Nonlinear finite element analysis of ‘flanged’ ferrocement beams

NONLINEAR FINITE ELEMENT ANALYSXS OF ‘FLANGED’ FERROCEMENT BEAMS A. R. BIN-OMAR,t H. H. &DEL-RAHMAN$ and G. J. AL-SuLAItuMll’ tfkpartment of Civil Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, spudi Arabia $&partment of Civil Engineering, King AbduJ-Azix Univtrsity, P.O. Box 9027, Jeddah 21413, Ssudi Arabia

(Received 24 Frbruury 1988)

A&r&act--A computational model based on the Timoshenko beam finite element formulations is developed using quadratic isoparamctric &ments with three degrees of freedom. This modci is capable of tracing the entire flexural behavior of ferrocement beams of I-type and box type using a layered approach under monotonicaIly increasing loads. Ferrccement is modelled as a single material whose properties represent the integrated response of its constituents; mortar and wire-mesh. The model thus allows for cracking, yielding and fracturing of fcrroccment in tension and yielding and mushing of ferrocemcnt in compression. A nonlinear problem is solved by using full and modified Newton-Raphson incremental iterative algorithms. The validity of the proposed analytical mode1 is assmscd by comparing the numerical results with the available experimental rest&s. Comparisons are made for three types of ferrocements with two, three and four layers of wire-mesh. The effect of the presence of skeletal steel is also investigated.

NOTATJON initial or uncrackcd Yormg’s modulus tangent or cracked Young’s modulus initial rigidity reduced rigidity cracking stress yielding stress in tension compressive strength of mortar shear modulus moment cracking strain yielding strain in tension ultimate compressive strain ultimate tensile strain curvature Poisson’s ratio

INTRODUCTJON The last decade has shown considerable progress in research towards developing ferrocement as an efficient building material. The thin skeleta1 size of ferrocement and its light weight coupled with its high strength can be easily obtained even without skilled labor. These qualities for ferrocement in general make it a potential material for remotely located industrial buildings, small housings, warehouses which always have the problems of transportation of materials and nonavailability of skilled labor. Ferrocement flanged beams with I or hollow box sections can be used in flexure, supporting roofing systems. A survey of literature shows that ferrocement can be analyzed by one of three approaches. In the first approach, mathematical models were developed on the lines of conventional reinforced concrete theory [I, 21. Other investigators used the plasticity

approach (9. In recent years, the advent of electronic computers and sophisticated methods of analysis, such as the finite element method, has made it possible to develop numerical models for the complex behavior of ferrocement elements. A recent example of the use of finite elements in modelling ferrocement is a paper by Prakhya and Adidam [4], who analyzed ferrocement slabs using rectangular heterosis elements. The study reported here has the primary objective of developing a computational model for the full range of flexural behavior of ferrocement flanged beams of I-type and box shaped type from the start of loading up to the total failure. The main features of the model are as follows. A Timoshenko beam fotmulation as given in (51 is adopted which is efficient in terms of accuracy, since it takes transverse shear effect into account. as well as computational simplicity, as it only requires finite elements with C(0) continuity [6]. Improved quadratic isoparametric finite elements [6] are used. A layered approach is chosen to model the behavior across the beam cross-section in which skeletal steel, if it exists, will be smeared into an equivalent steel layer. Cracking is modelled by smeared (distributed) cracking system. The modelling of ferrocement material adopted.in this work is primarily based on a simplistic idealization, in which ferrocement is treated as a single material whose properties represent the integrated response of the ferrocement constituents, i.e. mortar and wire-mesh reinforcement, rather than treating each constituent as a separate material, thus avoiding complicated formulations. This approach is different 581

