A Review of Three Common Concrete Multiaxial Strength Criteria from 2010 to 2020

Abstract

Strength criterion is fundamental to the study of the strength assessment and design of engineering structures under complex stress conditions. Through experiments with a range of concretes, a large number of multiaxial strength criteria have been proposed to date, with many new developments from 2010 to 2020.Even so, there has been minimal comparative analysis of the research results of different strength criteria. This paper presents a comparative analysis of the strength criteria of ordinary concrete, high-performance concrete and lightweight aggregate concrete from the past 10 years. First, the classic concrete strength criterion is summarized. Then, the research progress associated with the multiaxial strength criterion from 2010 to 2020 is reviewed. The accuracy of the concrete strength criterion proposed during this period is analysed and compared with the collected multiaxial strength experimental data of 777 groups of concrete using a statistical method. The analysis shows that the six-parameter strength criterion for ordinary concrete proposed by Ding-Wu, the three-parameter strength criterion based on the Willam-Warnke criterion for high-performance concrete proposed by He-Song and the six-parameter strength criterion for lightweight aggregate concrete proposed by Liu-Song are in good agreement with the experimental data collected.

Appendices

Appendix 1 The strength of different normal–strength concrete (NSC) under different stress states

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
25.6	− 10.5	− 24	− 95.8	23.03	0	0	− 23.03
[35,36,37]	− 11.8	− 52.7	− 105.4	[41, 42]	− 5.79	− 5.79	− 53.1
	− 10	− 69.3	− 92.4		− 5.79	− 11.59	− 57.93
	1.5	− 1.5	− 6		− 5.79	− 17.38	− 53.1
	1.4	− 2.9	− 5.7		− 5.79	− 23.17	− 65.17
	1.4	− 4.3	− 5.7		− 5.79	− 28.97	− 67.59
	0.9	0	− 18.2		− 5.79	− 40.55	− 60.34
	1.4	0	− 14.5		− 5.79	− 47.86	− 56.83
	1.9	0	− 6.3		− 2.9	− 50.69	− 50.69
	2.4	0	− 4.9		− 8.69	− 14.48	− 72.41
	0	− 7.7	− 31		− 8.69	− 8.69	− 62.76
	0	− 15.9	− 31.7		− 8.69	− 67.59	− 77.24
	0	− 22.5	− 30		− 8.69	− 20.28	− 79.66
	0	− 29.1	− 29.1		− 8.69	− 31.86	− 77.24
	− 5.3	− 68.6	− 68.6		− 8.69	− 37.66	− 82.07
	− 10	− 85	− 85		− 8.69	− 55.72	− 81.52
	− 16.7	− 106.4	− 106.4		− 8.69	− 60.83	− 82.07
	− 30.8	− 149.8	− 149.8		− 8.69	− 72.41	− 72.41
	0.5	− 12.4	− 12.4		− 2.9	− 2.9	− 41.03
	1	− 8.7	− 8.7		− 2.9	− 8.69	− 53.1
	1.7	− 6.9	− 6.9		− 2.9	− 24.14	− 57.93
	2.7	0	0		− 2.9	− 20.28	− 55.52
	0	0	− 25.6		− 2.9	− 31.86	− 57.93

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
19.29					− 2.9	− 37.24	− 55.52
[38]	− 0.31	− 7.73	− 30.9		− 2.9	− 49.31	− 57.93
	− 0.34	− 17.01	− 34.02		− 2.9	− 53.1	− 53.1
	− 0.33	− 24.94	− 33.25		− 2.9	− 2.9	− 43.45
	0.66	− 1.65	− 6.61		− 5.79	− 63.79	− 65.17
	0.6	− 2.99	− 5.97
	0.59	− 4.4	− 5.87	26.93	0	0	− 26.93
	0	− 6.39	− 25.56	[41]	− 0.62	− 50.69	− 50.69
	0	− 13.44	− 26.87		− 0.62	− 45.86	− 45.86
	0	− 19.73	− 26.3		− 1.24	− 49.66	− 53.1
	0.49	0	− 9.8		− 1.24	− 50.69	− 50.69
	0.68	0	− 6.8		− 2.9	− 62.76	− 62.76
	1.07	0	− 4.28		− 1.45	− 1.45	− 41.52
	1.31	0	− 2.62		− 2.9	− 2.9	− 41.76
	0.53	− 5.33	− 5.33		− 4.83	− 5.24	− 56.28
	0.46	− 9.17	− 9.17		− 7.24	− 7.24	− 71.24
	0	− 70.3	− 70.3		− 1.45	− 21.72	− 58.9
	0	− 63.84	− 63.84		− 2.9	− 36.21	− 66.14
	0	− 3.75	− 41.42		− 2.9	− 28.97	− 68.83
	− 7.6	− 27.018	− 114.76

