Prestressed Beams

Abstract

After a historical note about the origin of prestressing and its expected effects on RC elements, this chapter presents the main features of the two technologies, one based on the pretensioning and the other on the postensioning of the steel tendons, including the effects of prestressing losses. A discourse on the tendon profile in the beams is developed to orientate the deign choices. The resistance calculations of the current prestressed sections are eventually presented, concluding with the specific analysis of tendons anchorage and stresses diffusion. In the final section three calculation examples are shown related one to a precast pretensioned floor element, one to a precast post-tensioned beam and the last one to a flanged beam provided with a cast-in situ upper slab.

The original version of this chapter was revised: For detailed information please see Erratum. The erratum to this chapter is available at 10.1007/978-3-319-52033-9_11

Appendix: Data on Prestressing

10.1.1 Chart 10.1: Prestressing and Instantaneous Losses

Beams in pressed reinforced concrete.

Symbols

P _o :

initial prestressing (at concrete decompression)

N _po = P _o cos ϕ :

axial component of P _o

V _po = P _o sin ϕ :

transverse component of P _o

${P}_{\text{o}}^{\prime }$ :

prestressing at tendons tensioning

${N}_{\text{po}}^{\prime } = {P}_{\text{o}}^{\prime } \cos \,{\phi }$ :

axial component of $P_{o}^{\prime }$

${V}_{\text{po}}^{\prime } = {P}_{\text{o}}^{\prime } \sin \,{\phi }$ :

transverse component of $P_{o}^{\prime }$

ϕ:

angle of the tendon on the beam axis

e:

tendon eccentricity (positive towards the bottom)

${M}_{\text{g}}^{\prime }$ :

bending moment due to self-weight

${V}_{\text{g}}^{\prime }$ :

shear due to self-weigth

A _p :

area of prestressing reinforcement

E _p :

elastic modulus of prestressing reinforcement

see also Charts 3.4 and 6.1.

Initial Stresses “i”

(post-tensioned tendons)

$$ \begin{array}{*{20}l} {{\sigma }_{\text{ci}} = - \frac{{{N}_{\text{po}}^{\prime } }}{{{\bar{A}}_{\text{i}} }} - \frac{{{M}_{\text{g}}^{\prime } - {N}_{\text{po}}^{\prime } {\bar{e}}}}{{{\bar{I}}_{\text{i}} }}{\bar{y}}_{\text{c}} } \hfill & {\text{upper edge}} \hfill \\ {{\sigma }_{{\overline{G} {i}}} = - \frac{{{N}_{\text{po}}^{\prime } }}{{{\bar{A}}_{\text{i}} }}\quad {\tau }_{{\overline{G} }} = \frac{{{V}_{\text{g}}^{\prime } - {V}_{\text{po}}^{\prime } {\bar{e}}}}{{{\bar{z}}{b}_{{\overline{G} }} }}} \hfill & {\text{centroid}} \hfill \\ {{\sigma }_{\text{ci}}^{*} = - \frac{{{N}_{\text{po}}^{\prime } }}{{{\bar{A}}_{\text{i}} }} + \frac{{{M}_{\text{g}}^{\prime } - {N}_{\text{po}}^{\prime } {\bar{e}}}}{{{\bar{I}}_{\text{i}} }}{\bar{e}}} \hfill & {\text{tendon level}} \hfill \\ {{\sigma }_{\text{ci}}^{\prime } = - \frac{{{N}_{\text{po}}^{\prime } }}{{{\bar{A}}_{\text{i}} }} + \frac{{{M}_{\text{g}}^{\prime } - {N}_{\text{po}}^{\prime } {\bar{e}}}}{{{\bar{I}}_{\text{i}} }}{\bar{y}}_{\text{c}}^{\prime } } \hfill & {\text{lower edge}} \hfill \\ {{\sigma }_{\text{pi}} = \frac{{{P}_{\text{o}}^{\prime } }}{{{A}_{\text{p}} }}} \hfill & {\text{tendon}} \hfill \\ \end{array} $$

with positive tensile stresses where

$$ {\bar{A}}_{\text{i}} = {A}_{\text{c}} + {\alpha }_{\text{e}} {A}_{\text{s}} \quad {\alpha }_{\text{e}} = {E}_{\text{s}} /{E}_{\text{c}} $$

and with ${\bar{I}}_{\text{i}} ,\,{\bar{y}}_{\text{c}},\,{\bar{y}}_{\text{c}}^{\prime },$ … calculated similarly to the homogenized concrete section with passive reinforcement.