582

A. R. BIN-OHAR er al.

from that adopted by Prakhya and Adidam [4] in which mortar and mesh reinforcement were treated separately. The present approach is expected to be much more convenient for planar elements and thin shells. In brief, the model caters for cracking of mortar, yielding of wire-mesh in tension, rupture of wiremesh in tension, yielding and crushing of mortar in compression. The developed computer program (FCBl) implements the well known combined incrementaliterative Newton-Raphson algorithm and its invariants, i.e. initial stress and modified NewtonRaphson methods [5]. for the solution of the resulting system of nonlinear equations. The validity of the developed model is assessed by comparing the numerical results with the available experimental results. Comparisons are made with different ferrocement materials with two, three and four layers of wire-mesh reinforcement and two types of flanged cross-sections, oiz. I-type and box-type. The presence of skeletal steel is also investigated. Numerical experiments are conducted to study the effect of discretization on the numerical solutions. In these experiments finite element mesh size and number of layers in the cross-section are varied. Parameters that are used to study the flexural behavior of ferrocement beams include the following: deflections, curvatures, ultimate load and cracking patterns. Experimental results required to verify the developed model are obtained from a recent study by Al-Sulaimani et al. [7] and Rafeeqi [8]. It should be noted that the developed model handles only beams subjected to short-term monotonic loading. Other effects such as shrinkage, creep and cyclic loading are not taken into account.

ITNIlE

ELEMENT

FORMULATION

The finite element model is based on Timoshenko beam formulation and utilizes the quadratic isoparametric element with reduced integration. The beam cross-section is divided into a number of layers; each layer can have a different state of stress. In this way the gradual spread of nonlinearity across the beam thickness can be accounted for. Each node has three degrees of freedom, viz. axial displacement, vertical displacement and rotation. For a detailed description of the formulation the reader is referred to [5,9]. MATERIAL

MODEL OF FERROCEMEWT

The uniaxial stress-strain relationship adopted in this study is shown in Fig. 1. The trilinear stressstrain idealization of ferrocement behavior in tension was first presented by Walkus [IO] and later many investigators (1 I-131 observed the same behavior. Thus the ferrocement behavior in tension can be idealized into four distinct stages as follows.

(9 Elastic stage The incremental axial stress-strain in this stage is given as Af = E, AE. (ii) Cracking stage This stage starts when the axial stress reachesI,, the tensile strength of mortar. Cracking is taken into account by altering the value of E, thus in this case Af = E, AE.

Tonsion E-0

I

wrtnr

Comprarrlon

Fig. 1. Typical stress-strain

relationship

diagram of ferrocement. (Note: Diagram not to scale.)

Nonlinearfinite element analysis of ‘ibgd’

fcrmcemmt beams

Fig. 2. Testing set-up. Wire-mesh yielding stage (iii)

MoaHling of skeletal steel behavior

As the axial stress reaches fr, the wire-mesh starts to yield. No strain hardening is assumed and plastic flow continues until the tensile strain reaches a limiting value E,,.

The skeletal steel, if it exists is modelled by smearing the steel bars into equivalent steel layers which are assumed to be capable of transmitting axial forces only. A uniaxial elastic-perfectly plastic model for steel layers is used in the present work. It must be noted here that perfect bond is assumed to always exist between the wire-mesh reinforcement or skeletal steel and mortar.

(iv) Wire-mesh rupture When the axial strain reaches E,,, rupture of wire mesh is assumed to occur and ferrocement stress is reduced to zero. In compression an elastic-perfectly plastic idealization is adopted as shown in Fig. 1. The plastic flow in compression starts at an axial stress of fC and continues until an ultimate strain value of E,, is reached. Afterwards ferrocement is assumed to crush losing all its stiffness and stresses are reduced to zero. The shear stress is calculated as

The factor 1.2 is taken to account for warping in the cross-section [S]. The shear stress is assumed to remain elastic until crushing occurs where the shear stress is then reduced to zero. Poisson ‘s ratio

In finding shear modulus G the classical expression G = E/2(1 + V) is used with E as the modulus of elasticity and v as Poisson’s ratio. An average value of 0.2 has been used in the design and analysis of concrete [14, IS]. Since there is no information available regarding the Poisson’s ratio for ferrocement the same value of 0.2 is also assumed in this work.

