Lecture Chp-9&10 Columns

Lecture GoalsDefinitions for short columnsColumns

Analysis and Design of Short ColumnsGeneral InformationVertical Structural membersTransmits axial compressive loads with or without momenttransmit loads from the floor & roof to the foundationColumn:

Analysis and Design of Short ColumnsGeneral InformationColumn Types:TiedSpiralCompositeCombinationSteel pipe

Analysis and Design of Short ColumnsTie spacing h (except for seismic)tie support long bars (reduce buckling)ties provide negligible restraint to lateral expose of coreTied Columns - 95% of all columns in buildings are tied

Analysis and Design of Short ColumnsPitch = 1.375 in. to 3.375 in.spiral restrains lateral (Poissons effect)axial load delays failure (ductile) Spiral Columns

Analysis and Design of Short ColumnsElastic Behavior An elastic analysis using the transformed section method would be:For concentrated load, Puniform stress over sectionn = Es / EcAc = concrete areaAs = steel area

Analysis and Design of Short ColumnsElastic Behavior The change in concrete strain with respect to time will effect the concrete and steel stresses as follows:Concrete stressSteel stress

Analysis and Design of Short ColumnsElastic Behavior An elastic analysis does not work, because creep and shrinkage affect the acting concrete compression strain as follows:

Analysis and Design of Short ColumnsElastic Behavior Concrete creeps and shrinks, therefore we can not calculate the stresses in the steel and concrete due to acting loads using an elastic analysis.

Analysis and Design of Short ColumnsElastic Behavior Therefore, we are not able to calculate the real stresses in the reinforced concrete column under acting loads over time. As a result, an allowable stress design procedure using an elastic analysis was found to be unacceptable. Reinforced concrete columns have been designed by a strength method since the 1940s. Creep and shrinkage do not affect the strength of the member.Note:

Behavior, Nominal Capacity and Design under Concentric Axial loadsInitial Behavior up to Nominal Load - Tied and spiral columns. 1.

Behavior, Nominal Capacity and Design under Concentric Axial loads

Behavior, Nominal Capacity and Design under Concentric Axial loadsFactor due to less than ideal consolidation and curing conditions for column as compared to a cylinder. It is not related to Whitneys stress block.Let Ag = Gross Area = b*h Ast = area of long steel fc = concrete compressive strength fy = steel yield strength

Behavior, Nominal Capacity and Design under Concentric Axial loadsMaximum Nominal Capacity for Design Pn (max) 2.r = Reduction factor to account for accidents/bendingr = 0.80 ( tied )r = 0.85 ( spiral )ACI 10.3.6.3

Behavior, Nominal Capacity and Design under Concentric Axial loadsReinforcement Requirements (Longitudinal Steel Ast) 3.- ACI Code 10.9.1 requires Let

Behavior, Nominal Capacity and Design under Concentric Axial loads3.- Minimum # of Bars ACI Code 10.9.2 min. of 6 bars in circular arrangement w/min. spiral reinforcement.min. of 4 bars in rectangular arrangementmin. of 3 bars in triangular ties Reinforcement Requirements (Longitudinal Steel Ast)

Behavior, Nominal Capacity and Design under Concentric Axial loads3.ACI Code 7.10.5.1 Reinforcement Requirements (Lateral Ties) # 3 bar if longitudinal bar # 10 bar # 4 bar if longitudinal bar # 11 bar # 4 bar if longitudinal bars are bundledsize

Behavior, Nominal Capacity and Design under Concentric Axial loads3.Reinforcement Requirements (Lateral Ties) Vertical spacing: (ACI 7.10.5.2)

Behavior, Nominal Capacity and Design under Concentric Axial loadsArrangement Vertical spacing: (ACI 7.10.5.3)

Behavior, Nominal Capacity and Design under Concentric Axial loadsExamples of lateral ties.

Behavior, Nominal Capacity and Design under Concentric Axial loadsACI Code 7.10.4 Reinforcement Requirements (Spirals ) 3/8 dia. (3/8 f smooth bar, #3 bar dll or wll wire)sizeclear spacing between spirals 3 in.ACI 7.10.4.31 in.

