2.5 Reinforced structures
2.5.5 Calculation models for prestressed members
Depending on the prestressing system, the applied prestress force (P) is reduced due to elastic deformation of the structure. This elastic loss ( Pel) occurs if a system where the jack is mounted on a pre-tensioning frame is used (Figure 36). The prestress force (P0) present in the member is given by equation (19) below. Most post-strengthening methods for concrete
do not have elastic losses because the prestressing jack is attached to the concrete and the elastic deformation occurs whilst building up the prestress force P (Figure 37).
= (19)
Where:
P0 = prestress force present in the member
P = prestress force applied to the system (jacking force) Pel = elastic prestress force loss
Figure 36: Sketch of prestressing a member using a pre-tensioning frame or similar device where the jack is mounted on a frame, and therefore elastic loss occurs
Figure 37: Sketch of prestressing where the jack is mounted on the member and therefore no elastic loss occurs
According to Thomsing [111], the prestress force and the moment due to eccentricity act on the conceptual cross section. The moment, due to eccentric prestressing, is determined by the multiplication of the prestress force with the eccentricity.
Stress in rod and member
P Stress in rod but no stress in member
Jacking and bonding of the lamella to the member
Releasing of the jack
P0<P P0<P
P Stress in rod and member
Stress in rod and member P0
Releasing of the jack
Jacking and bonding of the lamella to the member
=P P0=P
Figure 38: Determination of the prestress in the case of eccentric prestressing [111]
Triantafillou and Deskovic [113] investigated the maximal-anchoring force for adhesively-bonded prestressed CFRP-lamellas, assuming concrete or adhesive failure. The analyses they present assume linear-elastic materials, and the governing deformation mode in the adhesive layer as shear. The stress-strain relationship is assumed as a bilinear model where the first part is linear-elastic, followed by a perfect plastic plateau until failure (Figure 39).
Figure 39: Shear stress-strain relationship for epoxy adhesive as used by Triantafillou and Deskovic
The shear stress in the elastic domain of the adhesive can by determined using equation (21) [113].
( ) = ,
2
sinh( ) (20)
= 1
+4
(21)
Where:
(x) = shear stress due to prestressing in the adhesive as a function of x
el,a = maximal-elastic shear strain (Figure 39) Ga = shear modulus of the adhesive
lel = length of the elastic domain ha = thickness of the adhesive layer hcf = thickness of the CFRP-lamella
Ecf = modulus of elasticity of the CFRP-lamella hs = height of the substrate
x = position over the length of the beam Es = modulus of elasticity of the substrate
The shear stress in the area where the adhesive has passed its elastic limits is equal to the ultimate shear stress of the adhesive ( u,a) (Figure 39).
Triantafillou and Deskovic assume that equation (22) below for the shear deformation is valid for the whole length of the reinforced beam. This assumption is consistent with the common-ly-held hypothesis that the strains in the inelastic regime are approximately described by the same relationship characterising elastic response (e.g. bending beam).
( ) = ,
2
sinh( ) (22)
Where:
(x) = shear strain due to prestressing in the adhesive as a function of x
el,a = maximal-elastic shear strain (Figure 39) lel = length of the elastic domain
= is given by equation (21)
The equations presented by Triantafillou and Deskovic [113] lead to the following shear stress and strain distribution:
Figure 40: Shear stress and strain distribution (immediately before delaminating due to prestress force) in the adhesive layer towards the end of the beam, calculated using the equations presented by Triantafillou and Deskovic [113]
For the ultimate anchorage load, the shear strain at the end of the beam is equal to the ulti-mate shear strain ( u,a). This condition allows the calculation of the length of the elastic do-main (lel) (equation (23)).
=
2 + + 4
2 (23)
=2 ,
, 2 (24)
Where:
u,a = ultimate shear strain of the adhesive (Figure 39)
el,a = maximal-elastic shear strain of the adhesive (Figure 39) lel = length of the elastic domain
= is given by equation (21)
Based on the shear stress distribution in the elastic domain, the normal stress in the CFRP can be determined. Triantafillou and Deskovic assume the normal stress distribution in the inelastic zone as linear. Knowing the normal stress distribution in the CFRP allows the calcu-lation of the ultimate pre-tension force (P0,u) (equation (25)).
, = , 2 + ( )
2 (25)
Where:
P0,u = ultimate pre-tension force EAcf = tension stiffness of the CFRP ha = thickness of the adhesive layer
el,a = maximal-elastic shear strain of the adhesive (Figure 39) lel = length of the elastic domain
= is given by equation (21)
Furthermore, Triantafillou and Deskovic [113] present an equation to calculate the ultimate pre-tension force (P0,u), assuming concrete failure (equation (26)):
, = , ( )
4 + + ,
2 (26)
Where:
P0,u = ultimate pre-tension force EAcf = tension stiffness of the CFRP ha = thickness of the adhesive layer
el,a = maximal-elastic shear strain of the adhesive (Figure 39) lel = length of the elastic domain
= is given by equation (21)
Triantafillou and Deskovic [114] investigated the maximal-anchor resistance for prestressed CRFP-laminates on European beech (Fagus Silvatica), using adhesive bond, and assuming timber shear failure. The shear stress-strain behaviour of wood is assumed as bilinear with an elastic start followed by an ideal plastic branch (Figure 39). The equations for timber fail-ure are the same as presented for adhesive failfail-ure [113] (equations (20) to (25)), except the stress-strain relationship of wood has to be used instead of the one of the adhesive. In case the CFRP-lamella is not as wide as the timber beam equation (21), it has to be modified (equation (27)). Acf = cross section area of the CFRP-lamella Ecf = modulus of elasticity of the CFRP-lamella
bs = width of the substrate
AEs = tension stiffness of the substrate (area multiplied by the module of elasticity)
Triantafillou and Deskovic [113, 114] report that the theoretical investigations were satisfying-ly verified with experiments using rather small and clear specimens. Furthermore, they tested three timber beams in bending in order to determine the influence of prestressed CFRP-lamellas on the bending capacity [114]. They conclude that, in order to avoid sudden col-lapse, the members have to be designed to yield in the compression first before fail by ten-sile fracture.