摘要
云冈石窟为中国三大石窟之一,位于山西大同市城西16km的武周山南麓,东西绵延lkm。该石窟开凿于公元5世纪的北魏时期。云冈石窟现存造像5万余尊。它以工程浩大,佛像气势雄伟而闻名于世。云冈石窟历史久远、内容丰富、雕刻精细,是我国雕刻艺术的宝库,具有重要的文化、艺术、宗教、历史价值。1961年由国务院公布为第一批全国重点文物保护单位。
云冈石窟中第9、10窟是一组双窟,是云冈石窟中殿堂风味最浓的两个窟。这两窟以其前殿有列柱开门,为云冈一大特点。两窟的前殿各有两根露明通顶石柱,柱呈八角形,柱面各刻十层佛龛,柱下刻须弥座,座置于柱础上,柱础为大象。可惜现今柱身外面雕刻及柱础象头部分皆风化殆尽,已不能显示当年的富丽景象。随着长时间的风化破坏,两窟立柱整体强度令人堪忧,立柱失稳后将直接影响洞窟的整体稳定性。
因此,为了得到9、10窟的稳定现状和变形破坏发展规律,为加固设计提供科学依据,本文采用有限差分软件FLAC3D,对9、10窟岩体稳定性进行了三维数值模拟分析,结合强度折减理论计算出石窟在自重作用下的稳定性系数,并进行了在地震作用下的石窟立柱应力、位移、速度和加速度响应分析。经过计算分析得到以下几点认识：
1.由FLAC3D三维有限差分计算结果知,在天然重力作用下,模型的最大不平衡力随着时步迅速从4.5MPa降低到零并收敛,共用了6457时步,说明模型在天然自重状态下稳定性较好。并且发现,模型中绝大多数部位塑性应变为零,说明第9、10窟整体稳定性好,没有发生破坏。
2.云冈9、10窟位移场基本呈对称分布,模型总位移方向以垂直向下并沿窟体向前倾斜为主。最大位移为1.28mm,发生在4根立柱顶端及山体顶部,表明此处的稳定性相对较差,变形较大,应加强监测和进行加固保护。
3.模型在自重应力场下,总体主应力从上往下逐渐增大,最大主应力在窟顶以上区域,以铅直方向为主,在近坡面一带发生偏转,方向与坡面方向一致；在靠近窟壁处最大主应力近似与窟壁平行,最小主应力近似与窟壁垂直,向坡体内逐渐恢复正常,符合斜坡应力分布规律。
4.本文基于强度折减法理论,用Fish语言自编程序,计算得模型整体稳定性系数R=1.45。R=1.45>1.3,说明目前模型整体处于稳定状态。但是随着风化、掉块的进一步加剧,岩石的强度会进一步降低,当降低到一定程度后,会影响立柱及石窟的稳定性。故建议对9、10窟前的4根立柱做长期监测,并进行必要的粘接、支撑、加固等保护措施。
5.石窟立柱在小震作用下,剪应力较小,剪应力方向为负,指向窟外,表明立柱向外剪切破坏可能性较大,立柱平均剪应力为68.2KPa,立柱2的剪应力最大,最大剪应力为115.8KPa。立柱没有塑性区出现,没有发生破坏。立柱的响应位移、速度和加速度都比较小。4根立柱的平均位移为2.75cm,平均速度为0.85cm,平均加速度为0.10 m/s2,而且立柱上部、中部和下部的位移、速度、加速度基本一致,相差很小。
6.石窟立柱在中震作用下,平均最大剪应力值达到114.5 KPa。立柱4的剪应力最大,最大剪应力为111.9 KPa。剪应力方向为负,指向窟外。立柱下部内侧发生了剪切破坏,并且立柱的位移、速度和加速度响应相对较大。4根立柱的平均位移为7.34cm,平均速度为2.31cm,平均加速度为2.38m/s2。立柱上部位移、速度和加速度小于立柱下部的位移、速度和加速度,这说明立柱下部已经发生塑性破坏。
7.石窟立柱在大震作用下,平均最大剪应力值达到327.1 KPa。立柱3的剪应力最大,最大剪应力为4.7 MPa。剪应力方向为负,指向窟外。立柱下部全部发生了剪切破坏,立柱的位移、速度和加速度响应很大。4根立柱的平均位移为19.71cm,平均速度为6.55cm,平均加速度为4.32m/s2。立柱上部位移、速度和加速度小于立柱下部的位移、速度和加速度。
8.云冈9、10窟立柱在小震作用下应力、位移、速度和加速度响应较小,没有破坏；在中震和大震作用下应力、位移、速度和加速度响应较大发生了剪切破坏。基本达到了“小震不坏、中震可修和大震不倒”的设防要求,但是在中震和大震情况下立柱发生了破坏,影响石窟稳定性,故建议采取粘结、支撑或锚固等适当的措施进行加固并加强监测,从而更好的保护云冈石窟。
Yungang Grottoes are one of China's three major grottoes, which are located in the south of Mount Zhouwhu,16km away from the west of Datong city of Shanxi province. The grottoes were dug in the 5th century AD in the Northern Wei dynasty,5 million statues existed in the Yungang Grottoes now. They are known as huge and Impressive st(?)es. Yungang Grottoes are the sculpture art treasure house of China because of long history, rich contents and fine carving, with important cultural, arti(?)ic, religious, historical value. They were announced as the first batch of national key cultural,elics protection units by the State Council in 1961. included as "World Heritage(?)t in 2001.
