Concrete Frames University of Port Harcourt, P.M.B. 5323 ...ijens.org/Vol_12_I_02/127502-3636-IJCEE-IJENS.pdf · the shear strength response of infilled frames with openings up to

International Journal of Civil & Environmental Engineering IJCEE-IJENS Vol: 12 No: 02 53

127502-3636 IJCEE-IJENS © April 2012 IJENS I J E N S

Modified Stiffness Matrix Method for Macro-Modeling of Infilled Reinforced Concrete Frames

T.C. Nwofor

Department of Civil and Environment Engineering

University of Port Harcourt, P.M.B. 5323 Port Harcourt,

Rivers State, Nigeria e-mail: [email protected]

ABSTRACT

In this paper, two kinds of models are used in order to validate a basic stiffness method for the macro-modeling of infilled frames. Previous numerical modeling techniques were faced with several complexities like the existence of plane of weak in the mortar joints and material non-homogeneity, which limited the real non-linear micro-modeling of infilled frames. The new explicit two dimensional finite element method, which is one of the models used in this work is used to study the behaviour of masonry infilled reinforced concrete frames and also considers the effect of the size of openings which is often ignored by most designers. A second model which is basically a macro-modeling technique which uses the stiffness matrix method to analyze an equivalent one-strut model used to replace the infilled panel is also used in this work, and results obtained validated against that of the micro-modeling procedure. It was observed that the stiffness matrix method for macro-modeling of infilled frames can quickly and effectively model the shear strength response of infilled frames with openings up to a failure load.

Keywords: Infilled frame, infill panel, equivalent one strut model, stress and displacement

1.0 INTRODUCTION

In many countries, situated in seismic regions, reinforced concrete frames are infilled by

brick masonry panels. Although the infill panels significantly enhance both the stiffness

and strength of the frame, their contribution is often not considered mainly because of the

lack of knowledge of the composite behavior of the frame and the infill. However, exten-

sive experimental research [1]-[4] and semi-analytical investigations [5], [6] have been

made. Recently, it has been shown that there is a strong interaction between the infill

masonry wall and the surrounding frame.

Attempts at the analysis of infilled frames since the mid 1950s have yielded several

analytical models. For a better understanding of the approach and capabilities of each



model it may be convenient to classify them into macro- and micro- models based on

their complexity. The basic characteristic of a macro- (or simplified) model is that they

try to encompass the overall (global) behavior of a structural element without modeling

all the possible modes of local failure. Micro- (or fundamental) models, on the other

hand, model the behavior of a structural element with great detail trying to encompass all

the possible modes of failure.

Since the first attempts to model the response of the composite infilled frames structures,

experimental and conceptual observations have indicated that a diagonal strut with

appropriate geometrical and mechanical characteristics could possibly provide a solution

to the problem. In 1958, Polyakov [7] suggested the possibility of considering the effect

of the infilling in each panel as equivalent to a diagonal bracing and this suggestion was

later taken up by Holmes [8] who replaced the infill by an equivalent pin-jointed diagonal

strut made of the same material and having the same thickness as the infill panel and a

width equal to one third of the infill diagonal length. Another set of researchers [9], [10]

related the width of the equivalent diagonal strut to the infill/frame contact lengths using

an analytical equation which has been adapted from the equation of the length of contact

of a free beam on an elastic foundation subjected to a concentrated load [11]. Based on

the frame/infill contact length, alternative proposals for the evaluation of the equivalent

strut width have been given [12], [13].

Also efforts were made to stimulate the response of infilled frames under earthquake

loading by taking into account stiffness and strength degradation of the infills [14]. They

proposed to model each infill panel by six compression-only inclined struts. Three

parallel struts are used in each diagonal direction and the off diagonal ones are positioned

at critical locations along the frame members. The advantage of this strut configuration

over the single diagonal strut is that it allows the modeling of the interaction between the

infill and the surrounding frame.

Even when it is a well known fact that infill walls have openings, recent research has

concentrated on simple cases of infill wall without openings. It is pertinent to note that

the direct influence of infill walls to the shear strength of the structural frame is greatly



reduced when the structure is subjected to cyclic or lateral loading as can be seen under

real earthquake situations. Useful experimental findings [15], [16] showed considerable

reduction in the shear response of infilled frames under cyclic loading. Analytical study

of infill frame with opening is limited and has little comparison due to the different

materials used and the different type of openings.

