A NUMERlCAL MODEL FOR THERMAL BOWING OF MASONRYWALLS

M. Dhanasekar1, V. Chandrasekaran2 and S.J. Grubits3

1. ABSTRACT

Understanding of bowing characteristics of masonry walls subjected to fire will enhance the ana1ysis of insulation, integrity and structural adequacy failure modes of the wall. This paper describes a thermo-structural coupled finite element model based on 1ayered thin shell element which is currently under development. The results of the initial simplified structural analysis model in predicting the thermal bowing of masonry walls are included in the papeL The model is verified using the experimental results reported by Shields et al22.

2. INTRODUCTION

Masonry walls, load bearing or otherwise, function as a barrier to the spread of fire in buildings. The loss of insulation of the wall increases the temperature of the unexposed face and subsequently increases the probability of fire incidence on the adjacent enclosures. With the loss of integrity of the wall, flames and smoke spread to the adjacent enclosures and retard fire fighting activities and cause panic in the minds of the entrapped occupants. The collapse of the wall causes se vere damage to the building and local instability to the structure. Ali the three modes of failure of masonry walls are recognised in most building codes and limit values of duration specified for each case.

The limit values are used in selecting the thickness or the slenderness ratio of masonry

Keywords: Thermal Bowing; Fire; Masonry; Walls; Finite Element.

1 Lecturer, Department of Civil Engineering & Building, University of Central Queensland, Rockhampton Qld 4702, Australia.

2 Principal Research Scientist, Fire Technology, Division of Building Construction and Engineering, CSIRO, North Ryde NSW 2113, Australia.

3 Program Manager, Fire Techno10gy, Division of Building Construction and Engineering, CSIRO, North Ryde NSW 2113, Australia.

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walls in the structural design phase without paying any regard to the actual performance of the wall to the imposed fire load. The design tables of masonry codes are based on the lower bound values of the experimental results which exhibited marked variability.

Masonry walls when subjected to fire on one face, bow either towards or away from the fire depending on its boundary conditions. This thermal bowing is a complex phenomenon as it depends on the composite and complex material properties which vary with temperature. Development of cracks during bowing further complicates the processo Understanding of bowing process is, therefore, extremely important before deciding on the fire endurance of walls against insulation, integrity and structural collapse failures. This paper addresses the issue of thermal bowing of masonry walls.

3. STUDIES ON FIRE AND THERMAL BEHAVIOUR OF STRUCTURAL ELEMENTS

The performance of structural elements subjected to fire has been investigated by a number of researchers in recent years with the major emphasis on reinforced concrete frames and slabs4,11 ,17. Finite element modelling based on three dimensional continuum elements using substructuring for frames and layered plate elements for slabs were used. However for masonry walls subjected to fire, only a plane strain model is available in the literature5. For environmental temperature loads (freezing -thawing) plane strain and plane stress models were used by Anand and Rahman and Ibrahim and Suter respectivelyl,IO. In this paper thin layered semi loof shell element is used for the analysis of thermal bowing of masonry walls.

The issues related to fire modelling have been elaborated by Grubits and Chandrasekaran6. Hendry in a private communication has addressed the need for the rational methods of analysis of masonry walls subjected to fire and acknowledged the requirement of significant amount of research in the are a 7. This paper includes some initial findings of an ongoing research project taken up in the direction.

Several mo deI and full scale wall panels have been tested for fire endurance limits8, 13,

22,24. ASTM E-1l9 or AS 1530.4 fire time-temperature curves have been typically used in which the temperature increased to approximately 800' C in about 25 minutes and then gradually increased to 1200'C in 6 hours2 . Bowing of a wall has been experimentaly investigated by Shields et al22. The fire endurance limits for the various types of walls tested were included in Lawrence and Gnanakrishnan 13. Design information on fire resistance of masonry walls was reported recently by Johnstone and Pearson with reference to the provisions AS 37003, 12.

For predicting the behaviour of structural elements at elevated temperature, the properties of materiaIs at the appropriate temperatures are necessary. MateriaIs should ideally be tested while the desired temperature leveI is kept constant. Such type of testing requires expensive equipment. Hence specimens of material have been subjected to elevated temperature in a furnace, cooled and then tested at ambient temperature in the laboratory in the past by Neville for Concrete and Bremner et al for Concrete Masonry Units l6, 24. Compressive, shear bond and tensile bond strengths of clay brick masonry at elevated temperature are being tested at the University of Central Queensland. Initial results show that the strength properties increase up to 200'C and then decrease.