The allowable stresses for the verification in service can be deduced from Charts 2.2 and 2.3.

Losses Due to Friction

(post-tensioned tendons)

$$ {P} = {P}_{1} {e}^{{ - {\mu} ({\psi } + {\alpha }{s})}} $$

with

P ₁ :

tension applied at end 1 (active side)

P :

force in the tendon at the abscissa s from end 1

s :

abscissa of the considered section (in m)

ψ :

progressive angle in rad between 1 and s (absolute value)

μ :

friction coefficient in the duct

α :

unintentional unit angular deviation

Without more accurate data, for tendons against metallic ducts without rust one can assume:

$$ \begin{aligned} & \mu = 0.20\quad {\text{for wires or strands}} \\ & \mu = 0. 3 5\quad {\text{for smooth bars}} \\ & \mu = 0.65\quad {\text{for ribbed bars}} \\ & \alpha = 0.01\quad {\text{rad/m}} \\ \end{aligned} $$

Elastic Losses

(pre-tensioned tendons)

$$ \begin{array}{*{20}l} {{\sigma }_{\text{ci}} = - \frac{{{N}_{\text{po}} }}{{{A}_{\text{i}} }} - \frac{{{M}_{\text{g}}^{\prime } - {N}_{\text{po}} {e}}}{{{I}_{\text{i}} }}{y}_{\text{c}} } \hfill & {\text{upper edge}} \hfill \\ {{\sigma }_{\text{Gi}} = - \frac{{{N}_{\text{po}} }}{{{A}_{\text{i}} }}\quad {\tau }_{\text{G}} = \frac{{{V}_{\text{g}}^{\prime } - {V}_{\text{po}} }}{{{zb}_{\text{G}} }}} \hfill & {\text{centroid}} \hfill \\ {{\sigma }_{\text{ci}}^{*} = - \frac{{{N}_{\text{po}} }}{{{A}_{\text{i}} }} + \frac{{{M}_{\text{g}}^{\prime } - {N}_{\text{po}} {e}}}{{{I}_{\text{i}} }}{e}} \hfill & {\text{tendon level}} \hfill \\ {{\sigma }_{\text{ci}}^{\prime} = - \frac{{{N}_{\text{po}} }}{{{A}_{\text{i}} }} + \frac{{{M}_{\text{g}} - {N}_{\text{po}} {e}}}{{{I}_{\text{i}} }}{y}_{\text{c}}^{\prime } } \hfill & {\text{lower edge}} \hfill \\ {{\sigma }_{\text{pi}}^{\prime } = {\sigma }_{\text{po}} + {\alpha }_{\text{e}}^{\prime } {\sigma }_{\text{ci}}^{*} } \hfill & {{\text{tendon}} ({\sigma }_{\text{po}} = {P}_{\text{o}} /{A}_{\text{p}} )} \hfill \\ \end{array} $$

with

$$ {A}_{\text{i}} = {A}_{\text{c}} + {\alpha }_{\text{e}} {A}_{\text{s}} + {\alpha }_{\text{e}}^{\prime } {A}_{\text{p}} \quad {\alpha }_{\text{e}}^{\prime } = {E}_{\text{p}} /{E}_{\text{c}} $$

and with ${I}_{\text{i}} ,\,{y}_{\text{c}},\,{y}_{\text{c}}^{\prime },$ … calculated similarly to the homogenized concrete section. With passive and active reinforcement (with cos ϕ ≅ 1, N _po ≅ P _o).

