Factors influencing hysteresis characteristics of concrete dam deformation

Transcription

1 Water Science and Engineering 2017, 10(2): 166e174 HOSTED BY Available online at Water Science and Engineering journal homepage: Factors influencing hysteresis characteristics of concrete dam deformation Jia-he Zhang a, Jian Wang a, *, Li-sha Chai b a College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing , China b Lixi Barrage Irrigation Management Office of Guangzhou City, Guangzhou , China Received 22 April 2016; accepted 5 December 2016 Available online 14 March 2017 Abstract Thermal deformation of a concrete dam changes periodically, and its variation lags behind the air temperature variation. The lag, known as the hysteresis time, is generally attributed to the low velocity of heat conduction in concrete, but this explanation is not entirely sufficient. In this paper, analytical solutions of displacement hysteresis time for a cantilever beam and an arch ring are derived. The influence of different factors on the displacement hysteresis time was examined. A finite element model was used to verify the reliability of the theoretical analytical solutions. The following conclusions are reached: (1) the hysteresis time of the mean temperature is longer than that of the linearly distributed temperature difference; (2) the dam type has a large impact on the displacement hysteresis time, and the hysteresis time of the horizontal displacement of an arch dam is longer than that of a gravity dam; (3) the reservoir water temperature variation lags behind of the air temperature variation, which intensifies the differences in the horizontal displacement hysteresis time between the gravity dam and the arch dam; (4) with a decrease in elevation, the horizontal displacement hysteresis time of a gravity dam tends to increase, as the horizontal displacement hysteresis time of an arch dam is likely to increase initially, and then decrease; and (5) along the width of the dam, the horizontal displacement hysteresis time of a gravity dam decreases as a whole, while the horizontal displacement hysteresis time of an arch dam is shorter near the center and longer near dam surfaces Hohai University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Concrete dam; Displacement; Hysteresis; Temperature; Analytical solution 1. Introduction The measurement and interpretation of dam deformation data is an important part of dam safety evaluation and has been studied by various researchers (Li, 1992; Wu et al., 2007a, 2007b; Zhang et al., 2008; Jesung et al., 2009; Gu et al., 2011). For a concrete dam, the total deformation is generally divided into various components, including a hydrostatic component, a thermal component, and a time-dependent component (or irreversible component) (Bonaldi et al., 1977; Fanelli and Giuseppetti, 1982; Wu, 2003; Gu and Wu, 2006; This work was supported by the the National Natural Science Foundation of China (Grant No ) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. * Corresponding author. address: wang_jian@hhu.edu.cn (Jian Wang). Peer review under responsibility of Hohai University. Xu et al., 2014). The thermal component, which is the topic of this paper, is the deformation caused by the change of the outside temperature. In the last decade, a few studies have explored the hysteresis characteristics of concrete dam deformation. Xu et al. (2012) pointed out that the variation of the thermal displacement component lags behind the variation of the air temperature, and this lag is known as the hysteresis time. According to previous interpretations, the reason for the hysteresis time is the low heat conduction velocity in concrete (Troxell and Davis, 1956; Neville, 1963; Li et al., 1996). However, in practice we find that the hysteresis time of the horizontal displacement of different dams may differ significantly even if the dam width and thermal conductivities are similar. The displacement hysteresis time of a narrow arch dam is often longer than that of a wide gravity dam. The low heat conduction velocity in concrete alone cannot fully explain / 2017 Hohai University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( creativecommons.org/licenses/by-nc-nd/4.0/).

