2019 Scienceline Publication

Journal of Civil Engineering and Urbanism

Volume 9, Issue 2: 07-16; March 25, 2019

ISSN-2252-0430

DOI: https://dx.doi.org/10.29252/scil.2019.jceu2

Effective Moment of Inertia of Single Spanned Reinforced

Concrete Beams with Fixed Beam-Column Joints

Lekan Makanju Olanitori

Federal University of Technology, Akure, Department of Civil Engineering, Akure, Ondo State, Nigeria

^Corresponding author’s Email: lekanmakanjuola@yahoo.com

ABSTRACT

( )

Of prime importance in the determination of the deflection of beams is the calculation of the moment of inertia

of the beam, the value of which changes along the span length from ꢀ _ꢁꢀ for uncracked sections to ꢀ _ꢂꢃꢀ for cracked

sections. From literature, many experimental works have been carried out on simply supported beams with varying

concrete characteristic strengths and percentages of reinforcement. However, none was on beam with fully or

( )

partially restrained ends. Hence the focus of this research work is to determine the effective moment of inertia

ꢄ

of a cracked L-section of reinforced concrete beam with full end restrained. Three existing models for determining

“ _ꢄꢀ were used in the estimation of the deflection of the beam, and these existing models were modified in order to

get a proposed model that gives a more accurate prediction of the deflection. At service load of 9.81 kN/m, the

estimated deflections using the values of from the existing three models and the proposed model were 2.01 mm,

ꢄ

9.81 mm, 2.68 mm and 8.37 mm respectively, while the actual deflection was 8.14 mm. From these results, the

proposed model predicts more accurately the deflection of the L-beam than the three existing models, however, it is

recommended that further research should still be carried out on reinforced concrete beams with fixed beam-column

joints, in order to get a model that can predict more accurately, the effective moment of inertia ꢀ _ꢄꢀ for other types

of beams such as rectangular and T-beams.

Keywords: Deflection, Uncracked Moment of Inertia; Cracked Moment of Inertia; Effective Moment of Inertia;

Cracking Moment; Elastic Modulus of Concrete.

INTRODUCTION

at service loads which will be compared with the

permissible deflection from the codes. One of the major

factors that affects the deflection of flexural members is

the effective moment of inertia.

Reinforced concrete structures must satisfy both the

ultimate and serviceability limit states. The serviceability

limit states of crack widths, deflections and excessive

vibrations are of prime importance. Historically,

deflections and crack widths have not been a problem for

reinforced concrete building structures (Wight and

MacGregor, 2009 ). The introduction of high-strength

concrete, high-strength reinforcing bars, coupled with

more precise computer-aided design softwares, the limit-

state serviceability design, has resulted in lighter and more

material-efficient structural elements and systems. This in

turn has necessitated better control of short-term and long-

term behavior of concrete structures at service loads (ACI

435R, 2000 ).

In practice, deflection control is based on the

deemed – to – fit provisions of the codes (Nkuma, 2013 ).

However, damages such as cracks have been noted on

partition walls of buildings resulting from excessive

deflection of slabs and beams even when the serviceability

requirements for deflection based on these deemed-to-fit

provisions of the code were satisfied (Nkuma, 2013 ).

Hence there is the need to estimate the expected deflection

The deflection of a flexural member is a function of

the support conditions, applied load and span, and the

flexural rigidity of the member. The majority of the

building codes do not concern themselves with

computations of deflections but rather with attempting to

provide minimum values of flexural rigidity. Deflection of

reinforced concrete flexural members is controlled by

reinforcement ratio limitations, minimum thickness

requirements, and span/deflection ratio limitations.

The minimum thickness provisions of American

Concrete Institute (ACI 435R, 2000 ) for deflection control

are contained in Table 2.4 of ACI 435R (2000), while the

basic span-effective depth ratios provisions of the BS

8110 are contained in Table 3.9 of BS 8110-1 (1997). The

allowable computed deflections specified in ACI 318

(2005) for one-way systems are reproduced in Table 2.5

of ACI 435R (2000), where the span-deflection ratios are

provided for a simple set of allowable deflections.

Before cracking, the entire cross section is stressed

by load. The moment of initial of this section is called the

To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9

(2): 07-16. www.ojceu.ir

ꢏ_ꢂꢃꢆ ^ꢲ^ꢳ^ꢊ^ꢴꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓꢌꢌ(ꢕ)

un-cracked moment of inertia, which is usually, the gross

un-cracked transformed moment of initial for the concrete

section( _ꢁ). As the load is increased, flexural cracking

occurs when the moment exceeds the cracking moment.

The corresponding moment of inertial for this cracked

section is referred to as the cracked moment of

ꢵ

ꢶ

ꢙ ꢆ ꢘꢓꢠꢝꢬꢌꢷ ꢌ ꢙ^ꢅꢌꢏꢸꢞ

Eq. (6)

√

ꢂ

ꢃ

ꢂ

Where:

ꢙ = modulus of rupture; ꢙ^ꢅ= concrete grade;

ꢃ

ꢂ

ꢷ_ꢂꢆ ꢔꢓꢘ for normal density concrete (2325 to 2400

kg/m³), 0.85 for semi low-density (1765 to 2325 kg/m³)

and 0.75 for low-density concrete (1445 to 1765 kg/m³).

For continuous members, ACI 318 (2005) stipulates

thatꢌ _ꢄ, may be taken as the average values obtained from

Eq. (7)) for the critical positive and negative moment

sections. For prismatic members, _ꢄ, may be taken as the

value obtained at mid-span for continuous spans (ACI

(

)

inertial . The deflection of a beam is calculated by

ꢂꢃ

integrating the curvatures along the length of the beam

(Wight and MacGregor, 2009 ). For an elastic beam, the

ꢈ

curvature, ^ꢅꢆ ꢇ , is calculated as ꢇ ꢆ _ꢉꢊ, where EI is the

ꢃ

flexural stiffness of the cross section. When the

integration is completed it can be seen that the deflection

of a member is a function of the span length, support or

end conditions, the type of loading and the flexural

stiffness, ꢋ . In general the elastic deflection for non-

cracked members can be expressed as Eq. (1):

ꢏꢐ^ꢑ

435R, 2000 ). If the average effective moment of inertia

ꢄ

is to be used, then according to ACI 318 (2005), the

following expression should be used:

ꢌ ꢆ ꢘꢓꢕ _ꢄ(ꢨ)ꢩ ꢘꢓꢝꢕ( _( )ꢩꢌ

)

Eq. (7)

( )

ꢄ ꢑ

ꢄ

ꢄ ꢅ

ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢍ ꢆ ꢎ

ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢔ)

ꢋ_ꢂ _ꢁ

Where the subscripts m, 1, and 2 refer to mid-span,

and the two beam ends, respectively.

Where, k is a factor depending on the degree of

fixity of the support, ꢏ is maximum moment, ꢐ is clear

The value of _ꢄ, can also be affected by the type of

loading on the member (Al-Zaid, 1991 ), i.e. whether the

load is concentrated or distributed. Furthermore, Al-Zaid

et al (1991) experimentally showed that the power m in

the effective moment of inertia expression is affected by

the loading conditions of a beam and the load level

(M_a/M_cr).

In their study, Al-Shaikh and Al-Zaid (1993)

revealed that Branson’s model underestimated the

effective moment of inertia of all test specimens. The

underestimation of I_ewas approximately 30% in the case

of a heavily reinforced member and 12 % for a lightly

reinforced specimen. Beyond the previously observed

behaviour of a reinforced concrete member subjected to a

mid-span concentrated load (Al-Shaikh and Al-Zaid

1993), it is obvious that reinforcement ratio affects the

accuracy of Branson’s model especially when the member

is heavily reinforced and that the value of m decreases as

the reinforcement ratio (ρ) of a concrete beam increases.

