Analysis of the nodal stresses in pile caps

Pile caps can be dimensioned using, preferably, plastic models (strut-and-tie) and models based on the flexion theory. In order to analyze the behavior of the stresses in the lower and upper nodal regions of the cap, a theoretical analysis of the experimental results found by several researchers was made. There was a discrepancy in the results obtained and, as a result, a critical analysis carried out and a new methodology for the verification of the nodal stress near the upper zone, based on the multiaxial behavior of the concrete, is suggested.

Keywords:
pile caps; strut-and-tie model; nodal stress

Authorship

M. A. TOMAZ

Universidade Federal de Uberlândia, Faculdade de Engenharia Civil, Uberlândia, MG, Brasil; eng.matomaz@gmail.com delalibera@ufu.brUniversidade Federal de UberlândiaBrazilUberlândia, MG, BrazilUniversidade Federal de Uberlândia, Faculdade de Engenharia Civil, Uberlândia, MG, Brasil; eng.matomaz@gmail.com delalibera@ufu.br

R. G. DELALIBERA

J. S. GIONGO

Universidade de São Paulo, Escola de Engenharia de São Carlos, Departamento de Engenharia de Estruturas, São Carlos-SP, Brasil. jsgiongo@sc.usp.br vitorfrg@gmail.comUniversidade de São PauloBrazilSão Carlos, SP, BrazilUniversidade de São Paulo, Escola de Engenharia de São Carlos, Departamento de Engenharia de Estruturas, São Carlos-SP, Brasil. jsgiongo@sc.usp.br vitorfrg@gmail.com

V. F. GONÇALVES

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables

Figures (28)
Tables (33)

Tested model	Pile number	Height (cm)	Length (cm)	Pile section (cm×cm)	Distance between piles (cm)	Column section (cm×cm)	Column reinforcement rate (%)	θ (º)	f_c,exp (MPa)
B1-1	2	23	50	10×15	31	15×15	-	56.02	21.50
B2-1	2	23	50	10×15	31	15×15	-	26.02	21.50
B1-2	2	23	50	10×15	32	15×15	-	55.18	15.00
B2-2	2	23	50	10×15	32	15×15	-	55.18	15.00
B1-A	2	23	50	10×15	32	15×15	-	55.18	32.30
B2-A	2	23	50	10×15	32	15×15	-	55.18	32.30
B1-B	2	20	50	10×15	32	15×15	-	51.34	32.00
B2-B	2	20	50	10×15	32	15×15	-	51.34	32.00
B1-4A	2	20	50	10×15	32	15×15	-	51.34	29.50
B2-4A	2	20	50	10×15	32	15×15	-	51.34	29.50
B1-4B	2	20	50	10×15	32	15×15	-	51.34	27.80
B2-4B	2	20	50	10×15	32	15×15	-	51.34	27.80
B1-4C	2	20	50	10×15	32	15×15	-	51.34	22.20
B2-4C	2	20	50	10×15	32	15×15	-	51.34	22.20
D1	2	21	50	10×15	35	15×15	-	50.19	22.90
D2	2	21	50	10×15	35	15×15	-	50.19	22.90
F1	2	20	50	10×15	40	15×15	-	45.00	23.60
F2	2	20	50	10×15	40	15×15	-	45.00	23.60
E1	2	20	50	10×15	45	15×15	-	41.63	19.50
G1	2	20	50	10×15	45	15×15	-	41.63	24.30

Tested model	F_r. (kN)	F_u,exp (kN)	R_est (kN)	R_cc (kN)	R_st (kN)
B1-1	368,669	508,165	254,083	306,394	171,230
B2-1	318,849	508,165	254,083	306,394	171,230
B1-2	199,280	348,741	174,371	212,412	121,301
B2-2	199,280	348,741	174,371	212,412	121,301
B1-A	348,741	474,301	237,151	288,889	164,974
B2-A	348,741	747,301	373,651	455,169	259,931
B1-B	348,741	727,373	363,687	465,746	290,949
B2-B	318,849	727,373	363,687	465,746	290,949
B1-4A	298,920	667,589	333,795	427,466	267,036
B2-4A	318,849	667,589	333,795	427,466	267,036
B1-4B	308,884	627,733	313,867	401,945	251,093
B2-4B	318,849	627,733	313,867	401,945	251,093
B1-4C	249,100	498,201	249,101	319,004	199,280
B2-4C	298,920	498,201	249,101	319,004	199,280
D1	229,172	508,165	254,083	330,741	211,735
D2	229,172	508,165	254,083	330,741	211,735
F1	229,172	478,723	239,362	338,508	239,362
F2	209,244	478,273	239,137	338,190	239,137
E1	169,388	368,669	184,335	277,460	207,376
G1	199,280	458,345	229,173	344,950	257,819