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
	− 7.6	− 28.842	− 116.28	36.1	0	0	− 36.1
	− 7.6	− 53.2	− 125.78	[41]	− 0.62	− 47.1	− 48.28
	− 7.6	− 56.24	− 131.48		− 0.62	− 51.66	− 51.1
	− 7.6	− 80.56	− 132.24		− 1.24	− 60.34	− 61.31
	− 7.6	− 85.12	− 140.98		− 1.24	− 50.69	− 51.24
	− 7.6	− 113.62	− 154.28		− 2.9	− 67.59	− 68.83
	− 7.6	− 142.12	− 142.12		− 1.45	− 1.45	− 54.55
	− 7.6	− 135.28	− 135.28		− 2.9	− 2.9	− 59.38
	− 15.2	− 30.134	− 144.78		− 1.45	− 21.72	− 63.24
	− 15.2	− 34.276	− 153.9		− 2.9	− 28.97	− 72.9
	− 15.2	− 61.56	− 164.92		− 4.83	− 5.24	− 64
	− 15.2	− 70.68	− 175.56		− 4.83	− 4.83	− 73.1
	− 15.2	− 93.1	− 167.2		− 1.45	− 26.55	− 66.41
	− 15.2	− 102.6	− 180.88		− 2.9	− 28.97	− 68.83
	− 15.2	− 124.26	− 178.98
	− 15.2	− 156.94	− 156.94	9.37	0	0	− 9.37
	− 22.8	− 30.43	− 164.16	[10]	− 3.9	− 3.87	− 37
	− 22.8	− 30.97	− 170.24		− 2.98	− 3.87	− 29.79
	− 22.8	− 35.87	− 171		− 4.39	− 8.51	− 45.29
	− 22.8	− 35.56	− 179.36		− 5.7	− 14.26	− 46.61
	− 22.8	− 68.78	− 188.86		− 4.55	− 18.82	− 46.13
	− 22.8	− 106.78	− 197.98		− 4.23	− 17.14	− 43.25
	− 22.8	− 102.22	− 201.78		− 5.05	− 26.57	− 52.86
	− 22.8	− 102.22	− 210.52		− 15.69	− 15.22	− 74.97
	− 22.8	− 136.04	− 210.52		− 12.42	− 11.54	− 58.15
	− 22.8	− 143.64	− 207.48		− 14.79	− 14.1	− 67.28
	− 22.8	− 144.02	− 188.1		− 29.03	− 28.65	− 96.11
	− 22.8	− 181.26	− 181.26

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
	− 22.8	− 177.84	− 177.84	10.74	0	0	− 10.74
	− 22.8	− 175.18	− 175.18	[10]	− 5.05	− 9.46	− 45.17
	0.82	− 4.1	− 4.1		− 4.88	− 14.26	− 49.02
	0.94	− 3.14	− 3.14		− 4.72	− 23.85	− 47.57
	− 0.29	− 29.41	− 29.41		− 4.72	− 36	− 48.05
	− 2.4	− 47.93	− 47.93		− 4.23	− 32.64	− 43.25
	0	− 25.9	− 25.9		− 3.9	− 34.08	− 38.44
	1.71	0	0		− 3.74	− 37.12	− 38.44
	− 0.25	− 2.52	− 25.17		− 18.47	− 27.37	− 91.3
	− 1.83	− 1.83	− 36.69		− 17.49	− 25.93	− 86.02
	− 0.21	− 0.21	− 20.82		− 18.31	− 36.48	− 91.3
	0	0	− 19.29		− 17.82	− 43.03	− 84.58
28.88				34.2
[39]	− 12.92	− 64.2	− 128.4	[44, 64]	1.2	0	− 23.56
	− 10.9	− 81.6	− 108.8		1.98	0	− 12.97
	1.43	− 1.43	− 5.83		2.3	0	− 9.31
	1.37	− 2.74	− 5.48		3.14	0	0
	1.32	− 3.96	− 5.28		0	− 10.75	− 43.01
	0	− 8.78	− 35.12		0	− 22.75	− 45.49
	0	− 27.24	− 36.32		0	− 31.88	− 42.5
	0.95	0	− 19.04		0	− 40.7	− 40.7
	1.41	0	− 14.06		− 13.56	− 34.25	− 134.66
	1.85	0	− 7.23		− 14.33	− 72.05	− 141.82
	2.03	0	− 4.06		− 12.56	− 93.15	− 126.78