Accordingly, they proposed the following equation for m:

span length, ꢋ_ꢂis elastic modulus of concrete and gross

ꢁ

moment of inertia of the section (ACI 435R, 2000 ). The

elastic modulus of concrete ꢋ_ꢂcan be estimated using

equation (2) below (Wight and MacGregor, 2009 ):

ꢅ

(

)

ꢋ_ꢂꢆ ꢕꢖꢗꢘꢘꢘ√ꢙ ꢌꢌꢌꢌꢌꢌ ꢚꢛꢜ ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢝꢞ)

ꢂ

ꢑ

ꢅ

(

)

ꢌꢋ_ꢂꢆ ꢟꢗꢖꢝꢕꢓꢠꢟ√ꢙ ꢌꢌꢌꢌꢌꢌ ꢡꢢꢣꢣ ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢝꢤ)

ꢂ

For the reinforced concrete beam, however, three

different values of must be considered depending on

section condition. When the section is un-cracked, the

value of is equal to _ꢁ. The value of ꢆ is used when

ꢂꢃ

the beam section is fully cracked. For the beam with

partially cracked section the value of must be taken as

ꢄ

(Akmaluddin and Thomas, 2006 ).

Branson (1965) used Eq. (3) to express the transition

from to that is observed in experimental data:

ꢁ

ꢂꢃ

ꢨ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢆꢌꢥ

ꢄ

ꢧ ꢌ _ꢁꢩꢌ*ꢔ ꢪꢌꢥ

ꢧ +ꢌ _ꢂꢃꢌꢫꢌ _ꢁꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓꢌꢌ(ꢬ)

ꢏ_ꢦ

Branson’s effective moment of inertia expression

(Equation 3), which averages the moments of inertia of

the un-cracked and fully-cracked portions of a concrete

beam, is adopted by ACI 318 (2005), which set the value

of m to 3 to obtain an average moment of inertia for the

entire span of a beam and this is expressed as Eq. (4).

ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢣ ꢆ ꢬ ꢪ ꢘꢓꢹꢺꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢹ)

Nonetheless, different studies (Scanlon et al., 2001;

Gilbert, 1999 and Gilbert, 2006 ) indicated that Branson’s

model constantly overestimates the moments of inertia of

reinforced concrete beams with low reinforcement ratios

(ρ < 1%), which causes underestimation of the deflections.

Bischoff (2005) found out that the underestimation of the

moments of inertia and deflections of lightly-reinforced

concrete beams by the Branson’s approach is caused by

the overestimation of the tension stiffening of concrete.

According to the analytical study carried out by Bischoff

(2005), the tension-stiffening component in Branson’s

method depends on the applied load level (M_a/M_cr) and on

the ratio of the gross moment of inertia to the cracked

moment of inertia (I_g/I_cr) of the beam, which varies

ꢭ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢌ ꢆꢌꢥ

ꢄ

ꢧ ꢌ _ꢁꢩꢌ*ꢔ ꢪꢌꢥ

ꢧ +ꢌ _ꢂꢃꢌꢫꢌ _ꢁꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓꢌꢌ(ꢟ)

Where:

ꢮ_ꢯꢰꢌ = Cracking moment; ꢮ_ꢦꢌ = Maximum service

load moment (un-factored) at the stage for which

deflections are being considered; _ꢁꢌ = Gross moment of

inertia of section; ꢱ_ꢯꢰꢌ = Moment of inertia of cracked

transformed section

And

To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9 (1):

07-16. www.ojceu.ir

( )

of a cracked L-section of reinforced

concrete beam with beam-column joint fixed.

inversely with the reinforcement ratio (ρ). Branson’s

expression provides accurate estimates for reinforced

concrete beams with reinforcement ratios greater than 1%,

which corresponds to an I_g/I_crratio of 3. For lower

reinforcement ratios (I_g/I_cr> 3), the member response

estimated by Branson’s approach is stiffer than the actual

response, resulting in the under prediction of the

deflections (Kalkan, 2013 ).

of inertia

ꢄ

2.0 MATERIAL AND METHODS

2.1 Materials. The materials used for this work is a

square, single panel, reinforced concrete space framed

model with beam-column joint fixed, constructed from

micro-concrete (using sand from borrowed pit), loading

box, laterite as the loading material, and dial gauges.

Figure 1 shows a typical model. The loading box which

measured 1m x 1m x 3.0 m was placed on top of the

model, and three dial gauges were placed at the centre of

the slab and centres of two adjacent beams. Manually,

known weight of laterite using head-pan was poured into

the loading box and readings of the dial gauges for

deflections of the beams and slab were taking at every

1.84 kN load of laterite. The process continued until

collapse occurred.

Bischoff (2005) presented the application of the

method to the in-plane bending behavior of reinforced

concrete beams and developed the following effective

moment of inertia expression, which is a weighted

average of the flexibilities of the un-cracked and cracked

portions of a reinforced concrete beam:

ꢨ

ꢔ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢔ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢔ

ꢆ ꢥ

ꢧ ꢌ ꢩꢌ*ꢔ ꢪꢌꢥ

ꢧ +ꢌ ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓꢌꢌ(ꢻ)

ꢐ_ꢄ

ꢁ

ꢂꢃ

A value of 2 was proposed for the power m in

Equation (9), based on the deflection equation given in

Eurocode 2 (CEN, 2002). The use of m = 2 assures that

the tension-stiffening contribution in the model is only

dependent on the applied load level (M_a/M_cr), as explained

by Bischoff (2005) and Bischoff (2007), in detail.

Consequently, the tension-stiffening model becomes

independent from the gross-to-uncracked moment of

inertia ratio (I_g/I_cr) and the reinforcement ratio (ρ) of the

beam.

In their work, Ammash and Muhaisin (2009),

presented a new form of the effective moment of inertia

model by enhancement of Branson's model, taking into

account the effect of several factors such as type of

loading, shear deformations, reinforcement ratio. The

models resulted from their studies were compared with

(experimental results, Branson's model results, and results

of other models). The results of the model give best

agreement with experimental results than Branson's and

the other models. The results showed that the effective

moment of inertia reduced by about 27% for span to depth

ratio of (20 to 5) due to shear deformation effects and

gives good agreement with the experimental results for all

types of cross section.

Also, Ammash et al. (2018), proposed a model for

estimating deflection. This model takes into consideration

parameters such as grade of concrete, loading conditions

and type of reinforcement. The results of the proposed

model showed a better agreement with the experimental

studies when compared to ACI equation and other models

from literature. The maximum difference in deflection

results from the proposed model and actual deflection

from experimental work was between 1% and 10%.

From literature, extensive research work have been

carried out on the determination of effective moment of

inertia for simply supported beams, however none was on

beam with fully or partially restrained ends. Hence the

aim of this research is to determine the effective moment

Figure 1: Square space framed model. (a): Schematic

figure of the Square RC (Space Framed Model); (b):

Schematic figure of the Square RC Space Framed (Model

with the Loading Box)

2.1.1 Cement. Ordinary Portland Cement (OPC)

obtained from Larfarge Cement, Ewekoro, Abeokuta,

Ogun State of Nigeria, was used in this study. The OPC

used complies with Type I Portland cement as in ASTM

C150-02a (2002).

2.1.2 Soil. The fine aggregate was collected from a

borrowed pit from Akure metropolis, while the coarse

aggregate was purchased from JCC Quarry, along Akure-

Owo road. From Figure 2, the coefficient of curvature Cc

for sand is 2.90 and the coefficient of uniformity C_ufor

sand is 1.06. These values indicate that the sand is well

To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9 (1):

07-16. www.ojceu.ir

graded, since it is within the satisfactory range of 2 and 3

for coefficient of curvature, as specified by the British

Standard Institution (BS 812-103.1, 1985 ).