Column reinforcement rate (%)	1%	2%	3%
Square columns	0.8	1.0	1.2
Very wide columns	0.35	0.42	1.0

Criteria	CCC	CCT	CTT or TTT
Blévot and Frémy [2]	1.40 f_c; upper nodal area (for pile caps on two piles) 1.75 f_c; upper nodal area (for pile caps on three piles) 2.10 f_c; upper nodal area (for pile caps on four piles) f_c; lower nodal zone (for pile caps in any situation)
Schlaich and Schäfer [14]	1.10 f_c	0.80 f_c	0.80 f_c
Fusco [15]	2/9 f_c	0.50 f_c	0.50 f_c
ABNT NBR 6118:2014 [2]	0.85 f_c	0.72 f_c	0.60 ff_c
EHE-1998 [16]	3.00 f_c	0.70 f_c	0.70 f_c
ACI-2014 [17]	0.85 f_c	0.68 f_c	0.51 f_c
CEB-fib [18]	0.85 f_c	0.60 f_c	0.60 f_c
CEB-fib [19]	1.00 f_c	0.75 f_c	0.75 f_c
Triple stress state	f_ck + 4 f_ctk	-	-

Active stresses (kN/cm²)
Tested model	Blévot and Frémy [2]		Fusco [15]		Schlaich and Schäfer [14]
Tested model	σ_zni	σ_zns	σ_zni	σ_zns	σ_zni	σ_zns
B1-1	2.46	3.28	1.21	0.36	1.55	2.26
B2-1	2.46	3.28	1.21	0.36	1.55	2.26
B1-2	1.73	2.30	0.83	0.26	1.07	1.55
B2-2	1.73	2.30	0.83	0.26	1.07	1.55
B1-A	2.35	3.13	1.13	0.35	1.46	2.11
B2-A	3.70	4.93	1.78	0.55	2.30	3.32
B1-B	3.98	5.30	1.73	0.59	2.34	3.23
B2-B	3.98	5.30	1.73	0.59	2.34	3.23
B1-4A	3.65	4.87	1.59	0.54	2.15	2.97
B2-4A	3.65	4.87	1.59	0.54	2.15	2.97
B1-4B	3.43	4.58	1.49	0.51	2.02	2.79
B2-4B	3.43	4.58	1.49	0.51	2.02	2.79
B1-4C	2.72	3.63	1.19	0.40	1.60	2.21
B2-4C	2.72	3.63	1.19	0.40	1.60	2.21
D1	2.87	3.83	1.21	0.43	1.66	2.26
D2	2.87	3.83	1.21	0.43	1.66	2.26
F1	3.19	4.26	1.14	0.47	1.70	2.13
F2	3.19	4.25	1.14	0.47	1.70	2.13
E1	2.78	3.71	0.88	0.41	1.40	1.64
G1	3.46	4.62	1.09	0.51	1.74	2.04
Limit stresses (kN/cm²)
Blévot and Frémy [2]		Fusco [15]		Schlaich and Schäfer [14]		ABNT NBR 6118:2014 [2]		EHE-1998 [16]		ACI 318-14 [17]		CEB-fib 1990 [18]		CEB-fib 2010 [19]		Triple stress state
σ_zni	σ_zns	σ_zni	σ_zns	σ_zni	σ_zns	σ_zni	σ_zns	σ_zni	σ_zns	σ_zni	σ_zns	σ_zni	σ_zns	σ_zni	σ_zns	σ_zns
2.15	3.01	1.08	1.48	1.72	2.37	1.55	1.83	1.51	6.45	1.72	2.15	1.29	1.83	1.61	2.15	3.08
2.15	3.01	1.08	0.48	1.72	2.37	1.55	1.83	1.51	6.45	1.72	2.15	1.29	1.83	1.61	2.15	3.08
1.50	2.10	0.75	0.33	1.20	1.65	1.08	1.28	1.05	4.50	1.20	1.50	0.90	1.28	1.13	1.50	2.23
