Numerical Evaluation of the Thermo-Mechanical Response - hikari

15 Pages·2013·0.64 MB·English

Adv. Theor. Appl. Mech., Vol. 6, 2013, no. 1, 33 - 47
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/atam.2013.356

Numerical Evaluation of the Thermo-Mechanical

Response of Shallow Reinforced Concrete Beams

Mazen Musmar

Al-ahliyya Amman University
Civil Engineering Department, Amman, Jordan
[email protected]

Copyright © 2013 Mazen Musmar. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.

Abstract

A precise understanding of the thermo-mechanical response of shallow
reinforced concrete beams is necessary to be able to design the proper sections for
the shallow flexural elements, that could serve their intended purpose, in terms of
safety and serviceability requirements, keeping in my mind the dominant use of
ribbed slabs with concealed shallow beams in many countries.
The study involves building a finite element structural model of a shallow
reinforced concrete beam for the evaluation of the structural performance and
thermal cracking at different temperatures. Material nonlinearity is taken into
account because of the changes in material properties experienced in fire. The
more complicated aspects of structural behaviour in fire conditions, such as
thermal expansion, transient state strains in the concrete, cracking or crushing of
concrete, yielding of steel are modelled.

Keywords: Structural modeling, fire resistance, shallow reinforced concrete
beams
34 Mazen Musmar

1. Introduction

Fire causing high temperature is a serious potential risk to buildings.
Improving fire resistance requires a proper knowledge of the responses of
different construction materials including concrete and steel. It also demands
understanding the material damage mechanisms, and investigation of structural
responses for buildings subjected to fire attack.
Zhaohui H, Ian W., and Roger J. [2006] stated that in terms of reinforced
concrete construction, design is still based on simplistic methods which have been
developed from standard fire tests that do not necessarily represent real building
behaviour. This makes it very difficult, if not impossible, to determine the level of
safety achieved in real concrete structures, or whether an appropriate level of
safety could be achieved more efficiently.
Alternative to existing methods of determining the fire resistance of building
structures is their fire behaviour modelling by means of computer aided design.
According to Vadim Kudryashov, Nguyen Thanh Kien, and Aleksandr Lupandin,
[2012], the use of specialized computer codes can significantly increase the
profitability of the project works and increase their efficiency. The weakness in
case of modelling is that the degradation of material properties is simplified.
A number of researchers have developed structural modelling approaches
such as Lie and Celikkod [1991], who developed a model for the high temperature
analysis of circular reinforced concrete columns. Huang and Platten [1997]
developed planar modelling software for reinforced concrete members in fire.
According to Kasper [2009], the reinforced concrete beams possess high
resistance to high temperature, high resistance to thermal shock, and strong
resistance against fire action when compared with steel beams. The main
disadvantage is the very low concrete tensile strength.
This study involves the aspects connected with structural modelling and the
numerical evaluation of thermal stresses and deformations induced by thermal
gradient affecting a shallow reinforced concrete beam.
The first law of thermodynamics states that thermal energy is conserved.
Specializing this to a differential control volume, ANSYS [ 2009]:
⎡∂T ⎤
  ρc=⎢ +{V}T{L}T⎥+{L}T{q}=&q&&
⎣ ∂t ⎦
(1)
Where:
ρ= Density
c= Specific heat 
T =Temperature
t=time
Numerical evaluation of thermo-mechanical response 35

⎧ ∂ ⎫
⎪ ⎪
∂X
⎪ ⎪
⎪ ∂ ⎪
{L}=⎨ ⎬= vector operator 
∂Y
⎪ ⎪
∂
⎪ ⎪
⎪⎩∂Z ⎪⎭

⎧V ⎫
x
⎪ ⎪
{V}=⎨V ⎬= velocity vector for mass transport of heat 
y
⎪ ⎪
V
⎩ ⎭
z

{q}= heat flux vector 
&q&&=heat generation rate per unit volume

Next, Fourier’s law is used to relate the heat flux vector to the thermal
gradients:
                                   {q}=−[D]{L}T                                        (2) 