2.1.3 Water. For this experimental study, tap water

was used to produce the space framed structures. The

water/cement (W/C) ratio used for the research work was

0.55.

2.1.4 Reinforcement. Mild steel reinforcement with

characteristic strength of 250 N/mm², was used for this

experimental work.

BS SIEVES :

100

0.001

CLAY

0.01

MEDIUM

0.1

100

FINE

COARSE

COARSEFINE MEDIUMCOARSECOBBLBEOULDER

FINE MEDIUM

SILT

SAND

GRAVEL

STONE

0.002

0.06

600

Particle size in mm

Figure 2: Particle size distribution curve for fine aggregate

3.0 Calculation of Immediate Deflection

3.1

Beam

Specifications

and

Strength

Characteristics. The space frame models were designed

in accordance with the requirements of BS8110 – 1

(1997). The dimensions of the model were:

Slab: 1000mm x 1000mm x 50mm thick; Beam:

75mm x 1000mm; Column: 75mm x 75mm. Column

height = 1000 mm; ꢼ_ꢯꢽꢆ ꢖꢌꢾꢢꢿꢿ^ꢑꣀꢌꢼ_ꣁꢆ ꢝꢕꢘꢾꢢꢿꢿ^ꢑ.

From design the required area of reinforcement

(ꣂ_ꣃꢃꢄ꣄) was 9.6 mm², and the provided area of

reinforcement (ꣂ_{ꣃꣅꢃ꣆꣇}) was 2R6 bars with 56.6 mm²area.

Deflection check according to BS 8110 -1 (1997) was

satisfactory.

Figure 3: Stress distribution across the rectangular beam

section.

3.2 Estimation of Deflection for beams Under

Estimated Service Load. The beam from the square

spaced framed was considered as L – beam. The beam

supports un-factored dead and live loads of 0.48 kN/m and

6.38 kN/m respectively. It was built of materials with

strength characteristic f_cu= 7 N/mm²for concrete, f_y= 250

N/mm²for steel and concrete density ꢷ ꢆ ꢝꢬꢖꢘꢌꢎ꣖ꢢꢣ^ꢭ,

E_c= 12.5 *10³N/mm².

From Eq. (2b), and Figure 3, we have:

ꢅ

ꢭ

ꢑ

꣈

ꢋ_ꢂꢆ ꢟꢗꢖꢝꢕꢓꢠꢟ ꢙ ꢆ ꢟꢗꢖꢝꢕꢓꢠꢟ꣉ꢖ ꢆ ꢔꢝꢓꢕ ꣊ ꢔꢘ ꢡꢢꢣꢣ ꢌꢌꢌ

ꢂ

ꢏ_꣋꣌ꢆ ꢘꢓꢹꢕꢌꢎꢡꢣꢗ ꣍꣎꣏꣐꣏ꢌꢏ_꣋꣌ꢌꢜꢛꢌꢣ꣑ꢣ꣏꣒꣓ꢌ꣑ꢙꢌ꣐꣏ꢛꢜꢛ꣓ꢞ꣒꣔꣏ꢌ꣑ꢙꢌ꣓꣎꣏ꢌꢤ꣏ꢞꢣꢌ

3.2.1 Check if the beam has cracked at service

loads. Compute _ꢁfor the un-cracked L-section (ignore the

effect of the reinforcement for simplicity):

Since the beam- column joint is designed and

detailed as fixed joint, the maximum bending moment at

the mid-span is:ꢌꢌꢏ ꢆ ꣍ꢐ^ꢑꢢꢔꢝ

Substituting for all the relevant parameters, the

estimated ultimate load ꣍_꣕ꢆ ꢔꢘꢓꢝꢌꢎꢡꢢꢣ and the estimated

Flange width for L-section ꢤ_ꢲꢫ ^{꣗ꢄꢦꢨꢌꣃꣅꢦ꣘ꢌ꣙ꢄ꣘ꢁ꣚꣛}ꢩ ꢤ_꣜

ꢆ

ꢅꢑ

^ꢅ꣝꣝꣝ꢩ ꢖꢕꢌ= 158 mm – according ACI 318-11 (2005).

ꢅꢑ

ꢔꢘꢓꢝ

service load ꣍_ꣃꢆ

⁄

ꢆ ꢠꢓꢬꢹꢌꢎꢡꢢꢣ.

ꢔꢓꢠ

To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9 (1):

07-16. www.ojceu.ir

ꢑ

꣢

ꢕ꣍ꢐ^꣢

[( ) ( )]

ꢔꢠꢟꢓꢕꢹ ꢩ ꢖꢹꢓꢔꢬ ꢩ ꢝꢘꢟꢓꢕꢝ ꢩ ꢕꢕꢟꢓꢠꢻ ꢔꢘ

ꢁ

ꢆ ꢆ ꣞ _{꣆꣜꣘ꢌꢦ꣟꣠ꣃ}ꢩ ꣞ꣂ꣡̅

ꢆ

ꢵ

꤅ꢆ

ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢔꢔ)

꣣ꢌꢌꢌꢌꢌ ꢆ ꢔꢘꢘꢔꢓꢻꢖꢌ꣤ꢌꢔꢘ^꣢ꢣꢣ^꣢

ꢬꢹꢟꢋ_ꢂ _ꢄ

ꢁ

꤆꣎꣏꣐꣏ꢗ ꢋ_ꢂꢆ ꢔꢝꢓꢕꢌꢎꢡꢢꢣꢣ^ꢑꢗ ꣍_꤇ꢆ ꢘꢓꢟꢹꢌꢎꢡꢢꢣꢣ^ꢑꢗ ꣍_꣙

ꢆ ꢠꢓꢬꢹꢌꢎꢡꢢꢣꢣ^ꢑꢗ ꢐ ꢆ ꢔꢣꢗ ꢆ _ꢄꢌ

Where:

Deflection due to dead load can be computed as shown

below:

꣞

ꢆꢌ _{꣆꣜꣘ꢌꢲ꣙ꢦ꣘ꢁꢄ}ꢩ

꣆꣜꣘ꢌ꣜ꢄ꣗

꣆꣜꣘ꢌꢦ꣟꣠ꣃ

ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌ

ꢆ ꢔꢠꢟꢓꢕꢹꢌ꣤ꢌꢔꢘ^꣢ꢌꢞ꣒꣥ꢌ

ꢆ ꢖꢹꢓꢔꢬꢌ꣤ꢌꢔꢘ^꣢

꣆꣜꣘ꢌꢲ꣙ꢦ꣘ꢁꢄ

꣆꣜꣘ꢌ꣜ꢄ꣗

ꢕ꣍ꢐ^꣢

ꢕꢌ꣤ꢌꢘꢓꢟꢹꢌ꣤ꢌꢔꢘꢘꢘ^꣢

ꢬꢹꢟꢌ꣤ꢌꢔꢝꢓꢕꢌ꣤ꢌꢕꢘꢹꢹꢌ꣤ꢌꢔꢘ^ꢭꢌ꣤ꢌꢔꢘ^ꢭꢌ

꤅_ꢨꢦ꣟

ꢆ

ꢆ ꢘꢓꢘꢻꢹꢝꢌꢣꢣ

ꢬꢹꢟꢋ_ꢂ

i. Determine the flexural cracking moment from

Eq. (5):

ꢏ_ꢂꢃꢆ ^ꢲ ꢊ

ꢄ

ꢳ ꢴ

Deflection due to estimated dead and live load at

service can be computed as shown below:

ꢵ

ꢶ

Where ꢙ ꢆ ꢘꢓꢠꢝꢬꢌꢷ ꢌ ꢙ^ꢅꢌꢏꢸꢞ and ꢷ_ꢂꢆ ꢔ for

√

ꢂ

ꢃ

ꢂ

ꢕ꣍ꢐ^꣢

ꢕꢌ꣤ꢌꢠꢓꢬꢹꢌ꣤ꢌꢔꢘꢘꢘ^꣢

ꢬꢹꢟꢌ꣤ꢌꢔꢝꢓꢕꢌ꣤ꢌꢕꢘꢹꢹꢌ꣤ꢌꢔꢘ^ꢭꢌ꣤ꢌꢔꢘ^ꢭꢌ

꤅_ꢨꢦ꣟

ꢆ

ꢆ ꢔꢓꢬꢔꢌꢣꢣ

normal concrete. Using Eq. (6):

ꢬꢹꢟꢋ_ꢂ

ꢄ

ꢙ ꢆ ꢘꢓꢠꢝꢬꢌꢷ_ꢂꢌ√ꢙ^ꢅꢆ ꢘꢓꢠꢝꢬꢌ꣤ꢌꢔꢌ꣤ꢌ꣉ꢖ ꢆ ꢔꢓꢠꢕꢌꢡꢢꢣꢣ^ꢑ

ꢃ

ꢂ

4. RESULTS AND DISCUSSIONS

4.1 Results

In the positive moment region,

ꢔꢓꢠꢕ꣤ꢔꢘꢘꢔꢓꢻꢖ

ꢏ_ꢂꢃ

ꢆ

ꢆ ꢟꢘꢓꢝꢬꢌꢡꢓ ꢣꢣ ꢆ ꢘꢓꢘꢟꢘꢝꢬꢌ꣤ꢌꢔꢘ^꣦ꢭꢌꢎꢡꢣ

ꢟꢔꢓꢘꢻ

꣍ꢐ^ꢑ

ꢔꢝ

The deflections measured from the experiment are

presented in column 5 of Table 1. Estimated mid-span

deflection can be estimated using Eq. (12). The results are

presented in column 4 of Table 1.

꣧꣨꣩꣪꣫꣪꣬꣭ꢌꢿ꣨ꢿ꣭꣮꣫ꢌ꣯꣫ꢌꢿ꣪꣰ ꢪ ꣩꣱꣯꣮ ꢆ

ꢘꢓꢟꢹꢌ꣤ꢌꢔ^ꢑ

ꣲ꣭꣯꣰ꢌꣳ꣨꣯꣰ꢌꢿ꣨ꢿ꣭꣮꣫ ꢆꢌ

ꢆ ꢟꢘꢓꢘꢌ꣤ꢌꢔꢘ^꣦ꢭꢌꢎꢡꢣꢌꢌꢌ(꣔꣐ꢞ꣔ꢎ꣏꣥)

ꢔꢝ

(

ꢑ

)

ꢘꢓꢟꢹ ꢩ ꢠꢓꢬꢹ ꣤ꢌꢔ

ꣴ꣏ꢞ꣥ꢌꢚꢐꣵꢛꢌꢐꢜꣶ꣏ꢌꢐ꣑ꢞ꣥ꢌꢐ꣑ꢞ꣥ꢜ꣒꣖ꢌ

ꢆ ꢘꢓꢕꢖꢌꢎꢡꢣꢌ(꣔꣐ꢞ꣔ꢎ꣏꣥)

ꢔꢝ

ꢕ꣍ꢐ^꣢

ꢕꢌ꣤ꢌ꣍ꢌ꣤ꢌꢔꢘꢘꢘ^꣢

ꢬꢹꢟꢌ꣤ꢌꢔꢝꢓꢕꢌ꣤ꢌꢕꢘꢹꢹꢌ꣤ꢌꢔꢘ^ꢭꢌ꣤ꢌꢔꢘ^ꢭꢌ

꤅_ꢨꢦ꣟

ꢆ

ꢆ ꢘꢓꢝꢘꢕ꣍ꢌꢣꢣꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢔꢝ)

ꢬꢹꢟꢋ_ꢂ

ꢄ

Therefore, it will be necessary to compute and

ꢂꢃ

ꢄ

Column 1 of Table 1 shows the slab load, while

at the mid-span.

column 2 shows the equivalent beam load. The estimated

deflection is far lesser than the actual deflection. The

ultimate beam load is 14.72 kN/m, and the service load for

live load is about 66.67% of the ultimate load and this

equals 9.81 kN/m. From Figure 4, the corresponding

estimated deflection for service load of 9.81 kN/m is 2.01

mm, while the corresponding actual deflection is 8.14

mm. From Table 1, column 5, the actual deflection

increases as the load increases. Also from Figure 4, the

actual load deflection curve is not linear and deflection

increases as load increases. At the beam ultimate load of

14.72 kN/m, the deflection curve flattened out and

collapse of the space framed structure collapsed. Column

4 of Table 1 and Figure 4, show the estimated deflection,

which increases with the load and linear throughout.

Column 6 of Table 1 shows by how much the actual

deflection exceeded the estimated deflection. Between

loads 0.46 kN/m and 2.30 kN/m, the estimated deflection

is greater than actual deflection. As the load increases,

between 0.46 kN/m and 2.30 kN/m, the gap between the

estimated and the actual deflection reduces. As from 2.3

kN/m, actual deflection is greater than estimated

deflection, and this deflection increases as the load

increased. The percentage increase of actual deflection

over the estimated one starts at 16.61% for load 2.76

kN/m and 614.05% for load 14.72 kN/m at failure. At the

service load of 9.81 kN/m, the gap between the estimated

and actual deflection is 305%.

iv. Compute ꣷ_꣸꣹at mid-span

Taking the compression zone to be rectangular:

ꢂꢃ

ꢆ ꣞ _{꣆꣜꣘ꢌꢦ꣟꣠ꣃ}ꢩ ꣞ꣂ꣡̅^ꢑꢌ

ꢭ

ꢆ (ꢔꢝꢬꢓꢘꢝꢌ꣤ꢌꢔꢘ ) ꢩ ꢖꢖꢖꢓꢟꢹ ꢩ ꢟꢔꢹꢖꢓꢟꢻ ꣤ꢔꢘ ꢆ ꢕꢘꢹꢹ꣤ꢔꢘ^ꢭꢌꢣꢣ^꣢

(

)

ꢂꢃ

3.2.2 Compute immediate dead-load + live load

deflection. When the live load is applied to the space

frame, the beam moments will increase, leading to

increased flexural cracking at the mid-span. As a result, I_e

will decrease.

i. Compute ꣷ_꣺at mid-span.

Because ꢏ_ꣃꣅꢦ꣘= 0.57 kNm, is greater than ꢏ_ꢂꢃ

40.23 x 10^-6kNm, hence the section is cracked and

ꢄ

must be determined by using Eq. (4).

ꢭ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢌ ꢆ ꢥ

ꢄ

ꢧ _ꢁꢩ *ꢔ ꢪ ꢥ

ꢧ +

ꢂꢃ

꣍ꢐ^ꢑ

ꢔꢝ

ꢠꢓꢬꢹꢌ꣤ꢌꢔ^ꢑ

ꢔꢝ

ꢏ_ꢦ

ꢆ

ꣀ ꣍꣎꣏꣐꣏ꢌ꣍ ꢆ ꣵ꣒ꢙꢞ꣔꣓꣑꣐꣏꣥ꢌꢐꢜꣶ꣏ꢌꢐ꣑ꢞ꣥ꣀꢌꢏ_ꢦ

ꢆ

ꢆ ꢘꢓꢕꢬꢌꢎꢡꢣ

Therefore:

ꢭ

꣢꣝ꢓꢑꢭꢌ꣟ꢌꢅ꣝^ꣿ꤀

꣝ꢓ꤁ꢭ

ꣻ^ꢈ_ꢈꣾ ꢆ ꣻ

ꣾ ꢆ ꢖꢕꢓꢻꢔꢌ꣤ꢌꢔꢘ

꣼ꢳ

꣦꤂

ꢭ

(

)

꤃ ꢘ

ꣽ

We have:

ꢆ ꢘꢌ꣤ꢌꢔꢘꢘꢔꢓꢻꢖꢌ꣤ꢌꢔꢘ

꣢

ꢭ

ꢔ ꢪ ꢘ ꣤ꢕꢘꢹꢹ꣤ꢔꢘ ꢆ ꢕꢘꢹꢹꢌ꣤ꢌꢔꢘ^ꢭꢌꢣꢣ^꣢ꢆ ꤄

(

)

ꢄ

ꢩ

ꢂꢃ

ꢁ

ii. Compute estimated immediate dead plus live-

load deflection.

