Analytical prediction of nonlinear behaviors of beams post-tensioned by unbonded tendons considering shear deformation and tension stiffening effect

Figures & data

Figure 1. Behavior of the concrete in tension and compression.

Figure 2. Bilinear stress–strain diagram for both prestressing and non-prestressing steel.

Figure 3. Stress–strain distribution of the cross-section before and after cracking.

Figure 4. Trial-and-error procedure for the section analysis.

Figure 5. Comparison of the experimental mid-span deflection to the analytical model.

Figure 5. Continued

Figure 5. Continued.

Figure 5. Continued.

Figure 6. Typical element strain distribution between adjacent cracks (CEB–FIB Model Code 1990).

Figure 7. Relationship between coefficient of determination – ${\mathrm{R}}^2$ and contribution factors – $\alpha$, $\beta$.

Figure 8. Proposed tension stiffening factor at post-yielding region for unbonded post-tensioned beam.

Figure 9. Procedure for the shear deformation computation.

Figure 10. Deflection computation procedure in all loading stages.

Table 1. Decription of the beam specimens

				Tendon Properties					Rebar Properties
		fck		Ap	fpy	fpu	fpe		Ast	Asc	fsy	fsu
BEAM SERIES	Beam designation	(MPa)		(mm^2)	(MPa)	(MPa)	(MPa)		(mm^2)	(mm^2)	(MPa)	(MPa)	Note
Pan Z et al. (2014)b=250 mm; h=450 mm;L=4000 mm; a=1000 mm;Monotonic loadingStirrup8@200, fy-stirrup=469 MPa	L4 (RC beam)	37		N/A	N/A	N/A	N/A		1901	760	366	366	Verification of shear effect on deflection
Hussien et al. (2012)b=160 mm; h=340 mmL=4000 mm; a=1000 mmCyclic loadingTrapezoidal profile	B7	43		99	1674	1860	1067		157	157	470	610	Regression
	B8	72		99	1674	1860	1067		157	157	470	610	Regression
	B9	95		99	1674	1860	1067		157	157	470	610	Regression
Harajly (1990)b=127 mm; h=228.6 mmL=3048 mm; a=1016 mmMonotonic loadingStraight profile	PP2R3-3	43.16		38.7	1245	1482	952		155	N/A	361	416	Regression
	PP3R3-3	43.16		77.4	1200	1427	883		226	N/A	445	512	Regression
Joseph et al. (1962)b=152.4 mm; h=307.3 mmL=2743 mm; a=914.4 mmMonotonic loadingStraight profile	OW.34.075	37.44		116	1372	1731	837		142	142	329	329	Regression
	OW.34.141	32.82		193	1372	1731	858		142	142	330	330	Regression
	OW.34.155	35.3		231	1372	1731	841		142	142	331	331	Regression
	OS.33.092	25.65		96.1	1372	1731	848		142	N/A	331	331	Regression
	OS.33.238	22.96		231	1372	1731	834		142	N/A	328	328	Regression
	OS.34.042	33.99		58.7	1475	1758	814		142	N/A	328	328	Regression
	OS.34.138	32.2		195	1475	1758	841		142	N/A	332	332	Regression
	OS.34.152	22.41		136	1372	1731	827		142	N/A	325	325	Regression
	OS.34.282	22.96		272	1475	1758	807		142	N/A	337	337	Regression
Tao and Du (1985)b=160 mm; h=280 mmL=4200 mm; a=1400 mmMonotonic loadingStraight profile	A6	30.6		157	1465	1790	854		462	N/A	400	400	Regression
	A8	33.1		58.8	1465	1790	894		462	N/A	400	400	Testing
	A9	33.1		157	1465	1790	920		804	N/A	395	395	Testing

Figure 11. Typical beam specimen dimensions.

Table 2. Comparison of load capacity and deflection at maximum load between test result and analytical model