10.1.2 Chart 10.2: Losses Due to Steel Relaxation

The tension loss due to steel relaxation, calculated after t hours from the load application in relation to the initial stress σ _pi, is given by

$$ {\rho }_{\tau } = {\rho }({\tau }) = \frac{{\Delta {\sigma }_{{\text{p}{\tau }}} }}{{{\sigma }_{\text{pi}} }} = {\rho }_{\text{l}} {\tau }^{{0.75(1 - {r})}} $$

with τ = t/1000 and r = σ _pi/f _ptk. The loss at 1000 h (τ = 1) is given by

$$ {\rho }_{1} = \bar{\rho }_{1} {c}({r}) $$

Where $\bar{\rho }_{1}$ is the loss measured at the time above for an initial stress corresponding to r = 0.7 and an average temperature of 20 °C. For the following product classes

• Class 1—Ordinary wires and strands

• Class 2—Stabilized wires and strands

• Class 3—Prestressing bars

In the absence of test results, one can assume

$$ \begin{aligned} \overline{\rho }_{1} & = 8.0\% \quad {\text{for}}\,{\text{class}}\,1 \\ \overline{\rho }_{1} & = 2.5\% \quad {\text{for}}\,{\text{class}}\,2 \\ \overline{\rho }_{1} & = 4.0\% \quad {\text{for}}\,{\text{class}}\,3 \\ \end{aligned} $$

and

$$ \begin{array}{*{20}l} {{c}({r}) = \left[ {\frac{{{r} - 0.4}}{0.3}} \right]^{{\frac{4}{3}}} } \hfill & {{\text{for class}}\,1} \hfill \\ {{c}({r}) = \left[ {\frac{{{r} - 0.5}}{0.2}} \right]^{{\frac{4}{3}}} } \hfill & {{\text{for classes}}\,2\,{\text{and}}\,3} \hfill \\ \end{array} $$

The final value of the loss ${\rho }_{\infty } = \Delta {\sigma }_{{\text{p}\infty }} /{\sigma }_{\text{pi}}$ can be calculated with τ = 500 (≈57 years). The table gives the final loss for the different values of the initial stress.

$\frac{{{\sigma }_{\text{pi}} }}{{{f}_{\text{ptk}} }}$	$100\Delta {\sigma }_{{\text{p}\infty }} /{\sigma }_{\text{pi}}$			$\frac{{{\sigma }_{\text{pi}} }}{{{f}_{\text{ptk}} }}$	$100\Delta {\sigma }_{{\text{p}\infty }} /{\sigma }_{\text{pi}}$
	Class 1	Class 2	Class 3		Class 1	Class 2	Class 3
0.40	0.00
0.42	3.23			0.66	32.25	9.06	14.49
0.44	7.41			0.67	32.36	9.37	15.00
0.46	11.59			0.68	32.43	9.65	15.45
0.48	15.50			0.69	32.43	9.90	15.84
0.50	19.01	0.00	0.00	0.70	32.39	10.12	16.19
0.52	22.09	1.09	1.74	0.71	32.29	10.31	16.49
0.54	24.71	2.50	3.99	0.72	32.15	10.47	16.75
0.56	26.90	3.90	6.25	0.73	31.98	10.60	16.96
0.58	28.67	5.22	8.35	0.74	31.76	10.71	17.14
0.60	30.06	6.40	10.24	0.75	31.51	10.79	17.27
0.61	30.62	6.94	11.10	0.76	31.22	10.86	17.37
0.62	31.10	7.44	11.90	0.77	30.91	10.90	17.43
0.63	31.49	7.90	12.64	0.78	30.57	10.92	17.47
0.64	31.81	8.32	13.31	0.79	30.21	10.92	17.47
0.65	32.06	8.71	13.93	0.80	29.82	10.90	17.45

10.1.3 Chart 10.3: Experimental Evaluation of Relaxation

If the experimental value $\rho_{1exp}$ of the tension loss due to relaxation is available, measured at 1000 h from the load application for an initial stress $\Delta {\sigma }_{{\text{i}\,\exp }} = {r}_{\exp } {f}_{\text{ptk}}$, the final loss $\Delta {\sigma }_{{\text{p}\infty }} = {\rho }_{\infty } {\sigma }_{\text{pi}}$ can be evaluated with

$$ {\rho }_{\infty } = {\kappa} \bar{\rho }_{1} $$

where

$$ \begin{aligned} & {\kappa } = {c}({r})500^{{0.75(1 - {r})}} \\ & \bar{\rho }_{1} = {\rho }_{1\exp } /{c}({r}_{\exp } ) \\ \end{aligned} $$

(see Chart 10.2 for the expression c(r)).