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4. STUDIES ON NUMERICAL MODELLING OF MASONRY W ALLS

Masonry is a complex medium for numerical analysis. The complexity in the analysis is predorninantly due to the influence of mortar joints acting as planes of weakness. The type of units and design of wall system are other important factors contributing to the complexity.

Finite element models based on interconnected brick and mortar joints have been used by Page in 1978 and Rots and Laurenco more recently in 1993 18,21. Bricks were modelled as a continuum while the mortar joints were modelled using interface joint elements. Simple shear and tension stress limited failure to more complex fracture energy based failure criteria were used for the interface elements in these models. For the analysis of large walls panels, these models were of lirnited use only. Continuum elements encompassing brick and mortar joints were developed for the analysis of large infill and shear wall panels 19. Sophisticated more complex failure criterion for masonry elements has been used in the mode!. Papa and Nappi developed a material model using homogenisation technique which allowed for the development of cracking in the homogenised media based on the principIes of damage mechanics20 . A comprehensive review of the recent advances in plane stress analysis models for masonry has been presented by Middleton et al lS.

The failure mechanism of hollow block masonry is significantly different to that of the solid masonry. Three dimensional continuum element models have historically been used in the analysis of hollow masonry. The demand for computational power was very high in those analyses. Syed Ahmed and Shrive presented a model for the analysis of hollow block masonry prisms subjected to Uni axial Compression23 . A three dimensional network of thin shell elements were used in the modelling of face and web shells of the masonry. Thin shell elements are also used in the analysis presented in the current paper.

A new, yet powerful numerical simulation model based on Distinct Element Method (DEM) has been applied to predict the cracking of masonry concrete walls subjected to dynarnic and impulsive loads by Meguro et a1 14. The DEM allow the modelling of masonry units as discrete elements interconnected by mortar springs. This mo deI (granular element system) is used to simulate the entire wall with the initial conditions of load and boundary condition defined. Dynamic equations of motion are solved to transfer the load from one grain element system to the other. The material parameters such as the mass and damping coefficients are required for each granular element assembly. The accuracy of the results of the modellargely depend on these properties which are significantly affected by the workmanship effects and are difficult to estimate.

5. FINITE ELEMENT MODEL FOR THERMAL BOWING OF MASONRY W ALLS

Under the fire loading on one face, masonry walls bow out of plane. Free standing cantilever type walls bow away from the source of fue while the walls with support on top and/or sides bow towards the fire. In order to predict the bowing, elements possessing out of plane displacement and appropriate rotational degrees of freedom are required. WhiIe most thin or thick pIate and shell elements could satisfy this requirement, they demand material modelling in terms of bending moment related failure criterion. Such criteria are not available for masonry at elevated temperatures and are difficult to establish.

It has been shown in the past that the transverse shear stresses in the thickness direction of wall do not affect the in plane behaviour of the plates and shells9. To account for the variation of flexural stresses, the thickness of the member is divided into a number of layers. Each layer in such a model will be in a state of plane stress. Well established

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plane stress deformation and failure models of masonry can then be used in the analysis. Thin semi loof shell element of LUSAS finite element system offers all the facilities defined in the paragraph 25. It is a doubly curved isoparametric element which is numerically integrated using 5 point Gauss rule in its plane and an explicit 5 point Newton - Cotes rule along its thickness direction. The element maps the wall in XY plane and bow in the Z direction due to the thermal gradient (dT/dZ).

The element used in modelling of the wall has eight nodes with each node possessing three translational dof and the loof points in between the midside and comer nodes allow for rotations of the through the thickness normais.

A fully coupled thermo-structural analysis is under development which allow for the variation of thermal and strength properties of masonry with reference to change in temperature. The thermal analysis of the wall for a defined time step is initially performed and the results of the temperature within the body of the wall read by the structural analysis programo Well established Newton - Raphson nonlinear iterative procedure is used until the stresses are converged within a prescribed tolerance without causing failure of any integration point (both Gauss and Newton - Cotes). For these integration points which violate the failure criterion, the thermal properties are also modified and the thermal program re evaluates the temperature distribution. The process is repeated for additional incremental temperature controlled by the incrementai time step. The model requires parallel processing facility and hence is being developed in V AX-VMS system. A PC based simplified finite element model is only included in the paper. The simplified model inherently assume steady state thermal analysis. This allow for the incrementai increase in temperature and thermal gradient through the thickness of the wall to be directly specified.

