Ruđer Bošković Institute - Influence of Rotational … · Web viewDetailed modelling of the column connections was preformed with SAP2000 using brick elements with incompatible

Influence of Rotational Stiffness between Column Elements on Global Stability of Historical Constructions

J. Atalić, D. Lazarević, K. Fresl

Faculty of Civil Engineering, University of Zagreb, Croatia

[email protected]

Abstract

Historical constructions are of exceptional value for all countries and every single issue must be treated with great care and respect.

During several analysis in Dubrovnik‘s old town we have come across specific difficulties related to traditionally made column

contacts. The importance of these contacts seems insignificant and we have usually ignored them. Unfortunatelly, numerous fractures

around column elements and many restoration traces proved that we have a serious recurring problem. Preliminary analysis showed

several fundamental deficiences in column connection properties like insufficient dowel anchoring, stress concentration and small

rotational stiffness. The contacts are methodicaly made to safely transmit only uniform compression and shear stresses. We have

included some of these properties in analysing stress distribution and lines of thrust. Results showed that a different treatment of

stone column connections causes significantly different stress distribution. If we ignore or depreciate the real connection properties

we can seriously overestimate safety factor. The reduction of a safety factor is mostly generated by low level of rotational stiffness.

Column contacts cannot compensate eccentrical force and fracture are continuosly advancing. Therefore, small rotational stiffness of

contact properties is becoming even smaller until eventually they act as a hinge. This is a dangerous state for construction because we

can easily endanger global stability of construction. Significance of understanding and proper evaluation of rotational stifness

beetween column elements is crucial for the proper evaluation of such construction systems.

1. Introduction

We have made numerous experimental and numerical investigations on different historical constructions but these traditionally made

stone column contacts had intrigued us particularly. Usually we have ignored them and their influence on construction system but

numerous characteristic fractures and many restoration traces around column contacts had warned us about this recurring problem.

Contact properties are predicted to safely transmit only uniform compression and shear stresses. Unfortunately, we encountered many

situations where unqualified people had intervened on the structures. Removed buttresses, a decrease of dead load and similar

situations had violated essential ideas of bearing structures and induced a significant eccentrical load. Column contacts with small

rotational stiffness cannot compesate for this situation and the final results are fractures near column elements. Continuous fracture

propagation is making small rotational stiffness of contact even smaller acting eventually closing as a hinge. If we ignore real

connection properties, we seriously overestimate the safety factor and even endanger the global stability of construction. We will

prove our statements with numerous numerical analyses and propose possible possible ways of reconstruction.

Figure 1: Fractures of column elements

2. Problem description

On traditionally used vault bearing system we have found many specific fractures on columns, their bases and capitals (Figure 1.).

We could also see many restoration traces and insets of different nuances and styles. Discussions with colleagues from various fields

and detailed historical surveys gave us insights into many damages and even collapses of bearing systems throughout history. After

preliminary investigation we concluded that we are dealing with serious recurring problem. Therefore we have made numerous

observations, experiments and calculations to establish the fundamental cause of the columns contact fractures.

The logical hypothesis about large relative displacements was not confirmed by observations, geotechnical testing or geodetic

measurements. However, it is only conditionally discarded because of the short measurement time and possibility of the discontinued

ground displacement caused by seismic activities. The hypothesis of high compression stresses is also discarded by experimental

analysis. Experiments showed a low level of compression stresses and various calculations proved this statements. Therefore we

were forced to seek problem somewhere else.

Examinations of specific fractures around column connections and several cracked columns (Figure 1.) revealed some contact

properties. These traditionally made contacts between column elements are made with great care and outstanding accuracy. Namely,

carefully smoothed stone areas of contact are joined with iron dowel which is centrically placed in a bit larger hole and finally filled

with lead. These contacts are assumed to safely transmit only uniform compression and shear stresses.

Preliminary analysis showed that insufficient dowel anchoring (Figure 2a.) is one of the fundamental deficiencies. Therefore, dowel

can safely transmit only a transverse and small pulling force. It is a consequence of small anchoring length and the lack of

adhesiveness caused by week mechanical properties of lead, specially by its affinity to creep. Obviously, described connection cannot

transmit significant bending moments and its rotational stiffness is very small.

Unfortunately, connections that we found were usually eccentrically loaded for numerous reasons such as: eccentrically connected

elements, disrupted bearing system, missing or ineffective tie rods, etc. Even small horizontal displacements cause a rotation of

column elements that make problem at contacts because of the minimal ability to compensate rotations. The leaning on the edge of

connection results in significant pressure on a small contact area. Stress concentration exceeds compression limits and declining of

compression trajectories makes unwanted tension stresses (Figure 2a.). The outcomes are fractures near connected areas, splitting of

columns and capitals and therefore a complete loss of bending stiffness. The connection is still able to transmit compression and

transversal force and therefore can be defined as a hinge. However, if we treat connections as pure hinges, the bearing system will be

unstable. Only three hinges, on one vertical, are enough to make the system unstable (Figure 2b.).

Figure 2: Behaviour of contacts

Moreover, discarded issues must be added: tensile stress in stone due to the pressure of corroded dowels, slender columns, small

vertical load, numerous interventions in construction system, present activities on structure, temperature changes, earthquakes etc.

…………

Figure 3: Measured fracture activities

3. Numerical strategies

We can conclude that great compressive strength of stone is not essential for bearing capacity. On the contrary, bearing system

primarily depends on stone tensile strength which ensures small rotational stiffness of joints. Without it the static system would be

practically a mechanism. To prove these radical statements, we made numerous numerical models.