Behavior, Nominal Capacity and Design under Concentric Axial loadsReinforcement Requirements (Spiral) Spiral Reinforcement Ratio, rs

Behavior, Nominal Capacity and Design under Concentric Axial loadsReinforcement Requirements (Spiral) ACI Eqn. 10-5where

Behavior, Nominal Capacity and Design under Concentric Axial loads4.Design for Concentric Axial Loads (a) Load CombinationGravity:Gravity + Wind:andetc.Check for tension

Behavior, Nominal Capacity and Design under Concentric Axial loads4.Design for Concentric Axial Loads (b) General Strength Requirementf = 0.65 for tied columnsf = 0.7 for spiral columnswhere,

Behavior, Nominal Capacity and Design under Concentric Axial loads4.Design for Concentric Axial Loads (c) Expression for Designdefined:

Behavior, Nominal Capacity and Design under Concentric Axial loadsor

Behavior, Nominal Capacity and Design under Concentric Axial loads* when rg is known or assumed:* when Ag is known or assumed:

Example: Design Tied Column for Concentric Axial Load

Example: Design Tied Column for Concentric Axial LoadDesign tied column for concentric axial loadPdl = 150 k; Pll = 300 k; Pw = 50 k fc = 4500 psi fy = 60 ksiDesign a square column aim for rg = 0.03. Select longitudinal transverse reinforcement.

Example: Design Tied Column for Concentric Axial LoadDetermine the loadingCheck the compression or tension in the column

Example: Design Tied Column for Concentric Axial LoadFor a square column r = 0.80 and f = 0.65 and r = 0.03

Example: Design Tied Column for Concentric Axial LoadFor a square column, As=rAg= 0.03(15.2 in.)2 =6.93 in2Use 8 #8 bars Ast = 8(0.79 in2) = 6.32 in2

Example: Design Tied Column for Concentric Axial LoadCheck P0

Example: Design Tied Column for Concentric Axial LoadUse #3 ties compute the spacing < 6 in. No cross-ties needed

Example: Design Tied Column for Concentric Axial LoadStirrup design Use #3 stirrups with 16 in. spacing in the column

Behavior under Combined Bending and Axial LoadsUsually moment is represented by axial load times eccentricity, i.e.

Behavior under Combined Bending and Axial LoadsInteraction Diagram Between Axial Load and Moment ( Failure Envelope )Concrete crushes before steel yieldsSteel yields before concrete crushesAny combination of P and M outside the envelope will cause failure.Note:

Behavior under Combined Bending and Axial LoadsAxial Load and Moment Interaction Diagram General

Behavior under Combined Bending and Axial LoadsResultant Forces action at Centroid( h/2 in this case )Moment about geometric center

Columns in Pure TensionSection is completely cracked (no concrete axial capacity)Uniform Strain

ColumnsStrength Reduction Factor, f (ACI Code 9.3.2)Axial tension, and axial tension with flexure. f = 0.9Axial compression and axial compression with flexure. Members with spiral reinforcement confirming to 10.9.3 f = 0.70Other reinforced members f = 0.65(a)(b)

ColumnsExcept for low values of axial compression, f may be increased as follows:when and reinforcement is symmetricand

ds = distance from extreme tension fiber to centroid of tension reinforcement.Then f may be increased linearly to 0.9 as fPn decreases from 0.10fc Ag to zero.

Column

ColumnsCommentary:Other sections:f may be increased linearly to 0.9 as the strain es increase in the tension steel. fPb

Design for Combined Bending and Axial Load (Short Column)Design - select cross-section and reinforcement to resist axial load and moment.

Design for Combined Bending and Axial Load (Short Column)Column TypesSpiral Column - more efficient for e/h < 0.1, but forming and spiral expensiveTied Column - Bars in four faces used when e/h < 0.2 and for biaxial bending

1)2)

General ProcedureThe interaction diagram for a column is constructed using a series of values for Pn and Mn. The plot shows the outside envelope of the problem.

General Procedure for Construction of IDCompute P0 and determine maximum Pn in compressionSelect a c value (multiple values)Calculate the stress in the steel components.Calculate the forces in the steel and concrete,Cc, Cs1 and Ts.Determine Pn value.Compute the Mn about the center.Compute moment arm,e = Mn / Pn.

General Procedure for Construction of IDRepeat with series of c values (10) to obtain a series of values.Obtain the maximum tension value.Plot Pn verse Mn.Determine fPn and fMn.Find the maximum compression level.Find the f will vary linearly from 0.65 to 0.9 for the strain values The tension component will be f = 0.9

Example: Axial Load vs. Moment Interaction DiagramConsider an square column (20 in x 20 in.) with 8 #10 (r = 0.0254) and fc = 4 ksi and fy = 60 ksi. Draw the interaction diagram.

Example: Axial Load vs. Moment Interaction DiagramGiven 8 # 10 (1.27 in2) and fc = 4 ksi and fy = 60 ksi

Example: Axial Load vs. Moment Interaction DiagramGiven 8 # 10 (1.27 in2) and fc = 4 ksi and fy = 60 ksi[ Point 1 ]

Example: Axial Load vs. Moment Interaction DiagramDetermine where the balance point, cb.

