Research Article Mathematical Model for Analysis of Uniaxial ...

Research ArticleMathematical Model for Analysis of Uniaxial and BiaxialReinforced Concrete Columns

Mohammed Salem Al-Ansari and Muhammad Shekaib Afzal

Department of Civil and Architectural Engineering Qatar University PO Box 2713 Doha Qatar

Correspondence should be addressed to Muhammad Shekaib Afzal shekaibqueduqa

Received 7 August 2020 Revised 4 November 2020 Accepted 11 November 2020 Published 25 November 2020

Academic Editor Faiz U A Shaikh

Copyright copy 2020 Mohammed Salem Al-Ansari and Muhammad Shekaib Afzal is is an open access article distributed underthe Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any mediumprovided the original work is properly cited

is paper presents amathematical model for the analysis of reinforced concrete (RC) uniaxial and biaxial columnsis proposedmodel is a quick and faster approach for the analysis and design of reinforced concrete rectangular columns without goingthrough the interaction charts procedure as well as other iterative methods for the computation of required axial load capacity (Pc)and moment capacity (Mc) A simplified flow chart has also been developed to find the required column capacity using thismathematical model Eight uniaxial columns (C-1 to C-8) and seven biaxial columns (CB-1 to CB-7) are analysed in this studyEach column is analysed having different steel reinforcement ratios (ρ) with different loading conditions In addition the studiedcolumns are subjected to both tension and compression failurese detailed examples for both uniaxial and biaxial columns (onefor each case) are also presented in this study e studied columns are also analysed using computer software spColumn eaverage variation of the mathematically computed values to the finite element software is not more than 10 showing promisingcomputational results

1 Introduction

Columns are the vertical compression members whichtransmit loads from the upper floors to the lower levels andto the soil through the foundations [1] Based on the po-sition of the load on the cross section columns are clas-sified as concentrically loaded (Figure 1) or eccentricallyloaded columns (Figure 2) Eccentrically loaded columnsare subjected to moments in addition to axial force emoments can be converted to a load P and eccentricitieseX and eY emoments can be uniaxial as in the case whentwo adjacent panels are not similarly loaded such ascolumns A and B in Figure 3 [2] A column is considered asbiaxially loaded when the bending occurs about the x- andy-axis such as in the case of corner column C in Figure 3 Ina recent study [3] Al-Ansari and Afzal also presented ananalytical model for generating interaction diagram chartsfor biaxial columns

e strength of reinforced concrete columns is normallyexpressed using interaction diagrams to relate the design

axial load 2emptyPn to the design bending momentemptyMn [4 5]Each control point on the column interaction curve (emptyPn minus

emptyMn) represents one combination of design axial loademptyPn and design bending momentemptyMn corresponding to aneutral-axis location (Figure 4) [6]

Extensive studies have been carried out on the inter-action diagrams (uniaxial and biaxial columns) of reinforcedconcrete (RC) rectangular columns [6ndash12] Several studieshave also been performed on providing numerical ap-proaches for the analysis and design of reinforced concretecolumns Furlong et al [13] provided an overview of theanalysis and design of reinforced concrete columns sub-jected to biaxial bending ey reviewed several methods ofanalysis that use traditional design methods and comparedtheir results with the obtained data from physical tests ofnormal strength concrete columns subjected to short-termaxial loads and biaxial bendingrsquos ey concluded that theelliptic load contour equation [14] and the reciprocalequation [15] are the simplest to use as they do not requirecomplicated calculations

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8868481 13 pageshttpsdoiorg10115520208868481

Chen et al [16] proposed an iterative numerical methodfor rapid section analysis and design of short concretecomposite columns subjected to biaxial bending Wang andHsu [17] proposed the numerical method approach for thedetermination of load-moment curvature relationship forshort and slender columns is numerical method ap-proach is also applicable for columns made of differentmaterials and shows good agreement with the differentexperimental results obtained in their study

Whitney [18] andHsu et al [19] providedmajor researchstudies on numerical method approaches Whitney sug-gested an approximate equation to estimate the nominalcompressive strength of columns subjected to compressionfailure Hsu in different research projects [10 17 20 21] alsopresented the results of experimental and analytical studieson the strength and deformation of biaxially loaded shortand tied columns with Lminus channel and T-shaped crosssections In another study Hsu [22] suggested a general

P

Figure 1 Concentrically loaded columns

P

x-axisy-axis Mx

My

(a)

Px-axisy-axis

eyex

(b)

Figure 2 Eccentrically loaded column

A

C

B

Figure 3 Uniaxially and biaxially loaded column

2 Advances in Civil Engineering

equation for the analysis and design of reinforced concreteshort and tied rectangular columns

is study proposes a mathematical model to analyseand design the uniaxial and biaxial columns based on ACIbuilding code of design [23] is model is a quick andeasy approach for analysing and designing the reinforcedrectangular columns without going through the interac-tion charts for the computation of the required axial loadcapacity (Pc) and moment capacity (McxMcy) A sim-plified flow chart has also been developed to find therequired column capacity using the proposed modelapproach e previous research studies of the mathe-matical model approach are limited to columns havingcompression failure only is study includes the nu-merical examples of columns using the proposed math-ematical model approach for both compression andtension failure cases is relatively new approach willalso be useful to the undergraduate and graduate studentsas well as researchers to calculate the required columncapacities using this approach in their research-relatedactivities

Numerical examples for the selected reinforced concretecolumns (uniaxial and biaxial columns) are also illustratedto check the adequacy of this proposed model Eight uniaxialcolumns (C-1 to C-8) and seven biaxial columns (CB-1 toCB-7) are analysed in this studyese columns are analysedhaving different steel reinforcement ratios (ρ) differentvalues of steel yield strength (fy) concrete compressivestrength (fc

prime) and different load capacity conditionsMoreover the results obtained from this proposedmodel arecompared with computer software spColumn 2016 [24]

2 Mathematical Model Formulation ACICode Design

e stress and strain distribution of a rectangular columnsection (uniaxial column) for the calculation of Pn andMn isgiven in Figure 5

e resultant force PN is equal to the summation of allinternal forces

PN CCon minus Ts + CS (1)