BEAM SERIES	LOAD CAPACITY					DEFLECTION AT MAXIMUM LOAD
	Mesured load (kN)	Analytical model(proposed factor D)			Analytical model(original factor D)	Mesured def. (mm)		Analytical model(proposed factor D)		Analytical model(original factor D)
		Load (kN)	Ratio	Load (kN)	Ratio		Def. (mm)	Ratio	Def. (mm)		Ratio
B7	140.0	133.3	0.95	134.7	0.96	70.0	67.2	0.96	101.4		1.45
B8	147.0	136.1	0.93	137.4	0.93	69.4	61.4	0.88	91.6		1.32
B9	154.0	139.4	0.91	140.9	0.92	52.4	69.7	1.33	105.9		2.02
PP2R3-3	32.1	35.8	1.11	35.8	1.11	65.5	74.5	1.14	78.5		1.20
PP3R3-3	58.6	60.4	1.03	61.1	1.04	29.0	33.2	1.15	46.2		1.60
OW.34.075	75.4	72.1	0.96	81.2	1.08	26.0	23.4	0.90	36.2		1.40
OW.34.141	95.9	90.9	0.95	108.9	1.14	14.7	15.4	1.05	28.3		1.93
OW.34.155	104.5	100.2	0.96	121.4	1.16	13.3	14.1	1.06	26.9		2.03
OS.33.092	84.4	79.8	0.94	91.6	1.09	20.6	21.1	1.02	34.0		1.65
OS.33.238	126.3	120.4	0.95	133.7	1.06	16.2	16.7	1.03	24.5		1.51
OS.34.042	56.8	55.4	0.98	55.8	0.98	41.0	38.0	0.93	40.8		1.00
OS.34.138	92.3	91.1	0.99	107.5	1.16	13.2	13.4	1.01	24.9		1.89
OS.34.152	74.6	65.6	0.88	71.7	0.96	17.6	16.4	0.93	24.3		1.38
OS.34.282	103.7	95.9	0.92	102.9	0.99	12.1	13.3	1.10	19.2		1.58
A6	99.0	95.3	0.96	99.5	1.00	42.0	38.3	0.91	50.2		1.20
A8	80.2	78.4	0.98	78.4	0.98	63.9	54.5	0.85	62.7		0.98
A9	141.0	130.3	0.92	132.3	0.94	39.0	38.1	0.98	44.8		1.15
Average ratio:			0.96		1.03			1.01			1.49
Mean deviation:			0.03		0.07			0.09			0.27

Note: Ratio = Analytical / Measured.

Table 3. Drucker–Prager coefficients for concrete damage plasticity model

Parameters	Value
Dilation angle	30
Eccentricity	0.1
fb0/fc0	1.16
K	0.67
Viscosity parameter	0.003

Table 4. Interaction setup between each element in numerical model

Interaction type	Interaction behavior	Setup
Stirrup to concrete	Embedded constraint	Default
Longitudinal rebar to concrete	Tie constraint	Default
Unbonded tendon to concrete	Tangential behavior	Frictionless
	Normal behavior	Hard contact

Figure 12. Strain observation in the numerical model.

Table 5. Strain comparisons between simple parabola–rectangle concrete model and modified Kent & Park concrete model

${\mathbf{f}}_{\mathbf{c}\mathbf{k}}$	Strain reaching peak stress		Ultimate compressive strain
	${\epsilon }_{\mathbf{c}1}$ (ABAQUS)	${\epsilon }_{\mathbf{c}2}$ (MATLAB)	${\epsilon }_{\mathbf{c}\mathbf{u}1}$ (ABAQUS)	${\epsilon }_{\mathbf{c}\mathbf{u}2}$ (MATLAB)
$30\mathbf{M}\mathbf{P}\mathbf{a}$	0.0022	0.002	0.0035	0.0035
$40\mathbf{M}\mathbf{P}\mathbf{a}$	0.0023	0.002	0.0035	0.0035
$50\mathbf{M}\mathbf{P}\mathbf{a}$	0.00245	0.002	0.0035	0.0035
$55\mathbf{M}\mathbf{P}\mathbf{a}$	0.0025	0.0022	0.0032	0.0031
$60\mathbf{M}\mathbf{P}\mathbf{a}$	0.0026	0.0023	0.003	0.0029
$70\mathbf{M}\mathbf{P}\mathbf{a}$	0.0027	0.0024	0.0028	0.0027
$80\mathbf{M}\mathbf{P}\mathbf{a}$	0.0028	0.0025	0.0028	0.0026
$90\mathbf{M}\mathbf{P}\mathbf{a}$	0.0029	0.0026	0.0029	0.0026

(1) Modified Kent & Park concrete model using in FE model

(2) Simple parabola–rectangle concrete model using in analytical model

Figure 13. Strain comparison of unbonded beam B7 between proposed analytical model and numerical model.

Figure 14. Strain comparison of unbonded beam B8 between proposed analytical model and numerical model.

Figure 15. Continued.

Figure 15. Continued.

Figure 15. Continued.

Figure 15. Continued.

