The effect of prestress force magnitude on the modal properties (frequency, damping and mode-shape) of uncracked prestressed concrete structures is something that has been widely debated among researchers to date, as out- lined in previous sections.
The effect of pre- and post-tensioning force magnitude on the natural bending frequencies of cracked prestressed concrete structures is something that is more established, and widely agreed upon. Saiidi et al. (1994) report an increase in natural frequency with increasing post-tensioning force. As pointed out by Bruggi et al. (2008) the tests carried out by Saiidi et al. (1994) were conducted on cracked beam sections only. Uncracked sections
were not tested. Williams and Falati (1999) present a formula to calculate the average effective second moment of area of a cracked concrete cross section. The effect of crack closure in accordance to this method is that it increases the effective second moment of area of the cross section, and subsequently the natural bending frequencies. Hop (1991) agrees, reporting a decrease in natural bending frequencies with increased cracking, and states that increasing the prestressing force acts as to close the cracks, stiffen the section and increase the natural bending frequencies of the beam sections. Grace and Ross (1996) also report a decrease in girder stiffness leading to a decrease in natural frequency also attributing it to cracking in the cross section. Unger et al. (2006) state that a loss in post-tensioning increases the appearance of cracks which reduces the bending stiffness and subsequent
natural frequencies of the system. De Roeck (2003) concurs, stating that“a
loss of pre-stress will result in a measurable change in eigenfrequencies only
if it is accompanied by originating cracks.” Hamed and Frostig (2004) also
report that large cracking damage yields drastic reduction in the natural frequencies of cracked prestressed concrete beams.
Pavic et al. (2001) agree that“concrete cracking and excessive static de-
flection in a prestressed post-tensioned slab can be overcome to a large extent
by the careful choice of the amount and location of the prestress” but ar-
gue that“no amount of prestressing, however, will significantly improve the
floor dynamic behaviour since this is governed largely by slab stiffness, mass and damping on which different levels of prestressing do not have major in-
fluence.” Dall’Asta and Dezi (1996) consider it is possible to determine the
prestressing force by measuring the natural frequency of a PSC structure in its cracked state only. Rodr´ıguez et al. (2010) acknowledges this fact in rela- tion to post-tensioned concrete wind turbine towers, stating that uncracked
towers “maintain their original stiffness and frequency of oscillation” but
once the towers are cracked and the cracks have been decompressed,“any
perturbations of the tower will only mobilise a smaller stiffness, an effect that will be shown by the vibration frequencies.”
There is however, disagreement when it comes to the effect of prestress force on the damping ratios for cracked prestressed concrete structures. Kato and Shimada (1986) state that there are hardly any changes in damp- ing values, whereas Blakeley et al. (1970) report damping of PSC structures to be lower than that of RC structures, quoting damping ratios of 1-2%
before cracking of the member and 3-6% after cracking.
2.14. Conclusions
It is clear from this discussion that there is a lack of agreement between researchers in the field as to the effect that prestress force magnitude has on the natural bending frequency of uncraked prestressed concrete beams. There are currently three distinct arguments;
1. The natural vibration frequency of prestressed concrete structures tends to decrease as the magnitude of the prestressing force is in- creased (Chan and Yung, 2000; Dai and Chen, 2007; Dall’Asta and Leoni, 1999; Law and Lu, 2005; Miyamoto et al., 2000; Raju and Rao, 1986; Tse et al., 1978).
2. The natural vibration frequency of prestressed concrete structures is unaffected by prestress force magnitude (Hamed and Frostig, 2006). 3. The natural vibration frequency of prestressed concrete structures
tends to increase as the magnitude of the prestressing force is in- creased (Hop, 1991; Kim et al., 2004; Saiidi et al., 1994; Zhang and Li, 2007).
A satisfactory mathematical model is yet to be formulated predicting the change in natural frequency with increasing prestressing force, however, the model suggested by Kim et al. (2004) has the best agreement with experimental results. It should be noted that the magnitude of the predicted change in natural frequency is very small for practical ranges of prestressing force, and could be considered to be negligible. Furthermore, as highlighted by Ho et al. (2012), recently the interest on variability of dynamic properties of bridges (i.e. natural frequency, mode shape, damping ratio) caused by environmental effects such as temperature, humidity, wind and other factors is increasing. Studies such as those conducted by Cornwell et al. (1999) and Peeters and De Roeck (2001) report frequency differences in the ranges of 6% and 14-18% respectively due to normal environmental changes. The change in natural frequency due to prestress loss may be negligible as a result. Furthermore, no statistical significance tests have yet been conducted on the data to determine whether the observed changes in natural frequency
with prestress force magnitude is simply due to random variation or whether it is systematic. The majority of experimental studies conducted indicate that the natural frequency of prestressed concrete structures increases with increasing prestressing force magnitude (Ho et al., 2012; Hop, 1991; Lu and Law, 2006; Saiidi et al., 1994; Williams and Falati, 1999; Zhang and Li, 2007; Zhang et al., 2012).
The effect of prestress force magnitude on the natural frequency of cracked prestressed concrete structures is something that is more widely established and agreed upon. Authors agree that for cracked pre- and post-tensioned concrete structures the effect of the post-tensioning force is to close cracks, stiffen the section and therefore increase the natural bending frequencies of the cracked pre- and post-tensioned concrete structures (Saiidi et al., 1994; Bruggi et al., 2008; Williams and Falati, 1999; Hop, 1991; Grace and Ross, 1996; Unger et al., 2006; De Roeck, 2003; Hamed and Frostig, 2004; Pavic et al., 2001; Rodr´ıguez et al., 2010).
The problem of the effect of prestress force magnitude on the natural bending frequency of prestressed concrete beams has implications in many fields, such as in prestressed bridge design, post-tensioned floor slabs, and even post-tensioned concrete wind turbine tower design. Prestress force magnitude decreases over time due to concrete creep, steel relaxation, an- chorage pull-in and other effects. The natural bending frequency of pre- stressed concrete bridges and post-tensioned concrete wind turbine towers may change over time according to the effect of prestress loss. It is impor- tant for structural engineers to be able to account for any subsequent change in natural frequency over time due to prestress loss, as it may become a crit- ical design parameter affecting the long term performance of the structure. Furthermore, if the change in natural frequency with prestress force magni- tude can be determined, this can be related to the well established Eurocode 2 (British Standards Institute, 2004) formula for predicting prestress loss with time, and subsequently a model for the variation in natural frequency over time due to prestress loss could be formulated.