Diagonal Tension Cracking Strength and Risk of RC Deep Beams

Abstract

This study focuses on the shear serviceability of simple and continuous reinforced concrete deep beams. The test results of 81 simple deep beams (i.e., simply supported single-span deep beams) and 29 continuous deep beams, for which their diagonal tension cracking loads were reported, were collected from existing studies. On this basis, the diagonal tension cracking mechanism is discussed, and four existing models for diagonal tension cracking are evaluated. The evaluation results show the existing models fail to accurately reflect the influences of the main design parameters (including the shear span-to-effective depth ratio and main tensile reinforcement ratio) on the diagonal tension cracking strength. Therefore, a new equation for the diagonal tension cracking strength for simple and continuous deep beams is proposed. The proposed equation is verified to be superior to the existing models, showing an average value and a coefficient of variation for tested-to-predicted diagonal tension cracking strength ratios of 1.02 and 0.21, respectively. On the other hand, a probabilistic analysis is conducted to evaluate the diagonal tension cracking risk under service load, showing that 35% of deep beams exhibited a diagonal tension cracking load that is less than the service load, which indicated that diagonal cracks easily occur in RC deep beams under service loads.

1. Introduction

Reinforced concrete (RC) simple deep beams (SDBs) and continuous deep beams (CDBs) are widely used for bridge cap beams, building transfer girders, and pile supported foundations. The shear strength of RC deep beams has been extensively studied by authors [1,2,3] and other researchers [4,5,6,7,8,9,10,11,12,13,14,15,16]. The strut-and-tie mechanism of simple and continuous deep beams was identified in experimental studies [4,6], and shear strength evaluations were carried out based on strut-and-tie or truss models [1,2,3,5,7,8,9,10,11,12,13,14,15,16]. On the other hand, the ultimate shear serviceability of deep beams is also of interest to researchers and engineers.

Unlike the ultimate shear strength, studies on the serviceability (such as diagonal tension cracking load and crack width under service load) of RC deep beams are limited. Due to the tied-arch mechanism, a deep beam can still carry considerable additional load after diagonal cracking. Furthermore, because of the ratio of diagonal tension cracking strength to ultimate shear strength decreases with the decrease in the shear span-to-effective depth ratio [17], the risk of diagonal tension cracking at service loads for deep beams is higher compared to that of slender beams. Birrcher [18] reported that diagonal cracks had been observed in several RC bent caps in service and two had to be retrofitted in a costly manner. Existing models [18,19,20,21] to calculate the diagonal tension cracking load are developed on the basis of limited test results. Different influencing parameters are considered by these models, which results in inconsistent predictions even in the same design condition.

In the present study, a state-of-the-art test database for diagonal tension cracking load of 110 simple and continuous deep beams is collected to evaluate the existing diagonal tension cracking models, and an improved equation for diagonal tension cracking load is proposed. Moreover, a probabilistic analysis is conducted to evaluate the diagonal tension cracking risk under service loads.

2. Database for Diagonal Tension Cracking Strength Evaluation

In order to study the diagonal tension cracking strength of SDBs and CDBs, 81 SDBs [4,18,22,23,24,25] and 29 CDBs [4,26,27,28] were collected and evaluated. Table 1 presents the database for the evaluation. The following criteria are considered in both the SDBs and CDBs: (1) the shear span-to-effective depth ratio a/d ≤ 2.0; (2) dimensions of loading and supporting plates are reported; (3) specimens are reported to fail in shear, and the reported ultimate load is less than 1.1 times of the prediction of by the strut-and-tie model (STM) according to ACI 318-14 when main tensile reinforcements are yielded [2,15]. The last criterion is to ensure that the collected specimens failed in shear tests because the diagonal tension cracking strength and subsequent diagonal cracks are not critical to flexural failure-controlled specimens. The main longitudinal reinforcement for all the beams in the database are deformed steel bars, while deformed or round steel bars were used for distributed web reinforcements.

Table 1. Database for diagonal tension cracking strength evaluation of simple and continuous deep beams.