With a view to demonstrating the suitability of the layered isoparametric semi loof shell elements for this type of analysis, it was decided to further simplify the complexities of the material model section. Bricks and joints were therefore modelled separately. It was possible to use, therefore, the standard two dimensional failure envelope in 0"1 - 0"2 plane commonly adopted for the analysis of concrete. The shape of the failure envelope is largely reported in the literature and hence not included in the paper. The definition of the envelope is based on the specification of compressive and tensile strengths. If a stress state at an integration point exceeds the specified tensile strength, the point is assumed to crack in a direction normal to the direction of the tensile stress. The material parallel to the direction of the crack is allowed to carry further stresses while the stresses normal to the direction of crack is reduced as govemed by a linear decay model specified. Some shear stress is allowed to be retained by the cracked zone. A linear decay in shear stiffness from the cracking strain to the ultimate strain is specified to account for the effect. In the material model, in addition to the non linearities due to cracking, the non linearities caused by the change in temperature are also included. A table of material properties at different reference temperature leveis are specified for the purpose. The Young' s modulus of elasticity, Poisson ' s ratio, coefficient of thermal expansion, compressive strength, tensile strength, normal stress decay parameter and shear retention parameter are the key properties which are included in the material mode!. As the shell element is divided into several layers in the thickness direction, each layer utilises a different material property set due to difference in the temperature levei and the levei of cracking.

The non linear analysis is repeated until it converges as per Euclidian Residual norm criterion. In this criterion the second norm (2 norm) of the residual force vector is divided by the 2 norm of the extemalload vector (including reactions) and expressed as a percentage. Up to 5% error is tolerated.

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6. VALIDA TION OF THE FINITE ELEMENT MODEL

6.1 Experimental Investigation

Thermal bowing of a mo dei masonry wall tested for the British Standard fire BS476 was presented by Shields et al in the 8th rn 2 MaC22. The results of the test have been used in this paper as the basis of validation of the simplified finite element model described in the previous section.

The model wall was a square panel of size of 1m x 1m constructed from 105mm x 50mm x 30mm calcium silicate bricks laid in 5mm thick 1:5 (cement:sand) mortar. The thickness of the wall was 50mm. There were 28 courses (bed joints) and each course contained 9 bricks . The wall panel was constructed on a firm steel base. The test panel with its steel based was placed within a steel frame and the base was bolted to the frame. The top and the two vertical sides of the wall had gaps and were filled with a ceramic insulating material of unspecified stiffness.

The panel was instmmented with five thermo couples and three displacement transducers. The thermo couples were arranged in a layer some 700mm away from the bottom at 12.5mm centres in the thickness direction. Displacement transducers were positioned at 650mm, 830mm an 980mm from the bottom and were denoted by C, B and A respective1y.

One face of the test wall panel was subjected to fire , the temperature of which was controlled within a furnace. The test set up is explained in detail in reference 22 and is not repeated here.

6.2 Finite Element Analysis

Eight noded isoparametric layered thin semi loof shell elements have been used to model the bricks and mortar assuming perfect bond. Due to symmetry about the vertical axis, only half the wall panel was analysed. In order to keep the aspect ratio of the shell elements within two, each brick was required to be subdivided into 15 elements, the head joints into 3 elements and the bedjoint covering one brick length into 5 elements. Such a model needed 17978 nodes and 5883 elements. Non linear analysis of such a huge model required an expensive hardward and was not considered. The mesh was made coarser by relaxing the aspect ratio of some elements to be as high as 10. This model consisted of 2950 nodes and 935 elements and still was large for non linear analysis on PCs. However a linear analysis was mn and the results of which were compared with yet another coarse mesh developed on the wall panel assumed to have been built with pseudo bricks of size 21 Omm x 60mm x 50mm. The properties of such bricks and mortars were derived using the homogenisation technique presented by Middleton et ai as each pseudo brick and mortar contained several real bricks and mortar joints l5 . The size of the problem in this way could be significantly reduced. This model contained 860 nodes and 261 elements. Non linear analysis consisting of eight increments of thermalloading has been carried out using the mode!. A maximum of five iterations were allowed within each increment for achieving the Euclidean residual norm of 5%. A total clock time of 8 hours 20 minutes was required for one complete mn on a PC486 DX33.