Section cracks

Zero stressin tension reinforcement

isins1 = 0

Pure compression

Pn Mn

Tension-controlled limitTension-controlled limit

εs1 = ndash25isiny

Pure bending

Pure tension

Maximum axial loadϕPn(max)

fs1 = ndash05 fyisins1 = ndash05 isiny

Balanced failure and compression controlled limit

fs1 = ndashfyisins1 = ndashisiny

Tens

ion

Com

pres

sion

ϕPn ϕMnAxi

al lo

ad P

n and

ϕP n

(kip

s)

Moment Mn and ϕMn (kip-)

Figure 4 Control points for column interaction curve (emptyPn minus emptyMn)

Advances in Civil Engineering 3

Similarly the resultant moment MN is equal to thesummation of all internal moments

MN Mconc + MT + MCS (2)

e following steps revealed the calculation of the re-quired internal forces and internal moments for a rectan-gular uniaxial RC column

21 Plain Concrete Section e Internal concrete com-pressive force (Cconc ) is computed as

Cconc 085fcprimeba( 1113857

Cconc 085fcprime bβc

(3)

where Cconc internal concrete compression forcefcprime compressive concrete strength b column widtha depth of the compression stress blockβ 085 minus 0008(fc

prime minus 30)ge 065 and c distance from ex-treme compression fiber to the neutral axis

Referring to Figure 5 the moment about the midpoint ofthe section (Mconc) can be computed as

Mconc Cc d minusa

2minus dPrime1113874 1113875

Mconc 085fcprime ba d minus

a

2minus dPrime1113874 1113875

(4)

where h column total depth dPrime ((h2) minus dprime) d columneffective depth (h minus dprime) and dprime distance from extremecompression fiber to centroid of top reinforcing steel

22 Tension Steel Section e internal tensile force Ts iscomputed as

Ts Asfy (5)

where As area of tensile steel reinforcement and fy yieldstress of reinforcing steele internal moment MT is

MT Asfy dPrime (6)

23 Compression Steel Section e internal compressiveforce Cs is computed as [25]

Cs Asprime fsprime( 1113857

Cs Asprime fsprime minus 085fc

prime( 1113857(7)

where Asprime area of compression steel reinforcement and

fsprime fy (if the compression steel yields)e internal moment MT is

MT Asprime fsprime minus 085fc

prime( 1113857 d minus dprime minus dPrime( 1113857 (8)

3 Mathematical Model Analysis

e following steps should be revealed to calculate the designaxial load and moment capacity of the required rectangularRC column section Columns may be subjected to tensionfailure or compression failure depends on the balancedeccentricity value (eb)

Cs = Asprime fsprime

Cc = 085fcprimeab

Ts = As fs

Pn

Mn

(d) (e)

hd

b 085fcprime

fs

a = β1c

d ndash a2PC

c

dprime

dPrime

εc = 0003

εsprime

c a2

As

Asprime

εs = fsEs

fsprime

(a) (b) (c)

Figure 5 Calculation of Pn and Mn for a given strain distribution (a) Section (b) Strain (c) Stress (d) Internal forces (e) Resultant forces


eb Mb

Pb

(9)

Mb Ccb d minusab

2minus dPrime1113874 1113875 + Cs d minus dprime minus dPrime( 1113857 + Ts times dPrime( 11138571113874 1113875 times 10minus 3

(10)

Pb Cs + Ccb minus Ts (11)

where ab β times cb cb (600 times d600 + fy)Ccb (085 times fc

prime times ab times b) times 10minus 3 and Cs Asprime(fsprime minus 085fc

prime)times 10minus 3 (if the compression steel yields then fs

prime fy)

31TensionFailureAnalysis Tension failure will occur whenthe balanced eccentricity value (eb) is less than load ec-centricity (e) Substituting the values of Cc Cs and Ts inequation (1) and solving for (a) will be a second-degreeequation [14]

PN CCon minus Ts + CS(Equation (1))

Aa2

+ Ba + C 0(12)

where A (085 times fcprime times b2) B 085 times fc

prime times b times (eprime minus d)C Asprime(fsprime minus 085fc

prime)(eprime minus d + dprime) minus fyAseprime and eprime e + dPrime(eprime e + d minus (h2)whenAs As

prime)Solve for (a) to get

a minus B plusmn

B2

minus 4AC1113968

2A (13)

Substitute the value of a in equation (3) to calculate Cc

and from equations (5) and (7) to compute Ts and Cs valuesese obtained values are substituted in equation (1) forPN CCon minus Ts + CS

PN 1eprime

Cc d minusa

21113874 1113875 + Cs d minus dprime( 11138571113874 1113875 (14)

MN PN times e (15)

e column axial load capacity andmoment capacity cantherefore be computed as

Pc emptyPN (16)

Mc emptyMN (17)

(whereempty is the column reduction factor having the value of065)

32 Compression Failure Analysis Compression failure willoccur when the balanced eccentricity value (eb) is biggerthan the load eccentricity (e) Substituting the values ofCc Cs and Ts in equation (1) and solving for (a) will be acubic equation [14]

Aa3

+ Ba2

+ Ca + D 0 (18)



prime)(eprime minus d + dprime) + 600Aseprime andD minus 600Aseprimeβprimedwhere eprime e + dPrime (eprime e + d minus (h2)

whenAs Asprime)

Once the values of A B C andD are calculated thevalue of a can be determined by the trial method or directlyby using MATLAB or any scientific calculator Moreoverthe cubic equation can also be solved using different nu-merical methods for example Newton Raphson MethodAfter getting the required value of (a) similar equationsfrom (14) to (18) (as mentioned in the Tension FailureAnalysis) should be used to get the required value of columnaxial load capacity (Pc) and moment capacity (Mc)

e following flow chart (Figure 6) can be followed tofind the required capacity of the rectangular uniaxial columnsection