Beam Type	Authors and Reference	Specimen No.	b	h	d	a	l	a/d	l/d	ρb	ρt	ρl	ρv	ρh	fc′	Vcr,t	Vu,t
			mm	mm	mm	mm	mm			%	%	%	%	%	MPa	kN	kN
Simple deep beams	Moody et al. [24]	III-24a	178	610	533	813	2438	1.52	4.57	2.72	−	2.72	−	−	17.8	89	298
		II-24b	178	610	533	813	2438	1.52	4.57	2.72	−	2.72	−	−	20.6	111	305
		III-27a	178	610	533	813	2438	1.52	4.57	2.72	−	2.72	−	−	21.4	100	349
		III-27b	178	610	533	813	2438	1.52	4.57	2.72	−	2.72	−	−	22.9	111	358
		III-28a	178	610	533	813	2438	1.52	4.57	3.34	−	3.34	−	−	23.3	111	305
		III-28b	178	610	533	813	2438	1.52	4.57	3.34	−	3.34	−	−	22.4	100	342
		III-25a	178	610	533	813	2438	1.52	4.57	3.45	−	3.45	−	−	24.3	122	269
		III-25b	178	610	533	813	2438	1.52	4.57	3.45	−	3.45	−	−	17.2	100	291
		III-26a	178	610	533	813	2438	1.52	4.57	4.25	−	4.25	−	−	21.7	133	422
		III-26b	178	610	533	813	2438	1.52	4.57	4.25	−	4.25	−	−	20.6	111	398
		III-29a	178	610	533	813	2438	1.52	4.57	4.25	−	4.25	−	−	21.7	133	391
		III-29b	178	610	533	813	2438	1.52	4.57	4.25	−	4.25	−	−	25.0	133	438
		III-30	178	610	533	813	2438	1.52	4.57	4.25	−	4.25	0.53	−	25.4	111	480
		III-31	178	610	533	813	2438	1.52	4.57	4.25	−	4.25	0.95	−	22.4	111	510
	Rogowsky et al. [4]	BM1/1.0S	200	1000	950	925	2000	0.97	2.11	0.94	−	0.94	−	−	26.1	350	602
		BM1/1.5S	200	600	535	925	2000	1.73	3.74	1.13	−	1.13	−	−	42.4	240	303
		BM1/1.0N	200	1000	950	925	2000	0.97	2.11	0.94	−	0.94	0.15	−	26.1	350	602
		BM1/1.5N	200	600	535	925	2000	1.73	3.74	1.13	−	1.13	0.19	−	42.4	240	303
	Tan et al. [28]	I-1/0.75	110	500	443	375	1750	0.85	3.95	2.58	−	2.58	−	−	56.3	80	500
		II-1/1.00	110	500	443	500	2000	1.13	4.52	2.58	−	2.58	−	−	77.6	110	255
		III-1/1.50	110	500	443	750	3000	1.69	6.78	2.58	−	2.58	−	−	77.6	70	185
		II-2N/1.00	110	500	443	500	2000	1.13	4.52	2.58	−	2.58	1.43	−	77.6	100	520
		III-2N/1.50	110	500	443	750	3000	1.69	6.78	2.58	−	2.58	1.43	−	77.6	90	335
		I-3/0.75	110	500	443	375	1750	0.85	3.95	2.58	−	2.58	−	1.59	59.2	120	560
		I-4/0.75	110	500	443	375	1750	0.85	3.95	2.58	−	2.58	−	1.59	63.8	140	580
		II-3/1.00	110	500	443	500	2000	1.13	4.52	2.58	−	2.58	−	1.59	78.0	100	390
		II-4/1.00	110	500	443	500	2000	1.13	4.52	2.58	−	2.58	−	1.59	86.3	70	330
		II-5/1.00	110	500	443	500	2000	1.13	4.52	2.58	−	2.58	−	3.17	86.3	130	470
		III-3/1.50	110	500	443	750	3000	1.69	6.78	2.58	−	2.58	−	1.59	78.0	80	200
		III-4/1.50	110	500	443	750	3000	1.69	6.78	2.58	−	2.58	−	1.59	86.3	100	190
		III-5/1.50	110	500	443	750	3000	1.69	6.78	2.58	−	2.58	−	3.17	86.3	100	265
	Oh and Shin [23]	H4100	130	560	500	250	2000	0.50	4.00	1.56	−	1.56	−	−	49.1	285	642
		H4200	130	560	500	425	2000	0.85	4.00	1.56	−	1.56	−	−	49.1	137	401
		H4500	130	560	500	935	2000	1.87	4.00	1.56	−	1.56	−	−	49.1	75	112
		N4200	130	560	500	425	2000	0.85	4.00	1.56	−	1.56	−	−	23.7	96	265
		H41A0	120	560	500	250	2000	0.50	4.00	1.29	−	1.29	0.13	−	50.7	111	347
		H43A0	120	560	500	625	2000	1.25	4.00	1.29	−	1.29	0.13	−	50.7	103	214
		U41A0	120	560	500	250	2000	0.50	4.00	1.29	−	1.29	0.13	−	73.6	220	438
		H41A1	120	560	500	250	2000	0.50	4.00	1.29	−	1.29	0.13	0.23	50.7	206	398