6.2. 1 Material Properties

Calcium silicate is an autoclaved concrete. The strength properties of the brick both at ambient and elevated temperatures are contained in Neville l6 . Coefficient of thermal expansion of aggregates and cement paste are also given in Neville. A model for the variation of the coefficient of expansion with reference to change in temperature is contained in Gnanakrishnan and Lawther5.

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The properties of bricks and mortar used in the analysis at room temperature (21"C) is presented in Table 1. The variation of the properties expressed as a percentage of the corresponding properties at room temperature is shown in Figure 1.

The pseudo brick, head joint and bed joint were homogenised as per the equations given in Appendix 2 of reference 15. The properties in the x and y directions were then averaged out as they were not significantly different. The average stiffness properties of the pseudo bricks, bed joint and head joint mortars were 22609 MPa, 15876 MPa and 15431 MPa respectively. The Poisson's ratio ofpseudo brick was worked out to be 0.20 and both pseudo mortar systems were 0.25. The strength and expansion of the pseudo bricks and mortar were not modified.

TABLE 1 - PROPERTIES OF BRICKS AND MORTAR AT 21"C

l. 2. 3. 4 . 5.

PROPERTY Compressive strenath (MPa) Tensile strength (MPa) Young ' s modulus of elasticity (MPa) Poisson's Ratio Coefficient of thermaJ expansion

~ 2007""--------,.._---, .;. 180 c ~ 160

~14O " .€ 120

!1:~~~~~~~~~-4 'C ~ 60

~ 40 " ..

. ~ 20

~ 01n~~~~~~~~~n; o § ~ g ~ ~ § ~ ~ ~

Temperature [Degree Celcius 1

BRICK MORTAR 25.0 10.0 2.50 1.00

25000. 10000. 0.20 0.25

9.7 x 10-6 9.8 x 10-6

-*- Coefficient of thermal eXp3/1SlOn

--*- Tensile Strength

-a- Poisson's ratio

-+- Modulus of Elasticity, Compressive Strengtb

FIGURE 1 - EFFECT OF TEMPERATURE ON MATERIAL PROPERTIES

6.2.2 Boundary Conditions

The boundary conditions of the wall panel significantly affect its thermal bowing characteristics. From the description of the experimental set up in reference 22, the input data on boundary conditions was prepared. The bottom of the wall was rigidly cast with a steel beam. This allowed for the translational degrees of freedom at the bottom of the wall to be restrained. The rotational degrees of freedom at loof points were, however, allowed free as the horizontal bed joints in unreinforced masonry could not effectively be fixed against rotations3. The vertical sides and top of the wall had ceramic insulation fillers . The fillers were initially assumed as springs in their respective directions with the spring constant proportional to the contributing area (calculated as the product of the thickness of wall x centre to centre distance of the adjacent nodal spacings). A modulus of IN/mm3 was assumed for the purpose. The springs in the top of the wall were assumed to act only in the vertical direction and the springs in the sides were assumed to act in the horizontal direction. All other degrees of freedom were left free. The modulus was found to be insensitive to the displacement in the z direction. In the nonlinear analysis reported in the paper, therefore, the vertical degrees of freedom of the top and the horizontal degrees of freedom of the sides of the wall were simply arrested.

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6.2.3 Loading

Temperature and thermal gradient in the thickness direction of the wall were directly input at ali the nodes of the wall. The temperature profile across the thickness of the wall reported in reference 22 indicated distinct nonlinearity. However it was not possible to input the profile as thin shell element could accommodate only a linear variation. Hence from the values of the temperature on the hot and cold faces, a linear gradient was worked out and input in the analysis. The therrnal loading used in the analysis is contained in Table 2.