2 Jia-he Zhang et al. / Water Science and Engineering 2017, 10(2): 166e this phenomenon. Zhang et al. (2015) derived analytical solutions of the hysteresis time of thermal deformation for a cantilever beam and an arch ring and pointed out that the dam type has a large impact on the hysteresis time of dam thermal deformation. However, their study was only based on theoretical analysis and few factors were considered. Therefore, it is necessary to study the factors influencing the displacement hysteresis time from additional perspectives. This study mainly focused on the effects of dam type, reservoir water temperature, and spatial position on the displacement hysteresis time as determined with the analytical method and numerical simulation. Some case studies have also been examined to further verify the validity of conclusions. 2. Quasi-stable mean temperature and equivalent linearly distributed temperature difference in concrete dam Since the heat conduction in a concrete dam is mainly determined by the temperature at the upstream and downstream surfaces of the dam, which can be approximated by one-dimensional heat conduction along the thickness of an infinite slab, this principle can be applied in the following analysis. Assuming that the thickness of the infinite slab is l, and the surface temperature changes according to the cosine function with different amplitudes and phases, the heat conduction equation can be expressed as vt vt ¼ a v2 T vx 2 The boundary conditions are as follows: A2 cos½uðt εþš x ¼ 0 T ¼ A 1 cosðutþ x ¼ l T is the temperature, a is the thermal diffusivity, t is time, A 1 is the amplitude of the boundary temperature at x ¼ l, A 2 is the amplitude of the boundary temperature at x ¼ 0, u is the circular frequency of temperature change, and ε is the hysteresis time of the boundary temperature at x ¼ 0 relative to the boundary temperature at x ¼ l. In thermal deformation and stress analysis, temperature is typically divided into three parts: the mean temperature T m, linearly distributed temperature difference T d, and nonlinearly distributed temperature difference T n. For a free slab whose deformation is unconstrained, T m and T d will produce tensile deformation and rotation, respectively, while T n will only cause thermal stress and have no effect on the deformation. Therefore, only T m and T d need to be considered in the hysteresis analysis for concrete dams. They can be computed as follows (Zhu, 1999): ð1þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k m ¼ 1 2ðchz 0 cos z 0 Þ z 0 chz 0 þ cos z 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi k d ¼ a 2 1 þ b2 1 rffiffiffiffiffi u z 0 ¼ l 2a a 1 ¼ 6 sinðuq mþ z 2 0 k m b 1 ¼ 6 1 cosðuq z 2 m Þ 1 k 0 m q m ¼ 1 p sin z0 tan 1 u 4 shz 0 q d ¼ 1 b1 u tan 1 ¼ 1 cosðuqm Þ k m a 1 u tan 1 ð10þ sinðuq m Þ According to Eqs. (2) and (3), q m and q d represent the hysteresis times of the mean temperature T m and linearly distributed temperature difference T d, respectively. For the sake of simplicity, we first assume that ε in Eqs. (2) and (3) is 0 (the effect of ε will be analyzed in section 4.2), and then calculate the relationship between the hysteresis times q m and q d and the slab thickness l for different thermal diffusivities a, as shown in Fig. 1, the thermal diffusivity ranges from 0.05 to 0.15 m 2 /d, covering the common range of concrete thermal diffusivity. The circular frequency is u ¼ 2p/365. That is, the cycle of external ambient temperature is 365 days. The following can be seen in Fig. 1: (1) q m > q d when the slab thickness is the same, indicating that the hysteresis time of the horizontal displacement of an arch dam is longer than that of a gravity dam, which will be discussed in detail below. (2) The hysteresis time q m of the mean temperature increases rapidly with the slab thickness, and then decreases slightly, and remains nearly unchanged at a value of 0.25p/u when the slab thickness is greater than 15 m. ð4þ ð5þ ð6þ ð7þ ð8þ ð9þ T m ¼ 1 2 k mfa 1 cos½uðt q m ÞŠ þ A 2 cos½uðt q m εþšg ð2þ T d ¼ 1 2 k dfa 1 cos½uðt q d ÞŠ þ A 2 cos½uðt q d εþšg ð3þ Fig. 1. Relationship between slab thickness and hysteresis times of average temperature and linearly distributed temperature difference.