Figure 4: 3D model with tetrahedal elements

Full 3D model of complete structure, with tetrahedral finite elements, was made using FEAP 7.4 (Figure 4.) software combined with

GiD 6.1.2a for data input and output. Into the basic continuous elastic model we introduced several original routines to describe real

connection behaviour. Program codes were supplemented with exclusion of overstressed finite elements and the search of minimal

energy in given direction (line search) with the bisection method to improve the convergence of incremental Newton – Raphson

technique. Fracture criterion of material is determined by the modified theory of normal stresses. The theory was originally defined

by Galileo and Rankine. It foresees the failure of fragile material in the case of tensile stretching. Fracture criterion is defined by the

expression:

where where σ1 and σ3 are the greatest and the smallest stresses in the Gauss point of the finite element. In our calculations, the

positive value stands for tensile, and the negative value stands for compressive stress. The values are defined as:

where ft and fk are tensile and compressive strengths of material in uniaxial experiment. Results confirmed our assumptions of

contacts and construction behaviour. We can clearly monitor stress concentration and exclusion of overstressed elements.

Construction is trying to compensate for the new situation but fracture propagation is decreasing rotational stifness and thus the

safety factor. Eventually, construction is becoming unstable.

f (σ1 , σ 3 )=(σ1−σ a )

n

σ s+(σ3−σ a )

n

σs

σ s=f t +f k

2σ a=f t− f k

2

Figure 5: Detailed model with solid elements and GAP’s for tension exlusion

Detailed modelling of the column connections was preformed with SAP2000 using brick elements with incompatible modes and

links able to exclude tension stresses (Figure 5.). Fractures in vault bearing system are ignored and links were used only on column

connections including properties of lead and dowel. Even this simple approach was sufficient to obtain rotations of contacts elements

(GAP opening) and overstressing. Positions of overstressed elements are exactly matched with located fractures.

Other numerous models of various complexities were made to adjust to the problem for common use. We used same gap elements for

contact behaviour on simple 2D model of the characteristic section (Figure 6.). On this model we can easily follow GAP opening and

its influence on stress distribution and lines of thrust. The figures are showing compression stresses with lines of thrust (black line).

The effect of included contact properties on lines of thrust can be compared with assumed position of lines of thrust (red line).

Position of lines of thrust were and still are crucial data for the analysis of historical constructions. This simple and approximate

approach shows the difference that cannot be ignored.

Figure 6: 2D model of charasteristic section

3D frame and shell model of complete structures (Figure 7.) were finally made for analysing different values of rotational stifness.

Rotational stifness (Figure 8.) is included in model on column contacts. By changing the level of rotational stifness we have

significantly changed all results.

Figure 7: 3D frame and shell model

Figure 8: Rotational stifness

We have presented various models and all of them proved the importance and need for the proper evaluation of column connections.

The amount of rotational stiffness was decisive for the understanding of carrying mechanism, finding correct distribution of stresses

and assessment of global stability of structure. It is hard to present some rule for the analysis of column contacts due to various

contact properties and construction systems. But the best warning for a serious approach can be the results on one of the most

valuable structures in Croatia, the Rector's Palace in Dubrovnik. Safety factor dropped from 10 for standard continuous elastic model

to 1.4 for models based on presented assumptions.

4. Conclusion

Presented data emphasize the importance of correct analysis of column connections along with complexities and difficulties. Slender

columns are usually marked as the weakest link in historical constructions. If we cannot secure hypotheses of uniform stress

distribution, we have at least to understand connections behaviour. Our analysis proved that by ignoring or depreciating real

connection properties we have significantly different stress distribution, lines of thrust and overestimated safety factor. Even more,

the progress of characteristic columns fractures can easily endanger the global stability of construction.

Finally, we have tested the possibility of repairing columns with thin layer of lead placed between two connected areas. The material

such as lead will enable small rotations and keep connected areas away from large stress concentrations. Lead is already used to fill

dowel hole and the additional horizontal thin layer is an appropriate and almost invisible intervention.

5. References

1. G.N. Pande, J. Middleton, B. Kralj, Eds, Computer Methods in Structural Masonry – 4, Proceedings of the Fourth International

Symposium on Computer Methods in Structural Masonry, E & FN Spon, London, 1997.

2. P. Roca, J.L. Gonzales, A.R. Mari, E. Onate, Eds, Structural analysis of historical constructions. Possibilities of numerical and

experimental techniques, CIMNE, Barcelona, 1997.

3. P. Roca, J.L. Gonzales, E. Onate, P.B. Lourenco, Eds, Structural analysis of historical constructions II. Possibilities of numerical

and experimental techniques, CIMNE, Barcelona, 1998.

4. P.B. Lourenco, P. Roca, Historical Constructions. Possibilities of numerical and experimental techniques, Proceedings of the 3rd

International Seminar, University of Minho, Guimaraes, 2001.

5. D. Lazarević, J. Dvornik, K. Fresl, Analiza oštećenja atrija Kneževa dvora u Dubrovniku, Građevinar, 2004, 56, 601-612

6. D. Lazarević, J. Dvornik, K. Fresl, M. Rak, Numerical analysis of damages of the rector's palace atrium in Dubrovnik, in: J. Radić,

V. Rajčić, R. Žarnić, Eds., Heritage Protection – Construction Aspects, Dubrovnik, Croatia, 2006, 109-116.