APPLICATIONS

Test specimens

Experimental results obtained by Al-Sulaimani et al. [7] and Rafeeqi [8] are used to assess the validity of the proposed numerical model. Details of the testing set-up are shown in Fig. 2. All beams have the same overall span of 1200 mm and are tested under two third point loads. The tested beams are divided into four groups according to the beam cross-section and whether skeletal steel is provided; these are I-beams without skeletal steel (I), I-beams with skeletal steel (IS), box-beams (B) and box-beams with skeletal steel (BS). Each group has three beams of different ferrocement types. These are made of two-, three- and four- wire-mesh layer types. A beam in any group is designated by the number of wire-mesh layers followed by the group designation. Material properties

The ferrocement material properties adopted in the present model are as given in Table 1. For skeletal steel, the experimental stress-strain diagram in tension is idealized as elastic-perfectly plastic with a yield stress value of 620 MPa.

Table 1. Properties of krocement Fet-rocement material layers

Initial modulus E,(N/mm:)

Tangent modulus E,(N,‘mm’)

2 3 4

18,000 19,500 23.000

615 900 1100

(7.81

Cracking stress X (N/mm?

3.53

Yielding stress /,(Nlmm’) 4.14 6.21 8.28

584

A. It.

BIN-U

Finite element discrekation

The finite element discretization is shown in Fig. 3. The beam is typically divided into ‘14 elements. Because of symmetry, only one half of the beam is considered (Fig. 3a). The beam cross-section is divided into a typical number of 16 ferrocement layers with the dimensions shown in Fig. 3b. It must be noted here that box-sections are idealized as equivalent I-sections, shown in Fig. 4a. Since the behavior of the beams is mainly flexural, this idealization may be justified. For beams provided with skeletal steel, three bars of 4mm diameter are provided in each flange. These are smeared into two equivalent steel layers as shown in Fig. 4b. Results

The load (P)-central deflection curves predicted by the present model are compared with the experimental ones in Figs 5-g for groups I, IS, B and BS respectively. As can be seen from these figures, a good correlation is achieved between the experimental and analytical load-central deflection curves for I- and

ypfcal

box-beams without skeletal steel (Figs 5 and 7). However, for beams with skeletal steel, groups IS and BS, the correlation is not that good. It is believed that such discrepancies are due to the adopted idealization of skeletal steel behavior as elastic-perfectly plastic. A closer idealization of the stress-strain diagram to the experimental curve may result in improvement of the analytical results of groups IS and BS. Table 2 summarizes the analytical and experimental ultimate loads. A study of Table 2 shows that apart from beam 21, the ultimate loads predicted by the present model are very close to the experimental values for beams without skeletal steel. However, the differences between experimental and analytical values are higher for beams with skeletal steel. This may also be attributed to the adopted idealization of skeletal steel behavior. It is worth noting here that the results obtained by the present mode1 are conservative and on the safe side. Figures 9a and 9b show central moment-curvature curves for beams in groups (I) and (B), predicted by the present model. The initial and reduced rigidities

I

P/2

3-node

elcmnt

J=

CIal.

Hidaid.

node i

f--$

1

6 # 100 = 1

25

L 1

1

(a)

L 1

L

L 1

1

Mesh of 7 element in one-half

of the beam

:

t

L 1

844 # 5 20.875 m mm thick mm thick

1

I

1 156 =

(b)

Fcrrocencne

layer

syatam

Fig. 3. Finite element discretization of ferrocement beams

9

Nonlinear bite

element malyris of aged’

fmoamcnt

585

bums

Y 20

1 17

20

1 Dt f 20

156

k

not drawn

156

b

4

Ha.

Note:

1 60

to scale.

4 in

All diaenaiona

P.

Fig. 4a. Equivalent section for box-beam.

v=-l . . . ._

..-

.-...---. 93.5 mp

~

.

.

Skeletal

.

T -

- 0.242 IID

93.5 m _.__

f

k-=--l 3 $6a

3l$6mn

f

steel

.

EL

L = 0.262 lllD

Equivalent weared ateel layers Fig. 4b. Skeletal steel representation.

Ultimata

24.0 31.5 39.5

Loadr

(kN)

18.5 29.5 50.2

-

ICY

YOOLL(2

LATB.

+

ICY

YODCL(J

LAVB.

-6 _

ICY YODCL(4 CXCII.