3 168 Jia-he Zhang et al. / Water Science and Engineering 2017, 10(2): 166e174 (3) The hysteresis time q d of the linearly distributed temperature difference increases with the slab thickness l. 3. Analytical solutions of displacement hysteresis time For the sake of simplicity, we consider the horizontal displacement of a gravity dam and an arch dam, respectively, so as to derive the analytical solutions of the displacement hysteresis time and analyze their characteristics Analytical solution of displacement hysteresis time for gravity dam From a structural perspective, a gravity dam can be treated as a cantilever beam. Thus, a triangular cantilever beam is considered, with a height of H and a bottom width of B, and its deformation characteristics are calculated when one side is subjected to variable temperature, as shown in Fig. 2. According to the principal of virtual work, the total displacement of the gravity dam can be obtained by integrating the deformation for each micro-segment: u ¼!du ¼! Mk þ Nε þ Qg ds ð11þ u is the total displacement; k, ε, and g are the curvature, axial strain, and shear strain of the micro-segment, respectively; M, N, and Q are the bending moment, axial force, and shear force of the micro-segment caused by the virtual unit load, respectively; and ds is the area element. As for a point P with a height of h, the horizontal displacement u x caused by temperature load is calculated as follows: u x ¼! h ðh zþ at Xn d h z i dzza T di Dz i ð12þ 0 L L i n is the number of partitions, and T di, z i, L i, and Dz i are the linearly distributed temperature difference, height, dam width, and micro-segment height of the ith layer, respectively. According to Eq. (12), the horizontal displacement of a gravity dam is proportional to the linearly distributed temperature difference. Substituting Eq. (3) into Eq. (12), and after trigonometric transformation (product to sum formula), Eq. (12) can be rewritten as! u x ¼ a Xn g 1i cosðutþþ a Xn g 2i!sinðutÞ ð13þ g 1i ¼ k di h z i L i Dz i fa 1 cosðuq di Þ A 2 cos½uðq di þ ε i ÞŠg ð14þ g 2i ¼ k di h z i L i Dz i fa 1 sinðuq di Þ A 2 sin½uðq di þ ε i ÞŠg ð15þ k di, q di, and ε i are the k d, q d, and ε values of the ith micro-segment. After the trigonometric transformation (sum to product formula) of Eq. (13), we can obtain u x ¼ r cos u t q 0 d ð16þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2! u 2 X n r ¼ a t g 1i þ Xn g 2i 0 q 0 d ¼ 1 u tan 1 X n g 2i X n g 1i 1 C A ð17þ ð18þ Eq. (18) provides the solution of the horizontal displacement hysteresis time q 0 d at point P Analytical solution of displacement hysteresis time for an arch dam An arch dam is a spatial, statically indeterminate structure. Its structural response can be divided into an arch effect and a beam effect, in which the arch effect plays the dominant role. Accordingly, we take a monolayer arch ring to perform dam deformation analysis according to a pure arch method. According to Eq. (11) and the symmetricity conditions under which the rotation angle and the tangential displacement at the arch crown are both zero, the axial force N and the bending moment M at the arch crown caused by the temperature loads T m and T d can be obtained, and the horizontal displacement (in the radial direction) caused by T m and T d can be computed. The horizontal displacement is h u x ¼ T 1 m h 2 ð19þ h 1 ¼ arð1 cos 4Þ 4R 2 AG cos 2 4 þ 24R 2 AG cos 4 sin 4 þ R 2 A I G þ EIk 0 4 sin 4 þ I þ R 2 A G þ EIk Fig. 2. Calculation of gravity dam deformation. 4R 2 AG ð20þ