The immediate dead plus live-load deflection can be

estimated at the mid-span using Eq. (11) below.

From the above, used in the computation of the

estimated deflection is grossly inaccurate. Since the

estimated deflection is lesser than actual deflection,

ꢄ

indicates that is over estimated.

ꢄ

To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9 (1):

07-16. www.ojceu.ir

Figure 4: Estimated and Actual Load and deflection curve at the beam center

Table 1: Load, deflection

꤅

꣦꤅

꤈꤉ꤊ

S/N

Slab Load

Load on Beam

Estimated deflection

Actual Deflection

^ꤋꤌꤊ꣤ꢔꢘꢘꤍ

꤅

ꤋꤌꤊ

(kN/m²)

(kN/m)

꤅_ꢉꤎꤏ(mm)

꤅_ꤐꤑꤏ(mm)

1.84

3.68

5.52

7.36

0.46

0.92

1.38

1.84

2.30

2.76

3.22

3.68

4.14

4.60

5.06

5.52

5.98

6.44

6.90

7.36

7.82

8.28

8.74

9.20

9.66

10.12

10.58

0.094

0.189

0.283

0.377

0.472

0.566

0.660

0.754

0.849

0.943

1.037

1.132

1.226

1.320

1.415

1.509

1.603

1.697

1.792

1.886

1.980

2.075

2.169

0.00

0.25

0.35

0.45

0.66

0.82

0.96

1.22

1.56

2.01

2.42

2.74

3.97

4.86

5.84

6.34

6.89

7.25

7.68

7.99

8.45

8.89

-100.0

-11.66

-2.70

-4.66

16.61

24.24

27.32

43.70

65.43

9.20

11.04

12.88

14.72

16.56

18.40

20.24

22.08

23.92

25.76

27.60

29.44

31.28

33.12

34.96

36.8

93.83

113.78

123.49

200.76

243.46

287.01

295.51

306.01

304.58

307.21

303.54

307.23

309.87

38.64

40.48

42.32

To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9 (1):

07-16. www.ojceu.ir

Table 1: Cont’d

S/N

Slab Load

Load on Beam

Estimated deflection

Actual Deflection

꤅

꣦꤅

(kN/m²)

(kN/m)

(mm) ꤅_ꢉꤎꤏ

(mm) ꤅_ꤐꤑꤏ

^ꤋꤌꤊ꣤ꢔꢘꢘꤍ

꤈꤉ꤊ

꤅

ꤋꤌꤊ

44.16

46.0

11.04

11.50

11.96

12.42

12.88

13.34

13.8

14.26

14.72

2.263

2.358

2.452

2.546

2.640

2.735

2.829

2.923

3.018

9.25

308.75

340.63

369.00

402.36

430.68

457.95

487.13

516.15

544.47

574.62

614.05

10.39

11.59

12.79

14.01

15.26

16.61

18.01

19.45

20.36

21.55

47.84

49.68

51.52

53.36

55.2

57.04

58.88

ꢕ꣍ꢐ^꣢

꣍

4.2 Determination of Experimental Effective

Moment of Inertiaꢌꣷ_{꣺(ꤒꤓꤔ)}

ꢆ

ꢆ ꢔꢘꢟꢔꢓꢠꢖ ꣊

꣊ ꢔꢘ^ꢭꢣꢣ^꣢ꢌꢌꢌꢌꢌꢋꢒꢓ (ꢔꢬ)

ꢄ(ꢉꤛꤜ)

ꢬꢹꢟꢋ_ꢂ꤅_ꤐꤑꤏ

꤅_ꤐꤑꤏ

The deflection at mid span, of the beam is calculated

Using Eq. (13),

is determine and presented in

ꢄ(ꤘꤙꤚ)

ꤕ

꤁꣜꣙

using equation (11) repeated as follow: ꤅_ꢨꢦ꣟ꢆ _{ꢭꤖ꣢ꢉ ꢊ}

Column 4 of Table 2.

꣼ ꤗ

At service load of 9.81 kN/m and actual deflection

The experimental effective moment of inertia,

ꢄ(ꤘꤙꤚ)

8.14 mm, the experimental effective moment of inertia,

can be worked out using Equation (13) by substituting

is 1255.38 x 10³mm⁴.

ꢄ(ꤘꤙꤚ)

(

)

estimated deflection ꤅_ꢉꤎꤏwith measured deflection

(

)

꤅_ꤐꤑꤏas given by Equation (13).

Table 2: Determination of

ꢄ(ꤘꤙꤚ)

ꤦ_꣺ꢪ ꤦ_{꣺(ꤠꤡꤢ)}

ꣷ_{꣺(ꤠꤡꤢ)}

ꣷ_ꤥ

ꤣꤣ

ꣷ_꣺

ꤣꤣ

Load on Beam

Actual Deflection

ꤧꤨꤩꤩꤍ

S/N

ꤤ

(mm) ꤅_ꤝꤞꤟ

(

)

(kN/m)

(

)

(

)

ꤦ_{꣺(ꤠꤡꤢ)}

ꤣꤣ

1001.97 x 10⁴

5088 x 10³

0.46

0.92

1.38

1.84

2.3

0.00

0.25

0.35

0.45

0.66

0.82

0.96

1.22

1.56

2.01

2.42

2.74

3.97

4.86

5.84

6.34

6.89

7.25

7.68

7.99

8.45

8.89

9.25

10.39

11.59

12.79

14.01

15.26

16.61

18.01

19.45

0.00

5750.02 x 10³

5476.21 x 10³

5324.09 x 10³

4356.07 x 10³

4090.46 x 10³

3993.07 x 10³

3534.85 x 10³

3071.59 x 10³

2622.31 x 10³

2376.04 x 10³

2273.43 x 10³

1689.76 x 10³

1478.91 x 10³

1312.79 x 10³

1284.84 x 10³

1251.82 x 10³

1255.75 x 10³

1247.83 x 10³

1259.39 x 10³

1247.54 x 10³

1239.69 x 10³

1243.25 x 10³

1152.96 x 10³

1074.94 x 10³

1011.54 x 10³

957.65 x 10³

910.61 x 10³

865.45 x 10³

824.78 x 10³

788.35 x 10³

-11.51

-7.09

-4.43

16.80

24.39

27.42

43.94

65.65

94.03

114.14

123.80

201.11

244.04

287.57

296.00

306.45

305.18

307.75

304.00

307.84

310.43

309.25

341.30

373.33

403.00

431.30

458.75

487.90

516.89

545.40

2.76

3.22

3.68

4.14

4.60

5.06

5.52

5.98

6.44

6.90

7.36

7.82

8.28

8.74

9.20

9.66

10.12

10.58

11.04

11.50

11.96

12.42

12.88

13.34

13.8

14.26

14.72

To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9 (1):

07-16. www.ojceu.ir

1. Distributed load =1.25; 2. Two point load =1.0 and 3.

Concentrated load = 0.75

4.3 Proposed Model

The model of Branson (1965) can be modified in the

form of Equation (14).