The cave 9 and cave 10 is a set of double-Cave in the caves, with the most concentrated flavor of the hall style. Each of the caves has two pillars in the front Hall. Pillars are octagonal cylinders and engraved with ten layers of the shrines.Unfortunately, outside of the pillars and the bases have been weathered now and can not display the splendid scene. With the long period of weathering damage, the overall strength of the two pills is very weak, and the pills will directly affect the instability of the overall cave.
Therefore, in order to get the present stability and deformation law of 9,10 Cave, providing the scientific basis for the reinforcement design, we use the finite difference software FLAC3D, analyzing on the stability of the rock of Cave 9 and 10, combining the strength reduction theory to calculate the stability coefficient in stable system, moreover, making the seismic effect of the stress and displacement of the pills. Through the simulation and calculation we find some Conclusions as follows:
1.By the calculations of FLAC3D we find that, under the natural gravity, the maximum unbalanced force with the time step of the model is reduced to zero rapidly from 4.5Mpa and convergence, but only the 6457 time step to achieve convergence.It shows the model has the good stability under the natural Weight state.
2.The total displacement direction of model is vertical downward and tilted forward. The largest displacement occurred in the top of the 4 pillars and mountain, the largest displacement is 1.28mm.lt shows that the stability of rock here is relatively poor, deformation is large，should be reinforced to strengthen the monitoring and protection. Cave 9 and cave 10 displacement field distribution is basicly symmetric.
3.Under the natural gravity, the overall principal stress increases gradually from top to bottom, the maximum principal stress direction in the Grottoes area above is mainly vertical, deflect to according with slope direction in the area near the slope;The maximum principal stress is approximately parallel with the cave wall.the minimun principal stress is approximately vertical with the cave wall near the cave wall，which gradually return to normal near the slpe body, in line with the slope stress distribution.
4.Based on the strength reduction theory nrogramming with the Fish language, calculating the overall stability coefficient R. R= 1.45> 1.3. It shows that the current model is stable. However, with the further intensifion of the weathered and flaked to a certain extent, the bottom of pills will be in shear failure firstly. So we suggested make the long-term monitoring on the pills, and make the necessary bonding, supporting，reinforcement and other protective measures to prevent the further weathering.
5.Under the small earthquakes, shear stress of the pills is small, the shear stress direction is negative, pointing to outside of cave. The average shear stress is 687 KPa. The max shear stress of pill 2 is largest, which is 115.8 KPa. Pills are not damaged, there is no plastic zone. So the response of displacement, velocity and acceleration are relatively small. The average displacement of 4 pillars is 2.75 cm.a(?) rage speed is 0.85cm. average acceleration is 0.10 m/s2. The upper, middle and lower part the displacement, velocity and acceleration of pills are basically the same, the difference is small.
6.Under the medium earthquake, the max average shear stress is 114.5 KPa. The shear stress of pill 4 is largest.which is 111.9 KPa. The shear stress direction is negative, pointing to outside of cave. Shear failure occurs in the inside of lower part of the pills. The displacement, velocity and acceleration of pills are larger. The average displacement of 4 pillars is 7.34cm, average speed is 2.31cm, average acceleration is 2.38 m/s2. The upper part the displacement, velocity and acceleration of pills are smaller than middle and lower part the displacement, velocity and acceleration of pills.
7.Under the large earthquake, the max average shear stress is 327.1 KPa. The shear stress of pill 3 is largest, which is 4.7 MPa. The shear stress direction is negative, pointing to outside of cave. Shear failure occurs in the whole of lower part of the pills. The displacement, velocity and acceleration of pills are largest. The average displacement of 4 pillars is 19.71 cm, average speed of 6.55 cm, average acceleration is 4.32m/s2. The upper part the displacement, velocity and acceleration of pills are smaller than middle and lower part the displacement, velocity and acceleration of pills.
8.Under small earthquake,the responses of stress, displacement, velocity and acceleration of the pillars of Caves 9 and 10 of Yungang Grottoes are small, there is no damage; Under medium and large earthquake, the responses of stress, displacement, velocity and acceleration of the pillars are larger, pills are damaged in Shear failure. It basiclly meet the requirement of "small earthquake not damage, medium earthquake repairable and large earthquake not down",but the pills are damaged under the medium and large earthquakes.So it should be taken to strengthen the monitoring and bonding, such as Bonding, supporting or anchoring and other appropriate reinforcement measures to better protect the Yungang Grottoes.
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