4 Jia-he Zhang et al. / Water Science and Engineering 2017, 10(2): 166e h 2 ¼ 2R 2 AG cos 2 4 þ I þ R 2 A G EIk 0 4 sin 4 cos 4 þ I þ R 2 A G þ EIk R 2 AG ð21þ E and G are the elastic modulus and the shear modulus of the material, respectively; A and I are the area and the moment of inertia of the cross-section, respectively; 4 is half of the central angle of the arch ring; R is the radius of the arch ring; k 0 is a dimensionless parameter related to the crosssection shape, which is 1.2 for a rectangular cross-section. According to Eqs. (19) through (21), the horizontal displacement of the arch ring is proportional to the mean temperature T m, but is independent of the linearly distributed temperature difference T d. The shear displacement has the same characteristics. The horizontal displacement hysteresis time of the arch dam is identical to the hysteresis time of the mean temperature T m, namely q m in Eq. (2). 4. Factors influencing hysteresis time of dam thermal deformation 4.1. Influence of dam type on displacement hysteresis time The displacement hysteresis time of an arch dam is generally longer than that of a gravity dam due to the difference in structural response. The horizontal displacement hysteresis time of a gravity dam depends on the linearly distributed temperature difference T d, as the horizontal displacement hysteresis time of an arch dam largely depends on the mean temperature T m. For the purpose of illustration, the horizontal displacement hysteresis times of a gravity dam and an arch dam with the same height H of 100 m were calculated according to Eqs. (18) and (9), respectively, and the corresponding curves are shown in Fig. 3. The bottom widths B of the gravity dam and arch dam were set at 70 m and 35 m, respectively, both crest widths were 8 m, and the thermal diffusivity of the concrete was assumed to be 0.1 m 2 /d. Furthermore, ε in Eqs. (2) and (3) was set at 0, i.e., the effect of hysteresis time of the boundary temperature was not considered (the effect of ε on displacement hysteresis time will be analyzed in the next section). Thus, the values of A 1 and A 2 did not affect the calculation results of hysteresis time. It can be seen from Fig. 3 that the horizontal displacement hysteresis time of an arch dam is longer than that of a gravity dam even if the width of the arch dam is less than that of a gravity dam. It is noted that the beam effect of an arch dam is not considered in this analysis. Actually, the horizontal displacement hysteresis time of an arch ring might be slightly reduced with consideration of the beam effect. However, since the arch effect plays a dominant role, the basic conclusion that the horizontal displacement hysteresis time of an arch dam is longer than that of a gravity dam is still true (this will be verified in subsection 4.3.2) Influence of water temperature on displacement hysteresis time In the previous analysis, we have assumed that ε ¼ 0in Eqs. (2) and (3). In reality, for a dam in operation, the upstream surface is in contact with the reservoir water, and the variation of the reservoir water temperature lags behind that of the air temperature, i.e., ε > 0. Therefore, it is necessary to further analyze the influence of the water temperature hysteresis time on the displacement hysteresis time of the gravity dam and the arch dam. For this analysis, we still use the infinite slab in section 2 as an example, assuming that the amplitudes of the water temperature and the air temperature are A 2 and A 1, respectively, and the lag time is ε. Through trigonometric transformation, Eqs. (2) and (3) can be rewritten as T m ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k m r 2 1 þ r2 2cos u t q 0 m ð22þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T d ¼ k d r 2 1 þ r02 2 cos u t q 0 d r 1 ¼ A 2 sinðuεþ r 2 ¼ A 1 þ A 2 cosðuεþ ð23þ ð24þ ð25þ r 0 2 ¼ A 1 A 2 cosðuεþ q 0 m ¼ q m þ m m ð26þ ð27þ Fig. 3. Comparison of horizontal displacement hysteresis times of gravity dam and arch dam. q 0 d ¼ q d m d m m ¼ 1 sinðuεþ u tan 1 A 1 =A 2 þ sinðuεþ ð28þ ð29þ m d ¼ 1 sinðuεþ u tan 1 ð30þ A 1 =A 2 sinðuεþ The variables m m and m d reflect the influence of water temperature hysteresis on the displacement hysteresis time. The parameters m m and m d mainly depend on ε, A 1, and A 2 rather than the slab thickness l. Fig. 4 shows the relationships