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
	− 9.84	− 98.6	− 98.2	23.32
	1.67	− 6.68	− 6.68	[45]	0.87	− 2.24	− 4.48
	1.09	− 10.8	− 10.8		0.82	− 3.56	− 4.58
	0.58	− 11.9	− 11.9		0	− 7.15	− 28.6
	0	− 33.3	− 33.3		0	− 15.44	− 30.87
	2.34	0	0		0	− 22.29	− 29.72
	− 11.26	− 28.15	− 112.6		0.72	0	− 14.34
	0	0	− 28.88		1.02	0	− 10.26
38					1.41	0	− 5.62
[39]	0	0	− 38		1.61	0	− 3.22
	0	− 8.25	− 44.84		0	− 27.98	− 27.98
	0	− 2.91	− 44.46		1.85	0	0
	0	− 13.26	− 56.62		0.87	− 4.08	− 4.08
	0	− 14.71	− 61.56		0.59	− 5.8	− 5.8
	0	− 26.14	− 57.76		0.28	− 6.9	− 6.9
	0	− 38	− 61.56		0.89	− 1.13	− 4.86
	0	− 36.06	− 68.78		− 22.8	− 173.28	− 173.28
	0	− 51.3	− 69.16		− 22.8	− 22.8	− 151.62
	0	− 58.14	− 70.68		− 30.4	− 38.38	− 190.38
	− 30.4	− 189.24	− 188.86		− 30.4	− 74.48	− 207.1
	− 30.4	− 39.52	− 199.5		− 30.4	− 114.38	− 213.56
	− 30.4	− 77.9	− 225.34		− 30.4	− 154.66	− 227.62
	− 38	− 41.04	− 206.34		− 30.4	− 193.04	− 192.28
	− 38	− 121.98	− 237.5		0	0	− 23.32
	− 38	− 169.86	− 247

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
24.51				22.35	0	0	− 22.35
[38]	2.32	0	0	[70]	0	− 27.41	− 27.41
	0	0	− 24.51		− 325.34	− 325.34	− 325.34
	0	− 8.69	− 34.74		− 5.45	− 5.57	− 52.83
	0	− 18.61	− 37.22		− 7.15	− 17.53	− 72.87
	0	− 26.84	− 35.78		− 7.82	− 21.38	− 71.77
	0	− 33.52	− 33.52		− 6.81	− 36.15	− 69.82
	0.67	0	− 13.4		− 6.08	− 45.77	− 61.65
	0.92	0	− 9.2		− 5.73	− 56.96	− 57.23
	1.45	0	− 5.8		− 37.73	− 38.29	− 150.36
	1.81	0	− 3.62		− 45.47	− 55.89	− 188.01
	− 2.11	− 2.11	− 42.14		− 46.57	− 135.71	− 187.28
	− 0.28	− 0.28	− 27.62		− 41.73	− 163.47	− 166.59
	− 0.31	− 3.1	− 31.02		− 25.05	− 83.35	− 83.96
	− 0.34	− 8.55	− 34.19	39.0
	− 0.36	− 18.06	− 36.12	[46]	0	0	− 39
	− 0.38	− 28.85	− 38.46		0	− 9.75	− 48.75
	− 0.35	− 35.45	− 35.45		0	− 24.96	− 49.92
	− 2.7	− 53.93	− 53.93		0	− 45.63	− 45.63
	0.92	− 2.31	− 9.24		1.57	0	− 31.43
	0.87	− 4.36	− 8.71		2.29	0	− 22.93
	0.69	− 5.18	− 6.9		2.96	0	− 14.78
	0.61	− 6.13	− 6.13		2.98	0	− 9.95
	0.51	− 10.27	− 10.27		4.29	0	0
	1.19	− 5.96	− 5.96		4.56	0.91	0
	1.46	− 4.85	− 4.85		4.1	2.05	0