T ABLE 2 - THERMAL LOADING DATA

INCREMENT #1 #2 #3 #4 #5 #6 #7 #8

Central temp CC) 21 " 35.5 60.5 86.5 110.5 135.5 160.5 185.5

Gradient (C/mm) O 0.58 1.58 2.58 3.58 4.58 5.58 6.58

Fire temp CC) 2 1' 50" 100" 150" 2()()" 250" 300" 350"

6.3 ResuIts

The experimental resuIts reported by Shields et aI contained two distinctly differing nonlinear thermal bowing of the model wall. In the initial part of the fire loading the top displacement transducer (A) recorded a negative deflection indicating bowing away from fire while the transducer C recorded a positive deflection (bowing towards the fire) and transducer B recorded negligible deflection. This phenomenon continued up to fire temperature of approximately 250°C. With further increase in temperature, all transducers showed positive deflection indicating bowing of the wall towards the fire. The rate of increase in deflection of the three points A, Band C were identical. The absolute value of A at any given time continue to be lower than that of B and C.

Shields et ai have attributed the initial mode of thermal bowing to drying shrinkage without reference to the structural behaviour. The finite element analysis reported in the section was able to predict the interesting initial behaviour of the wall. The deflections of gauge points A, B and C predicted by the finite element analysis along with the experimental results of Shields et al are presented in Figure 2.

450,--------------------,

400

~350

a 300

" ~250 " o -;;-200 .3 E 150 o­E ~ 100

50

O;,~"~~n_rrrr~rrrl -5 o 5 10 15

Disolacement rmml

___ Gauge A (Experimental)

~ Gauge B (Experimental)

--*- Gauge C (Experimental)

__ Gauge A (FEM)

--8- Gauge B (FEM)

-G- Gauge C (FEM)

FIGURE 2 - VARIA TION OF BOWING OF SOME SELECTED POINTS WITH INCREASE IN TEMPERATURE

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Had the wall been restrained fuIly against translation and rotation at its bottom, cantilever bowing mode characterised by the Z-translation of the points A, B and C away from the fire would have resulted. With the rotational degrees of freedoms unrestrained, the wall behaved in a similar way as the experimental wall panel. The results indicate that the assumptions on the boundary conditions and the use of layered semi loof sheIl elements are realistic. The difference in the actual values of the deflections between the finite element and experimental predictions may be due to the differences in the properties of the materials used in the analysis and the experimento The relatively coarser mesh and the pseudo brick concept used in the modeIling could have also contributed to the variation.

The later part of the bowing of the wall has been dictated by the influence of the boundary conditions preventing the translation of the sides of the wall and the resulting friction between the ceramic filler material and the wall. The occurrence of the condition was acknowledged by Shields et a1 22 . In the absence of the frictional characteristics at the vertical boundaries of the waIl, the Z - translational degrees of freedom were also arrested after the fourth increment of loading. The change in boundary condition altered the behaviour of the wall and the displacements at A, B and C showed the tendency of the wall to bow towards fire.

The bowing profiles of the wall at four selected fire temperature are presented in Figure 3. The waIl, in the figure has been viewed in the YZ plane. The effect of varying boundary conditions is evident from the figure. Contours defining the bowing of waIl in Z direction at two of the four selected temperatures are presented in Figure 4. The nature of therrnal bowing could be appreciated with the help of Figures 3 and 4.

MYSTRO' '1.0 -2 OATE ' 2'-'2-93

BOWING EXAGGERATIO N 15% I NCREMENT "2 Z TRANSLATION

AT TOP & SIDES F IRE 050deg.C FREE

I NCREMEN T "4 ;W Z TRANSLATI ON F I RE 150d C: AT TOP & SI DES ego I FREE

H O T

H O T

y

, AT TOP & S IDES FIRE 350d C AT TOP & SIDES I NCREMENT "'6 ' f:Z TRANSLATION I NCREME NT "8 , Z TRANSLAT I ON

FIRE 250deg.C RESTRAINED ego 'J RESTRAINED

H H m O O T , T

,

7"---_____ -"'X

T IT LE ' THER MAL BOWING OF A MOOEL MASONRY WALL ISh i . ld , . , ai)

FIGURE 3 - FINITE ELEMENT PREDICTION OF BOWING PROFIELS OF THE MASONRY W ALL

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MYSTRO' 11.0 -2 DATE ' 23-12-93

CONTOURS DF OZ CDNTOURS DF A - 1 . 500 A -6.000 8 - 1 .000 8 -3 . C - O. 5000 C O. O O. O O 2. E O. 2500 E 4. F O. 5000 F 6.

G 8.

