## 1 Introduction  
Concrete is widely used in the construction industry because of its many advantages, such as its low-cost [1], high strength, good plasticity [2], and good durability. With the development of the construction industry, ordinary strength concrete cannot meet the needs of actual projects. At the same time, ultra-high strength concrete (UHSC), with its superior mechanical properties and durability, has been studied and produced [3]. UHSC is a type of concrete with a low water/binder ratio and various additives. The most common additives include mineral admixtures and fibrous materials. With nanotechnology's continuous development, more and more nano-mineral materials are being applied in the construction industry, such as nano-calcium carbonate (nano-CaCO\({}_{3}\)) [4; 5], nano-titanium dioxide (nano-TiO\({}_{2}\)) [6; 7], nano-silica (nano-SiO\({}_{2}\)) [8; 9], and so on. Due to its high level of activity and very high specific surface area, nano-SiO\({}_{2}\) (nS) is often integrated into the construction of cement-based materials to obtain a superior performance, compared with traditional materials [10; 11]. Most researchers have paid attention to the properties of UHSC-containing nS, such as their mechanical properties, durability, and microstructure, for example, and the study of a nano-silica modified low water/binder ratio (w/b = 0.2) UHSC can be traced back to [PERSON] et al. [11]. A series of experiments confirmed that the optimal mixing amount of nS obtained is 3.74 wt.%. [PERSON] et al. [12] investigated the effect of nS on the durability of UHSC. By comparing the accelerated corrosion test results of UHSC with and without nS, it was found that UHSC containing nS has better corrosion resistance. nS can effectively reduce the corrosion rate of steel bars in UHSC. [PERSON] et al. [13] evaluated the performance of UHSC with nS by thermogravimetric analysis, mercury intrusion porosimetry, and scanning electron microscopy. After replacing the cement amount of 3 wt.%, the cement paste exhibited the best performance. UHSCs blended with nano-silica alone were studied [11; 13], and more researchers began to focus on the addition of at least two kinds of nano-particles to UHSCs. [PERSON] et al. [14] studied the properties of UHSC containing nS and nano-limestone. For a denser microstructure and high mechanical properties of UHSC, it is necessary to dope the UHSC matrix with nS and nano-limestone, which added 1 wt.% and 3 wt.%, respectively. [PERSON] et al. [15] studied the effects of different nano-CaCO\({}_{3}\) and nS contents on the performance of UHSC. The study showed that the optimal dosages for nano-CaCO\({}_{3}\) and nS are 1.6%-4.8% and 0.5%-1.5%, respectively. It was also found that nS provides more strength before the age of 7 days, and the 7-28 days strength increased due to the nano-CaCO\({}_{3}\). Compared to studies on the effect of nS on the mechanical properties and microstructure of UHSCs, there have been relatively few studies on the effect of nS on the autogenous shrinkage of UHSCs. All UHSC designs are based on a low water-cement ratio. Although such a configuration can obtain high strength, UHSC also shows exceptionally large autogenous shrinkage, thereby increasing the early cracking risk of the substrate. In short, shrinkage is the unavoidable volume deformation of UHSC in the process of gradual hardening. It is a disadvantage that cannot be ignored in the construction engineering phase of UHSC. The effect of nS on the autogenous shrinkage of concrete has been studied extensively. [PERSON] et al. [16] studied the properties of self-compacting pastes mixed with nS. Their results showed that nS increases the total shrinkage due to the initiation of the pozzolanic reaction. [PERSON] et al. [17] studied the properties of composite cement pastes with the addition of nS and silica fume and found that both nS and silica fume promote the increase of autogenous shrinkage. They also found that in early-age shrinkage, nS plays a more active effect than silica fume. [PERSON] et al.