5 170 Jia-he Zhang et al. / Water Science and Engineering 2017, 10(2): 166e174 of ε with m m and m d for different A 1 /A 2 values. In situ measurements show that ε is generally less than 90 days. Within this scope, m d is more sensitive to the change of ε than m m. Numerous measured data of the reservoir water temperature indicate that A 1 > A 2 and ε is less than six months, so, according to Eqs. (22) through (30), m m > 0 and m d > 0. This means that, due to the influence of the reservoir water temperature, the hysteresis time of mean temperature increases from q m to q m þ m m, while the hysteresis time of the linearly distributed temperature difference decreases from q d to q d m d. With reference to the analysis above, it is concluded that, due to the lag of the water temperature variation behind the air temperature variation, the hysteresis time of the horizontal displacement of an arch dam is extended to q m þ m m, while that of a gravity dam is shortened to q d m d when the influence of ε is considered. That is, the hysteresis characteristic of reservoir water temperature intensifies the differences between the hysteresis times of horizontal displacement of gravity and arch dams Variation of displacement hysteresis time with elevation A 2 and ε in Eqs. (2) and (3) at different elevations in a dam are different. The elevation h of a point affects the upper limit of the integral of the cantilever beam (for a gravity dam or for the beam effect of an arch dam) during the calculation of displacement hysteresis time. Therefore, it is necessary to study the variation of displacement hysteresis time with elevation Variation of displacement hysteresis time with elevation in gravity dam For the sake of simplicity, we assume that ε in Eqs. (2) and (3) is 0; that is, the hysteresis effect of water temperature is not considered provisionally. Two points P 1 and P 2 on the upstream dam surface are selected, as shown in Fig. 5. We first calculate the horizontal displacement u x1 of the lower point P 1 using Eq. (12): u x1 ¼! h 1 ðh 1 zþ at d 0 L dz ð31þ According to the differential mean value theorem, the equation above can be substituted with the following: u x1 ¼ðh 1 x 1 Þ ah 1 Lðx 1 Þ T dðx 1 Þ ð32þ x 1 2[0, h 1 ]. According to Eq. (3) and the trigonometric transformation, can be expressed as u x1 u x1 ¼ f 1 cos½uðt q d1 ÞŠ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 1 ¼ a g 2 1 þ g2 2 q d1 ¼ 1 g2 u tan 1 g 1 ð33þ h 1 x g 1 ¼ k 1 d1 Lðx 1 Þ h 1fA 1 ðx 1 Þcos½uq d ðx 1 ÞŠ A 2 ðx 1 Þcos½uq d ðx 1 Þþ uεðx 1 ÞŠg h 1 x g 2 ¼ k 1 d1 Lðx 1 Þ h 1fA 1 ðx 1 Þsin½uq d ðx 1 ÞŠ A 2 ðx 1 Þsin½uq d ðx 1 Þþ uεðx 1 ÞŠg Similarly, for the higher point P 2, the integration interval is divided into [0, h 1 ] and [h 1, h 2 ], and following the procedure similar to the one described above, we obtain u x2 ¼ ðh 1 x 1 Þ ah 1 Lðx 1 Þ T dðx 1 Þ þ ðh 2 x 2 Þ aðh 2 h 1 Þ T d ðx Lðx 2 Þ 2 Þ ¼ f 1 cos½uðt q d1 ÞŠ þ f 2 cos½uðt q d2 ÞŠ ð34þ x 2 2[h 1, h 2 ] qffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2 ¼ a g 2 3 þ g2 4 Fig. 4. Relationships of ε with m m and m d. q d2 ¼ 1 g4 u tan 1 g 3