The following tables give the values of the factors κ(r) and 1/c(r _exp), respectively, for the different product classes as defined Chart 10.2.

${\kappa} = {\kappa} ({r})$	${r} = {\sigma }_{\text{pi}} /{f}_{\text{ptk}}$
	0.60	0.65	0.70	0.75	0.80
Class 1	2.962	3.175	4.048	3.938	3.728
Class 2 and 3	2.561	3.482	4.048	4.350	4.361

$1/{c}({r}_{\exp } )$	${r}_{\exp } = {\sigma }_{{\text{pi}\,\exp }} /{f}_{\text{ptk}}$
	0.60	0.65	0.70	0.75	0.80
Class 1	1.717	1.275	1.000	0.814	0.681
Class 2 and 3	2.519	1.467	1.000	0.743	0.582

10.1.4 Chart 10.4: Total Prestressing Losses

With respect to the initial values calculated with the formulas of Chart 10.1, the stresses in the materials are subject to the following long-term losses.

Symbols

ΔP _ρ :

loss due to steel relaxation

ΔP _s :

loss due to concrete shrinkage

ΔP _v :

loss due to concrete creep

ΔP :

total prestressing loss

ΔN _p = ΔP cos ϕ :

axial component of ΔP

ΔV _p = ΔP sin ϕ :

transverse component of ΔP

see also Chart 10.1.

Long-Term Losses

(with cos ϕ ≅ 1, ΔN _p ≅ ΔP)

Without more precise data, the following final values can be assumed.

${\Delta }{P}_{\rho } = {A}_{\text{p}} {\Delta} {\sigma }_{{{\rho }\infty }}$	loss due toi relaxation (Chart 10.2)
${\Delta }{P}_{\text{s}} = {A}_{\text{p}} {\Delta} {\sigma }_{\text{p}}^{\prime } = {A}_{\text{p}} {E}_{\text{p}} {\varepsilon }_{{\text{cs}\infty }}$	loss due to shrinkage (Chart 1.5)
$\Delta {P}_{\text{v}} = {A}_{\text{p}} \Delta {\sigma }_{\text{p}}^{\prime \prime } = {A}_{\text{p}} {\alpha }_{\text{e}}^{\prime } {\phi }_{\infty } {\sigma }_{\text{ci}}^{*}$	loss due to creep (Chart 1.14)

Total loss

$$ \Delta {P} = \Delta {P}_{\rho } + \Delta {P}_{\text{s}} { + }\Delta {P}_{\text{v}} = {A}_{\text{p}} \Delta {\sigma }_{\text{p}} $$

In order to take into account the interaction with the losses due to shrinkage and creep, the one due to relaxation can be reduced with the coefficient 0.8.

Variations of stresses (positive in tension)

$$ \begin{array}{*{20}l} {\Delta {\sigma }_{\text{c}} = + \frac{{\Delta {N}_{\text{p}} }}{{{A}_{\text{i}} }} - \frac{{\Delta {N}_{\text{p}} {e}}}{{{I}_{\text{i}} }}{y}_{\text{c}} } \hfill & {\text{upper edge}} \hfill \\ {\Delta {\sigma }_{\text{G}} = + \frac{{\Delta {N}_{\text{p}} }}{{{A}_{\text{i}} }}\quad \Delta {\tau }_{\text{G}} = + \frac{{\Delta {V}_{\text{p}} }}{{{zb}_{\text{G}} }}} \hfill & {\text{centroid}} \hfill \\ {\Delta {\sigma }_{\text{c}}^{\prime } = + \frac{{\Delta {N}_{\text{p}} }}{{{A}_{\text{i}} }} + \frac{{\Delta {N}_{\text{p}} {e}}}{{{I}_{\text{i}} }}{y}^{\prime }_{\text{c}} } \hfill & {\text{lower edge}} \hfill \\ {\Delta {\sigma }_{\text{c}}^{*} = + \frac{{\Delta {N}_{\text{p}} }}{{{A}_{\text{i}} }} + \frac{{\Delta {N}_{\text{p}} {e}^{2} }}{{{I}_{\text{i}} }}} \hfill & {\text{tendon level}} \hfill \\ {\Delta {\sigma }_{\text{p}}^{\prime } = - \Delta {\sigma }_{\text{p}} + {\alpha }_{\text{e}}^{\prime } \Delta {\sigma }_{\text{c}}^{*} } \hfill & {\text{tendon}} \hfill \\ \end{array} $$