Experimental investigation by Benjamin and Williams [17] on the lateral stiffness of

infilled frames with openings showed a 50% reduction of the ultimate strength in infilled

frames having an opening at the center of the infill with dimensions proportional to the

infill dimensions by a ratio of one is to three.

Also experimental investigation into the effect of opening positions on the behaviour of

infilled frames with or without shear connectors was carried out [18]. It was observed

that opening at either end of the loaded diagonal of an infilled frame without connectors

reduces its shear strength about 75%. For infilled frame with shear connectors the

reduction in shear strength was about 60-70% as compared with infilled frame with a

solid panel. The reduction of strength in both cases is as a result of the centrally loaded

square opening.

The main purpose of this research is to model the shear resistance of lateral loaded

infilled reinforced concrete frame structure which accounts for the effect of openings in

the infill panel using the finite element method as an analytical tool and also to propose a

nonlinear macro-model for lateral load analysis of masonry infilled reinforced concrete

frame structure. We should note that in most cases, door and window openings are

provided in masonry infill panels to make up for functional and ventilation requirements

of buildings. Considering these openings which are the true representation of masonry

infilled structure adds complexity and difficulties in analysis. The presence of these

openings would tend to reduce the lateral strength and stiffness of the infilled frames.

However this reduction in strength has not been considered especially in the macro-

models mentioned in this review. Hence the models were only applicable to the analysis

of solid masonry infilled frames. It has been strongly emphasized by most Emergency

Management Agencies [19], that the effect of masonry infilled frames with or without

openings should be considered in the estimation of seismic vulnerability of existing

framed buildings. In most cases the strength and stiffness of infilled frames with



opening is based on the micro modeling of a composite infilled frame structure. Hence

the development of a useful macro-model to predict the shear strength response in terms

of the lateral load carrying capacity and component of the internal forces at ultimate load

of infilled frames with opening is also necessary.

2.0 FINITE ELEMENT MODELING OF INFILLED FRAME STRUCTURE

The developed finite element model would be employed here to simulate the in-plane

behaviour of masonry-infilled frames tested by previous researchers. Detailed

experimental results of the specimens have been summarized [15]. Among several

specimens tested under lateral load in their investigation specimen WC3 would be singled

out for this present investigation. The specimen is a single panel of 3600mm long by

2800mm high masonry infilled frame with a 0.8 x 2.2m central opening. This particular

specimen would correspond to structural model MIP04 used in this work. The basic

method was to allow a horizontal load increase up to failure load and applied at the upper

Conner of the Reinforced concrete infilled frame. From the foregoing the values

obtained from experimental test would be compared with result of finite element analysis

on the micro model in order to validate the model.

2.1 Development of Element Stiffness and Stress Displacement Matrix

For the purpose of this study the finite element method of analysis for a continuum would

be used. Basic triangular elements shall be used and the formulation adopted is the

displacement approach. In using this method the model displacements are the basic

unknown, while the stresses and strain are assumed to be constant for each element. The

finite element method of analysis used in this paper would involve voluminous numerical

works which would be considerably simplified by matrix formulation of the whole

problem, hence very suitable for computerization.

For plane elasticity problem the elastic matrix denoted by [D] can be expressed as



+

−−

−−

=

)1(200

011

011

yx

y

yxxy

y

yxxy

xyy

yxxy

yxx

yxxy

x

vE

vv

E

vv

vE

vv

vE

vv

E

D (1)

where Ex and Ey are the modulli of elasticity in the x and y direction respectively, Vxy

and Vyx are the poisson’s ratio in the xy and yx plane respectively.

The element stiffness matrix [Ke] would be a 6 x 6 matrix for the plane elasticity triangle,

because there exist two degree of freedom (DOF) at each node of the triangular element

(see Figure 2), hence the Nodal force vector [Fe] can be related to the displacement vector

as in equation 2.

{ } [ ]{ }eee KF δ= (2)

(a) (b)

Figure 2: (a) Nodal displacements vector (b) Nodal force vectors displayed in the

Cartesian co-ordinate system.