4 Numerical Examples for Uniaxial Columns

Eight reinforced rectangular columns (C-1 to C-8) havingdifferent column sizes are analysed using the numericalmethod approach ese columns are having different re-inforcement ratios (ρ) in addition to different failure typesboth tension and compression failures e design inputload data for these columns are illustrated in Table 1

e above eight columns C1 to C8 are analysed using themathematical model approach to find the required values ofaxial load capacity Pc and moment capacity Mc Moreoverthese values are also compared with the computer softwarespColumn e results obtained are depicted in Table 2

ese above columns are also analysed with differentavailable methods Whitneyrsquos 1st approximation method[18] Whitneyrsquos second approximation method [18] and themethod provided by HSU [19] ese available methods areonly available for the columns having the compressionfailureere are no examples available for the columns withthe tension failure cases e results comparison is men-tioned in Table 3

41 Detailed Numerical Example for Column C-4 (400times 400)

Input Data Figure 7

Pu 400 kNMu 100 kNmiddotmfcprime 30MPa

fy 415MPaAs 1000mm2

Asprime 1000mm2

dprime 80mmϕ 065

Solution

(1) Finding the value of e (MuPu)

100400 250mm(2) cb (600 times d600 + fy) (600 times 320600 + 415)

18916mm(3) ab β times cb 085 times 18916 160788mm


(4) Ccb 085fcprimeabb 085 times 30 times 160788 times 400

164times103 kN(5) Cs As

prime(fy minus 085fcprime) 1000

(415 minus 085(30)) 3895times102 kN(6) Ts As fy 1000 times 415 415times102 kN(7) Pb Cs + Ccb minus Ts 1615times103 kN

(8) Mb (Cc(d minus (ab2) minus dPrime) + Cs(d minus dprime minus dPrime)+(Ts times dPrime))dPrime ((h2) minus dprime) 120mmMb 2927 times 102 kNmiddotm

(9) eb (MbPb) 1813mm lt e (250mm) (TEN-SION Failure)

Start

cb = ((600 times d)(600 + fy))

ab = β times cb

Ccb = 085f primecabbCs = Aprimes ( f primes ndash 085f primec)

(If the compression steel yields then f primes = f y)Ts = As fy

Pb = Cs + Cc ndash TsMb = (Ccb (d ndash (ab2) ndash dPrime) + Cs (d ndash dprime ndash dPrime) + (Ts times dPrime))

eb = (MbPb)

e gt ebYes

Tension failureCompressionfailure

Aa3 + Ba2 + Ca + D = 0 Aa2 + Ba + C = 0

No

Findingthe valueof (a) for

cubicEquation

Findingthe valueof (a) for quadraticequation

Check condition of steel yieldingεs gt εy rarr fs = fyεs lt εy rarr fs ne fy

A = 0425 times f cprime times bB = 085 times f cprime times b times (eprime ndash d)C = Asprime (f sprime ndash 085f cprime)(eprime ndash d + dprime) + 600 AseprimeB = ndash600 Aseprimeβprimed

A = 0425 times f cprime times bB = 085 times f cprime times b times (eprime ndash d)C = Asprime ( f sprime ndash 085fcprime)(eprime ndash d + dprime) + fyAseprime)

dPrime = ((h2) ndash dprime)

eprime = e + dPrimeCc = 085fcprimeab

PN = (1eprime)(Cc (d ndash (a2)) + Cs (d ndash dprime))Pc = OslashPN

MN = PN times eMc = OslashMN

End

Figure 6 Flow chart of the mathematical model for the rectangular uniaxial rectangular column


Table 1 Uniaxial column input data

Column identifier Pu (kN) Mu (kNmiddotm) As (mm2) Asprime (mm2) fcprime (MPa) fy (MPa) ϕ dprime (mm) e (mm)

C1 (200times 400) 300 60 400 400 30 300 07 80 200C2 (200times 400) 200 50 400 400 20 300 07 60 250C3 (300times 500) 800 200 1000 1000 30 415 07 75 250C4 (400times 400) 400 100 1000 1000 30 415 065 80 250C5 (300times 450) 1300 1574 1530 1530 25 300 065 75 121C6 (200times 400) 400 20 402 402 30 300 07 60 50C7 (300times 500) 600 100 1500 1500 30 415 065 90 167C8 (400times 800) 1000 200 2000 2000 30 415 065 80 200

Table 2 Uniaxial column design results

Column identifier eb (mm) Failure conditionMathematical model spColumn

Pc (kN) Mc (kNmiddotm) Pc (kN) Mc (kNmiddotm)

C1 1407 Tension 3776 755 3783 756C2 1415 Tension 2348 587 2474 6184C3 233 Tension 1033 258 1036 259C4 1818 Tension 6503 1626 650 163C5 222 Compression 1310 158 1425 1725C6 138 Compression 1191 595 1810 9052C7 276 Compression 1495 250 1677 2795C8 364 Compression 3606 722 3975 795

Table 3 Column design results comparison with different methods

Column identifier Failure conditionMathematical model Whitney 1st

approximationWhitney 2nd

approximation HSU

Pc (kN) Mc (kNmiddotm) Pc (kN) Mc (kNmiddotm) Pc (kN) Mc (kNmiddotm) Pc (kN) Mc (kNmiddotm)C1 Tension 3776 755 mdash mdash mdash mdash mdash mdashC2 Tension 2348 587 mdash mdash mdash mdash mdash mdashC3 Tension 1033 258 mdash mdash mdash mdash mdash mdashC4 Tension 650 162 mdash mdash mdash mdash mdash mdashC5 Comp 1310 158 1580 191 1285 155 1580 191C6 Comp 1191 595 1877 94 1125 56 1878 94C7 Comp 1495 250 1677 280 1482 247 1677 280C8 Comp 3606 722 4300 860 3582 716 4300 860

400

320

400

As = 1000

PC

Asprime = 1000

80

120

Figure 7 Column C-4


(10) Finding the value of (a) using the QuadraticEquationAa2 + Ba + C 0A 0425 times fc

prime times b 51 times 103B 085 times fc

prime times b times (eprime minus d) where (eprime e + dPrime 370mm)B 51 times 105C Asprime(fy minus 085fc

prime)(eprime minus d + dprime) minus fyAseprime minus 1029 times 108a (minus B +