10.1.5 Chart 10.5: Stages of Verification of Stresses

The main stages of verification of stresses in the materials of a beam prestressed off-site and then transported to its final position are listed below. The verification in particular concern:

• the maximum compressions in concrete to avoid an excessive propagation of microcracking (see Chart 2.2);

• the maximum tensions in steel, both passive and prestressed to avoid yielding (see Chart 2.3);

• the maximum tensions in concrete for the verification of decompression and formation of cracks (see Chart 2.2);

• the range of tensions in steel measured from decompression of concrete for the verifications of cracks opening (see Chart 2.16)

The verifications related to cracking refer to the criteria of Chart 2.15. For the stresses due to shear see Charts 4.1, 4.5 and 4.6. The stresses in the section of the beam are assumed to be calculated with the load combinations related to the serviceability limit states (see Chart 3.2).

The stages of verification of stresses can be articulated in various ways, depending of the type of structure, the construction methods and the refinement of the calculation itself. The main ones, listed below, are illustrated with the schemes of the following figure and represent the incremental actions to be progressively cumulated to the previous ones. A separate case is the transient condition (a′) of lifting that ends with no further developments.

Initial Stage (a): “Precompression”

(with self-weight g ₁)

The initial stresses σ _ci, … are calculated with the formulas of Chart 10.1; the prestressing losses of the scheme (b) will then be evaluated based on these values.

Transient Stage (a′): “Lifting”

(change of supports and dynamic effects)

Stresses are calculated with the same formulas of Chart 10.1 where the internal forces $M_{\text{g}}^{\prime }$ and $V_{\text{g}}^{\prime }$ due to the self-weight refer to the new configuration of supports with the action α _dg₁. Without more accurate calculations, the dynamic effects can be calculated in the most unfavourable way with the coefficient

$$ {\alpha }_{\text{d}} = 1 \pm 0.15 $$

The verifications under minimum loads are usually carried on this stage.

Incremental stage (b): “Losses”

The stress losses with respect to the initial values are calculated with the formulas of Chart 10.4.

Incremental Stage (c): “Permanent”

The effects of permanent loads g ₂ are calculated with

$$ \begin{array}{*{20}l} {\Delta {\sigma }_{\text{c}} = - \frac{{{M}_{\text{g}}^{\prime \prime } }}{{{I}_{i} }}{y}_{\text{c}} } \hfill & {\text{upper edge}} \hfill \\ {\Delta {\tau }_{\text{G}} = \frac{{{V}_{\text{g}}^{\prime \prime } }}{{{zb}_{\text{G}} }}} \hfill & {\text{centroid}} \hfill \\ {\Delta {\sigma }_{\text{p}} = {\alpha }_{\text{e}} \frac{{{M}_{\text{g}}^{\prime \prime } }}{{{I}_{i} }}{e}} \hfill & {\text{tendon}} \hfill \\ {\Delta {\sigma }_{\text{c}}^{\prime } = + \frac{{{M}_{\text{g}}^{\prime \prime } }}{{{I}_{i} }}{y}_{\text{c}}^{\prime } } \hfill & {\text{lower edge}} \hfill \\ \end{array} $$

The combination (a) + (b) + (c) for the verification under permanent loads is deduced at this point.