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
31.7					3.16	3.16	0
[40]	− 5	− 5	− 51	30.03
	− 10	− 10	− 69.3	[46]	0	0	− 30.03
	− 15	− 15	− 86.9		0	− 7.63	− 38.14
	− 20	− 20	− 106.3		0	− 19.97	− 39.94
	− 25	− 25	− 123.2		0	− 34.95	− 34.95
	− 30	− 30	− 147.2		1.23	0	− 24.62
42.7					1.43	0	− 14.32
[40]	− 10	− 10	− 7.64		1.9	0	− 9.49
	− 15	− 15	− 103		2.07	0	− 6.91
	− 20	− 20	− 113.4		2.88	0	0
	− 30	− 30	− 144		3	0.65	0
48.6					3.12	1.56	0
[40]	− 5	− 5	− 97.5		2.88	2.88	0
	− 10	− 10	− 119.4	21.14
	− 15	− 15	− 148.3	[33]	0	0	− 21.14
	− 20	− 20	− 169.4		1.93	0	0
	− 5	− 15	− 116		1.66	0	− 26.78
	− 5	− 20	− 126.8		1.34	0	− 27.73
	− 10	− 20	− 136.5		1.22	0	− 29.82
	− 10	− 25	− 146.5		0.93	0	− 27.15
	− 10	− 30	− 163.3		0.83	0	− 30.23
	− 15	− 25	− 171	45.45[33]

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
20[32]					0	0	− 45.45
	0	0	− 20		2.67	0	0
	− 5.2	− 5.2	− 52.43		2.33	0	− 50.13
	− 7.36	− 18.4	− 73.6		2.2	0	− 58.65
	− 7.18	− 21.53	− 71.77		1.99	0	− 60.99
	− 7.02	− 35.1	− 70.2		1.78	0	− 60.37
	− 6.21	− 46.6	− 62.13		0.96	0	− 56.08
	− 6.17	− 61.71	− 61.71	27.61[33]
	− 37.34	− 37.34	− 149.35		0	0	− 27.61
	− 47	− 56.4	− 188.01		2.55	0	0
	− 46.82	− 93.64	− 187.28		2.28	0	− 38.45
	− 44.04	− 176.17	− 176.17		1.99	0	− 43.56
					1.76	0	− 42.67
					1.48	0	− 47.08
					1.23	0	− 38.29

Appendix 2 The strength of different high performance concrete (HPC) under different stress states

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
60.16	0	0	–60.16	44.13	0	0	–44.13
[85,86,87]	5.08	0	0	[94]	1.1	–2.21	–22.68
	2.2	0	–43.92		1.06	–5.38	–21.89
	3.23	0	–32.26		1.01	–10.33	–20.65
	4.28	0	–17.11		0.97	–14.70	–19.59
	4.73	0	–9.45		0.88	–17.74	–17.74
	4.86	0	–6.48		1.46	–1.46	–15.14
	2.23	–44.64	–44.64		1.41	–3.53	–14.17
	3.02	–30.18	–30.18		1.37	–6.88	–13.77
	3.76	–15.05	–15.05		1.28	–9.66	–12.93
	4.11	–11.73	–11.73		1.15	–11.47	–11.47
	4.19	–8.37	–8.37		1.41	–0.71	–7.10
	4.22	–5.63	–5.63		1.59	–1.32	–5.34
	4.23	–4.23	–4.23		2.03	–2.03	–4.10
	0	–66.78	–66.78		2.82	–2.82	–3.75
	–14.36	–38.78	–143.63		2.91	–2.91	–2.91
	–15.34	–64.43	–153.4		3.27	0	0
	–15.77	–82.03	–157.74
	–15.25	–117.45	–152.53	66.43	0	0	–66.43
	–13.16	–131.64	–131.64	[94]	1.33	–2.66	–29.43
	0	–13.52	–67.58		1.33	–6.51	–27.24
	0	–20.99	–69.97		1.26	–12.49	–26.31
	0	–30.33	–75.83		1.13	–17.21	–22.92
	0	–36.93	–73.85		1.06	–21.79	–21.79
	0	–51.28	–68.37		1.86	–1.86	–18.40
					1.73	–4.32	–17.21

