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By contrast, restraint forces are not required to maintain
the equilibrium but, due to obstructed deformation,
merely to meet the compatibility conditions. They are directly
proportional to the absolute system stiff ness. In suffi
ciently ductile systems, restraint forces can be reduced
to zero in the ultimate limit state.
Restraint force reduction by cracking and
plastic deformation
The numerical analyses of the load-bearing behavior under
the combined action of loads and restraint forces were
carried out using the non-linear method to determine
internal forces in accordance with DIN 1045-1 and
DIN Fachbericht 102. The non-linear model used was
verifi ed by the simulation of corresponding tests. In these
tests, the additionally applied restraint force had virtually
no eff ect on the limit load of the test beams [2].
The eff ects on the internal forces due to restraint are
then expressed as a linear diff erence in temperature ∆T M .
Fig. 1 shows the result of a parametric study carried out
for a reinforced concrete beam fi xed at both ends. Under
monotonically increasing loading, the restraint forces
strongly decrease during crack formation. They remain at
a roughly constant level once the fi nal cracking pattern
has been reached. The restraint forces are further reduced
by plastic deformation when the reinforcement begins to
yield until the limit load of the system has been reached.
This process is strongly dependent on the mechanical reinforcement
ratio or the relative height of the compression
zone x/d.
Prestressing shifts the restraint reduction to a higher
load level. In the case of an identical total mechanical reinforcement
ratio, the magnitude of the restraint reduction
that can be achieved in prestressed concrete is roughly
equivalent to that of reinforced concrete (Fig. 2).
Consideration of restraint in design
Numerical analyses carried out for bridges built in reinforced
and prestressed concrete revealed the same conditions
governing restraint reduction. The rules on restraint
reduction contained in DIN Technical Report 102 were
confi rmed as safe [2]. A more comprehensive design proposal
is included in [1] and [2].
BFT 02/2010
Podium 10
Fig. 2 Eff ect of prestressing on restraint reduction at a total mechanical reinforcement ratio of
v = 0.4 at the fi xing point [1].
Abb. 2 Einfl uss der Vorspannung auf den Zwangabbau bei einem mechanischen Gesamtbewehrungsgrad
von v = 0,4 an der Einspannung [1].
Durch eine Vorspannung verschiebt sich der
Zwangabbau auf ein höheres Lastniveau. Die Größe des
erreichbaren Zwangabbaus ist beim Spannbeton im Vergleich
zum Stahlbeton bei gleichem mechanischem Gesamtbewehrungsgrad
etwa gleich (Abb. 2).
Berücksichtigung des Zwangs bei der Bemessung
Numerische Untersuchungen an ausgeführten Stahlbeton-
und Spannbetonbrücken führten zu den gleichen
Gesetzmäßigkeiten für den Abbau des Zwangs. Die Regelungen
zur Abminderung des Zwangs im DIN-Fachbericht
102 wurden als konservativ bestätigt [2]. Ein weitergehender
Bemessungsvorschlag ist in [1] bzw. [2]
enthalten.
References/Literatur
[1] Maurer, R., Arnold, A.: Bemessung von Tragwerken aus Stahlbeton
und Spannbeton für eine kombinierte Beanspruchung aus
Last und Biegezwang. Bauingenieur , Springer Verlag, Oktober
2009
[2] Arnold, A.: Zum Einfl uss der Zwangschnittgrößen aus Temperatur
bei Tragwerken aus Konstruktionsbeton mit und ohne
Vorspannung. Dissertation, Heft 1 der Schriftenreihe Betonbau,
TU Dortmund, 2008
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