Incremental Stage (d): “Serviceability”

The effects of variable loads ψ _i q are calculated with

$$ \begin{array}{*{20}l} {\Delta {\sigma }_{\text{c}} = - \frac{{{M}_{\text{q}} }}{{{I}_{i} }}{y}_{\text{c}} } \hfill & {\text{upper edge}} \hfill \\ {\Delta {\tau }_{\text{G}} = \frac{{{V}_{\text{q}} }}{{{zb}_{\text{G}} }}} \hfill & {\text{centroid}} \hfill \\ {\Delta {\sigma }_{\text{p}} = {\alpha }_{\text{e}}^{\prime } \frac{{{M}_{\text{q}} }}{{{I}_{i} }}{e}} \hfill & {\text{tendon}} \hfill \\ {\Delta {\sigma }_{\text{c}}^{\prime } = + \frac{{{M}_{\text{q}} }}{{{I}_{i} }}{y}_{\text{c}}^{\prime } } \hfill & {\text{lower edge}} \hfill \\ \end{array} $$

The combination (a) + (b) + (c) + (d) for the verifications under serviceability loads is deduced at this point, with the combination coefficient ψ _i deduced from Chart 3.2 depending on the requirements (frequent or rare combination).

Tensile stresses have been assumed positive in what mentioned above.

10.1.6 Chart 10.6: Verification of Resistance in Bending

Section in prestressed concrete subject to uniaxial bending.

Symbols

M _Ed :

design value of applied bending moment

M _Rd :

design value of the resisting bending moment

A _s, A′_s :

areas of passive reinforcement in tension and compression

d, d′:

distance of reinforcement from the edge in compression (see figure)

A _p :

area of prestressing reinforcement

d _p :

distance of tendon from the edge in compression (see figura)

b :

width of the edge in compression (see figure)

b _w :

web width (see figure)

t :

flange thickness (see figure)

ε _po :

initial strain of prestressing reinforcement

see also Charts 2.2, 2.3 and 10.1.

Verifications with Resisting Tendon

(longitudinal tendon with initial strain ε _po)

Rectangular section

(or T-shaped with ${\bar{x}} \le {t}$)

$$ {M}_{\text{Rd}} = {A}_{\text{p}} {f}_{\text{pd}} \left( {{d}_{\text{p}} - {\bar{x}}/2} \right) + {A}_{\text{s}} {f}_{\text{yd}} ({d} - {\bar{x}}/2) + {A}_{\text{s}}^{\prime } {f}_{\text{yd}} ({\bar{x}}/2 - {d}^{\prime } ) \ge {M}_{\text{Ed}} $$

with

$$ {\bar{x}} = \left( {{A}_{\text{p}} {f}_{\text{pd}} + {A}_{\text{s}} {f}_{\text{yd}} - {A}_{\text{s}}^{\prime } {f}_{\text{yd}} } \right)/{bf}_{\text{cd}} $$

and with the limits

$$ \begin{array}{*{20}l} {{\varepsilon }_{\text{p}} = \frac{{{d}_{\text{p}} - {x}}}{x}{\varepsilon }_{\text{cu}} + {\varepsilon }_{\text{po}} } \hfill & {\left( {{\varepsilon }_{\text{yd}}^{*} \le {\varepsilon }_{\text{p}} \le {\varepsilon }_{\text{pd}} } \right)} \hfill \\ {{\varepsilon }_{\text{s}} = \frac{{{d} - {x}}}{x}{\varepsilon }_{\text{cu}} } \hfill & {\left( {{\varepsilon }_{\text{yd}} \le {\varepsilon }_{\text{s}} \le {\varepsilon }_{\text{sd}} } \right)} \hfill \\ {{\varepsilon }_{\text{s}}^{\prime } = \frac{{{x} - {d}^{\prime } }}{x}{\varepsilon }_{\text{cu}} } \hfill & {\left( {{\varepsilon }_{\text{yd}} \le {\varepsilon }_{\text{s}}^{\prime } } \right)} \hfill \\ \end{array} $$

where

$$ {x} \cong {\bar{x}}/0.8\quad {\varepsilon }_{\text{cu}} = 0.35\% \, ({up}\,{to}\,{\text{class}}\,{C5}0/{6}0) $$