		H41A2(2)	120	560	500	250	2000	0.50	4.00	1.29	−	1.29	0.13	0.47	50.7	257	490
		H41A3	120	560	500	250	2000	0.50	4.00	1.29	−	1.29	0.13	0.94	50.7	238	455
		N42B2(2)	120	560	500	425	2000	0.85	4.00	1.29	−	1.29	0.24	0.47	50.7	146	361
		N42C2(2)	120	560	500	425	2000	0.85	4.00	1.29	−	1.29	0.37	0.47	50.7	167	374
		U41A1	120	560	500	250	2000	0.50	4.00	1.29	−	1.29	0.13	0.23	73.6	253	542
		U41A2	120	560	500	250	2000	0.50	4.00	1.29	−	1.29	0.13	0.47	73.6	293	548
		U41A3	120	560	500	250	2000	0.50	4.00	1.29	−	1.29	0.13	0.94	73.6	196	547
		N42A2	130	560	500	425	2000	0.85	4.00	1.56	−	1.56	0.12	0.43	23.7	57	284
		N42B2	130	560	500	425	2000	0.85	4.00	1.56	−	1.56	0.22	0.43	23.7	128	377
		N42C2	130	560	500	425	2000	0.85	4.00	1.56	−	1.56	0.34	0.43	23.7	124	358
		H41A2(1)	130	560	500	250	2000	0.50	4.00	1.56	−	1.56	0.12	0.43	49.1	257	713
		H41B2	130	560	500	250	2000	0.50	4.00	1.56	−	1.56	0.22	0.43	49.1	220	706
		H41C2	130	560	500	250	2000	0.50	4.00	1.56	−	1.56	0.34	0.43	49.1	208	709
		H42A2(1)	130	560	500	425	2000	0.85	4.00	1.56	−	1.56	0.12	0.43	49.1	208	488
		H42B2(1)	130	560	500	425	2000	0.85	4.00	1.56	−	1.56	0.22	0.43	49.1	217	456
		H42C2(1)	130	560	500	425	2000	0.85	4.00	1.56	−	1.56	0.34	0.43	49.1	122	421
		N33A2	130	560	500	625	1500	1.25	3.00	1.56	−	1.56	0.12	0.43	23.7	97	228
		N43A2	130	560	500	625	2000	1.25	4.00	1.56	−	1.56	0.12	0.43	23.7	110	255
		N53A2	130	560	500	625	2500	1.25	5.00	1.56	−	1.56	0.12	0.43	23.7	69	207
		H31A2	130	560	500	250	1500	0.50	3.00	1.56	−	1.56	0.12	0.43	49.1	230	746
		H51A2	130	560	500	250	2500	0.50	5.00	1.56	−	1.56	0.12	0.43	49.1	206	655
	Brown et al. [25]	I-CL-8.5-0	152	762	686	730	3048	1.06	4.44	1.94	−	1.94	0.43	−	17.8	178	352
		II-N-F-5.8-3	457	457	406	657	3048	1.62	7.50	2.18	−	2.18	0.15	−	19.9	171	798
		II-N-F-5.8-8	457	457	406	657	3048	1.62	7.50	2.18	−	2.18	0.15	−	19.7	206	489
		II-N-F-4.6-8	457	457	406	652	2540	1.60	6.25	2.18	−	2.18	0.15	−	21.6	200	494
Zhang and Tan [22]	2DB50	80	500	459	500	1500	1.09	3.27	1.15	−	1.15	−	−	32.4	55	136
	3DB100	230	1000	904	1000	3000	1.11	3.32	1.20	−	1.20	−	−	29.3	220	672
	3DB70	160	700	642	700	2100	1.09	3.27	1.22	−	1.22	−	−	28.7	160	361
	2DB35	80	350	314	350	1050	1.11	3.34	1.25	−	1.25	−	−	27.4	45	85
	3DB35	80	350	314	350	1050	1.11	3.34	1.25	−	1.25	−	−	27.4	45	85
	2DB100	80	1000	926	1000	3000	1.08	3.24	1.26	−	1.26	−	−	30.6	110	242
	3DB50	115	500	454	500	1500	1.10	3.30	1.28	−	1.28	−	−	28.3	80	167
	2DB70	80	700	650	700	2100	1.08	3.23	1.28	−	1.28	−	−	24.8	80	156
	1DB100bw	230	1000	904	1000	3000	1.11	3.32	1.20	−	1.20	0.46	−	28.7	150	780
	1DB70bw	160	700	642	700	2100	1.09	3.27	1.22	−	1.22	0.42	−	28.3	90	429
	1DB35bw	80	350	313	350	1050	1.12	3.35	1.25	−	1.25	0.47	−	25.9	40	100
	1DB50bw	115	500	454	500	1500	1.10	3.30	1.28	−	1.28	0.33	−	27.4	80	187
Birrcher [18]	III-1.85-00	533	1067	980	1733	6475	1.77	6.60	2.31	−	2.31	−	−	21.2	436	1622
	I-03-2	533	1118	978	1728	6475	1.77	6.62	2.29	−	2.29	0.29	0.33	36.1	641	2531
	I-02-2	533	1118	978	1728	6475	1.77	6.62	2.29	−	2.29	0.20	0.20	27.2	538	2020