1.50	2.10	0.75	0.33	1.20	1.65	1.08	1.28	1.05	4.50	1.20	1.50	0.90	1.28	1.13	1.50	2.23
3.23	4.42	1.62	0.72	2.58	3.55	2.33	2.75	2.26	9.69	2.58	3.23	1.94	2.75	2.42	3.23	4.45
3.23	4.52	1.62	0.72	2.58	3.55	2.33	2.75	2.26	9.69	2.58	3.23	1.94	2.75	2.42	3.23	4.45
3.20	4.48	1.60	0.71	2.56	3.52	2.30	2.72	2.24	9.60	2.56	3.20	1.92	2.72	2.40	3.20	4.41
3.20	4.48	1.60	0.71	2.56	3.52	2.30	2.72	2.24	9.60	2.56	3.20	1.92	2.72	2.40	3.20	4.41
2.95	4.13	1.48	0.66	2.36	3.25	2.12	2.51	2.07	8.85	2.36	2.95	1.77	2.51	2.21	2.95	4.10
2.95	4.13	1.48	0.66	2.36	3.25	2.12	2.51	2.07	8.85	2.36	2.95	1.77	2.51	2.21	2.95	4.10
2.78	3.89	1.39	0.62	2.22	3.06	2.00	2.36	1.95	8.34	2.22	2.78	1.67	2.36	2.09	2.78	3.88
2.78	3.89	1.39	0.62	2.22	3.06	2.00	2.36	1.95	8.34	2.22	2.78	1.67	2.36	2.09	2.78	3.88
2.22	3.11	1.11	0.49	1.78	2.44	1.60	1.89	1.55	6.66	1.78	2.22	1.33	1.89	1.67	2.22	3.17
2.22	3.11	1.11	0.49	1.78	2.44	1.60	1.89	1.55	6.66	1.78	2.22	1.33	1.89	1.67	2.22	3.17
2.29	3.21	1.15	0.51	1.83	2.52	1.65	1.95	1.60	6.87	1.83	2.29	1.37	1.95	1.72	2.29	3.26
2.29	3.21	1.15	0.51	1.83	2.52	1.65	1.95	1.60	6.87	1.83	2.29	1.37	1.95	1.72	2.36	3.26
2.36	3.30	1.18	0.52	1.89	2.60	1.70	2.01	1.65	7.08	1.89	2.36	1.42	2.01	1.77	2.36	3.35
2.36	3.30	1.18	0.52	1.89	2.60	1.70	2.01	1.65	7.08	1.89	2.36	1.42	2.01	1.77	2.36	3.35
1.95	2.73	0.98	0.43	1.56	2.15	1.40	1.66	1.37	5.85	1.56	1.95	1.17	1.66	1.46	1.95	2.82
2.43	3.40	1.22	0.54	1.94	2.67	1.75	2.07	1.70	7.29	1.94	2.43	1.46	2.07	1.82	2.43	3.44
Limit stresses average
2.49	3.48	1.24	0.55	1.99	2.74	1.79	2.11	1.74	7.46	1.99	2.49	1.49	2.11	1.87	2.49	3.50

Tested model	Pile number	Height (cm)	Length (cm)	Pile diameter (cm)	Distance between piles x (cm)	Distance between piles y (cm)	Column section (cm×cm)	Column reinforcement rate (%)	θ_x (º)	θ_x (º)	f_c,exp (MPa)
A	4	60	-	20	156	90	30×30	0.891	37.57	53.13	24.80
B	4	60	-	20	156	90	30×30	0.891	37.57	53.13	24.80
C	6	60	260	20	90	90	30×30	0.891	30.81	53.13	27.10
D	4	60	-	20	156	90	30×30	0.891	37.57	53.13	30.30
E	4	60	-	20	156	90	30×30	0.891	37.57	53.13	41.10
F	4	60	-	20	156	90	30×30	0.891	37.57	53.13	30.30