T-shaped

(with ${\bar{x}} > {t}$)

$$ \begin{aligned} {M}_{\text{Rd}} = \, & {{atf}}_{{cd}} \left( {{d}_{\text{p}} - {t}/2} \right) + {b}_{\text{w}} {\bar{x}}{f}_{\text{cd}} \left( {{d}_{\text{p}} - {\bar{x}}/2} \right) + \\ & \,\, + {A}_{\text{s}} {f}_{\text{yd}} \left( {{d} - {d}_{\text{p}} } \right) + {A}_{\text{s}}^{\prime } {f}_{\text{yd}} \left( {{d}_{\text{p}} - {d}^{\prime } } \right) \ge {M}_{\text{Ed}} \\ \end{aligned} $$

with

$$ \begin{aligned} {\bar{x}} & = \left( {{A}_{\text{p}} {f}_{\text{yd}} + {A}_{\text{s}} {f}_{\text{yd}} - {A}_{\text{s}}^{\prime } {f}_{\text{yd}} - {\text{atf}}_{\text{cd}} } \right)/{b}_{\text{w}} {f}_{\text{cd}} \\ {a} & = {b} - {b}_{\text{w}} \\ \end{aligned} $$

and with the same limits of the rectangular section.

Verifications with Prestressing Force

(constant force P _o = E _p ε _po in place of the tendon)

Rectangular section

(or T-shaped with ${\bar{x}} \le {t}$)

$$ {M}_{\text{Rd}}^{*} = {b}\bar{x}{f}_{\text{cd}} ({y}_{\text{o}} - {\bar{x}}/2) + {A}_{\text{s}} {f}_{\text{yd}} ({d} - {y}_{\text{o}} ) + {A}_{\text{s}}^{\prime } {f}_{\text{yd}} ({y}_{\text{o}} - {d}^{\prime } ) \ge {M}_{\text{Ed}}^{*} $$

where

$$ \begin{aligned} {M}_{\text{Ed}}^{*} & = {M}_{\text{Ed}} - {N}_{\text{pd}} {e} \\ {N}_{\text{pd}} & = {P}_{\text{d}} \cos {\phi }\quad {P}_{\text{d}} = {\gamma }_{\text{P}} {P}_{\text{O}} \\ \end{aligned} $$

with

$$ {\bar{x}} = \left( {{N}_{\text{pd}} + {A}_{\text{s}} {f}_{\text{yd}} - {A}_{\text{s}}^{\prime } {f}_{\text{yd}} } \right)/{bf}_{\text{cd}} $$

and with the limits

$$ \begin{aligned} {\varepsilon }_{\text{s}} & = \frac{{{d} - {x}}}{x}{\varepsilon }_{\text{cu}} \quad \left( {{\varepsilon }_{\text{yd}} \le {\varepsilon }_{\text{s}} \le {\varepsilon }_{\text{sd}} } \right) \\ {\varepsilon }_{\text{s}}^{\prime } & = \frac{{{x} - {d}^{\prime } }}{x}{\varepsilon }_{\text{cu}} \quad \left( {{\varepsilon }_{\text{yd}} \le {\varepsilon }_{\text{s}}^{\prime } } \right) \\ \end{aligned} $$

where

$$ {x} \cong {\bar{x}}/0.8\quad {\varepsilon }_{\text{cu}} = 0.35\% \,({\text{up to class C}}50/60) $$

T-shaped section

(with ${\bar{x}} > {t}$)

$$ \begin{aligned} {M}_{\text{Rd}}^{*} & = {{atf}}_{{cd}} ({y}_{\text{o}} - {t}/2) + {b}_{\text{w}} {\bar{x}}{f}_{\text{cd}} ({y}_{\text{o}} - {\bar{x}}/2) + \\ & \quad +\, {A}_{\text{s}} {f}_{\text{yd}} ({d} - {y}_{\text{o}} ) + {A}_{\text{s}}^{\prime } {f}_{\text{yd}} ({y}_{\text{o}} - {d}^{\prime } ) \ge {M}_{\text{Ed}}^{*} \\ \end{aligned} $$

where

$$ \begin{aligned} & {M}_{\text{Ed}}^{*} = {M}_{\text{Ed}} - {N}_{\text{pd}} {e} \\ & {N}_{\text{pd}} = {P}_{\text{d}} \cos {\phi }\quad {P}_{\text{d}} = {\gamma }_{\text{P}} {P}_{\text{O}} \\ \end{aligned} $$

with

$$ \begin{aligned} {\bar{x}} & = \left( {{N}_{\text{pd}} + {A}_{\text{s}} {f}_{\text{yd}} - {A}_{\text{s}}^{\prime } {f}_{\text{yd}} - {{atf}}_{{cd}} } \right)/{b}_{\text{w}} {f}_{\text{cd}} \\ {a} & = {b} - {b}_{\text{w}} \\ \end{aligned} $$

and with the same limits of the rectangular section.

