International Journal of Recent Advances in Engineering & Technology (IJRAET)

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Moment Curvature Relationship For Structural Elements of RC

Building Using Matlab
1
Mohammed Fakhruddin Momin, 2Prashant Barbude, 3Kunal Bhagat, 4Prashant V. Muley
1,2,3,4
Department of Civil Engineering, Data Meghe College of Engineering

a strain profile. The strain profile can be fixed by

Abstract— The paper highlights the investigation of
moment-curvature relationships for rectangular shaped assuming the strain at the extreme compression fibre
reinforced concrete (RC) beam and column cross sections ‘ ’ and the neutral axis depth ‘kd’
for the reference reinforced concrete building structure
considered for Probabilistic Seismic Risk Evaluation. The
most popular model namely Modified Kent Park model is
used for concrete stress strain relationship since it is simple
and effective in considering the effects of confinement. The
module determines the moment curvature relationship for
beam and column of a user defined cross-section using
Matlab code. MATLAB code to obtain moment curvature
relation for beam and column.

Keywords: Moment-Curvature, Modified Kent Park

Method, Stress-Strain Relationship and Moment Curvature.

I. INTRODUCTION Fig 1.1

Corresponding to the assumed strain profile, the
Moment Curvature Relationship-Theoretical Moment
distribution of concrete stress over the compressed part of
Curvature Determination In order to perform nonlinear
the section is determined using the stress-strain curve for
analysis, whether static or dynamics one should
concrete. The strain in reinforcement at different levels
understand how the section of the structural member
can be determined from similar triangles of strain profile.
behaves and how the hinges are defined. In order to define
hinges, moment curvature relationship of RC.C section is For reinforcement bar i at a depth di, the strain si is given
used. Theoretical determination of moment curvature by equation.(1).
characteristics for reinforced concrete sections with
flexure and axial load is based on the following si = cm …………..(1)
assumptions (Park & Paulay, 1975).
The stresses corresponding to strains may then be found
1 Plane section before bending remain plane after from stress-strain curve for reinforcing steel. The steel
bending. compressive forces, Cs or tensile forces, Ts may be found
by multiplying steel stresses and the areas of steel.
2 Stress-strain curves for both concrete and steel are
known. To determine the concrete compressive force, Ccon and
its position from extreme compression fibre, γkd the stress
3Tension carrying capacity of concrete is neglected. block having width equal to mean stress, αfc’ and depth
kd. The stress block parameters α and γ are calculated so
The curvatures associated with a range of bending that the total compressive force Ccon and its point of
moments and axial loads may be determined using these application from extreme compression fiber, γkd are same
assumptions and from the requirements of strain for both the actual stress block and equivalent rectangular
compatibility and equilibrium of forces. stress block. The values of α and γ is different for different
Consider a reinforced concrete section. Let the section be levels of cm.
acted upon by an axial force ‘P’.Todetermine its
moment-curvature relationship, the first step is to assume The mean stress factor, α and the centroid factor, γ for any
strain at the extreme compression fiber can be determined
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ISSN (Online): 2347 - 2812, Volume-5, Issue -2, 2017
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 International Journal of Recent Advances in Engineering & Technology (IJRAET)
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for rectangular sections from the stress-strain relationship case, P=0. The moment M corresponding to the assumed
using equation no.(2) and equation no.(3) respectively. ᵋcm is then calculated. The corresponding curvature can
now be obtained. By carrying out the calculation for a
Area under stress-strain curve = range of ᵋcm values, the moment-curvature characteristics
can be plotted.
c d c= cm …………..(2)
Modified Kent and Park (KP) Model by Park et al.
(1982). Park et al. (1982) modified the original model by
making an allowance for the enhancement in the concrete
Therefore, ….………..(3)
strength and the peak strain due to confinement. The
increase in concrete strength was assumed to be equal to
The first moment of area about origin of area under pt fyh, where pt is the volumetric ratio of tie steel, and fyh
stress-strain curve . is the yield strength of tie steel.