6 Jia-he Zhang et al. / Water Science and Engineering 2017, 10(2): 166e Fig. 6. Changes of A 2 and ε with z 0. h 2 x g 3 ¼ k 2 d2 Lðx 2 Þ ðh 2 h 1 ÞfA 1 ðx 2 Þcos½uq d ðx 2 ÞŠ A 2 ðx 2 Þcos½uq d ðx 2 Þþuεðx 2 ÞŠg h 2 x g 4 ¼ k 2 d2 Lðx 2 Þ ðh 2 h 1 ÞfA 1 ðx 2 Þsin½uq d ðx 2 ÞŠ A 2 ðx 2 Þsin½uq d ðx 2 Þþuεðx 2 ÞŠg Since the hysteresis time of the linearly distributed temperature difference increases with the dam width and L(x 2 ) < L(x 1 ), the horizontal displacement hysteresis time q d2 at point P 2 is shorter than the horizontal displacement hysteresis time q d1 at point P 1, which means that the horizontal displacement hysteresis time increases gradually with a decrease in elevation. Using the trigonometric transformation, Eq. (34) can be rewritten as u x2 ¼ r cosfu½t ðq d1 DεÞŠg Fig. 5. Measuring point positions. ð35þ Dε ¼ q d1 q d2 > 0. Next, the variations in the amplitude and hysteresis time of water temperature are considered, and their influence on the displacement hysteresis time variation of a gravity dam with elevation is analyzed. According to numerous measured data of reservoir water temperature, the amplitude of reservoir water temperature A 2 tends to decline with an increase in the water depth z 0, as the hysteresis time of water temperature ε tends to increase with the water depth, as shown in Fig. 6. To consider the influence of the hysteresis property of water temperature, Eq. (16) was used to calculate displacement hysteresis times at different distances below the dam top of the gravity dam introduced in section 4.1. Fig. 7 shows that the hysteresis time of the horizontal displacement of a gravity dam tends to increase with the distance below the dam top for both carryover and non-carryover storage reservoirs when the water temperature hysteresis property is considered, which is consistent with the case without consideration of the hysteresis property of reservoir water temperature. For the sake of simplicity, the finite element method was used to analyze the displacement hysteresis time change of the gravity dam with the distance below the dam top, and the water temperature on the upstream dam surface was assumed to be consistent with that of a non-carryover storage reservoir. Fig. 8 shows the finite element mesh of the gravity dam. Fig. 9 shows the variation curve of the horizontal displacement hysteresis time of the gravity dam obtained from the analytical and FEM methods. As shown in Fig. 9, the finite element solution for the hysteresis time of the horizontal displacement of the gravity dam tends to increase with the distance below the dam top, which is consistent with the analytical solution Variation of displacement hysteresis time with elevation in arch dam The change in displacement hysteresis time with elevation in an arch dam is more complex than that in a gravity dam. This is because a gravity dam can be treated as a statically determinate structure (a cantilever beam), as an arch dam is a spatial, statically indeterminate shell structure, which means that there are more connections between each part of the arch dam. From a structural response standpoint, an arch dam can be separated into a series of independent arch rings and beams. The variation of the displacement hysteresis time of an arch dam with elevation has the same characteristics as those of both an arch ring and a beam (the latter is similar to a gravity dam). Specifically, the distribution of the displacement hysteresis time is related to the shape of the arch dam, and is affected by the width-to-height ratio of the dam and the widthto-height ratio of the river valley. Fig. 7. Variation of horizontal displacement hysteresis time of gravity dam with distance below dam top.