B2 minus 4AC

radic2A)

a 10059mm(11) Check if the tension steel has yielded

c (aβ) 11834mmεs ((d minus cc)) times 0003 0005 εy (fyEs)

0002εs gt εy steel yields there4 (fs fy)

(12) PN (1eprime)(Cc(d minus (a2)) + Cs(d minus dprime) times 10minus 3

Cc 085fcprimeab 1026 times 106 kN

PN 100055 kNMN PN times e 2501 kNmiddotm

(13) Pc emptyPN 65035 kN(14) Mc emptyMN 16256 kNmiddotm

5 Numerical Examples for Biaxial Columns

Seven reinforced biaxial rectangular columns (CB-1 to CB-7) having different column sizes are also analysed using theproposed model ese columns are also having differentreinforcement ratios (ρ) in addition to different failuretypes that is tension-tension compression-compressionand tension-compression failures e design input loaddata for these columns are illustrated in Table 4 ecolumn cross section subjected to biaxial bending is shownin Figure 8 A similar flow chart has to be adopted (asdiscussed in the uniaxial column sections) once for thecase of eccentricity in the x-direction (ex) and later for theeccentricity in the y-direction (ey) to obtain the requiredvalues of load capacities in x- and y-direction (emptyPxemptyPy)ese values are later used in Breslerrsquos formula [15](equation (19)) to find the value of Pc Moreover the Mcxand Mcy values can be found by using equations (21) and(22) accordingly

Pc 1

1emptyPx( 1113857 + 1emptyPy1113872 1113873 minus 1emptyPNmax( 1113857 (19)

emptyPNmax 08ϕ 085fprimec(Ag minus Ast) + fyAst( 1113857 (20)

where emptyPNmax maximum permissible column loadAst total area of steel and Ag

(Gross area of cross section) minus (sectional area of concretemember member)

e moments in the x- and y-direction can be found as

Mcx PC times euy (21)

Mcy PC times eux (22)

e above seven columns (CB-1 to CB-7) are analysedwith mathematical model approach to find the requiredvalues of axial load capacity Pc using reciprocal formulaMoreover the values of Pc are also compared with thecomputer software spColumn e results obtained aredepicted in Table 5

51 Detailed Numerical Example for Column CB-7(400times1200)

Input Data Figure 9

Pu 1500 kNMux 300 kNmiddotmMuy 300 kNmiddotmfcprime 20MPa

fy 300MPaAs 3080mm2

Asprime 3080mm2

dprime 60mmϕ 065

Solution (Solving for the X-direction)

(1) Finding the value of ey (MuxPu) 200mm(2) cb (600 times d600 + fy)

(600 times 1140600 + 300) 760mm(3) ab β times cb 093 times 760 7068mm(4) Ccb 085fc

primeabb

085 times 20 times 7068 times 400 481times 103 kN(5) Cs As


(300 minus 085(20)) 871times 102 kN(6) Ts As fy 3080 times 300 924times102 kN(7) Pbx Cs + Ccb minus Ts 4754 kN(8) Mbx (Cc(d minus (ab2) minus dPrime) + Cs(d minus dprime minus dPrime)+

(Ts times dPrime))dPrime ((h2) minus dprime) 540mmMbx 2155 times 103 kNmiddotm

(9) eby (MbxPbx) 453mm gt ey (200mm)

(COMPRESSION Failure)(10) Finding the value of (a) using the Cubic Equation

Aa3+Ba2 + Ca + D 0A 0425 times fc

prime times b times 10minus 3 34B 085 times fc

prime times b times (eprime minus d) times 105 where(eprime e + dPrime 740mm)B minus 272 times 103C Asprime(fsprime minus 085fc

prime)(eprime minus d + dprime) + 600Aseprime 1071 times 106D minus 600Aseprime

βprimed minus 145 times 109


c (aβ) 1016mmεs (d minus cc) times 0003 00028 εy (fyEs)


(12) PNX (1eprime)(Cc(d minus (a2)) + Cs(d minus dprime) times 10minus 3


PNX 7068 kNMNX PNX times ex 1414 kNmiddotm

(13) PCX emptyPNX 4594 kN


(14) MCX emptyMNX 91879 kNmiddotm

(Solving for the Y-direction) (Figure 10)

(1) Finding the value of ex (MuyPu) 200mm(2) cb (600 times d600 + fy)


primeabb 085 times 20times

2108 times 1200 43times103 kN(5) Cs As


(300 minus 085(20)) 871times 102 kN

(6) Ts As fy 3080 times 300 924times102 kN(7) Pby Cs + Ccb minus Ts 4248times103 kN(8) Mby (Cc(d minus (ab2) minus dPrime) + Cs(d minus dprime minus dPrime)+

(Ts times dPrime))dPrime ((h2) minus dprime) 140mmMby 6582 times 102 kNmiddotm

(9) ebx (MbyPby) 15494mm gt ex (200mm)

(TENSION Failure)(10) Finding the value of (a) using the Quadratic

EquationAa2 + Ba + C 0

Table 4 Biaxial column input data

Column identifier Pu (kN) Mux (kNmiddotm) Muy(kNmiddotm) As (mm2) As

prime (mm2) fcprime (MPa) fy (MPa) ϕ dprime (mm) ex (mm) ey (mm)

CB-1 (300times 600) 300 100 80 1232 1232 30 400 065 80 267 333CB-2 (200times 400) 200 40 20 6284 6284 20 300 07 40 100 200CB-3 (300times 300) 2500 250 120 1225 1225 30 400 065 70 48 100CB-4 (375times 500) 1700 200 100 2100 2100 30 415 065 60 59 118CB-5 (400times 500) 800 200 50 14138 14138 30 415 065 60 625 250CB-6 (350times 700) 400 60 40 1638 1638 20 300 065 45 100 150CB-7 (400times1200) 1500 300 300 3080 3080 20 300 065 60 200 200

(x-axis)