10.1.7 Chart 10.7: Anchorage of Tendons

End anchorages of unbonded post-tensioned or bonded pre-tensioned tendons.

Verification of Anchors of Post-Tensioned Tendons

The verification of concrete bearing can be set with

$$ {P}_{\text{d}} = {\gamma }_{\text{P}} {P} \le {P}_{\text{Rd}} $$

where the resisting value of prestressing force is given by

$$ {P}_{\text{Rd}} = {f}_{\text{cd}}^{*} {A}_{\text{o}} $$

with

$$ \begin{array}{*{20}l} {{A}_{\text{o}} = {a}^{\prime } {b}^{\prime } } \hfill & {\text{gross area of loaded print}} \hfill \\ {{a}^{\prime } ,\,{b}^{\prime } } \hfill & {\text{sides of the anchorage plate}} \hfill \\ \end{array} $$

and where

$$ {f}_{\text{cd}}^{*} = {f}_{\text{cdj}} \sqrt {{A}_{1} /{A}_{\text{o}} } $$

with

$$ \begin{array}{*{20}l} {{f}_{\text{cdj}} = {f}_{\text{ckj}} /{\gamma }_{\text{C}} } \hfill & {\text{design resistance at tensioning}} \hfill \\ {{A}_{1} = {ab}} \hfill & {\text{involved loaded surface}} \hfill \\ \end{array} $$

The sides a, b are calculated with

$$ {a} = {a}^{\prime } + 2{d}_{\text{a}} \le 3{a}^{\prime } \quad {b} = {b}^{\prime } + 2{d}_{\text{b}} \le 3{b}^{\prime } $$

based on the margins d _a, d _b of the loaded print with respect to the closest edges of the section. In any case, the surfaces involved are to be cut halfway to the closest adjacent plates.

Anchorage Length of Bonded Tendons

The end anchorage length of bonded pre-tensioned strand is calculated with

$$ {l}_{\text{b}} = \frac{\phi }{4}\frac{{{f}_{\text{pd}} }}{{{f}_{\text{bd}} }} $$

where ϕ is the diameter of the bar, wire or strand, f _pd = 0.9 f _ptk/γ _S is its design resistance, f _bd = β _b f _ctk/γ _C the design value of the bond resistance. Without more accurate measurements, it can be assumed

$$ \begin{array}{*{20}l} {{\beta }_{\text{b}} = 2.25} \hfill & {{\text{for}}\,{\text{wires}}\,{\text{and}}\,{\text{ribbed}}\,{\text{bars}}} \hfill \\ {{\beta}_{\text{b}} = {3}.00} \hfill & {{\text{for}}\,{\text{strands}}} \hfill \\ \end{array} $$

If the tendon is cut without a previous slow release, an uneffective end segment should be added, equal to

$$ {l}_{\text{o}} \cong 7{\phi } $$

It can be assumed that the effectiveness of the strand vary linearly within the bonded segment 0 ≤ x ≤ l _b:

$$ {f}_{\text{pd}} {x}/{l}_{\text{b}} $$

10.1.8 Chart 10.8: Additional Data

Prestressed elements in bending with or without shear reinforcement.

Verifications in Shear with Resisting Tendons

(bonded pre-tensioned strands along the edge in tension)

What given in Charts 4.1, 4.2, 4.5 and 4.6, respectively, for elements without shear reinforcement and for resistance, serviceability and cracking of beams with stirrups is valid.

Verifications in Shear with Prestressing Force

(Constant force in place of tendons)

What given in Chart 6.21 is valid, provided N _o = P _d cos ϕ is assumed and V _Ed is substituted with ${V}_{\text{Ed}}^{*} = {V}_{\text{Ed}} - {P}_{\text{d}} \sin \,{\phi}.$

Construction Rules and Support Details

What given in Chart 4.5 for shifting of moments, spacing and minimum shear reinforcement is valid, as well as what given in Chart 5.2 for support details, and what mentioned in Chart 6.22 for minimum longitudinal bonded reinforcement.