	II-03-CCC2021	533	1067	980	1733	6475	1.77	6.61	2.31	−	2.31	0.31	0.45	22.7	618	2224
	II-02-CCC1021	533	1067	980	1768	6475	1.80	6.61	2.31	−	2.31	0.20	0.19	31.9	587	1464
	III-1.85-02	533	1067	980	1733	6475	1.77	6.61	2.31	−	2.31	0.20	0.19	28.3	498	2171
	III-1.85-03	533	1067	980	1733	6475	1.77	6.61	2.31	−	2.31	0.29	0.29	34.4	609	1833
	III-1.85-03b	533	1067	980	1733	6475	1.77	6.61	2.31	−	2.31	0.31	0.29	22.8	507	2095
	III-1.2-02	533	1067	980	1131	6475	1.15	6.61	2.31	−	2.31	0.20	0.19	28.3	734	3763
	IV-2175-1.85-02	533	1905	1750	3054	6475	1.75	3.70	2.37	−	2.37	0.21	0.19	34.0	961	3394
	IV-2175-1.85-03	533	1905	1750	3054	6475	1.75	3.70	2.37	−	2.37	0.31	0.29	34.0	970	3745
	IV-2175-1.2-02	533	1905	1750	2002	6475	1.14	3.70	2.37	−	2.37	0.21	0.21	34.5	1165	5440
	M-03-4-CCC2436	914	1219	1016	1792	6475	1.76	6.37	2.93	−	2.93	0.31	0.27	28.3	1575	5018
Continuous deep beams	Rogowsky et al. [4]	BM7/1.0 T1	200	1000	975	940	2100	0.96	2.15	0.46	0.63	1.09	−	−	34.5	265	418
		BM7/1.5 T1	200	600	545	943	2100	1.73	3.85	0.92	1.12	2.04	−	−	30.4	150	222
		BM5/1.0 T1	200	1000	975	952	2100	0.98	2.15	0.46	0.63	1.09	0.60	−	36.9	410	875
		BM3/1.5 T1	200	600	545	941	2100	1.73	3.85	0.92	1.12	2.04	0.19	−	14.5	110	242
		BM8/1.5 T1	200	600	545	945	2100	1.73	3.85	0.92	1.12	2.04	0.19	0.12	37.2	170	339
	Yang et al. [26,27]	L5NN	160	600	555	217	600	0.39	1.08	0.97	0.97	1.94	−	−	34.9	244	456
		L10NN	160	600	555	520	1200	0.94	2.16	0.97	0.97	1.94	−	−	32.7	171	264
		H6NN	160	600	555	277	720	0.50	1.30	0.97	0.97	1.94	−	−	66.4	303	633
		H10NN	160	600	555	519	1200	0.93	2.16	0.97	0.97	1.94	−	−	69.5	228	372
		L5-40	160	400	355	115	400	0.32	1.13	1.01	1.01	2.02	−	−	33.0	183	408
		L5-72	160	720	653	276	720	0.42	1.10	1.10	1.10	2.20	−	−	33.0	285	492
		L10-40	160	400	355	317	800	0.89	2.25	1.01	1.01	2.02	−	−	32.7	79	202
		L10-72	160	720	653	640	1440	0.98	2.21	1.10	1.10	2.20	−	−	32.7	194	301
		H6-40	160	400	355	159	480	0.45	1.35	1.01	1.01	2.02	−	−	66.4	270	591
		H6-72	160	720	653	352	864	0.54	1.32	1.10	1.10	2.20	−	−	66.4	411	696.5
		H10-40	160	400	355	320	800	0.90	2.25	1.01	1.01	2.02	−	−	68.8	142	335
		H10-72	160	720	653	641	1440	0.98	2.21	1.10	1.10	2.20	−	−	68.8	252	392.5
		H10NS	160	600	555	518	1200	0.93	2.16	0.97	0.97	1.94	0.29	−	69.5	234	413
		L5NS	160	600	555	217	600	0.39	1.08	0.97	0.97	1.94	0.30	−	34.9	247	475
		L10NS	160	600	555	520	1200	0.94	2.16	0.97	0.97	1.94	0.30	−	32.7	165	348
		H6NS	160	600	555	280	720	0.50	1.30	0.97	0.97	1.94	0.30	−	66.4	300	683
		L5NT	160	600	555	218	600	0.39	1.08	0.97	0.97	1.94	0.60	−	34.9	230	512
		L10NT	160	600	555	518	1200	0.93	2.16	0.97	0.97	1.94	0.60	−	32.7	206	446
		H6NT	160	600	555	278	720	0.50	1.30	0.97	0.97	1.94	0.60	−	66.4	264	757
		H10NT	160	600	555	520	1200	0.94	2.16	0.97	0.97	1.94	0.60	−	69.5	224	637
		L5SS	160	600	555	218	600	0.39	1.08	0.97	0.97	1.94	0.30	0.35	34.9	247	607
		L10SS	160	600	555	520	1200	0.94	2.16	0.97	0.97	1.94	0.30	0.35	32.7	148	352
		H6SS	160	600	555	278	720	0.50	1.30	0.97	0.97	1.94	0.30	0.35	66.4	256	799
		H10SS	160	600	555	522	1200	0.94	2.16	0.97	0.97	1.94	0.30	0.35	69.5	232	492