Where:
⎡K 0 0 ⎤
xx
[D]=⎢ 0 K 0 ⎥ = conductivity matrix 
⎢ yy ⎥
⎢ 0 0 Kzz⎥
⎣ ⎦
K ,K ,K = conductivity in the element x, y, and Z directions
xx yy zz  
respectively.
Combining equation1 and equation 2 results in

⎛∂T ⎞
ρc⎜ + {V}T{L}T⎟={L}T([D]{L}T)+&q&&                                      (3) 
⎝ ∂t ⎠
Expanding Eqn. 3 to its more familiar form:
⎛∂T ∂T ∂T ∂T ⎞ ∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎞
ρc⎜⎜⎝ ∂t + Vx ∂x + Vy ∂y +Vz ∂z ⎟⎟⎠=&q&&+ ∂x⎜⎝Kx ∂x ⎟⎠+ ∂y⎜⎜⎝Ky ∂y ⎟⎟⎠+ ∂z⎜⎝Ky ∂z ⎟⎠
(4)
366 Maazen Musmmar

AAssuming thaat no heat ggeneration eexists in the hardened concrete, thee term &q&&
mmay be negleected.

2.. Finite element analysis

ANSYS is used to mmodel the bbeam havingg dimensionns as illustraated in Figuure
(11). In orderr to eliminaate the effeect of stirruups on stresss distributtiion, no sheear
reeinforcemennt is providded. This iis in conformance witth ACI [3118-11] articcle
111.4.6.1d; thaat allows noot to providde stirrups aas minimumm shear reinnforcement ffor
beeams with ddepths not ggreater thann 250 mm. LLongitudinaal reinforceement of 4##12
mmm is provided as botttom steel reeinforcemennt, giving aa total steeell area of 452
mmm2. Whenn performinng thermal analysis, cconcrete is representedd by Solidd70
ellement. Sollid70 will tthen be repplaced by SSolid65 wheen performmiing structural
annalysis.

FFigure 1: Detail of shalllow reinforcced concretee Beam detaail

Figure2: Finite elemeent modelingg
NNumerical evvaluation ooff thermo-mmechanical response 337

Figure 3: Structural Soolid65 and thhermal Solidd70 elements, representinng concrete.

A schemmatic of strucctural solid665 and therrmal solid700 representinng concretee is
shhown in figuure (3). Thee structural solid65 element models the nonliinnear responnse
off reinforcedd concrete, it models the concrette material based on aa constitutiive
mmodel for thee triaxial behavior of cooncrete afteer Williams and Warnkke [1975]. Itt is
caapable of pplastic deforrmation, crracking in tthree orthoggonal direcctions at eaach
inntegration pooint.

The criteerion for faiilure of conncrete due too a multi axxial stress iss expressed in
thhe form:
F
−SS ≥ 0 (5)
f
c

Wherre:

F: aa function of principal sstress state
S: faailure surfacce
f : unniaxial crusshing strenggth.
c

If equuation (5) is satisfied, thhe material will crack oor crush.
38 Mazen Musmar

Figure 4: Structural LINK180 and thermal LINK33 element geometry

Bottom steel reinforcement of 4#12mm bars with total steel area of 452 mm2
is provided. Steel reinforcement bars are represented by link33 thermal finite
elements during thermal analysis, switchable to link180 structural elements on
conducting structural analysis.

LINK33 is a uniaxial element with the ability to conduct heat between its
nodes. The element has a single degree of freedom, temperature, at each node. On
performing structural analysis LINK33 is replaced by LINK180. It is 3-D spar
uniaxial tension-compression element with three degrees of freedom at each node;
translations in the nodal x, y, and Z directions.