(y-a

xis)

bx = hy

hx = by

Muy

Mux

Figure 8 Biaxial column cross section

Table 5 Biaxial column design results

Column identifier Failure condition Mathematical model spColumnPc (kN) Pc (kN)

CB-1 Tension and tension 280 343CB-2 Tension and tension 262 228CB-3 Compression and compression 480 415CB-4 Compression and compression 2132 2035CB-5 Tension and compression 1291 1142CB-6 Compression and compression 2852 2805CB-7 Compression and tension 1993 2081


A 0425 times fcprime times b times 10minus 3 10200

B 085 times fcprime times b times (eprime minus d) times 105 where

(eprime e + dPrime 340mm)B 0C Asprime(fsprime minus 085fc

prime)(eprime minus d + dprime) minus fy Aseprime minus 262 times 108a 160227mm

(11) Check if the tension steel has yieldedc (aβ) 172287mmεs (d minus cc) times 0003 000292 εy (fyEs)


(12) PNY (1eprime)(Cc(d minus (a2)) + Cs(d minus dprime) times 10minus 3


PNY 3216 kNMNX PNY times ey 64325 kNmiddotm

(13) PCY emptyPNY 2091 kN(14) MCY emptyMNY 41812 kNmiddotm

Finding the value of Pc using Breslerrsquos equation (19)

Pc 1


where emptyPNmax 08 ϕ( 085fprimec (Ag minus Ast) + fyAst)

5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN

6 Validation of the Mathematical Model

In order to validate the proposed mathematical model ap-proach the model is validated with the existing experimental

results of columns subjected to uniaxial and biaxial loadingse experimental results data has been extracted from thetest results provided by HSU [22] Two uniaxial columns asprovided by Bresler (B-1 and B-2) and one biaxial column asprovided by Anderson and Lee (SC-4) are selected from theresearch article [22] to compare the results with themathematical model

Table 6 illustrates the experimental testing data providedby HSU e data and the results are provided in imperialunits (Kips-ft) units erefore they are converted to metricunits accordingly to compare the values with our results

Table 7 provides the experimental test results as well asthe validation of the test data with the proposed mathe-matical model e column capacity (Pc) results obtainedfrom the experimental data are quite close to the mathe-matical model results showing satisfactory computationalresults

7 Results and Discussions

e results obtained from the mathematical model approachfor both uniaxial and biaxial columns showed a safe andconservative column design method e results of eightuniaxial column sections (C-1 to C-8) using the proposedmodel are also compared with different available mathe-matical models provided by Whitneyrsquos 1st approximationmethod Whitneyrsquos second approximation method and themethod provided by HSU Columns C1 to C-4 were sub-jected to tension failure whereas columns C-5 to C-8 werethe compression failure cases e other three mathematicalstudies (Whitneyrsquos 1st approximation Whitneyrsquos second

1200

1140

400

As = 3080

PC

Asprime = 3080

60

540

Figure 9 Column CB-7 (X-X axis)

As = 3080

PCAsprime = 3080

400 340140

1200

Figure 10 Column CB-7(Y-Y axis)


Table 6 Experimental testing data [22]

Experimentalinvestigator

Column identifier with size (inx in) b (mm x mm)

As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)

Bresler (uniaxialcolumn) B-1 (6times 8) (152times 203) 124 (800) 37 (256) 535 (369) 175 (445) 6 (1524) 0


Anderson and Lee(biaxial column) SC-4 (4times 4) (102times102) 08 (516) 5435 (375) 456 (3146) 075 (19) 282 (7163) 282 (7163)

Table 7 Validation of experimental data [22]


Column identifier withsize (in x in) (mm x mm) ϕc

eb (in)(mm) Failure condition

Experimentalresults

Mathematicalmodel (PexpPMath)

Pc (kips) (kN) Pc (kips) (kN)

Bresler (uniaxialcolumn) B-1 (6times 8) (152times 203) 065 467

(1185) eblt ex (tension) 24 (107) 29 (132) 083

Bresler (uniaxialcolumn) B-2 (6times 8) (152times 203) 065 46 (117) ebgt ex

(compression) 60 (267) 59 (263) 101

Anderson and Lee(biaxial column) SC-4 (4times 4) (102times102) 065 27 (69) eblt ex (tension) 135 (60) 11 (49) 122

0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)

Mathematical modelspColumnWhitneyrsquos 1st approx

Whitneyrsquos 2nd approxHSU

Figure 11 Axial load capacity comparison for uniaxial columns (C1ndashC8)

0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)



Figure 12 Moment capacity comparison for uniaxial columns (C1ndashC8)


approximation and the method provided by HSU) are onlylimited to the case where the columns are subjected to thecompression failure only

ese studied columns (C-1 to C-8) are also analysedusing the computer software spColumn and the comparisonresults for axial load capacities (Pc) and moment capacities(Mc) are displayed in bar charts (Figures 11 and 12)

For the biaxial columns (CB-1 to CB-7) the axial loadcapacity results for mathematical model approach usingBreslerrsquos formula and the computer software spColumn aredisplayed in the bar chart (Figure 13)

e values of Pc obtained using the mathematical modelare quite close to the computer software results showingrelatively satisfactory computational results

8 Conclusion

In this study the mathematical model is presented toanalyse and design the uniaxial and biaxial columnswithout going through the column interaction charts tofind the required axial load capacities and moment ca-pacities A simplified flow chart has also been developed tosolve the required column section following the mathe-matical model steps

Eight (RC) uniaxial columns (C-1 to C-8) and seven(RC) biaxial columns (CB-1 to CB-7) are analysed in thisstudy ese columns are analysed having different steelreinforcement ratios (ρ) different values of steel yieldstrength (fy) concrete compressive strength (fc

prime) anddifferent load capacity conditions Moreover the studiedcolumns are subjected to both tension and compressionfailures