A suitable displacement function is chosen to define the displacement at any point in the

element. This is simply represented by two linear polynomials functions containing six

U2

U3

U1

1

2

3

V1

V2

V3

x

y

Fx2

Fx3

Fx1

1

2

3

Fy1

Fy2

Fy3

x

y



unknown coefficients ( )621, ααα L representing the six degrees of freedom in the case

of a plane triangular element.

++=++=

yxv

yxu

654

321

αααααα

(3)

The triangular element stiffness matrix [Ke] is represented by

{ } [ ] [ ][ ][ ]tBDBK Te ∆= (4)

where matrix [B] constraints constant linear dimensional values, ∆ represents area of

triangle and t represents the thickness of the triangular elements. It is simpler in practice

to perform the matrix multiplications of equation 4 numerically with the computer.

To determine the element stresses from the element nodal displacements, the

relationship below is considered where

( ){ } [ ] [ ] { }eBDyx δσ =, (5)

Where the stress-displacement matrix [H] equates to the product of matrix [D] and [B]

[H] = [D] [B]

( ){ } [ ] { }edHyx =,σ (6)

Where σ is the component of normal stress ( )yx σσ , and shear stress( )xyτ

2.2 Computer Program Formulation

In other to implement the finite element method, a computer programme for two

dimensional finite element analysis developed by the author would be used. The

computer programme is divided into two parts (subroutines). The first part consists of the

routines for the control numbers and data input modulus, the second part consists of

routines for tabulated nodal displacements and element stresses. The basic steps to obtain

the element stiffness matrix [Ke] and stress matrix [H] have already been discussed in

details and would involve voluminous numerical work, hence this processes were well

built up in the subroutines to take care of the overall analysis. The input data consists of

specifying the geometry of the idealized structure, its mechanical properties, the loading

and the support condition. The data also includes certain control numbers that would

help the efficiency of the program such as the total number of nodes and elements.

THE INPUT data for the micro-model is as follows:



(1) Nodal point coordinates in direction x and y for each node.

(2) Element properties: This includes the mechanical properties E and v for masonry

and concrete in two directions x and y, the thickness (t) of the structural model

and any other data defining each element and the structures as a whole.

(3) Boundary conditions: These consist of the restraints of the nodes of the supports

and the stiffness of the elastic supports

(4) Loading: Consists of the component of the lateral load placed at the top Conner of

the structure.

THE OUTPUT consists mainly of

(1) The components of displacements { }δ at each node in the directions x and y and

the maximum displacement (δmax) for a the model

(2) The stresses in each element as follows

(a) Component of normal stresses (σ) in the directions X and Y

(b) The shearing stress (τxy)

(c) The maximum shear stresses τmax

The bulk of the input data for the finite element micro-modelling of masonry infilled

structure will consist mainly of coordinates of nodal points and element properties. The

typical structural model for the validation of micro-model would consist of 216 elements

and 218 nodal points. Note that manual development of the mesh would entail

considerable expenditure of time and labour, hence a mesh generation subsidiary program

would aid the generation of this mesh and the corresponding coordinates of the nodal

points of the elements automatically. The output from this subsidiary programme

consists mainly of

• Coordinates x and y of all the nodal points numbered in a consecutive order.

• Element properties for all the triangular elements generated and numbered in a

specific order.

These outputs are then used as input data for the main programme.



2.3 Consideration of Size of Openings on the Shear Strength of Infilled Frame

Structure

In order to investigate the effect of the size of openings on the lateral strength and

stiffness of infilled reinforced concrete frames, a parametric study would be conducted

using the finite element analysis on the infilled brick masonry. The effect of the opening

size on the shear strength would be studied for values of parameters denoted by β and λm

which is defined as percentage of the opened area of the solid infill panel area and ratio

of the infill panel strength with openings to that without openings respectively.

Hence a number of one-story one-bay infilled structure with varying size of opening

would be analyzed using finite element method aided by the suitable computer

programme code. A typical structural micro model for the analysis is shown below

(figure 2 and 3) with a 30kN horizontal load acting at the top corner of the infilled

reinforced concrete frame structure.