Note: b is the beam width; span length l is the distance between the centers of adjacent supports; ρ_b and ρ_t are bottom and top main tensile reinforcment ratios, respectively; V_cr,t and V_u,t are the reported diagonal tension cracking strength and ultimate shear strength, respectively.

Figure 1 presents the variance of the dimensionless diagonal tension cracking strength $V_{cr,t}/\left(\sqrt{f_c\prime }bd\right)$ according to main design parameters, including the shear span-to-effective depth ratio a/d, the concrete cylinder compressive strength f_c′, main tensile reinforcement ratio ρ_l, ratio of distributed reinforcement passing through a diagonal strut ρ_T, and effective depth d. It should be noted that a is the length of the shear span where shear failure occurs. For CDBs in the database, the shear failure is concentrated in the inner shear span. Effective depth d is the distance from the bottom main longitudinal reinforcement centroid to the top of the beam cross section for both simple and continuous deep beams. The main tensile reinforcement ratio ρ_l of SDBs considers only main longitudinal reinforcement at beam bottom, while ρ_l of CDBs considers main longitudinal reinforcements at both beam top and bottom. The reason is that both the top and bottom reinforcements in tension form truss systems with diagonal struts in CDBs to carry external loads, as shown in Figure 2. Ratio ρ_T equals ρ_hsinθ + ρ_vcosθ, where ρ_h and ρ_v are the horizontal and vertical web reinforcement ratios, respectively, and θ is the inclination of the diagonal strut. The ratios ρ_h and ρ_v equal A_sh1/(s_hb) and A_sv1/(s_vb), respectively, where A_sh1 and A_sv1 are areas of one layer of horizontal and vertical web reinforcement, respectively, and s_h and s_v are spacings of horizontal and vertical web reinforcement, respectively. The strut inclination θ is determined according to [2].