= c d c = (1 - ) cm cd c Equivalent Compressive Stress Block Parameters

Stress block parameters for Modified Kent and Park

……..(4) model

The compressive force in concrete is given by equation Region AB: cm ≤ 0.002K

(5)
…………..(9)
Ccon = fc’bkd …………..(5)

Acting at a distance of γkd from the extreme compression

fiber. The force equilibrium equations can be written …………..(10)
using eqn (6) and eqn (7)
Region BC: 0.002K≤ cm ≤ 20,c
P= fc’bkd+ ………..(6)
(11)
M= fc’bkd ( )+ ( ) ..(7)

Where,
.(12)
n = number of reinforcement bars
Region CD: cm 20,c
fsi = Stress in ith bar
14)
Asi = Area of ith bar

D = Total depth of section

(15)
d = Effective depth of the section
Validation of moment-curvature matlab code using text
di = Depth of ith bar from extreme compression fiber book seismic design aids for nonlinear analysis of
reinforced concrete structure by (chandrasekaran,
The corresponding curvature is given by eqn (8). nunziante, serino, carannante)

For Beam:
…………..(8)

The theoretical moment-curvature relationship for a given

axial load level may be determined by incrementing the
concrete strain at the extreme compression fiber, ᵋcm. For
each value of ᵋcm the neutral axis depth kd that satisfies
the force equilibrium is found by adjusting kd until the
force equilibrium equation is satisfied. For flexure only
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 International Journal of Recent Advances in Engineering & Technology (IJRAET)
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Fig 1.2 Beam Section

Fig1.5 Graph given by (chandrasekaran, nunziante,

serino, carannante)

Fig1.3 Graph given by(chandrasekaran, nunziante,

serino, carannante)

Fig1.4 Graph obtain from matlab programming

Table 2. Comparison of various quatities between
Textbook and MATLAB code's Moment-Curvature
relationship for column.
Text book MATLAB code
1. Moment (kN-m) 270 268
Curvature(rad/m) 0.0006 0.00056
2. Moment (kN-m) 265 264
Curvature(rad/m) 0.00056 0.00054
Fig1.4 Graph obtain from matlab programming 3. Moment (kN-m) 260 262
Table 1. Comparison of various quatities between Curvature(rad/m) 0.00052 0.00050
Textbook and MATLAB code's Moment- Curvature II. CONCLUSION
relationship for beam.
Text book MATLAB code The author have carefully studied about the model given
1. Moment (kN-m) 270 268 by mander, kent park and modified kent park model. An
Curvature(rad/m) 0.024 0.026 analytical model is presented to simulate the moment
2. Moment (kN-m) 275 273 curvature behavior of reinforced concrete. Based on
Curvature(rad/m) 0.028 0.027 control of load increments, the algorithm proposed by
3. Moment (kN-m) 280 279 mander enables determination of moment –curvature
Curvature(rad/m) 0.027 0.025 –strain relationship with any geometry and material
4. Moment (kN-m) 285 282 properties up to the maximum capacity of the section ;
however with constant axial load or control of
Curvature(rad/m) 0.026 0.024
deformation increment ,this model can be used to
For Column: compute both the ascending and descending branches of
moment curvature curve. The MR relations play an
important part in the study of limit analysis of two or three
dimensional reinforced concrete frame.

The plastic hinge length must be considered to predict the

ultimate strength of RC beams where the plastic
deformation is concentrated at any location with narrow
range.
Fig 1.5 Column Section

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The simplified numerical analysis with the moment REHABILITATION OF BUILDINGS (FEMA
curvature relation still can be effectively used in the Publication 273). Washington, D. C., USA:
nonlinear analysis of RC beams and columns. Federal Emergency Management Agency.

The co-ordinates of moment curvature plot of [10] FEMA 356. (2000, November). Prestandard and
(chandrasekaran, nunziante, serino,carannante) fig1.3 for commentary for the seismic rehabilitation of
beam and fig1.5 for column were plotted in MATLAB and buildings. Washington, D. C.: American Society of
compared with the calculated plot (which was obtained Civil Engineers.
from the MATLAB code) as shown in fig1.4 and fig 1.6
for beam and column respectively. [11] Inel, M., & Ozmen, H. B. (2006). Effects of plastic
hinge properties in nonlinear analysis of reinforced
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