.

$$S_g=0.0660e^{\left(0.0732V_g\right)} \mathrm{with} {\mathrm{R}}^2=0.947$$

(3)

4. Verification of the Applicability of the Unified Equation

In this section, tests were performed on the established unified equation (Equation (3)) for the UPV and the strength evolution index to verify its applicability for estimating concrete strength. First, the aforementioned test data underwent self-verification. The specimens examined were NC1–5, NC6–20, FC1–10, and SC1–SC10. For concrete with a specific mixture proportion, dividing the UPV measured at a specific age by the terminal UPV yielded the UPV evolution index (V_g) of the concrete. By substituting V_g into Equation (3), the strength evolution index (S_g) of the concrete at that age was obtained. Multiplying S_g with the terminal strength of that mixture proportion yielded the estimation of the concrete strength. Terminal UPV and strength refer to the UPV and strength values at 28 days for normal concrete but at 70 days for pozzolanic concrete.

Figure 9 presents a comparison of the estimated strengths of all specimens and the strengths measured by performing compression tests. The errors between the strength estimations and actual measurements were generally in the range of ±15%, indicating that the unified equation (Equation (3)) could be applied for estimating concrete strength at different time points.

In addition to using the established relationship curve data to perform self-applicability tests, this study also collected the test data of numerous relevant studies [24,25,26,27,28] regarding UPV and strength values at different ages, which could be adopted to examine the applicability of the proposed unified equation. The collected data was for concrete with numerous paste volume ratios, coarse aggregate contents, and pozzolanic admixture contents; these mixture proportions are listed in

Table 7. Normal concrete mixture proportions collected from the relevant studies for verifying the applicability of the unified equation.

No.	w/cem	S/A (%)	Vpaste (%)	Mixture Proportion, kg/m3
				Water	Cement	Slag	Fly Ash	Super-Plasticizer	Fine Aggregate	Coarse Aggregate
PNC1	0.4	40	36	201	502	0	0	0	681	1027
PNC2	0.5			220	440	0	0	0
PNC3	0.6			235	392	0	0	0
PNC4	0.7			248	354	0	0	0
PNC5	0.4	40	42	234	585	0	0	0	608	926
PNC6	0.5			257	514	0	0	0
PNC7	0.6			275	458	0	0	0
PNC8	0.7			289	413	0	0	0
PNC9	0.4	42	30	163	419	0	0	4.19	750	1044
PNC10	0.6			195	327	0	0	0.82
PNC11	0.4	36	36	196	502	0	0	5.02	592	1044
PNC12	0.6			235	392	0	0	0.98
PNC13	0.4	49	30	163	419	0	0	4.19	881	914
PNC14	0.6			195	327	0	0	0.82
PNC15	0.4	44	36	196	502	0	0	5.02	723	914
PNC16	0.6			235	392	0	0	0.98
PPC1	0.36	44	31	161	460	0	0	4.60	785	1000

for normal concrete and

Table 8. Pozzolanic concrete mixture proportions collected from the relevant studies for verifying the applicability of the unified equation.

No.	w/cem	S/A (%)	Vpaste (%)	Mixture Proportion, kg/m3
				Water	Cement	Slag	Fly Ash	Super-Plasticizer	Fine Aggregate	Coarse Aggregate
PFC1	0.4	42	30	163	293	0	126	4.19	750	1044
PFC2	0.6			195	229	0	98	0.82
PFC3	0.4	36	36	196	352	0	151	5.02	592	1044
PFC4	0.6			235	275	0	118	0.98
PFC5	0.4	49	30	163	293	0	126	4.19	881	914
PFC6	0.6			195	229	0	98	0.82
PFC7	0.4	44	36	196	352	0	151	5.02	723	914
PFC8	0.6			235	275	0	118	0.98
PSC1	0.4	42	30	163	167	251	0	4.19	750	1044
PSC2	0.6			195	131	196	0	0.82
PSC3	0.4	36	36	196	201	301	0	5.02	592	1044
PSC4	0.6			235	157	235	0	0.98
PSC5	0.4	49	30	163	167	251	0	4.19	881	914
PSC6	0.6			195	131	196	0	0.82
PSC7	0.4	44	36	196	201	301	0	5.02	723	914
PSC8	0.6			235	157	235	0	0.98
PPC2	0.365	53	35	169	264	216	-	6.24	916	812

for pozzolanic concrete. In total, 34 mixture proportions were documented.