Tested model	Pile number	Height (cm)	Length (cm)	Pile section (cm×cm)	Distance between piles (cm)	Column section (cm×cm)	Column reinforcement rate (%)	θ (º)	f_c,exp (MPa)
B20A1/1	3	60	-	20	96	35×35	9.816	52.00	27.40
B20A1/2	3	60	-	20	96	35×35	9.816	52.00	33.00
B20A2	3	60	-	20	96	35×35	9.816	52.00	35.50
B20A3	3	60	-	20	96	35×35	9.816	52.00	37.90
B20A4	3	60	-	20	96	35×35	9.816	52.00	35.60
B30A1	3	60	-	30	96	35×35	9.816	52.00	31.00
B30A2	3	60	-	30	96	35×35	9.816	52.00	40.30
B30A3	3	60	-	30	96	35×35	9.816	52.00	24.50
B30A4	3	60	-	30	96	35×35	9.816	52.00	24.60

Tested model	Pile number	Height (cm)	Length (cm)	Pile section (cm×cm)	Distance between piles (cm)	Column section (cm×cm)	Column reinforcement rate (%)	θ (º)	f_c,exp (MPa)
B35P25E25 and 0	2	35	117.50	25×25	62.50	25×25	1.0053	45.00	40.60
B35P25E25 and 2,5	2	35	117.50	25×25	62.50	25×25	1.0053	45.00	40.60
B35P25E25 and 0_AswC	2	35	117.50	25×25	62.50	25×25	1.0053	45.00	32.80
B35P25E25 and 0A_sw0	2	35	117.50	25×25	62.50	25×25	1.0053	45.00	32.80
B35P25E25 and 0CG	2	35	117.50	25×25	62.50	25×25	1.0053	45.00	28.90
B45P25E25 and 0	2	45	117.50	25×25	62.50	25×25	2.7489	54.50	31.00
B45P25E25 and 5	2	45	117.50	25×25	62.50	25×25	2.7489	54.50	31.00
B45P25E25 and 0A_swC	2	45	117.50	25×25	62.50	25×25	2.7489	54.50	32.40
B45P25E25 and 0A_sw0	2	45	117.50	25×25	62.50	25×25	2.7489	54.50	32.40
B45P25E25 and 0CG	2	45	117.50	25×25	62.50	25×25	2.7489	54.50	28.90
B35P50E25 and 0	2	35	117.50	25×25	62.50	25×50	0.87965	53.10	35.80
B35P50E25 and 12.5	2	35	117.50	25×25	62.50	25×50	0.87965	53.10	35.10
B45P50E25 and 0	2	45	117.50	25×25	62.50	25×50	1.3745	61.80	35.80
B45P50E25 and 12.5	2	45	117.50	25×25	62.50	25×50	1.3745	61.80	35.10

Tested model	Pile number	Height (cm)	Length (cm)	Pile section (cm×cm)	Distance between piles (cm)	Column section (cm×cm)	Column reinforcement rate (%)	θ (º)	f_c,exp (MPa)
B110P125R1	2	40	110	12.5×12.5	60	12.5×12.5	5.12	56.30	30.47
B110P125R25	2	40	110	12.5×12.5	60	12.5×12.5	5.12	53.20	30.47
B110P125R4	2	40	110	12.5×12.5	60	12.5×12.5	5.12	43.90	30.47
B115P125R1	2	40	115	12.5×12.5	65	12.5×25	5.12	53.00	30.47
B115P125R25	2	40	115	12.5×12.5	65	12.5×25	5.12	49.30	30.47
B115P125R4	2	40	115	12.5×12.5	65	12.5×25	5.12	57.40	30.47
B120P125R1	2	40	120	12.5×12.5	70	12.5×37.5	4.267	55.70	30.47
B120P125R25	2	40	120	12.5×12.5	70	12.5×37.5	4.267	51.90	30.47
B120P125R4	2	40	120	12.5×12.5	70	12.5×37.5	4.267	55.20	30.47
B127P125R1	2	40	127	12.5×12.5	75	12.5×50	4.48	52.90	30.47
B127P125R25	2	40	127	12.5×12.5	75	12.5×50	4.48	49.60	30.47
B127P125R4	2	40	127	12.5×12.5	75	12.5×50	4.48	53.60	30.47