7 172 Jia-he Zhang et al. / Water Science and Engineering 2017, 10(2): 166e174 Two examples of single-curvature arch dams were analyzed. The dams, located in a rectangular river valley, had the same radius and central angle. The main shape parameters of the arch dam were as follows: the dam height was 100 m, the central angle was 110, the arch ring radius was 70 m, the crown width was 8 m, and the bottom widths were 20 m and 35 m for width-to-height ratios of 0.20 and 0.35, respectively. Three computation schemes were used for the analysis: (a) The pure arch scheme, in which only the arch effect was considered and the beam effect was ignored. The layers of the arch ring were independent of each other. (b) The pure beam scheme, in which only the beam effect was taken into account. In this scheme, a single beam in the arch crown profile was considered in the computation, and its hysteresis time calculation was similar to that of a gravity dam. This scheme had no real significance, and was mainly used for comparative purposes. (c) The monolith scheme, in which the dam was considered a monolith subject to both arch and beam effects. Both the analytical method and the finite element method were used to analyze scheme (a) and scheme (b), as scheme (c) was analyzed using only the finite element method because no analytical solution was available for scheme (c). The thermal diffusivities of the dam body and the bedrock were both set at 0.1 m2/d, and the water temperature on the upstream dam surface was assumed to be consistent with that in a non-carryover storage reservoir. Fig. 10 shows the finite element mesh adopted in scheme (c), the dam is a symmetric structure, and only half of the arch dam is considered in the modeling. This model can be modified and applied to scheme (a) by splitting the dam into independent layers of arch rings, or applied to scheme (b) by only picking the grids located at the crown cantilever (Wang and Liu, 2008). Fig. 11 shows the change in the horizontal displacement hysteresis time of an arch crown in the three schemes when the width-to-height ratios of arch dams are set at 0.20 and 0.35, respectively. The following can be seen from Fig. 11: (1) The analytical solutions for the pure arch scheme and the pure beam scheme are both similar to the results from the finite element solutions, which suggests that the analytical algorithm of displacement hysteresis time of the gravity dam and the arch ring proposed in the previous section is reasonable. (2) If the shape of the dam cross-section remains constant, the displacement hysteresis time of the pure beam is less than that of the pure arch, which verifies the conclusion drawn in the previous section that the dam type is one of the crucial factors affecting the displacement hysteresis time. (3) When the width-to-height ratio of the dam is lower, i.e., when the cross-section of the arch dam is narrower, the results of pure arch analysis match the results of the finite element method for a monolithic structure more closely. This is consistent with the load-sharing characteristics of the multi-arch-beam solution. (4) When the monolith scheme is used, the horizontal displacement hysteresis time of an arch dam first increases and then decreases with the increase of the distance below the dam top. This observation is based on analysis of a 100 m-height singlecurvature arch dam with a constant radius and central angle and located in a rectangular river valley. For different heights, variable radii, and variable central angles, the displacement hysteresis curves of hyperbolic arch dams are likely to be different. (5) The finite element calculation suggests that, due to the constraint effect of the bedrock, the displacement hysteresis Fig. 9. Variation of horizontal displacement hysteresis time with distance below gravity dam top calculated using different methods. Fig. 10. Finite element mesh adopted in scheme (c) (B/H ¼ 0.35). Fig. 8. Finite element mesh of gravity dam.