Table 9. The measured UPV and strength data collected for normal concrete.

No.	Day 1		Day 3		Day 4		Day 7		Day 9		Day 14		Day 28		Day 56
	S	V	S	V	S	V	S	V	S	V	S	V	S	V	S	V
PNC1	23	3900			48	4437	51	4477	55	4485	56	4500	59	4564
PNC2	14	3633			31	4252	39	4354	39	4386	43	4401	48	4456
PNC3	7	3245			16	3972	24	4092	25	4193	28	4241	31	4290
PNC4	3	2739			11	3800	15	3993	18	4040	20	4118	22	4198
PNC5	23	3862			45	4300	51	4389	51	4406	53	4430	62	4517
PNC6	11	3552			28	4147	34	4293	35	4295	40	4326	46	4389
PNC7	8	3310			18	3974	23	4115	24	4190	24	4246	29	4306
PNC8	4	3026			9	3680	14	3916	14	3968	15	4073	18	4159
PNC9	25	4011	37	4271			45	4470			49	4551	57	4624	62	4657
PNC10	10	3572	21	3945			26	4202			32	4278	36	4391	41	4433
PNC11	25	3936	35	4287			42	4430			43	4478	48	4542	49	4574
PNC12	10	3440	18	3971			24	4206			28	4245	34	4346	38	4279
PNC13	23	3990	37	4355			50	4459			57	4560	60	4592	67	4659
PNC14	7	3385	17	3904			24	4123			29	4225	31	4320	35	4389
PNC15	24	3915	36	4301			50	4430			54	4481	58	4552	58	4628
PNC16	6	3261	15	3898			23	4122			27	4223	33	4302	33	4381
PPC1	20	3967	36	4285			46	4469			46	4513	57	4619

and

Table 10. The measured UPV and strength data collected for pozzolanic concrete.

No.	Day 1		Day 3		Day 7		Day 14		Day 28		Day 56
	S	V	S	V	S	V	S	V	S	V	S	V
PFC1	17	3781	30	4261	36	4392	45	4465	47	4553	59	4660
PFC2	6	3304	12	3859	17	4012	21	4150	26	4256	34	4426
PFC3	16	3769	25	4154	36	4287	44	4378	47	4485	50	4547
PFC4	5	3271	9	3607	14	3885	16	4023	22	4154	26	4324
PFC5	15	3803	22	4217	35	4368	48	4464	58	4548	70	4641
PFC6	3	2835	8	3563	13	3788	16	3915	21	4084	26	4239
PFC7	13	3682	25	4156	32	4255	41	4388	43	4470	55	4531
PFC8	4	2962	8	3632	12	3856	18	4009	21	4143	27	4278
PSC1	14	3362	29	4194	41	4351	50	4466	51	4499	60	4624
PSC2	2	2617	9	3590	16	3886	22	4143	25	4239	29	4343
PSC3	5	3161	19	4005	29	4145	39	4296	44	4445	48	4523
PSC4	2	2811	9	3543	15	3903	22	4076	26	4184	30	4328
PSC5	8	3468	21	4151	41	4286	50	4414	56	4507	63	4645
PSC6	3	2689	10	3585	18	3904	28	4137	34	4288	39	4418
PSC7	4	3037	16	3960	29	4089	35	4297	44	4432	50	4536
PSC8	2	2672	7	3520	14	3782	19	3998	27	4179	32	4272
PPC2	14	3752	31	4172	46	4303	46	4440	54	4553	66	4592

summarize all the measured UPV and strength data pairs collected from the relevant studies for normal concrete and pozzolanic concrete, respectively, at various ages. It should be noted that the UPV and strength values at 56 days instead of 70 days for pozzolanic concrete were used as the terminal UPV and strength values required for the calculation of evolution index due to the lack of test data at 70 days.