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
46.09	–5.07	–20.28	–102.78		1.59	–8.10	–16.14
[47]	–5.07	–63.60	–106.47		1.53	–11.36	–15.15
	–4.61	–88.49	–93.1		1.46	–14.42	–14.42
	–14.29	–29.04	–144.72		2.26	–2.26	–11.23
	–16.59	–100.02	–166.85		2.52	–2.52	–8.37
	–13.37	–129.51	–136.43		2.79	–2.79	–5.51
	–30.42	–41.02	–204.64		3.12	–3.12	–4.12
	–34.57	–139.19	–231.83		3.19	–3.19	–3.19
	–27.65	–176.99	–186.2		4.45	0	0
					0	–7.91	–79.52
72.73	–7.27	–28.36	–144.01		0	–21.59	–86.29
[47]	–8	–100.37	–167.28		0	–44.84	–89.75
	–7.27	–141.82	–149.1		0	–66.03	–88.09
	–20.36	–40.73	–203.64		0	–80.51	–80.51
	–22.55	–134.55	–224.74		1.46	0	–33.15
	–20.36	–192.73	–202.92		1.86	0	–21.32
	–40	–53.09	–266.19		2.39	0	–10.89
	–40	–161.46	–268.37		2.99	0	–6.71
	–35.64	–226.92	–238.55		3.72	0	–4.25
					4.45	1.13	0
83.44	–8.34	–33.38	–167.71		5.18	2.59	0
[47]	–8.34	–102.63	–171.89		4.85	3.65	0
	–7.51	–148.52	–156.87		5.05	5.05	0
	–22.53	–45.89	–228.63
	–23.36	–141.01	–235.3	90.64	0	0	–90.64
	–20.86	–201.09	–211.94	[49, 94]	1.45	–2.99	–29.55
	–44.22	–58.41	–292.87		1.36	–6.89	–27.46
	–42.55	–268.68	–282.86		1.27	–13.05	–26.10
					1.18	–18.04	–24.02

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
61.17	0	0	–61.17		1.09	–22.30	–22.30
[47]	0	–3.67	–69.73		1.99	–1.99	–20.03
	0	–16.52	–78.30		1.9	–4.80	–19.03
	0	–34.26	–81.97		1.81	–9.15	–18.40
	0	–53.22	–85.64		1.81	–13.32	–17.77
	0	–70.35	–84.41		1.72	–17.22	–17.22
	0	–76.46	–77.07		2.72	–2.72	–13.69
					3.17	–3.17	–10.70
72.44	0	0	–72.44		3.44	–3.44	–6.89
[47]	0	–4.35	–90.55		3.72	–3.72	–4.89
	0	–18.83	–94.90		3.9	–3.90	–3.90
	0	–38.39	–96.35		5.17	0	0
	0	–55.78	–93.45
	0	–68.09	–85.48	47.07	0	–5.79	–58.04
	0	–76.79	–81.13	[46]	0	–15.72	–62.84
					0	–32.81	–65.62
94.2	0	0	–94.20		0	–47.68	–63.54
[47]	0	–5.65	–105.50		0	–57.10	–57.10
	0	–23.55	–113.04		1.18	0	–26.50
	0	–49.93	–119.63		1.6	0	–18.36
	0	–71.59	–113.98		2.12	0	–9.65
	0	–93.26	–112.10		2.59	0	–5.93
	0	–102.68	–103.62		3.06	0	–3.48
					3.58	0.89	0

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
60.15	4.27	0	0		3.67	1.84	0
[47]	2.29	0	–4.81		3.44	2.59	0
	2.41	0	–12.03		3.58	3.58	0
	1.86	0	–18.65		0	0	–47.07
	1.44	0	–28.87		3.53	0	0
	1.02	0	–40.90
	0	0	–60.15	87.96	0	–10.20	–102.39
				[46]	0	–27.80	–111.27
69.17	4.91	0	0		0	–57.88	–115.67
[47]	2.97	0	–3.46		0	–85.15	–113.56
	2.56	0	–4.84		0	–104.23	–104.23
	2.77	0	–13.83		1.58	0	–36.77
	2.21	0	–21.44		2.02	0	–22.78
	1.8	0	–35.97		2.55	0	–11.61
	1.11	0	–45.65		3.43	0	–7.83
	0	0	–69.17		4.4	0	–5.10
					5.19	0	0
87.44	5.16	0	0		5.45	1.32	0
[47]	4.63	0	–6.12		5.19	2.64	0
	3.24	0	–6.12		5.1	3.87	0
	2.89	0	–13.99		5.28	5.28	0
	2.62	0	–26.23		0	0	–87.96
	1.66	0	–33.23
	1.22	0	–49.84
	0	0	–87.44