To conduct properly this investigation the central opening of a one-bay infilled structure

is varied, but with particular interest on opening ratio of 0-25%. Structural models

tagged M1P01-M1P05 would be considered with each model having a particular

percentage opening in the infill panel.

Figure 2: Infilled reinforced concrete frame structure with central opening

2.5

3.0 0.3 0.3

0.3 30kN



Figure 3: Triangularly meshed micro-model ready for finite element analysis

Shear load-lateral displacement path obtained from finite element analytical modeling of

MIP04 model which is similar to the WC3 model can be compared against that obtained

from results of experimental data obtained by Dawe and Seah [15] as shown in figure 4 in

order to validate the finite element programme for the micro-modelling of masonry

infilled concrete frame structure with openings.

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30

L a tera l Displa c em ent (mm )

La

tera

l L

oa

d (

KN

)

This R es earch

Dawe and S eah (1989)

Figure 4: Lateral load-displacement curves

30kN

o o o



There was reasonable agreement between the experimental collapse loads of 285kN and

the numerical result of 295kN and a good correlation coefficient of cr = 0.9 was obtain

between the experimental and the numerical results.

The maximum value for shear stress τmax for the different models considered can be

obtained from the result of finite element analysis. These results show that the shear

stress τxy is a function of factors such as the applied lateral force, dimensions and the

elastic properties of the structure. Hence, a more general formula for shear strength and

ultimate shear load will consist of one which includes such variables.

In other to investigate the effect of opening size on the shear strength of masonry infilled

frames, a study was conducted for various values of the parameters denote β and λm as

defined previously. To this end the infilled frame structural models with central openings

denoted as MIP01, MIP02, MIP03, MIP04, MIP05 corresponding to 0-25% were

analyzed. The structural models are subjected to lateral loads which could be the result of

seismic forces, and a finite element analysis of the models carried out to determine the

effect of opening sizes on the lateral strength of masonry infilled frames. Here the

estimated shear strength factor λm (defined as the ratio of infilled panel strength with

opening to that without opening) is used for comparison of the numerical data obtained.

Figure 5 shows the variation of the shear strength reduction factor λm to the opening ratio

β. The results shows that an increase in the opening percent leads to a decrease in the

infilled frame shear strength. The shear strength decreased to about 75% for a bare

frame. It was also noticed that the shear strength reduction factor λm was practically

constant for an opening percentage of more that 55%. A comparison is made between the

results of this investigation with previous analytical results [20]. The favorably

agreement of this work with previous analytical study has further assisted in the

validation of the finite element model used in this work.



0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Ope ning ra tio (β) (β) (β) (β)

Sh

ea

r S

tre

ng

th r

ed

uc

tio

n f

ac

tor

(λλ λλm

)

Figure 5: Variation of shear strength reduction factor of infilled frame with opening ratio for a case of central opening.

From the foregoing, the values of shear strength reduction factor λm obtained from the

analysis can be plotted against the values of the opening percentage β. Hence a

consideration can be made using the shear strength factor λm to stimulate the equivalent

width of the compressed diagonal strut, for the macro-modeling of infilled frame

structure. A reasonable regression equation can be obtained relating λm to β for a case of

central opening of the compressed diagonal as

λm= 0.95 e0.03β (7)

3.0 MACRO-MODELLING OF INFILLED FRAMES

� A typical macro-model, which consists of a modified, one-structural model proposed

by the author, would be used to carry out this investigation. Here, the infill is

replaced with an equivalent pin jointed diagonal strut with mechanical property

correlated from that of the infill material. A three-strut model which consists of two

off diagonal struts which can be used for the nonlinear analysis of actual infilled

frames failing in corner crushing mode had been proposed [21]. However, this model

was not used to analyze infilled frames with openings. In order to consider openings

in the proposed macro model for this investigation, the equivalent diagonal strut area



is modified to account for the variation in these openings. Hence an equivalent

structure would be obtained by comparing with available numerical results obtained

in the previous section.