ꢭ

ꢻꢕ

ꢔꢘꢘꢔꢓꢻꢖ ꣊ ꢔꢘ^꣢

ꢕꢘꢹꢹ ꣊ ꢔꢘ^ꢭ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢁ

ꤪ ꢆ ꢕꢓꢝꢠ ꢪ ꢘꢓꢕꢝꢕ ꢥ

ꢺ^ꢅꢘꢓꢘꢔꢹꢹꢕ

ꢧ ꢆ ꢟꢓꢻꢝꣀꢌꢌꢌꢌꢌꢌꢌꤺ ꢆ

ꢆ

ꢆ ꢥ

ꢧ _ꢁꢩ *ꢔ ꢪ ꢥ

ꢧ + ꤪ _ꢂꢃꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢔꢟ)

ꢄ

ꢔꢟꢕ

ꢆ ꢔꢓꢻꢠ

ꢂꢃ

ꤻ

꤯

ꢔꢘꢘ

Where:

ꤹ ꢆ

ꢆ

ꢆ ꢘꢓꢕꣀ ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꤶ ꢆ

ꢆ

ꢆ ꢘꢓꢔ

ꢺ

ꢘꢓꢘꢬꢖꢖ

ꢔꢘꢘꢘ

ꤪ = Experimentally determined reduction factor.

ꢻꢓꢹꢔꢌ꣤ꢌꢔ^ꢑ

ꢅ

ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢏ_ꢦꢆ

ꢆ ꢘꢓꢹꢝꢌꢎꢡꢣ

(

)

(

)

ꢺ ꢩ ꤪꢺ ꣒

ꢘꢓꢘꢔꢹꢻ ꢩ ꢟꢓꢻꢝ ꣊ ꢘꢓꢘꢬꢖꢖ

ꢔꢝ

ꢷ ꢆ ꢙ ꢪ

ꢆ ꢔꢓꢝꢕ ꢪ

꣙

ꤺ

ꢔꢓꢻꢠ

ꢆ ꢔꢓꢔꢕ

Since we are interested in the deflection at the

service load, then I_exp(I_exp= 1255.38 x 10³) at the service

load level will be substituted in the Eq. (13) above.

)

꣢꣝ꢓꢑꢭ

꣝ꢓꤖꢑ꣊ꢅ꣝^꤀

ꢆ ꣻ

ꣾ^{(꣝ꢓꢅꤲꢅꢓꢅ꤁}꣊ ꢔꢘꢘꢔꢓꢻꢖ ꣊ ꢔꢘ^꣢ꢩ ꥃꢥ(ꢘꢓꢔ ꢩ ꢔꢓꢔꢕ) ꢪ

ꢄ

(

)

ꣻ_{꣝ꢓꤖꢑ꣊ꢅ꣝}_꤀ꣾ ^{꣝ꢓꢅꤲꢅꢓꢅ꤁}ꢧꥄ ꢕꢘꢹꢹ ꣊ ꢔꢘ^ꢭ꣊ ꢘꢓꢔ ꢩ ꢘꢓꢕ ꢆ ꢟꢓꢔꢔ ꣊ ꢔꢘ^꣦꤂꣊

꣢꣝ꢓꢑꢭ

ꢭ

(

)

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢌꢔꢝꢕꢕꢓꢬꢹ ꣊ ꢔꢘ^ꢭꢆ ꢥ

ꢧ _ꢁꢩ *ꢔ ꢪ ꢥ

ꢧ + ꤪ

ꢂꢃ

꣢

꣦꤂

ꢭ

[(

)

]

ꢔꢘꢘꢔꢓꢻꢖ ꣊ ꢔꢘ ꢩ ꢔꢓꢝꢕ ꢪ ꢟꢓꢔꢔ ꣊ ꢔꢘ ꢕꢘꢹꢹ ꣊ ꢔꢘ ꣊ ꢘꢓꢠ

Therefore

ꢆ ꢟꢓꢔꢔ ꣊ ꢔꢘ^꣦꤂꣊ ꢔꢘꢘꢔꢓꢻꢖ ꣊ ꢔꢘ ꢩ ꢔꢓꢝꢕ ꢪ ꢟꢓꢔꢔ ꣊ ꢔꢘ

꣢

꣦꤂

ꢭ

[(

)

]

ꢭ

ꢟꢘꢓꢝꢬꢌ꣤ꢌꢔꢘ^꣦꤂

ꢘꢓꢹꢝ

ꢄ

ꢏ

ꢌꢥ ^ꢂꢃꢧ ꢆ ꤫

ꢏ_ꢦ

꤬ ꢆ ꢟꢻꢓꢘꢠꢌ꣤ꢌꢔꢘ

꤃ ꢘ

꣦꤂

ꢭ

꣊ ꢕꢘꢹꢹ ꣊ ꢔꢘ^ꢭ꣊ ꢘꢓꢠ

(

)

We have:

ꢭ

[

]

ꢆ ꢟꢔꢓꢔꢹ ꢩ ꢔꢓꢝꢕ ꢕꢘꢹꢹ ꣊ ꢔꢘ ꣊ ꢘꢓꢠ ꢆ ꢬꢹꢔꢠꢓꢘꢟ ꣊ ꢔꢘ

ꢭ

ꢔꢝꢕꢕꢓꢬꢹ ꣊ ꢔꢘ^ꢭꢆ ꢘꢌ꣤ꢌꢔꢘꢘꢔꢓꢻꢖꢌ꣤ꢌꢔꢘ^꣢

ꢩ

ꢔ ꢪ ꢘ ꣤ꢕꢘꢹꢹꢌ꣤ꢌꢔꢘ ꤪ ꢆ ꢝꢔꢖꢘꢓꢹꢬꢌ꣤ꢌꢔꢘ ꤪꢌ

ꢄ

(

)

ꢕ꣍ꢐ^꣢

ꢕꢌ꣤ꢌ꣍ꢌ꣤ꢌꢔꢘꢘꢘ^꣢

ꢬꢹꢟꢌ꣤ꢌꢔꢝꢓꢕꢌ꣤ꢌꢬꢹꢔꢠꢓꢘꢟꢌ꣤ꢌꢔꢘ^ꢭꢌ꣤ꢌꢔꢘ^ꢭꢌ

1255.38 * 10³= 5088 x 10³α

꤅_ꢨꢦ꣟

ꢆ

ꢬꢹꢟꢋ

ꢄ

ꢔꢝꢕꢕꢓꢬꢹ

ꤪ ꢆ

ꢆ ꢘꢓꢝꢟ

ꢆ ꢘꢓꢝꢖꢬꢌ꣍ꢌꢣꢣꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢔꢖ)

ꢕꢘꢹꢹ

ꢕ꣍ꢐ^꣢

ꢬꢹꢟꢋ_ꢂꤪ

ꢕꢌ꣤ꢌ꣍ꢌ꣤ꢌꢔꢘꢘꢘ^꣢

ꢬꢹꢟꢌ꣤ꢌꢔꢝꢓꢕꢌ꣤ꢌꢘꢓꢝꢟꢌ꣤ꢌꢕꢘꢹꢹꢌ꣤ꢌꢔꢘ^ꢭꢌ꣤ꢌꢔꢘ^ꢭꢌ

꤅_ꢨꢦ꣟

ꢆ

ꢆ ꢘꢓꢹꢕꢬꢌ꣍ꢌꢣꢣꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢔꢕ)

ꢄ

Model 3: Bischoff’s Model (2005)

The proposed effective moment of inertia , is given by

( )

ꢄ

Model 1: Akmaluddin and Thomas Model (2006)

( )

the equation below:

ꢨ

The proposed effective moment of inertia , is

ꢔ

ꢏ_ꢂꢃ

ꢔ

ꢏ_ꢂꢃ

ꢏ_ꢦ

ꢔ

ꢄ

ꢆ ꢥ

ꢧ

ꢌ

ꢩꢌ*ꢔ ꢪꢌꢥ

ꢧ +ꢌ ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓꢌꢌ(ꢔꢖ)

given by the equation below:

ꢐ_ꢄ

ꢏ_ꢦ

ꢁ

ꢂꢃ

ꢌꢌꢌꢌꢌꢌꢌꢌꢌꢌ ꢆ _ꢂꢃꢄꢩ ꤭ _ꢁꢪ _ꢂꢃꢄ꤮꣏^ꢇ

Substituting for the values of ꢆ ꢔꢘꢘꢔꢓꢻꢖ꣤ꢔꢘ^꣢,

ꢂꢃ

ꢆ

ꢄ

ꢁ

ꢕꢘꢹꢹ꣤ꢔꢘ^ꢭ, ꢏ_ꢂꢃꢆ ꢟꢘꢓꢝꢬ, ꢏ_ꢦꢆ ꢘꢓꢕꢕꢕ꣤ꢔꢘ^꤂and m = 2, we

have:

Where:

ꢑ

ꢤ꣥^ꢭ

ꢖꢕ꣤ꢔꢘꢘ^ꢭ

ꢔꢝ

ꢔ

ꢐ_ꢄ

ꢔ

ꢟꢘꢓꢝꢬ

ꢔ

ꢟꢘꢓꢝꢬ

ꢘꢓꢹꢝ ꣊ ꢔꢘ^꤂

ꢔ

ꢆ ꢥ

ꢧ

꣊

ꢩ *ꢔ ꢪ ꢥ

ꢧ + ꣊

(

)

(

)

ꢆ

ꢘꢓꢔꢠꢔꢹ ꢩ ꢘꢓꢘꢟꢔꢹ꣒ꢺ

ꢆ

ꢘꢓꢔꢠꢔꢹ ꢩ ꢘꢓꢘꢟꢔꢹ꣤ꢔꢕ꣤ꢘꢓꢘꢘꢖꢕ

ꢂꢃꢄ

ꢘꢓꢹꢝ ꣊ ꢔꢘ^꤂

ꢔꢘꢘꢔꢓꢻꢖ ꣊ ꢔꢘ^꣢

ꢕꢘꢹꢹ ꣊ ꢔꢘ^ꢭ

ꢔꢝ

꣢

ꢭ

( )

ꢆ ꢘꢓꢔꢠꢔꢹ ꢩ ꢘꢓꢘꢘꢟꢖ ꣤ꢠꢝꢕ꣤ꢔꢘ ꢆ ꢔꢘꢟꢘꢓꢠꢬ꣤ꢔꢘ

ꢂꢃꢄ

ꢔ

[

]

ꢆ ꢘ ꢩ ꢔ ꢪ ꢘ ꣊

ꢆ

ꢕꢘꢹꢹ ꣊ ꢔꢘ^ꢭꢕꢘꢹꢹ ꣊ ꢔꢘ^ꢭ

ꢏ

꤯

ꢇ ꢆ ꢪ ꢥ ^ꢦꢧ ꢥ ꢧ ꢹꢓꢟꢖꢟ ꢪ ꢻꢓꢘꢠꢘꢖꢺ ꢩ ꢝꢓꢹꢟꢝꢺ

ꢂꢃ

ꢑ

ꢄ

(

)

ꢆ ꢕꢘꢹꢹ ꣊ ꢔꢘ^ꢭ

ꢏ_ꢂꢃ

꤯

ꢄ

ꢏ

ꢕ꣍ꢐ^꣢

ꢕꢌ꣤ꢌ꣍ꢌ꣤ꢌꢔꢘꢘꢘ^꣢

꤯

ꢆ ꤯ ꢥꢔ ꢪ ^ꢂꢃꢧ

ꢂꢃ

꤅_ꢨꢦ꣟

ꢆ

ꢏ_ꢦ

ꢬꢹꢟꢋ

ꢬꢹꢟꢌ꣤ꢌꢔꢝꢓꢕꢌ꣤ꢌꢕꢘꢹꢹꢌ꣤ꢌꢔꢘ^ꢭꢌ꣤ꢌꢔꢘ^ꢭꢌ

ꢆ ꢘꢓꢝꢘꢕ꣍ꢌꢣꢣꢌꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢔꢹ)ꢌ

ꢄ

ꢟꢘꢓꢝꢬ

ꢆ ꢔꢘꢘꢘ ꢥꢔ ꢪ

ꢧ ꢆ ꢻꢻꢻꢓꢻꢕꢌꢣꢣꢌꢌꢌꢌꢌꢌ

ꢘꢓꢹꢝ꣤ꢔꢘ^꤂

This is the same as that of Branson’s model.

ꢘꢓꢹꢝ꣤ꢔꢘ^꤂ꢻꢻꢻꢓꢻꢕ

ꢑ

(

)

ꢇ ꢆ ꢪ ꤫

꤬ ꢥ

ꢧ ꢹꢓꢟꢖꢟ ꢪ ꢻꢓꢘꢠꢘꢖ꣤ꢘꢓꢘꢬꢖꢖ ꢩ ꢝꢓꢹꢟꢝ꣤ꢘꢓꢘꢬꢖꢖ

ꢭ

ꢟꢘꢓꢝꢬ

ꢔꢘꢘꢘ

(

)

ꢇ ꢆ ꢪꢝꢘꢬꢹꢟꢓꢖꢹ ꢹꢓꢟꢖꢟ ꢪ ꢘꢓꢬꢟꢔꢠ ꢩ ꢘꢓꢘꢘꢟꢘ ꢆ ꢪꢔꢠꢕꢓꢹꢠ꣤ꢔꢘ

ꢆ ꢩ ꤭ _ꢁꢪ _ꢂꢃꢄ꤮꣏^ꢇꢆ ꢔꢘꢟꢘꢓꢠꢬ꣤ꢔꢘ^ꢭ

ꤰ

꣦ꢅ꤂꤁ꢓꤖ꤂꣟ꢅ꣝

ꢔꢘꢘꢔꢻꢓꢖ꣤ꢔꢘ^ꢭꢪ ꢔꢘꢟꢘꢓꢠꢬ꣤ꢔꢘ^ꢭ

ꤰ

꣏

4.5 Comparative Analysis of the Models

Branson’s model, model 1, model 2 and the

proposed model were used to estimate the deflection of

the beam. The results of the estimation were presented in

Table 3.

(

)

ꢄ

ꢩ

ꢂꢃꢄ

ꢭ

꣦ꢅ꤂꤁ꢓꤖ꤂꣟ꢅ꣝

ꢆ ꢔꢘꢟꢘꢓꢠꢬ꣤ꢔꢘ^ꢭꢩ ꢘ

(

)

꣏

ꢆ ꢔꢘꢟꢘꢓꢠꢬ꣤ꢔꢘ ꢩ ꢹꢻꢖꢻꢓꢘꢖ꣤ꢔꢘ

ꢄ

ꢆ ꢔꢘꢟꢘꢓꢠꢬ꣤ꢔꢘ^ꢭꢌꢣꢣ^꣢

ꢄ

ꢕ꣍ꢐ^꣢

ꢕꢌ꣤ꢌ꣍ꢌ꣤ꢌꢔꢘꢘꢘ^꣢

ꢬꢹꢟꢌ꣤ꢌꢔꢝꢓꢕꢌ꣤ꢌꢔꢘꢟꢘꢓꢠꢬꢌ꣤ꢌꢔꢘ^ꢭꢌ꣤ꢌꢔꢘ^ꢭꢌ

꤅_ꢨꢦ꣟

ꢆ

ꢆ ꢔꢓꢘꢌ꣍ꢌꢣꢣꢌꢌꢌꢌꢌꢌꢌꢋꢒꢓ (ꢔꢠ)

ꢬꢹꢟꢋ_ꢂ

ꢄ

At service load of 9.81 kN/m, the estimated

deflections using Branson’s / Bischoff’s Models, model 1,

model 2 and the proposed model are 2.01 mm, 9.81 mm,

2.68 mm and 8.37 mm respectively. The percentage

difference of these deflections to the actual deflection of

8.14 mm at the service load are 305%, - 17%, 204% and -

2.75% respectively. From the above, only the proposed

model, did not perform badly.