3. Boundary conditions

A temperature of 600 C is applied by convection with a film coefficient of 50
W/m2/C to the bottom face of the reinforced concrete beam, and also 25 C is
applied at the upper face. Figure 5 illustrates the thermal boundary conditions.
NNumerical evvaluation ooff thermo-mmechanical response 339

Figure5 : Thhermal boundaary conditionss

FFigure 6: Strructural bounddary conditionns

Accordinng to Moaaveni [20033], the nummerical calcculation forr temperatuure
diistribution iis carried oout by Ansyys utilizingg Galarkin finite elemeent techniqque
thhat is capaable to perrform heatt exchanger calculatioons where the thermmal
coonductivity of the struccture is taken into accouunt.

The shalllow reinforrced concrette beam is ppin supporteed at the lefft support aand
rooller supported at the riight, as showwn in figuree 6. The tottal beam lenngth is 1.90m,
annd the distaance betweeen supports is 1.80m, bbeam widthh is 0.3m annd total deppth
eqquals 0.15mm.

4.. Analysiss and disccussion off results

A nonlinnear transieent thermal structural analysis is carried ouut taking innto
acccount the tthermal deppendant propperties of thhe concretee as thermall conductivity
annd specific heat. The aanalysis is pperformed oon the resuult of the soolution of twwo
tyypes of probblems. Firsst time trannsient analyysis is carried out to ddetermine tthe
teemperature distributionn within thhe beam aas a functiion of timme. The fieeld
teemperature ddistributionfor the trannsient thermmal analysisis then appplied as a looad
too perform sttructural anaalysis. The obtained reesults of theermal analyssis are plotted
400 Maazen Musmmar

inn figures 7,88,9 regarding the tempeerature proffiles, the theermal flux vvector, and tthe
teemperature ddistribution in steel reinnforcementt.

Figure 77: Temperatuure profile

Figurre 8: Thermaal Flux
NNumerical evvaluation ooff thermo-mmechanical response 441

Figure 9: Temperaturre distribution in reinforcement bars.

Table 1 llists the theermal and sttructural finnite elementts that repreesent concreete
annd steel reiinforcementt in thermaal and strucctural analyysis. Nodal temperaturres
from the traansient theermal analyysis are appplied at aa specified time in tthe
suubsequent ssteady statee stress anaalysis. The change froom thermall to structural
annalysis is performed by switchingg the thermmal SOLID770 elementss to structural
SOOLID65 eleements, andd the thermaal LINK33 elements too the structuural LINK180
ellements. Table 1 lissts the afoorementioneed switchaable elemennts. Materrial
prroperties illuustrated in ttable 2 are uused in the aanalysis.
Thhe relationsship betweeen the temmpperature vaariation andd the assocciated thermmal
strrains is as ffollows:

ε = αΔT (6)
thermal
Wherre:
ε
thermaal: the thermal defformation
α: the thhermal coefficientt of expansion occ−1
ΔT: the tthermal gradient oc
42 Mazen Musmar

Table 1: Thermal and structural elements

Switchable elements Switchable elements

concrete Steel reinforcement
Element Thermal Structural Thermal Structural
Type Solid 70 Solid65 Link33 Link180
Number of
8 8 2 2
nodes
Number of
DOF per 1 3 2 3
node
Translations Translations
Nature Temperature in nodal Temperature in nodal
x,y,z x,y,z

The solid65 element models the nonlinear response of reinforced concrete, it
models the concrete material based on a constitutive model for the triaxial
behavior of concrete after Williams and Warnke.

Table 2: Material properties for concrete and steel

Material Concrete Steel reinforcement
Compressive strength 30 MPa
Tensile strength 3.78 MPa 420 MPa
Elastic modulus 25143 MPa 200000MPa
Poisson’s ratio 0.2 0.3
Density 2400 Kg/m3 7800 Kg/m3
Thermal conductivity (k) 1.2W/moc 60W/moc
Specific heat capacity ( c) 1000J/kgoc 500J/kgoc
Thermal expansion 1.2x10-5 /oc 1.08x10-5 /oc
coefficient α

Solid65 is capable of plastic deformation; cracking in three orthogonal
directions at each integration point. The cracking is modeled through an
adjustment of the material properties that is done by changing the element
stiffness matrices.