Figure 1. Relationship between dimensionless diagonal tension cracking strength V_cr,t/(√f_c′bd) and design parameters. (a) Shear span-to-effective depth ratio a/d; (b) concrete compressive strength f_c′; (c) main tensile reinforcement ratio ρ_l; (d) distributed reinforcement ratio ρ_T; (e) effective depth d.

Figure 2. Combination of strut-and-tie actions with bottom and top ties in CDBs (adapted from Rogowsky [29]).

From Figure 1, it can be seen that the diagonal tension cracking strength of the CDBs shows similar tendency with that of the SDBs. The diagonal tension cracking strength of both SDBs and CDBs increases significantly with the increase in f_c′ or decrease in a/d. On the other hand, the effect of distributed web reinforcement on the cracking strength is not significant. For the effect of ρ_l and d, further analysis is needed below, although the diagonal tension cracking strength shows slight downward overall trends with increases in ρ_l and d.

3. Diagonal Tension Cracking Strength Modeling and Evaluation

The diagonal tension cracking strength of deep beams can be evaluated by using existing models, such as the splitting model, empirical model, and flexural-shear cracking model. Figure 3 shows the STM based splitting model. In a bottle-shaped strut, lateral spreading of compression force generates tensile stresses perpendicular to the strut and initiates longitudinal splitting cracks near the ends of the strut. On the basis of the splitting model, Foster [19] assumed uniformly distributed transverse stress through the bursting region to obtain the splitting stress (Equation (1)), while Sahoo et al. [20] considered an idealized triangular stress distribution (Equation (2)):

$$V_{cr}=1.2\sqrt{{f^{\prime }}_c}{l^{\prime }}_{s}b\sin \theta$$

(1)

$$V_{cr}=0.56\sqrt{{f^{\prime }}_c}{l}_{s}b\sin \theta$$

(2)

where V_cr is the diagonal tension cracking strength predicted by models; l_s′ is the length of bursting zone in the strut defined by Foster [20], and it is slightly shorter than the strut length l_s (refer to Figure 3).

Figure 3. Splitting cracks of a bottle-shaped strut.