Model 2: Ammash and Muhaisin Model (2009)

(

)

(

)

ꤱꤲꤳ

ꢏ

ꢆ ꢥ ^ꢂꢃꢧ

꣊ _ꢁꢩ ꤴꤵ ꤶ ꢩ ꢷ ꢪ ꢥ ^ꢂꢃꢧ

ꤷꤸ _ꢂꢃ꣊ ꤶ ꢩ ꤹ

(

)

(

)

ꢄ

ꢏ_ꢦ

Where

ꢅ

ꢺ^ꢅ

ꢺ

ꢩ ꤪꢺ ꣒

꣥

ꢁ

ꤻ

꤯

(

)

ꤹ ꢆ ꣀ ꢌꢷ ꢆ ꢙ ꢪ

ꣀ ꢌꢌꤪ ꢆ ꢕꢓꢝꢠ ꢪ ꢘꢓꢕꢝꢕ ꢥ ꢧ ꣀ ꢌꢌꤺ ꢆ

ꢤ_꣜

ꢂꢃ

ꣀ ꢌꤶ ꢆ

꣙

ꢺ

ꤺ

ꢙ ꢪ ꢌꢼ꣯ꤼ꣫꣨ꤽꢌ꣰꣭꣱꣭꣮꣰ꢌ꣨꣮ꢌꣳ꣨꣯꣰꣪꣮ꤾꢌ꣫ꤿ꣱꣭ꢌ꣩ꥀꤼꥁꢌ꣯꣩ꥂ

꣙

To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9 (1):

07-16. www.ojceu.ir

Table 3: Load, deflection

DEFLECTION (mm)

S/N

Load on

Beam (kN/m)

Branson’s Deflection

Model, and Model 3

Model 1

Model 2

Proposed

model

Actual

(mm) ꤅_ꤐꤑꤏ

0.46

0.920

1.38

1.84

2.30

2.76

3.22

3.68

4.14

4.60

5.06

5.52

5.98

6.44

6.90

7.36

7.82

8.28

8.74

9.20

9.66

10.12

10.58

11.04

11.50

11.96

12.42

12.88

13.34

13.8

14.26

14.72

0.0943

0.1886

0.2829

0.3772

0.4715

0.5658

0.6601

0.7544

0.8487

0.9430

1.0373

1.1316

1.2259

1.3202

1.4145

1.5088

1.6031

1.6974

1.7917

1.8860

1.9803

2.0746

2.1689

2.2632

2.3575

2.4518

2.5461

2.6404

2.7347

2.8290

2.9233

3.0176

0.46

0.92

1.38

1.84

2.30

2.76

3.22

3.68

4.14

4.60

5.06

5.52

5.98

6.44

6.90

7.36

7.82

8.28

8.74

9.20

9.66

10.12

10.58

11.04

11.50

11.96

12.42

12.88

13.34

13.8

14.26

14.72

0.1256

0.2512

0.3767

0.5023

0.6279

0.7535

0.8791

1.0046

1.1302

1.2558

1.3814

1.5070

1.6325

1.7581

1.8837

2.0093

2.1349

2.2604

2.3860

2.5116

2.6372

2.7628

2.8883

3.0139

3.1395

3.2651

3.3907

3.5162

3.6418

3.7674

3.8930

4.0186

0.3924

0.7848

1.1771

1.5695

1.9619

2.3543

2.7467

3.1390

3.5314

3.9238

4.3162

4.7086

5.1009

5.4933

5.8857

6.2781

6.6705

7.0628

7.4552

7.8476

8.2340

8.6324

9.0247

9.4171

9.8095

10.2019

10.5943

10.9866

11.3790

11.7714

12.1638

12.5562

0.00

0.25

0.35

0.45

0.66

0.82

0.96

1.22

1.56

2.01

2.42

2.74

3.97

4.86

5.84

6.34

6.89

7.25

7.68

7.99

8.45

8.89

9.25

10.39

11.59

12.79

14.01

15.26

16.61

18.01

19.45

deflections, and therefore the beam is not satisfactory in

deflection.

5.0 CONCLUSION AND RECOMMENDATIONS

From the above, it is most likely that structures

which deflection criteria were based on span/effective

depth ratio is likely to fail in deflection, as there is the

possibility of the occurrence of damage in terms of cracks

of non-structural elements such as partition walls.

5.1 Conclusions

Branson’s / Bischoff’s Model and model 2 grossly

under estimated the deflection by 305% and 2043%

respectively, while model 1 grossly overestimated the

deflection by 17%, and the proposed model in this study

overestimated the deflection by just 2.75%. Therefore all

the existing models performed badly. From the above,

Branson’s / Bischoff’s Model and model 2 overestimated

the effective moment of inertia, while model l under

estimated the effective moment of inertia.

The beam was satisfactory using the span/effective

depth ratio. However, the actual deflection at service load

for this experimental work was 8.14 mm which exceeded

the maximum permissible computed deflections (ACI

318, 2005) of L/480, which equals 2.08 mm. Therefore,

non-structural elements, such as partition walls, supported

by such beams are likely to be damaged by large

5.2 Recommendation

Based on the above conclusions, the following

recommendations are made:

i. Research should be conducted on the effect of

concrete grade on the effective moment of inertia using

locally available materials.

ii. Effects of reinforcement percentage on the

effective moment of inertia using locally available

materials should be investigated.

iii. The span/effective depth ratio alone should not

be used in checking for deflection, rather this should be

complemented by actual deflection calculation.

To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9 (1):

07-16. www.ojceu.ir

Bischoff, P.H. (2007). Rational Model for Calculating

Deflection of Reinforced Concrete Beams and Slabs.

Canadian Journal of Civil Engineering, 34(8): 992-

1002.

DECLARATIONS

Acknowledgments

The author would like to thank the Department of

Civil Engineering, Federal University of Technology

Akure, Ondo State, for making available, the use of their

structural engineering laboratories.

Branson, D.E. (1965). Instantaneous and Time-Dependent

Deflections of Simple and Continuous Reinforced

Concrete Beams. HPR Report No. 7, Part 1, pp. 1-78,

Alabama Highway Department, Bureau of Public

Roads, Alabama.

Author’s contributions

BS 8110-1 (1997). Structural Use of Concrete – Part 1:

Code of Practice for design and construction. British

Standards Institution, London.

The author of this research paper have directly

participated in the planning, execution, or analysis of this

study and have read and approved the final version

submitted.

BS 812 -103.1 (1985). Testing aggregates – Part 103:

Methods for determination of particle size

distribution – Section 103.1 Sieve tests. British

Standards Institution, London.

Conflict of interest statement

I hereby states that, there is no conflict of interest

whatsoever with any third party.

CEN 2 (2002): Design of Concrete Structures – Part 1:

General Rules and Rules for Building. European

Committee for Standardization, Brussels, Belgium.

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To cite this paper: Olanitori LM (2019). Effective Moment of Inertia of Single Spanned Reinforced Concrete Beams with Fixed Beam-Column Joints. J. Civil Eng. Urban., 9 (1):

07-16. www.ojceu.ir