For the uniaxial columns the proposed mathematicalmodel results are also compared with the different availablenumerical approaches done byWhitneyrsquos 1st approximationWhitneyrsquos 2nd approximation and the method provided byHSU All of these three methods were formulated based onthe case of compression failure only ese uniaxial columnsare also analysed using the computer software spColumne results obtained showed that this proposed mathe-matical approach showed good agreement with the com-puter software spColumn showing relatively satisfactoryresults

e studied biaxial columns are subjected to differentfailure conditions that is tension-tension failure com-pression-compression failure and tension-compressionfailure Breslerrsquos formula was used to find the required ca-pacity (Pc) after finding the (Px)and(Py) from the math-ematical model approach e biaxial columns were alsoanalysed with the computer software e average variationof the mathematically computed values for biaxial columnsto the finite element software was not more than 10Moreover the results obtained for the columns subjected totension failure are quite close with the computer softwarespColumn Moreover this mathematical model has alsobeen validated with the existing experimental results con-ducted by HSU

In short this newly proposed mathematical model is agood and quick approach to analyse the reinforced concreteuniaxial and biaxial columns is model can also help thestudents and the academic researchers to find the columncapacities without going through the column interactioncharts and other long iterative approaches

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this article

References

[1] M S Al-Ansari and A B Senouci ldquoMATHCAD teachingand learning tool for reinforced concrete designrdquo Interna-tional Journal of Engineering Education vol 15 no 1pp 64ndash71 1999

[2] M S Al-Ansari M S Afzal and M S Afzal ldquoSimplifiedirregular column analysis by equivalent square methodrdquoJournal of Structural Engineering amp Applied Mechanics vol 2no 1 pp 36ndash46 2019

[3] M S Al-Ansari and M S Afzal ldquoSimplified biaxial columnInteraction chartsrdquo Engineering Reports vol 1 no 5 ArticleID e12076 2019

[4] J A Rodriguez and J D Aristizabal-Ochoa ldquoBiaxial inter-action diagrams for short RC columns of any cross sectionrdquoJournal of Structural Engineering vol 125 no 6 pp 672ndash6831999

[5] J G MacGregor J K Wight S Teng and I Paulus Rein-forced Concrete Mechanics and Design Vol 3 Prentice-HallUpper Saddle River NJ USA 1997

[6] Interaction Diagram-Tied Reinforced Concrete ColumnshttpswwwstructurepointorgpdfsInteraction-Diagram-Tied-Reinforced-Concrete-Column-Symmetrical-ACI318-14htm StructurePoint Skokie IL USA 2020

[7] P Andersen and H-N Lee A Modified Plastic lteory ofReinforced Concrete University of Minnesota MinneapolisMN USA 1951

[8] A Aas-Jakobsen ldquoBiaxial eccentricities in ultimate load de-signrdquo Journal Proceedings vol 61 no 3 pp 293ndash316 1964

0

500

1000

1500

2000

2500

3000

CB-1 CB-2 CB-3 CB-4 CB-5 CB-6 CB-7

Mom

ents

(KN

-M)

Mathematical modelspColumn

Figure 13 Axial load capacity comparison for biaxial columns(CB-1ndashCB-7)


[9] J F Fleming and S D Werner ldquoDesign of columns subjectedto biaxial bendingrdquo Journal Proceedings vol 62 no 3pp 327ndash342 1965

[10] C T T Wsu ldquoBiaxially loaded L-shaped reinforced concretecolumnsrdquo Journal of Structural Engineering vol 111 no 12pp 2576ndash2595 1985

[11] D A Ross and J Richard Yen ldquoInteractive design of rein-forced concrete columns with biaxial bendingrdquo JournalProceedings vol 83 no 6 pp 988ndash993 1986

[12] C-K Wang ldquoSolving the biaxial bending problem in rein-forced concrete by a three-level iteration procedurerdquo Com-puter-Aided Civil and Infrastructure Engineering vol 3 no 4pp 311ndash320 1988

[13] R W Furlong C-T T Hsu and S Ali Mirza ldquoAnalysis anddesign of concrete columns for biaxial bending-overviewrdquoStructural Journal vol 101 no 3 pp 413ndash422 2004

[14] M N Hassoun and A Al-Manaseer Structural Concretelteory and Design John Wiley amp Sons Hoboken NJ USA2020

[15] B Bresler ldquoDesign criteria for reinforced columns under axialload and biaxial bendingrdquo Journal Proceedings vol 57 no 11pp 481ndash490 1960

[16] S F Chen J G Teng and S L Chan ldquoDesign of biaxiallyloaded short composite columns of arbitrary sectionrdquo Journalof Structural Engineering vol 127 no 6 pp 678ndash685 2001

[17] G G Wang and C-T T Hsu ldquoComplete biaxial load-de-formation behavior or RC columnsrdquo Journal of StructuralEngineering vol 118 no 9 pp 2590ndash2609 1992

[18] C S Whitney ldquoPlastic theory of reinforced concrete designrdquoProceedings of the American Society of Civil Engineers vol 66no 10 pp 1749ndash1780 1942

[19] T T C Hsu and Y-L Mo Unified lteory of ConcreteStructures John Wiley amp Sons Hoboken NJ USA 2010

[20] C-T T Hsu ldquoChannel-shaped reinforced concrete com-pression members under biaxial bendingrdquo Structural Journalvol 84 no 3 pp 201ndash211 1987

[21] C-T T Hsu ldquoT-shaped reinforced concrete members underbiaxial bending and axial compressionrdquo Structural Journalvol 86 no 4 pp 460ndash468 1989

[22] C-T T Hsu ldquoAnalysis and design of square and rectangularcolumns by equation of failure surfacerdquo Structural Journalvol 85 no 2 pp 167ndash179 1988

[23] ACI Committee Building Code Requirements for StructuralConcrete (ACI 318-14) and Commentary American ConcreteInstitute Farmington Hills MI USA 2014

[24] SP Column v6 Design and Investigation of Reinforced Con-crete Column Sections StructurePoint Skokie IL USA 2016

[25] Z A Siddiqi Concrete Structures Part-1 Help Civil EngineerPublisher Lahore Pakistan 2020


Chen et al [16] proposed an iterative numerical methodfor rapid section analysis and design of short concretecomposite columns subjected to biaxial bending Wang andHsu [17] proposed the numerical method approach for thedetermination of load-moment curvature relationship forshort and slender columns is numerical method ap-proach is also applicable for columns made of differentmaterials and shows good agreement with the differentexperimental results obtained in their study