Appendix 3 The strength of normal–strength concrete(NSC) at different temperatures[63]

T/℃	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	T/℃	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
200 ℃	0	0	–32.60	400 ℃	0	0	–23.80
	–12.85	–32.01	–126.31		–11.58	–29.31	–114.37
	–13.65	–68.22	–135.42		–12.15	–60.43	–120.56
	–12.16	–90.25	–119.30		–11.17	–83.48	–110.35
	–9.53	–94.28	–95.67		–8.92	–87.56	–88.42
	0	–22.40	–44.80		0	–7.93	–32.70
	0	–31.73	–42.30		0	–17.30	–34.60
	0	–38.73	–39.50		0	–25.05	–33.40
	1.24	0	–12.30		0	–29.70	–29.70
	1.62	0	–8.79		0.81	0	–8.10
	1.78	0	–7.12		0.98	0	–5.21
	2.7	0	0		1.16	0	–4.25
	0	–10.30	–41.20		1.42	0	0
300 ℃	0	0	–30.05	500 ℃	0	0	–19.50
	–12.24	–31.05	–123.32		–11.03	–27.55	–107.81
	–13.35	–66.08	–130.41		–11.75	–59.36	–116.35
	–11.62	–87.21	–115.38		–10.73	–80.21	–106.40
	–9.41	–93.52	–92.12		–8.59	–86.31	–85.52
	0	–9.63	–38.50		0	–6.00	–24.00
	0	–20.75	–41.50		0	–13.75	–27.50
	0	–29.70	–39.60		0	–20.25	–27.00
	0	–35.20	–35.20		0	–25.20	–25.20
	1.13	0	–11.30		0.57	9	–5.85
	1.48	0	–6.85		0.74	0	–3.84
	1.61	0	–6.02		0.81	0	–3.25
	2.01	0	0		1.2	0	0

Appendix 4 The strength of high performance concrete (HPC) at different temperatures[77,78,87]

T/℃	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	T/℃	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
200 ℃	0	0	–63.96	400 ℃	–11.38	–11.38	–113.84
	4.81	0	0		–13.68	–36.95	–136.83
	2.07	0	–41.33		–14.57	–61.18	–145.66
	2.98	0	–29.75		–14.81	–77.02	–148.12
	3.7	0	–14.81		–14.13	–108.76	–141.25
	4.27	0	–8.54		–12.24	–122.36	–122.36
	4.57	0	–6.09
	1.94	–38.71	–38.71	500 ℃	0	0	–35.72
	2.75	–27.52	–27.52		1.82	0	0
	3.53	–14.12	–14.12		0.72	0	–14.30
	3.55	–10.15	–10.15		1.05	0	–10.49
	3.98	–7.96	–7.96		1.26	0	–5.04
	3.94	–5.25	–5.25		1.33	0	–2.65
	4.19	–4.19	–4.19		1.02	0	–1.36
	0	–67.48	–67.48		0.66	–13.11	–13.11
	–12.10	–12.10	–121.00		0.9	–9.04	–9.04
	0	–14.42	–72.10		1.06	–4.22	–4.22
	0	–20.69	–68.98		1.17	–3.34	–3.34
	0	–29.06	–72.65		1.43	–2.86	–2.86
	0	–36.80	–73.59		1.49	–1.98	–1.98
	0	–53.22	–70.96		1.52	–1.52	–1.52
	–14.77	–39.89	–147.74		0	–8.38	–41.91
	–15.01	–63.05	–150.11		0	–13.72	–45.73
	–15.53	–80.76	–155.31		0	–18.70	–46.74
	–14.60	–112.42	–146.00		0	–25.36	–50.72
	–12.38	–123.78	–123.78		0	–35.78	–47.70
					0	–42.26	–42.26