3.1 Modeling of Infilled Frames Adopting the One-Strut Model (OSM)

The analysis of the proposed model would be carried out using the stiffness matrix

method for pin-jointed bar elements. Where the stiffness matrix K for a bar element is

represented by

[ ]

−−−−

=

2

2

22

22

msymmetric

lml

mlmm

llll

L

AEk

mm

e (8)

where Fe represents the force vector and δe represents the displacement vector

considering two degrees of freedom at each end of the bar, the force vector and the

displacement vector can similarly be related in equation (2)

{ } [ ]{ }eee KF δ= (2)

An equivalent one-strut model for macro-modeling of infilled frame is shown in

figure 6.

Figure 6: One-strut model for masonry macro-modelling of infilled frame structure

L = 3.30m

2.65

m

Equivalent diagonal

strut

30kN



According to Saneinejad and Hobbs [22] the total equivalent diagonal region area is

simplified as

( )θαα

Cos

htA CC−

=1

(9)

Considering the resistance of the infilled panel (R) and the reinforced concrete frame to

the collapse load of the structure

H = h

MCosR Pj2

+θ (10)

R = m

Pj

f

RA

h

MhH

',

cos

2=

−θ

(11)

The equation can be modified to account for opening as

m

om

pjoo f

RA

h

MhHR

',

cos

2=

−=

θ (12)

From the foregoing the dimensionless parameter mλ relating to the effect of opening size

on shear strength capacity of infill panels can be defined as follows

pj

pjomom MhH

MhH

A

A

R

R

2

2

−−

===λ (13)

Hence in order to modify the equivalent diagonal area to account for openings, it is

expected that the regression equation relating the shear strength reduction factor (λ) to

the opening area ratio (β) be utilized

The modified equivalent diagonal region area in the infilled frames with a central

opening would given by

Am = λmAd (14)

Also carrying out this analysis it would be necessary to note the geometric properties of

the diagonal struts are functions of the length of contact between the wall and the column

α h and between the wall and beam α L [23]. Hence assuming a beam on elastic

foundation as proposed [9], [24] the following relationships have been obtained.

4

2

4

2 θπα

tSinE

hlE

m

cfh = (15)



and

4

2

4

θπα

tSinE

hlE

m

bf

L = (16)

Where Em, Ef = elastic moduli of the masonry wall and frame material respectively. t, h, l = thickness, height and length of the infill wall, respectively.

lc, lb = moments of inertia of the column and the beam of the frame respectively.

θ = tan-1

L

h

Hendry [23] also proposed the following equation to determine the equivalent or effective

strut width w, where the strut is assumed to be subjected to a uniform stress

22

2

1

2 hc ααω += (17)

Once the geometric and material properties of the struts are calculated, the stiffness

matrix method for bar elements can be employed to determine the stiffness of the infilled

frame, the internal forces and the deflections.

Considering the one strut macro-model in figure 6, the following geometric and material

properties can be deduced.

Infill wall:

Thickness t = 106mm

Elastic modulus Em = 11.152 x 103N/mm2

Frame:

Area of beam Ab = 90,000mm2

Area of column Ac = 90,000mm2

Moment of inertia of beam and column = Ib = Ic = 6.75 x 108mm4

Elastic modulus Ef = 2.9 x 104N/mm2

Where

== −−

0.3

5.2tantan 11

l

hθ = 39.80



From equation 15 and 16

4

60.79

4

2 SinE

hIE

m

cfh +

= πα = 1006.37mm

4

60.79

4

SinE

hIE

m

bfL +

= πα = 2012.80mm

Using equation 17

22

2

1

2 Lh

w αα += = 1125mm

Area of diagonal strut Ad =

2

w x 106mm = 119,205mm2

hence modified area of diagonal strut to account for the cases of opening

Am = Adλm

Where λm = 0.95e0.03β

Analyzing the frame using a classical methods of structural analysis in the stiffness

matrix for a two dimensional structure, maximum unknown horizontal deflection would

be obtained from the solution of the global structural matrix.

4.0 DISCUSSION OF RESULTS

The equivalent one strut system was used for the macro-modelling of infilled frames

using a classical method of structural analysis in the stiffness matrix method. Using this

model, the non-linear static behaviour of masonry-infilled frames was studied by

analyzing structural models MIP01-MIP05. The maximum horizontal displacement in

the frame was analyzed for by using a modified area for the equivalent strut from

equation 14 and the effective width of strut from equation 17. The value of displacement

obtained using this model is compared with that obtained previously with the micro

model.