Figure 10a presents a comparison between the UPV-based strength estimations of concrete at different ages and the strength measurements obtained from actual compression tests. Of the 203 entries of verification data, most achieved an error within ±15%; only 27 entries (13.3%) had an error greater than 15%. To determine the possible cause of the relatively large error in prediction, another comparison chart, Figure 10b, was created, which compares the estimation results of concrete among samples at 1~3 days old and among those at all ages. Figure 10b reveals that 21 entries (approximately 10.3%) with estimation errors greater than 15% were identified in the estimation results for concrete at ages within 3 days, and these errors were all relatively conspicuous. In other words, only 6 entries (3.0% of the total verification data) with prediction errors greater than 15% were identified for concrete samples at ages more than 3 days, and these errors were only slightly greater than 15%. This result indicates that using the proposed unified equation to estimate the strength of concrete at ages after 3 days can yield highly reliable estimations.

Further exploration indicated that estimations using the unified equation seem to have higher percentages of error in concrete specimens at ages less than 3 days. Concrete at ages between 1–3 days exhibits the highest hydration rate, and the influence of different water-to-cementitious material ratios and levels of pozzolanic material content on variations in concrete UPV and strength is also the most conspicuous during this period. Therefore, using the unified equation of the relationship between UPV and the strength evolution index for concrete in this period incurs a relatively high level of uncertainty.

For practical applications in construction, the execution of major construction procedures such as applying post tensioning or removing formwork supports requires the concrete to achieve a specific level of strength. The aforementioned construction procedures are usually carried out three days after the concrete is placed. At this time, the strength estimation accuracy and reliability of the unified equation are both high. In particular, in the actual application of the unified equation, only the terminal UPV and strength of the same batch of concrete need to be obtained; the mixture proportions of the concrete and the UPV–strength relationship curve are unnecessary. This method should be easy to execute for the nondestructive testing and quality control of concrete strength in construction projects.

Finally, it is necessary to reiterate that the innovation of this research is to find a unified equation to describe the relationship between the UPV and strength evolution index. For practical application, the terminal UPV and strength need to be known first. For a large construction project that takes several years, the terminal UPV and strength of the concrete used in the project can be measured at the early stage of construction (28-day or 70-day age of concrete). In the following years of construction, the unified equation proposed in this study can be used to predict the strength of concrete at different ages for the execution of major construction procedures and quality control.

5. Conclusions

The relationship between the UPV and strength of normal concrete is substantially influenced by the mixture proportions and the age of the concrete. In this study, regression analyses were conducted separately on the water-to-cementitious-material ratio and age, and the relationship curves thus obtained were mutually independent and scattered. Thus, no simplified single curve could be obtained for the relationship between concrete UPV and strength.

Based on the idea that differences in mixture proportion and material type are eventually reflected in the final properties of hardened concrete, this study proposes a method for characterizing the relationship between the evolution index of UPV and strength and aims to establish a unified equation to represent the relationship between the UPV and the strength evolution index. The following conclusions were drawn from the testing results:

• Converting the relationship between the UPV and strength into that of UPV and the strength evolution index yielded sufficiently convergent correlations for normal concrete with various mixture proportions and at different ages. Moreover, this convergence did not decrease when pozzolanic materials were added.
• This study established a unified equation for representing the relationship between the UPV evolution index and the strength evolution index of concrete. The UPV and strength values at 28 days for normal concrete and those at 70 days for pozzolanic concrete were used as the terminal UPV and strength values required for the calculation of the evolution index.
• The proposed unified equation could be used to estimate the strength of concrete at different ages. The estimation results indicated that almost all specimens over 3 days old achieved an error of less than ±15%; therefore, the proposed unified equation has substantial value for practical application in major construction procedures (e.g., applying post tensioning or removing formwork supports).