T/℃	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	T/℃	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
300 ℃	0	0	–61.64		–8.87	–8.87	–88.65
	4.14	0	0		–11.03	–29.77	–110.27
	1.86	0	–37.22		–11.31	–47.50	–113.09
	2.34	0	–23.41		–11.88	–61.76	–118.77
	3.11	0	–12.43		–11.60	–89.34	–116.03
	3.48	0	–6.97		–9.62	–96.16	–96.16
	3.81	0	–5.08
	1.68	–33.51	–33.51	600 ℃	0	0	–24.90
	2.22	–22.22	–22.22		1	0	0
	2.7	–10.80	–10.80		0.39	0	–7.82
	3.15	–9.00	–9.00		0.52	0	–5.20
	3.42	–6.85	–6.85		0.58	0	–2.33
	3.56	–4.74	–4.74		0.99	0	–1.98
	3.47	–3.47	–3.47		0.76	0	–1.01
	0	–66.69	–66.69		0.35	–7.00	–7.00
	–12.30	–12.30	–123.00		0.46	–4.65	–4.65
	0	–13.03	–65.16		0.61	–2.42	–2.42
	0	–21.21	–70.70		0.68	–1.93	–1.93
	0	–28.54	–71.35		0.73	–1.46	–1.46
	0	–37.59	–75.18		0.77	–1.02	–1.02
	0	–53.08	–70.77		0.81	–0.81	–0.81
	–12.30	–12.30	–123.00		0	–5.98	–29.90
	–13.89	–37.49	–145.85		0	–10.15	–33.83
	–16.05	–67.41	–160.50		0	–14.18	–35.46
	–15.69	–81.60	–156.92		0	–18.31	–36.61
	–15.41	–118.63	–154.07		0	–25.39	–33.85
	–12.82	–128.22	–128.22		0	–30.65	–30.65
					–6.79	–6.79	–67.86
400 ℃	0	0	–49.70		–7.74	–20.90	–77.41
	2.99	0	0		–8.99	–37.75	–89.89
	1.59	0	–31.80		–9.05	–47.07	–90.51
	2.08	0	–20.79		–8.77	–67.51	–87.67
	2.19	0	–8.75		–7.55	–75.46	–75.46
	2.26	0	–4.52
	2.5	0	–3.33
	1.07	–21.48	–21.48
	1.59	–15.91	–15.91
	1.84	–7.34	–7.34
	2.09	–5.98	–5.98
	2.22	–4.44	–4.44
	2.48	–3.30	–3.30
	2.52	–2.52	–2.52
	0	–11.41	–57.04
	0	–17.81	–59.36
	0	–24.67	–61.68
	0	–32.21	–64.41
	0	–42.21	–60.28
	0	–55.88	–55.88

Appendix 5 The strength of Lightweight aggregate concrete (LAC) under different stress states

Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$	Fc/MPa	$\sigma_{1}$	$\sigma_{2}$	$\sigma_{3}$
16.68	0	0	–16.68		0.86	0.15	–1.15
[50]	1.75	0	0		0.65	0.13	–1.30
	0	–21.35	–21.35		0.41	0.21	–4.10
	–2.82	–2.82	–28.17		0.6	0.6	–0.80
	–3.15	–7.87	–31.49		0.58	0.58	–2.30
	–3.10	–9.30	–31.00		0.71	–2.84	–2.84
	–3.42	–17.08	–34.15		1.97	–2.62	–2.62
	–3.18	–23.85	–31.80		1.72	–1.72	–1.72
	–3.03	–30.32	–30.32		1.75	1.75	1.75
	–11.30	–13.56	–45.19		1.23	0.62	0.62
	–12.08	–24.16	–48.31		1.84	0.46	0.46
	–10.98	–32.93	–43.90		0	–5.25	–20.98
	–9.77	–39.09	–39.09		0	–10.28	–20.56
	–15.20	–15.20	–50.66		0	–15.06	–20.08
	–15.32	–25.53	–51.05		0	–21.35	–21.35
	–14.21	–47.38	–47.38
	–29.43	–29.43	–58.86	25.5	–5.0	–10.0	–43.1
	–25.74	–38.60	–51.47	[36]	–5.0	–15.0	–49.5
	–25.66	–51.31	–51.31		–7.5	–10.0	–48.3
	–41.96	–41.96	–55.94		–7.5	–15.0	–56.9
	–38.03	–50.70	–50.70		–10.0	–15.0	–61.2
	–57.74	–57.74	–57.74		–10.0	–17.5	–62.3
	0.48	0	–9.69		–12.5	–15.0	–63.5
	1.25	0	–4.99		–3.0	–10.0	–45.7
	0.63	0	–6.33		–5.0	–10.0	–54.9
	1.8	0	–3.60		–5.0	–15.0	–65.5
	1.98	0	–2.64		–7.5	–15.0	–65.5
	–10.59	–10.59	–42.34		–10.0	–15.0	–75.9
	0.9	0.09	–0.90		–12.5	–15.0	–82.5