TITLE ' BOWING CONTOURS OF A MODE L MASONRY WALL I Sh ; o l d, ot ali

FIGURE 4 - CONTOURS OF THERMAL BOWING

7.SUMMARY

Understanding of bowing of masonry walls subjected to fIre on one face is extremely important as it could be further applied in the analysis of the insulation, integrity and structural adequacy failure modes of the wall. This paper described a nonlinear fInite element mo deI based on layered thin semi loof shell elements for the analysis of masonry walls subjected to fIre. The shell element initially mapped the wall in its plane (XY) and allowed the wall to bow out of plane (Z translation) on the application of fire loading. Stresses in each layer of the shell has only been monitored as the transverse shear in the thickness direction was assumed to have no effect on the inplane stresses in each layer. This allowed the use of plane stress failure criterion.

Bricks and mortar joints were modelled separately. Standard principal stress based two dimensional failure criterion generally employed for the concrete like materials was used. Variation of the strength and thermal expansion properties of bricks and mortar with reference to temperature was allowed for.

The fInite element model was verifIed using the results of a model wall tested by Shield et al for British Standard fIre22. The predicted and the experimental results were in reasonable agreement. The use of layered shell elements has therefore been regarded as justifIed.

8. ACKNOWLEDGMENT

The research project was funded by the UCQ URG 93/3 grant and a special grant from CSIRO. Further fInancial support is extended by the UCQ URG 94/035 research grant. Secretarial support was offered by Carmel Sneyd. Research Assistance was offered by Sivanesan Sritharan.

9. REFERENCES

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000 O 000 000 000 000

OZ

(2) AS1530.4 (1985), "Methods of Fire Tests on Building Materiais, Components and Structures", SSA Fire Test Code.

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Concrete Masonry Walls", Proc 4NAMC, California, pp 2601-2612. (9) Hu, H.T. and Schnolbrich, W.c. , (1991), "Nonlinear Finite Element Analysis of

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(10) Ibrahim, K.S . and Suter, G.T. (1990), "Finite Element Study of Thermal Stresses in Lowrise Concrete Masonry Walls", Proc 5NAMC, Illinios, pp 865-878.

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(13) Lawrence, S.J. and Gnanakrishnan, N. (1988), "The Fire Resistance ofMasonry Walls", NBTC Technical Record 531 , CSIRO, Australia.

(14) Meguro, K. , Iwashita, K, and Hakuno, M. (1988), "Fracture Tests on Masonry Concrete Elements by Granular Assembly Simulation", Proc 9th World Conference on Earthquake Engineering, Tokyo, pp V 1-183 to V 1-186.

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(16) Neville, A.M. (1981), "Properties of Concrete", ELBS, 3rd edition, pp 498-506. (17) Nizammuddin, Z. and Bresler, B. (1977), "Fire Response of Reinforced

Concrete Slabs", Report No. UCB FRG WP 77-1, University of Califomia. (18) Page, A.W., (1978), "Finite Element Model for Masonry", J of Structure

Division, ASCE, VI04, No 8, pp 1267-1285. (19) Page, A.W. , Kleeman, P.W. and Dhanasekar, M., (1985), "An Inplane Finite

Element Model for Brick Masonry", New Analysis Techniques for Structural Masonry, ASCE Structures Congress, Chicago, pp 1 -18.

(20) Papa, E. and Nappi, A. (1993) , "A Numerical Approach for the Analysis of Masonry Structures", Masonry Intemational, V7, No 1, pp 18-24.

(21) Rots, J.G. and Lourenco, P.B . (1993) , Fracture Simulation of Masonry Using Nonlinear Interface Elements", Proc 6NAMC, V2, Philadelphia, pp 983-993.

(22) Shields, T.J., O' Connor, D.J., Silcock, G.W.H. and Donegan, H.A., (1988), "Thermal Bowing of a Model Brickwork Panel", Pro c 8IB2Mac, Dublin, Ireland, pp 846-856.

(23) Syedahmed, E.Y. and Shrive, N.G. (1993), "A Model for FacesheU Bedded Hollow Masonry using 2D Finite Elements", Proc 6NAMC, VI, Philadelphia, pp 419-430.

(24) Bremmer, T.W., Holm, T.A and Valsangkar, A.J., (1990) , "Structural Integrity of Fire Walls at High Temperatures", Proc 5NAMC, Illinios, pp 879-888.

(25) FEA, (1993), "Lusas 11: A Finite Element System", Version 11, UK.

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