Birrcher [18] proposed an empirical equation (Equation (3)) for the cracking strength of a strut as a function of a/d, without consideration of the diagonal tension cracking mechanism. Moreover, Equation (3) was fitted according to test results of SDBs only, without considering CDBs.

$$0.17\sqrt{{f^{\prime }}_c}bd\le V_{cr}=\left(0.54-0.25a/d\right)\sqrt{{f^{\prime }}_c}bd\le 0.42\sqrt{{f^{\prime }}_c}bd$$

(3)

Kim and White [21] proposed a flexural-shear cracking model on the basis of flexural-shear cracks caused by local stress concentration after flexural cracking initiation. They considered that the local stress concentration is associated with the bond strength between concrete and longitudinal bars and the development of arch action in the shear span of the beam. Because Equation (4) is derived from simple beams, it is not applicable to CDBs.

$$V_{cr}=0.78{\left[\sqrt{{\rho }_b}{\left(1-\sqrt{{\rho }_b}\right)}^2\left(d/a\right)\right]}^{1/3}\sqrt{{f^{\prime }}_c}bd$$

(4)

Figure 4 compares the test results and predictions of Equations (1)–(4). Although different transverse stress distributions are assumed by Foster [19] and Sahoo et al. [20], the splitting models in Equations (1) and (2) overestimate the test results of SDBs and CDBs. Their tested-to-predicted diagonal tension cracking strength ratios V_cr,t/V_cr present average values of 0.43 and 0.67, respectively, and coefficients of variation (COVs) of 0.38 and 0.31, respectively. On the other hand, Birrcher’s empirical model [18] (Equation (3)) and Kim’s flexural-shear cracking model [21] (Equation (4)) predict the test results relatively well, showing lower values of COV (0.25 and 0.27, respectively). However, the inaccuracy of the predictions is still significant in deep beams with a/d ≥ 1.0, as shown in Figure 4.

Figure 4. Comparison between test results and predictions of existing diagonal tension cracking models. (a) Foster’s model [19]; (b) Sahoo’s model [20]; (c) Birrcher’s model [18]; (d) Kim’s model [21].

Considering l_s′sinθ = 0.8d in Equation (1) and l_ssinθ = 0.9d in Equation (2), the splitting models (i.e., Equations (1) and (2)) can be defined as a function of $\sqrt{f_c\prime }bd$. As shown in Figure 4, unlike Equations (3) and (4) showing better predictions, the effect of a/d on the diagonal tension cracking strength is not considered in the splitting models.

Figure 5 compares Equation (3) which neglects the main reinforcement, with Equation (4) which addresses the bottom main reinforcement of SDBs. The prediction by Equation (3) shows a increasing trend of V_cr,t/V_cr with an increase in main tensile reinforcement ratio ρ_l, while the prediction by Equation (4) presents a more consistent trend. It indicates that the diagonal tension cracking strength is affected by the main tensile reinforcement. In Equation (4), Kim’s model [21] based on flexural-shear cracking mechanism identifies the factors that affect the diagonal tension cracking load. However, the model considers some assumptions (such as the arch-like variation of internal moment arm length) only for simple beams. Furthermore, the majority of the specimens for model calibration are slender beams, which deteriorates model accuracy in deep beams.

Figure 5. Diagonal tension cracking strength according to longitudinal bar ratio. (a) Birrcher’s model [18]; (b) Kim’s model [21].

In this study, according to the identified design parameters, an equation with unknown constants (k₁, k₂, and k₃) was proposed: V_cr = k₁(a/d)^k2ρ_l^k3√f_c′bd. The constants are determined by regression analysis of the 110 existing deep beam test results in Table 1. The proposed diagonal tension cracking equation is defined as Equation (5).

$$V_{cr}=0.45{\rho }_l{}^{0.1}{\left(a/d\right)}^{-0.5}\sqrt{{f^{\prime }}_c}bd$$

(5)