Whitney [18] andHsu et al [19] providedmajor researchstudies on numerical method approaches Whitney sug-gested an approximate equation to estimate the nominalcompressive strength of columns subjected to compressionfailure Hsu in different research projects [10 17 20 21] alsopresented the results of experimental and analytical studieson the strength and deformation of biaxially loaded shortand tied columns with Lminus channel and T-shaped crosssections In another study Hsu [22] suggested a general

P

Figure 1 Concentrically loaded columns

P

x-axisy-axis Mx

My

(a)

Px-axisy-axis

eyex

(b)

Figure 2 Eccentrically loaded column

A

C

B

Figure 3 Uniaxially and biaxially loaded column










Section cracks


isins1 = 0

Pure compression

Pn Mn


εs1 = ndash25isiny

Pure bending

Pure tension





Tens

ion

Com

pres

sion

ϕPn ϕMnAxi

al lo

ad P

n and

ϕP n

(kip

s)










(3)




Mconc Cc d minusa

2minus dPrime1113874 1113875


a

2minus dPrime1113874 1113875

(4)



Ts Asfy (5)


MT Asfy dPrime (6)




prime( 1113857(7)








Cc = 085fcprimeab

Ts = As fs

Pn

Mn

(d) (e)

hd

b 085fcprime

fs

a = β1c

d ndash a2PC

c

dprime

dPrime

εc = 0003

εsprime

c a2

As

Asprime

εs = fsEs

fsprime

(a) (b) (c)



eb Mb

Pb

(9)

Mb Ccb d minusab


(10)





prime fy)



Aa2

+ Ba + C 0(12)





a minus B plusmn

B2

minus 4AC1113968

2A (13)



PN 1eprime

Cc d minusa

21113874 1113875 + Cs d minus dprime( 11138571113874 1113875 (14)

MN PN times e (15)


Pc emptyPN (16)

Mc emptyMN (17)



Aa3

+ Ba2

+ Ca + D 0 (18)




whenAs Asprime)








Input Data Figure 7


fy 415MPaAs 1000mm2

Asprime 1000mm2

dprime 80mmϕ 065

Solution











Start

cb = ((600 times d)(600 + fy))

ab = β times cb




eb = (MbPb)

e gt ebYes


Aa3 + Ba2 + Ca + D = 0 Aa2 + Ba + C = 0

No


cubicEquation









End













approximation HSU


400

320

400

As = 1000

PC

Asprime = 1000

80

120

Figure 7 Column C-4






B2 minus 4AC

radic2A)










Pc 1










Input Data Figure 9


fy 300MPaAs 3080mm2

Asprime 3080mm2

dprime 60mmϕ 065




primeabb



























(300 minus 085(20)) 871times 102 kN










(x-axis)

(y-a

xis)

bx = hy

hx = by

Muy

Mux

















Pc 1



5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN








1200

1140

400

As = 3080

PC

Asprime = 3080

60

540


As = 3080

PCAsprime = 3080

400 340140

1200






As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)






























Section cracks


isins1 = 0

Pure compression

Pn Mn


εs1 = ndash25isiny

Pure bending

Pure tension





Tens

ion

Com

pres

sion

ϕPn ϕMnAxi

al lo

ad P

n and

ϕP n

(kip

s)










(3)




Mconc Cc d minusa

2minus dPrime1113874 1113875


a

2minus dPrime1113874 1113875

(4)



Ts Asfy (5)


MT Asfy dPrime (6)




prime( 1113857(7)








Cc = 085fcprimeab

Ts = As fs

Pn

Mn

(d) (e)

hd

b 085fcprime

fs

a = β1c

d ndash a2PC

c

dprime

dPrime

εc = 0003

εsprime

c a2

As

Asprime

εs = fsEs

fsprime

(a) (b) (c)



eb Mb

Pb

(9)

Mb Ccb d minusab


(10)





prime fy)



Aa2

+ Ba + C 0(12)





a minus B plusmn

B2

minus 4AC1113968

2A (13)



PN 1eprime

Cc d minusa

21113874 1113875 + Cs d minus dprime( 11138571113874 1113875 (14)

MN PN times e (15)


Pc emptyPN (16)

Mc emptyMN (17)



Aa3

+ Ba2

+ Ca + D 0 (18)




whenAs Asprime)








Input Data Figure 7


fy 415MPaAs 1000mm2

Asprime 1000mm2

dprime 80mmϕ 065

Solution











Start

cb = ((600 times d)(600 + fy))

ab = β times cb




eb = (MbPb)

e gt ebYes


Aa3 + Ba2 + Ca + D = 0 Aa2 + Ba + C = 0

No


cubicEquation









End













approximation HSU


400

320

400

As = 1000

PC

Asprime = 1000

80

120

Figure 7 Column C-4






B2 minus 4AC

radic2A)










Pc 1










Input Data Figure 9


fy 300MPaAs 3080mm2

Asprime 3080mm2

dprime 60mmϕ 065




primeabb



























(300 minus 085(20)) 871times 102 kN










(x-axis)

(y-a

xis)

bx = hy

hx = by

Muy

Mux

















Pc 1



5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN








1200

1140

400

As = 3080

PC

Asprime = 3080

60

540


As = 3080

PCAsprime = 3080

400 340140

1200






As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)




























(3)




Mconc Cc d minusa

2minus dPrime1113874 1113875


a

2minus dPrime1113874 1113875

(4)



Ts Asfy (5)


MT Asfy dPrime (6)




prime( 1113857(7)








Cc = 085fcprimeab

Ts = As fs

Pn

Mn

(d) (e)

hd

b 085fcprime

fs

a = β1c

d ndash a2PC

c

dprime

dPrime

εc = 0003

εsprime

c a2

As

Asprime

εs = fsEs

fsprime

(a) (b) (c)



eb Mb

Pb

(9)

Mb Ccb d minusab


(10)





prime fy)



Aa2

+ Ba + C 0(12)





a minus B plusmn

B2

minus 4AC1113968

2A (13)