It can be seen from Figure 7a to 7e that the equivalent strut model was able to model the

ultimate load capacity of the masonry infilled frames with openings up to failure as a

very close agreement was seen between the micro and macro models with a general

corresponding average magnification factor of 1.1.



0

50

100

150

200

250

300

350

0 5 10 15 20 25

L a tera l Displa c eme nt δ δ δ δ (mm)

La

tera

l L

oa

d (

KN

)

Mic ro Model

Mac ro Model

(a)

0

50

100

150

200

250

300

350

0 5 10 15 20 25

L a tera l Displa ce ment δ δ δ δ (mm)

La

tera

l L

oa

d (

KN

)

Model Mic ro

Mac ro Model

(b)



0

50

100

150

200

250

300

350

0 5 10 15 20 25 30

L ate ra l Displa c em ent δδδδ (m m)

La

tera

l L

oa

d (

KN

)

Mic ro Model

Mac ro Model

(c)

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30

L a tera l Displa c em ent δδδδ (m m)

La

tera

l L

oa

d (

KN

)

Mic ro Model

Mac ro Model

(d)



0

50

100

150

200

250

300

350

0 5 10 15 20 25 30 35 40

L atera l D is placem ent d (m m)

La

tera

l L

oa

d (

KN

) Micro Model

Macro Model

(e)

Figure 7: Lateral load – displacement relations for different size of opening on the infill panel (a) MIP01 (b) MIP02 (c) MIP03 (d) MIP04 (e) MIP05

CONCLUSION

The macro-modeling technique can be used for the design of infilled frame with opening

by utilizing a modified area for the equivalent strut. Noting that in this work two kinds of

numerical modeling strategies were used to stimulate the in-plane non-linear behaviour of

infilled frames with openings, where the two dimensional finite element micro-model

developed for the inelastic non-linear analysis of masonry-infilled structure was validated

and used for the study of the effect of openings on the shear strength of the structure.

Furthermore application of this model may be require a lot of computational skill

especially for individuals that may not have useful analytical program software hence, an

equivalent one strut model was adopted and modified to investigate the nonlinear

behaviour of infilled frames with a central openings. Here a modified diagonal area

related against the shear strength reduction factor obtained from the different case of

opening sizes can be use improve the estimation of the equivalent strut to account for the

effects of openings when utilizing the macro-modeling technique. This model was used



in the study of one storey-one bay infilled frame structures, and the results obtained

compared favorably with that obtained from the finite element micro modeling technique.

Valuable extension of this study would include but not limited to the following,

(a) The utilization of the concept of equivalent strut to model the behavior of multi-

storey one-bay full and partial infilled frames.

(b) The macro-modeling technique should be extended to accommodate the effect of

position of openings on the non-linear analysis of infilled frames.

(c) Finally the extension of the macro-modeling technique to the response of infilled

frame to dynamic shear loads is also necessary in order to obtain results that

would be utilized by designers in real dynamic regimes.

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NOMENCLATURE

A = Area of loaded diagonal region of infill panel Cr = Correlation coefficient

Ex, Ey = Modulus of elasticity in x and y direction

{ }eF = Element force vector

h = Height of column H = Lateral load carrying capacity of solid masonry infilled frame

[H] = Triangular element stresses matrix Ho = Lateral load carrying capacity of masonry infilled frame with opening l = Length of beam K = Constant depending upon brick properties and brick-mortar joint

configuration; [Ke] = Triangular element/bar element stiffness matrix Mp = Plastic moment capacity of frame members n = Total number of data points R = Resistance of solid infill panel Ro = Resistance of infill panel with opening t = Thickness of infilled plane

{ }eδ = Element displacement vector

xyτ = Shearing stress component in rectangular coordinates

σx, σy = Normal component of stress in the x and y axes εx, εy = Strain in the x and y directions

β = Ratio of central window opening to infill panel area λm = Modification factor of diagonal region area δc = Ratio of column contact length to height of column θ = Tan-1 (h/l)

Documents

Concrete Frames University of Port Harcourt, P.M.B. 5323 ...ijens.org/Vol_12_I_02/127502-3636-IJCEE-IJENS.pdf · the shear strength response of infilled frames with openings up to