Figure 6 compares the test results with the predictions of the proposed model. The proposed method predicted the diagonal tension cracking strength of the test specimens with reasonable precision. The average value and COV of tested-to-predicted diagonal tension cracking strength ratios V_cr,t/V_cr are 1.02 and 0.21, respectively. In addition, the ratios present no significant upward or downward trend with the variation of main design parameters, including the shear span-to-effective depth ratio a/d, the concrete cylinder compressive strength f_c′, main tensile reinforcement ratio ρ_l, ratio of distributed reinforcement passing through a diagonal strut ρ_T, and effective depth d. Therefore, compared with existing models, the proposed model can better predict diagonal tension cracking strengths and can accurately reflect the influence of main design parameters on the diagonal tension cracking strength. It should be noted that the diagonal tension cracking strength of deep beams does not show a significant size effect within the scope of this study. To properly evaluate the impact of size effect on the diagonal tension cracking of deep beams, larger-scale experiments and more theoretical studies are still needed.

Figure 6. Tested-to-predicted diagonal tension cracking strength ratios V_cr,t/V_cr by proposed model according to main design parameters. (a) Shear span-to-effective depth ratio a/d; (b) concrete compressive strength f_c′; (c) main tensile reinforcement ratio ρ_l; (d) distributed reinforcement ratio ρ_T; (e) effective depth d.

4. Diagonal Tension Cracking Risk under Service Load

Birrcher [18] proposed an approach to estimate the service load of AASHTO LRFD as a function of the capacity of test specimens. According to this approach, the service load of ACI 318-14 [30] is calculated as 34% of the test results. The calculation details are presented in Appendix A. This value is regarded as a general representation of the service load on a deep beam [18]. It should be noted that the authors adopt ACI 318-14 instead of ACI 318-19 [31] because ACI 318-19 is the latest version of the ACI design code, and there are still very few deep beams designed according to it in practical engineering. In order to more reasonably estimate the diagonal tension cracking risk of deep beams under service load in practice, the STM in ACI 318-14 was adopted, which has not been substantially changed since it was first introduced into the ACI design code in 2002.

Figure 7 shows the ratios of the tested diagonal tension cracking strength to ultimate shear strength (V_cr,t/V_u,t) of the deep beam specimens in the database. The ratios range from about 0.2 to 0.8. Comparing with the prediction of service load level (=0.34V_t), 35% of deep beams exhibited the diagonal tension cracking load less than the service load (i.e., V_cr/V_t < 34%). This result indicates that diagonal cracks can easily occurr in RC deep beams under service load. Thus, the minimum web reinforcement provision (i.e., vertical and horizontal web reinforcement ratios shall be at least 0.25%) specified in Section 9.9.3.1 of ACI 318-14 [29] should be strictly followed. It was experimentally verified to be efficient for controlling the maximum diagonal crack width by Birrcher [18].

Figure 7. Ratios of diagonal cracking-to-ultimate shear strength according to shear span-to-effective depth ratio a/d.

5. Summary and Conclusions

In the present study, the diagonal tension cracking strength and risk of simple and continuous deep beams under service load were analyzed. To evaluate the validity of existing diagonal tension cracking models, their predictions are compared to existing deep beam test results collected by the authors. Moreover, a new model is proposed. The principal results are summarized as follows:

• The evaluation of the four existing diagonal tension cracking model based on the 110 collected deep beam tests shows that the models fail to accurately reflect the influences of main design parameters on the diagonal tension cracking strength. On the other hand, design parameters $\sqrt{f_c\prime }bd$ a/d, and ρ_l are identified as the influential parameters of the diagonal tension cracking strength.
• Using the design parameters, a new equation is proposed to better predict the diagonal tension cracking strength of deep beams including SDBs and CDBs. The average value and COV of tested-to-predicted diagonal tension cracking strength ratios are 1.02 and 0.21, respectively. Moreover, the proposed model is verified to accurately reflect the influence of main design parameters on the diagonal tension cracking strength of RC SDBs and CDBs.
• Under service load, diagonal cracks occur easily in RC deep beams. Thus, the minimum transverse reinforcement requirement prescribed in the ACI 318 design code should be strictly satisfied.