; ; ; ; . ; ; , . ;r ,, ;a ;r CENTRAL

OtfLECTlON

(mm)

Fig. 5. Load vs deflection for 21, 31 and 41.

LAVl.

9.

;x

A. R. BIN-OMNIet al.

.

.

.

.

I

CENTRAL

v

8

4. 9, I,

DEFLECTtOW

,I

(4 1,

(mm)

Fig. 6. Load vs deflection for 21s. 31s and 41s.

,*-

Ultimrto

_L

rcu

YOOCL(J

LYS)

10s.

s,

FEY

MOOEL(J

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LYI)

EXPLR.

2

’

7.-

6 I.

-

II

-

.P

-

Lords [ kN1 FF.n 27.0 26.0 38.0 37.0 53.0 48.5

Exp.

CENTRAL

DEFLECTION

(mm)

Fig. 7. Load vs deflection for 2B, 3B and 4B.

1 Ultimrto

Loads (kN)

4 4

CLY rtu

_

ElPLR.

uoalL(J

LLVI.)

YODCL(*

LAYl.)

CENTRAL

I

DEfLECTION

+-IL-II

(mm)

Fig. 8. Load vs deflection for ZBS, 3BS and 4BS.

Nonlinear finite element analysis of ‘flanged ferrocement beams Table 2. Comparison of ultimate loads Beam 21 31 41 2B 3B 24Bs 31s 41s 2BS 3BS 4BS

Experimental (kN)

FEM IkNJ

24.0 31.5 39.3 21.0 38.0 53.0 51.0 57.6 68.2 58.2 15.0 19.0

18.5 29.5 38.2 26.0 37.0 48.5 44.1 53.0 62.0 48.0 62.0 13.5

an example, Fig. 10 shows the material state for beam 41 near failure. Cracking zones and extent of cracks are compared in the same figure with the experimental pattern. It should be noted that only vertical flexural cracks are allowed in the present model.

Percent differena 23.0 :: 3:7 2.6 8.5 13.5 8.0 9.1 17.5 17.3 7.5

NUMERICAL

0.0

I

1

1.0

2.0

EXPERIMENTS

In order to study the effect of discretization on the analytical results, two numerical experiments are carried out. In the first experiment, the number of finite elements representing beam 41 is varied between eight, 14 and 26 elements, keeping all other parameters the same. Figure 11 depicts the load-central deflection response and the ultimate loads for the three mesh sizes. It can be noticed that the response is almost the same for the three mesh sizes and that a mesh of

obtained from these curves are compared in Table 3 with the corresponding experimental values. The present model provides a full description of the material state in all layers across the beam thickness at the integration points at any stage of loading. As

0.0 f

581

10-14 elements

may be sufficient.

In the second experiment the number of layers across the depth of beam 41 is varied. The emphasis

I

curv%c

1

*‘&lln)

1

1

5.0

6.0

Fig. 9a. Analytical moment vs curvature for l-type beams.

Fig. 9b. Analytxal moment vs curvature for B-type beams.

1

A. R. B~N-OMAM lr ul. Table 3. Initial sod red& Specimen type

EW? Experimental

rigidities from M-# curves EI,(lo”) Experimental

FEM

FEM

21 31

1.13 1.21

1.16 1.80

2.00 2.00

2.14 2.60

:: 3B 4B

0.90 0.96 1.00 1.45

1.40 1.20 1.35 1.90

2.00 2.13 2.20 3.00

1.60 1.62 1.70 1.90

Finite

I

ElmtIc ferntemmt

(both

in tenlion

ten. 6 camp.)

in tenaim

Fig. 10. Material state in beam (41) near failure.

CENTRAL

DEFLECTION

(mm)

Fig. II. Load vs deflection for beam 41 (coarse vs fine elements).

clewnt

of “ftpngcd’ferrocementbeams

Nor&tar bite ekrnertt a&&

CENTRAL

DEFLECtlOX

589

(nun)

Fig. 12. toad vs &tleeiion for kam 41 (coarse vs fine tayera).

here is not only on the total number of layers but also on the number of layers in the fianges and web. It is clear frem Fig_ 12 that almost the same load-central d&e&on response is obtained for the range of total number of layers between 10 and 24. In the meantime, it can be seen that while the reduction in the number of layers in the flanges results in a corresponding reduction in the ultimate load, the reduction in the number of iayers in the web has little influence on the ultimate load. This is obvious for beams failing in flexute.