PN 1eprime

Cc d minusa

21113874 1113875 + Cs d minus dprime( 11138571113874 1113875 (14)

MN PN times e (15)


Pc emptyPN (16)

Mc emptyMN (17)



Aa3

+ Ba2

+ Ca + D 0 (18)




whenAs Asprime)








Input Data Figure 7


fy 415MPaAs 1000mm2

Asprime 1000mm2

dprime 80mmϕ 065

Solution











Start

cb = ((600 times d)(600 + fy))

ab = β times cb




eb = (MbPb)

e gt ebYes


Aa3 + Ba2 + Ca + D = 0 Aa2 + Ba + C = 0

No


cubicEquation









End













approximation HSU


400

320

400

As = 1000

PC

Asprime = 1000

80

120

Figure 7 Column C-4






B2 minus 4AC

radic2A)










Pc 1










Input Data Figure 9


fy 300MPaAs 3080mm2

Asprime 3080mm2

dprime 60mmϕ 065




primeabb



























(300 minus 085(20)) 871times 102 kN










(x-axis)

(y-a

xis)

bx = hy

hx = by

Muy

Mux

















Pc 1



5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN








1200

1140

400

As = 3080

PC

Asprime = 3080

60

540


As = 3080

PCAsprime = 3080

400 340140

1200






As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)






















eb Mb

Pb

(9)

Mb Ccb d minusab


(10)





prime fy)



Aa2

+ Ba + C 0(12)





a minus B plusmn

B2

minus 4AC1113968

2A (13)



PN 1eprime

Cc d minusa

21113874 1113875 + Cs d minus dprime( 11138571113874 1113875 (14)

MN PN times e (15)


Pc emptyPN (16)

Mc emptyMN (17)



Aa3

+ Ba2

+ Ca + D 0 (18)




whenAs Asprime)








Input Data Figure 7


fy 415MPaAs 1000mm2

Asprime 1000mm2

dprime 80mmϕ 065

Solution











Start

cb = ((600 times d)(600 + fy))

ab = β times cb




eb = (MbPb)

e gt ebYes


Aa3 + Ba2 + Ca + D = 0 Aa2 + Ba + C = 0

No


cubicEquation









End













approximation HSU


400

320

400

As = 1000

PC

Asprime = 1000

80

120

Figure 7 Column C-4






B2 minus 4AC

radic2A)










Pc 1










Input Data Figure 9


fy 300MPaAs 3080mm2

Asprime 3080mm2

dprime 60mmϕ 065




primeabb



























(300 minus 085(20)) 871times 102 kN










(x-axis)

(y-a

xis)

bx = hy

hx = by

Muy

Mux

















Pc 1



5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN








1200

1140

400

As = 3080

PC

Asprime = 3080

60

540


As = 3080

PCAsprime = 3080

400 340140

1200






As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)




























Start

cb = ((600 times d)(600 + fy))

ab = β times cb




eb = (MbPb)

e gt ebYes


Aa3 + Ba2 + Ca + D = 0 Aa2 + Ba + C = 0

No


cubicEquation









End













approximation HSU


400

320

400

As = 1000

PC

Asprime = 1000

80

120

Figure 7 Column C-4






B2 minus 4AC

radic2A)










Pc 1










Input Data Figure 9


fy 300MPaAs 3080mm2

Asprime 3080mm2

dprime 60mmϕ 065




primeabb



























(300 minus 085(20)) 871times 102 kN










(x-axis)

(y-a

xis)

bx = hy

hx = by

Muy

Mux

















Pc 1



5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN








1200

1140

400

As = 3080

PC

Asprime = 3080

60

540


As = 3080

PCAsprime = 3080

400 340140

1200






As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)
































approximation HSU


400

320

400

As = 1000

PC

Asprime = 1000

80

120

Figure 7 Column C-4






B2 minus 4AC

radic2A)










Pc 1










Input Data Figure 9


fy 300MPaAs 3080mm2

Asprime 3080mm2

dprime 60mmϕ 065




primeabb



























(300 minus 085(20)) 871times 102 kN










(x-axis)

(y-a

xis)

bx = hy

hx = by

Muy

Mux

















Pc 1



5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN








1200

1140

400

As = 3080

PC

Asprime = 3080

60

540


As = 3080

PCAsprime = 3080

400 340140

1200






As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)


























B2 minus 4AC

radic2A)










Pc 1










Input Data Figure 9


fy 300MPaAs 3080mm2

Asprime 3080mm2

dprime 60mmϕ 065




primeabb



























(300 minus 085(20)) 871times 102 kN










(x-axis)

(y-a

xis)

bx = hy

hx = by

Muy

Mux

















Pc 1



5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN








1200

1140

400

As = 3080

PC

Asprime = 3080

60

540


As = 3080

PCAsprime = 3080

400 340140

1200






As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)





























(300 minus 085(20)) 871times 102 kN










(x-axis)

(y-a

xis)

bx = hy

hx = by

Muy

Mux

















Pc 1



5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN








1200

1140

400

As = 3080

PC

Asprime = 3080

60

540


As = 3080

PCAsprime = 3080

400 340140

1200






As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)

































Pc 1



5150 kN Pc (1(14594) + (12091) minus (15150))

1993 kN








1200

1140

400

As = 3080

PC

Asprime = 3080

60

540


As = 3080

PCAsprime = 3080

400 340140

1200






As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)

























As (In2)(mm2)

fcprime (Ksi)(MPa)

fy (Ksi)(MPa)

dprime(In)(mm)

ex (in)(mm)

ey (in)(mm)








Experimentalresults




(1185) eblt ex (tension) 24 (107) 29 (132) 083


(compression) 60 (267) 59 (263) 101


0

1000

2000

3000

4000

5000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Load

capa

city

(kN

)




0

200

400

600

800

1000

C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Mom

ents

(KN

-M)









8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)


























8 Conclusion







Data Availability




References









0

500

1000

1500

2000

2500

3000


Mom

ents

(KN

-M)






















Research Article Mathematical Model for Analysis of Uniaxial ...

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