Table

$\mathrm{L}$	- Beam length	${\epsilon }_{\mathrm{p}\mathrm{y}},{\epsilon }_{\mathrm{p}\mathrm{u}}$	- Yielding, ultimate strain of unbonded tendon, respectively
$\mathrm{M}$	- Internal bending moment	${\epsilon }_{\mathrm{s}\mathrm{m}}$	- Mean strain of reinforcement
${\mathrm{E}}_{\mathrm{c}\mathrm{m}}$	- Modulus of elasticity of concrete	${\epsilon }_{\mathrm{s}1}$	- Strain of reinforcement in uncracked concrete
${\mathrm{G}}_{\mathrm{e}}$	- Uncracked shear modulus	${\epsilon }_{\mathrm{s}2}$	- Strain of reinforcement in cracked section
${\mathrm{G}}_{\mathrm{f}}$	- Effective shear modulus at the flexure cracked region	${\epsilon }_{\mathrm{s}\mathrm{r}1}$	- Steel strain at the point of zero slip under cracking forces reaching ${\mathrm{f}}_{\mathrm{c}\mathrm{t}\mathrm{m}}$
${\mathrm{G}}_{\mathrm{c}\mathrm{r}}$	- Post-cracking shear modulus	${\epsilon }_{\mathrm{s}\mathrm{r}2}$	- Steel strain at crack under cracking forces reaching ${\mathrm{f}}_{\mathrm{c}\mathrm{t}\mathrm{m}}$
$\mathrm{Q}$	- Internal shear force	${\epsilon }_{\mathrm{c}\mathrm{p}}$	- Strain of concrete adjacent to unbonded tendon
$\mathrm{c}$	- Neutral axis depth	$\mathrm{\Delta }{\epsilon }_{\mathrm{p}}$	- Increasing strain of the unbonded tendon
${\mathrm{d}}_{\mathrm{s}}$	- Effective depth of tension rebar	${\mu }_{\mathrm{p}}$	- Poisson’s ratio of concrete
${\mathrm{d}}_{\mathrm{s}\mathrm{c}}$	- Effective depth of compression rebar	$\mu$	- Deflection ductility factor
${\mathrm{d}}_{\mathrm{p}}$	- Effective depth of unbonded tendon	${\sigma }_{\mathrm{c}\mathrm{t}}$	- Concrete tensile stress
${\mathrm{f}}_{\mathrm{c}\mathrm{k}}$	- Characteristic compressive cylinder strength of concrete	${\sigma }_{\mathrm{c}}$	- Concrete compressive stress
${\mathrm{f}}_{\mathrm{c}\mathrm{t}\mathrm{m}}$	- Mean tensile strength of concrete	${\sigma }_{\mathrm{s}}$	- Tensile stress of reinforcement
${\mathrm{f}}_{\mathrm{s}\mathrm{y}},{\mathrm{f}}_{\mathrm{s}\mathrm{u}}$	- Yielding, ultimate strength of reinforcement, respectively	${\sigma }_{\mathrm{s}\mathrm{r}1}$	- Steel stress in crack section, when first crack has formed
${\mathrm{f}}_{\mathrm{p}\mathrm{y}},{\mathrm{f}}_{\mathrm{p}\mathrm{u}}$	- Yielding, ultimate strength of unbonded tendon, respectively	${\sigma }_{\mathrm{s}\mathrm{r}\mathrm{n}}$	- Steel stress in the crack, when stabilized crack pattern has formed
$\mathrm{h}$	- Beam depth	$\delta$	- Beam deflection
${\epsilon }_{\mathrm{c}2}$	- Concrete compressive strain reaching peak stress	${\theta }_{\mathrm{f}}$	- Flexural rotation
${\epsilon }_{\mathrm{c}\mathrm{u}2}$	- Ultimate compressive strain in concrete	${\theta }_{\mathrm{s}}$	- Shear rotation
${\epsilon }_{\mathrm{s}\mathrm{y}},{\epsilon }_{\mathrm{s}\mathrm{u}}$	- Yielding, ultimate strain of reinforcement, respectively	${\theta }_{\mathrm{f}\mathrm{s}}$	- Flexural shear rotation

London: Thomas Telford. 1990. CEB – FIP Model Code for Concrete Structures. Comite Euro-international du Beton-Federation international. London, UK: Thomas Telford Ltd.

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Hussien, O. F., T. H. K. Elafandy, A. A. Abdelrahman, S. A. Baky, and E. A. Nasr. 2012. “Behavior of Bonded and Unbonded Prestressed Normal and High Strength Concrete Beams.” HBRC Journal 8 (3): 239–251. doi:https://doi.org/10.1016/j.hbrcj.2012.10.008.

 Google Scholar

Tao, X., and G. Du. 1985. “Ultimate Stress of Unbonded Tendons in Partially Prestressed Concrete Beams.” PCI Journal 30 (6): 72–91. doi:https://doi.org/10.15554/pcij.11011985.72.91.

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