Based on the results presented in this paper the foflowing conclusions may be drawn.

The developed computational model is adequate for predicting the entire nonlinear behavior of ferrosement flanged beams under flexme up to failure. In particular, the foad~effection refationships and the ultimate loads are determined within reasonable accuracy if good knowledge of the material properties is available. The idealization of box-section into an equivalent §ion used for d&ret&g the box-beams is found to give good results for box-boams without skeletal steel. When quadratic ¶metric beam elements are used, only a small number of elements (10-14) may be sufficient for accurate results. Four layers in each figure and 4-g layers in the

web are ~commended. work reported here is parr of an MS. research project carried out at tha Department of Civil Engineering. King Fahd University of Petroleum and

Acknow~kdgments-The

Minerals, Saudi Arabia. The financial assistance Provided by KFUPM through a macarcb assistantship to Mr BinChnar is gratefully acknowkdgod.

REFRRENCES

D. Alexander, Ferroccment in relation to the rcinforc&d concrete code. J. Ferrocemenr 9, 113-125 (1979). 2. M. A. Mansut and P. Paramasivam, Cracking b&&or and ultimate strength of fcrroccment in &xure. fioeeetfings of fhf &LX& I~~e~tio~~ S~~~~ on Ferraremetrt, January t985 (Edited by L. R. Aurbriaco er al.), pp. 47-59. IF1< (1984). 3. Shamsu). Huq and R. P. Pama, Ferrocamcnt in flexurc-analysis and design. J. Ferrocement 8, 169493 (197%). 4. K. V. G. Prakhya and S. R. Adidam, Finite element analysis of fc~~ent plates. J. Ferrocement 17, I.

313-320 (t987).

5. D. R. J. Owen and E. Hinton, Finite Efem*nts in Plasticity-Theory and Practice. Pineridge Frcss, Swansea, U.K. (1980). 6. E. Hinton and D. R. 3. Owen, An Inrroduction to Finite Element Computations. Pineridge Press, Swansea, U.K. fI979). 7. C. J. A~-Su~aimani,1. A. ~sunb~ and S. F. A. Rafecqi, Study of the Aexural behavior and design of ftrrocement I-beams and hollow box beams. Paper presented in AC1 Annual Convention, San Antonio, Texas, March 1987. a. S. F. A. Rafecqi, Fkxural behavior of ferrocemcru I-beams and box-beams. MS%T&is, King Fahd University of hetrokum and Minerals; Dhahran (I 987). 9. H. H. Abdtl-Rahman, Computationa models for the nonhnear analysis of ninfomad concrete flexural slab systems. Ph.D. Thesis, Dept. Civil Engineering, Wnivernity College of Swansea, U.K. (1982). 10. B. R. Walkus, &h&or of fcrroccment in bending. i. Srtun. Ertgng 3, I&125 (1979). II. S. P. Shah and P. N. BaJag;uru, Ferroocment. In Ben 3Mnforced C’uncretes (Edited by R. N. Swamyf, PP. l-51. Surrey University Press, U.K. (1984). r2. G. B. Barson, G. M. Sabnis and A. E., Naaman. Survey of mechanical properties of farmcement as a structural material. Publication SF-61, ACI, 9-24 (1979).

590

A. R. BIN-~

13. C. D. Johnston and S. G. Mattar, Ferrocementbehavior in tertsion and compression. &oc. ASCE, J. Strucr. Dir. 102, 875-889 (1976). 14. M. E. Tasuji, F. 0. Slate and A. H. Nitson, The behavior of plain concrete subject to biaxial stress. Research report No. 360, Department

et al. of Structural Engineering, Cornell University (1976). IS. E. W. Tedjiogucu, Nonlinear analysis of reinforced concrete beams by the finite element method. Ph.D. Dissertation submitted to State University of New York, Buffalo, NY (1980).