8 Jia-he Zhang et al. / Water Science and Engineering 2017, 10(2): 166e Fig. 12. Distribution of hysteresis time of dam horizontal displacement along dam width. Fig. 11. Variation of horizontal displacement hysteresis time with distance below arch dam top for different width-to-height ratios. time near the base of both the gravity dam and the arch dam decreases slightly Variation of displacement hysteresis time with dam width In the analysis described above, the measurement points were placed at the upstream dam surface without considering the displacement hysteresis time change along the dam width. In reality, due to the influence of transverse deformation, the displacement hysteresis time varies with the dam width as well. In this study, the effect of the dam width was analyzed with the finite element method. Fig. 12(a) through 12(c) show the distribution of horizontal displacement hysteresis time along the width at different elevations for the gravity dam and the arch dam (with width-toheight ratios of 0.35 and 0.20). The computational parameters were the same as those described in subsection 4.3. The computation results indicate the following: (1) Along the dam width, from upstream to downstream, the horizontal displacement hysteresis time of the gravity dam decreases, while the horizontal displacement hysteresis time of the arch dam is shorter near the center and longer near the dam surface. This is because the gravity dam deformation along the dam width is a combination of the deflection deformation caused by beam rotation and the transverse expansion. The beam rotation and transverse expansion are dependent on the changes of the linearly distributed temperature difference and average temperature, respectively. The arch dam deformation along the dam width is a combination of the expansion of the arch axis and the transverse expansion of the beam, with both components being dependent on the change in mean temperature. In addition, the influences of the shape of the dam cross-section and the gradual change of the reservoir water temperature with elevation also result in differences in horizontal displacement hysteresis times of gravity and arch dams along the dam width. (2) The horizontal displacement hysteresis time near the dam bottom is less than that in the upper part, which is related to the constraint created by the friction between the dam and the bedrock. The displacement hysteresis time at the center of the dam base is longer than that at the upstream and downstream sides. This is because the dam bottom deformation is dominated by the tensile deformation along the width, which depends on the heat conduction velocity and time. Therefore, the horizontal displacement hysteresis time at a point far away from the dam surface is relatively long. 5. Case studies In order to further verify the conclusions, especially the conclusion that dam type has a large impact on the hysteresis time of thermal deformation, different gravity and arch dams were analyzed, including the Shuikou Gravity Dam (whose maximum height is 101 m), Lijiaxia Hyperbolic Arch Dam (whose maximum height is 165 m), Baishan Gravity Arch Dam (whose maximum height is m), and Chencun Gravity Arch Dam (whose maximum height is 76.3 m). There are relatively complete and continuous observation records of displacement and environmental factors for all of these dams. Fig. 13 shows the variation of hysteresis time of dam horizontal displacement with the elevation in the dam. The following can be seen:

9 174 Jia-he Zhang et al. / Water Science and Engineering 2017, 10(2): 166e174 (4) Along the dam width, the hysteresis time of horizontal displacement of a gravity dam decreases from upstream to downstream, while the hysteresis time of horizontal displacement of an arch dam is shorter near the center and longer near the dam surfaces. Finally, it is noted that this study only examined horizontal displacement of the dam. To study vertical displacements, similar procedures can be followed. For example, the vertical displacement of a gravity dam is caused by vertical expansion and contraction and its hysteresis time is mainly dependent on the hysteresis time of the mean temperature, which is different from the hysteresis time for horizontal displacement. References Fig. 13. Distribution of hysteresis time of dam horizontal displacement. (1) The horizontal displacement hysteresis time presents an increasing tendency in the sequence of a gravity dam, gravity arch dam, and hyperbolic arch dam, which is consistent with the theoretical analysis. (2) The horizontal displacement hysteresis time increases with the distance below the dam top. The main reason is that the dam body thickness and the hysteresis time of reservoir water temperature increase with the distance below the dam top. However, due to the constraint effect of the bedrock, the hysteresis time near the base of the dam decreases slightly for some dams. 6. Conclusions The thermal deformation of a concrete dam changes periodically and its variation lags behind the air temperature variation. To analyze and explain the causes of this lag, known as the hysteresis time of the dam displacement, an extensive investigation was performed. The conclusions can be summarized as follows: (1) The dam type has a large impact on the hysteresis time of thermal deformation. In general, the hysteresis time of horizontal displacement for an arch dam is longer than that of a gravity dam. This is because the hysteresis time of the former is dependent on the mean temperature of the dam body, while that of the latter is more dependent on the linearly distributed temperature difference. (2) The water temperature variation in the reservoir lags behind the air temperature variation, which intensifies the differences between horizontal displacement hysteresis times of gravity and arch dams. (3) With an increase in the distance below the dam top, the horizontal displacement hysteresis time of the gravity dam tends to increase, as the horizontal displacement hysteresis time of the arch dam is likely to increase first and then decrease. In addition, due to the constraint of the bedrock, the horizontal displacement hysteresis time near the dam bottom is less than that in the upper part of the dam. 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