Example: Axial Load vs. Moment Interaction DiagramDetermine where the balance point, cb. Using similar triangles, where d = 20 in. 2.5 in. = 17.5 in., one can find cb

Example: Axial Load vs. Moment Interaction DiagramDetermine the strain of the steel

Example: Axial Load vs. Moment Interaction DiagramDetermine the stress in the steel

Example: Axial Load vs. Moment Interaction DiagramCompute the forces in the column

Example: Axial Load vs. Moment Interaction DiagramCompute the forces in the column

Example: Axial Load vs. Moment Interaction DiagramCompute the moment about the center

Example: Axial Load vs. Moment Interaction DiagramA single point from interaction diagram, (585.6 k, 556.9 k-ft). The eccentricity of the point is defined as

[ Point 2 ]

Example: Axial Load vs. Moment Interaction DiagramNow select a series of additional points by selecting values of c. Select c = 17.5 in. Determine the strain of the steel. (c is at the location of the tension steel)

Example: Axial Load vs. Moment Interaction DiagramCompute the forces in the column

Example: Axial Load vs. Moment Interaction DiagramCompute the forces in the column

Example: Axial Load vs. Moment Interaction DiagramCompute the moment about the center

Example: Axial Load vs. Moment Interaction DiagramA single point from interaction diagram, (1314 k, 351.1 k-ft). The eccentricity of the point is defined as

[ Point 3 ]

Example: Axial Load vs. Moment Interaction DiagramSelect c = 6 in. Determine the strain of the steel, c =6 in.

Example: Axial Load vs. Moment Interaction DiagramCompute the forces in the column

Example: Axial Load vs. Moment Interaction DiagramCompute the forces in the column

Example: Axial Load vs. Moment Interaction DiagramCompute the moment about the center

Example: Axial Load Vs. Moment Interaction DiagramA single point from interaction diagram, (151 k, 471 k-ft). The eccentricity of the point is defined as

[ Point 4 ]

Example: Axial Load vs. Moment Interaction DiagramSelect point of straight tension. The maximum tension in the column is

[ Point 5 ]

Example: Axial Load vs. Moment Interaction Diagram

Example: Axial Load vs. Moment Interaction DiagramUse a series of c values to obtain the Pn verses Mn.

Chart1

-610

13.466

37.6081052632

60.781

83.1231428571

104.7478181818

125.7486086957

151.2835

185.99

219.1383076923

250.9015555556

281.4281428571

310.846

339.266

366.7847096774

393.486625

417.0214545455

439.3495294118

461.2274285714

482.6926666667

503.7787027027

524.5154736842

544.9298461538

556.41

576.2497560976

585.0513011583

603.4528571429

635.6448837209

667.0304545455

697.6633333333

727.5926086957

756.8631914894

785.51625

813.5895918367

841.118

868.1335294118

894.6657692308

920.7420754717

946.3877777778

971.6263636364

996.4796428571

1020.9678947368

1045.11

1068.923559322

1092.425

1115.6296721312

1138.5519354839

1161.2052380952

1183.6021875

1205.7546153846

1227.6736363636

1249.3697014925

1270.8526470588

1292.1317391304

1313.2157142857

1321.1588169014

1341.8768333333

1362.4231232877

1382.8046486486

1403.028

1423.0994210526

1443.0248311688

1462.8098461538

1482.4597974684

1501.97975

1521.3745185185

1540.6486829268

1559.8066024096

1578.8524285714

1597.7901176471

1616.6234418605

1635.356

1653.9912272727

1672.5324044944

1690.9826666667

1709.345010989

1727.6223043478

1745.8172903226

1763.9325957447

1781.9707368421

1799.934125

1817.825072165

1835.6457959184

1853.3984242424

1867.9062021639

M (k-ft)

P (k)

Column Analysis

Sheet1

Concrete Column Analysis

fy60ksiEs29000ksi

fc4ksir1

400f0.7

Ast13.81b10.85

Ast22.54

Ast33.81

h20in

b20in

Strain of the Steel

c (in)S1S2S3CcCs1Cs2TPMMPe

2-0.00075-0.012-0.02325115.6-21.75-60-60-82.8675-152.4-228.6-348.26752150.73375179.2278125-348.2675-6.1755224073106.6882118339-207.3117798883

2.25-0.0003333333-0.0103333333-0.0203333333130.05-9.6666666667-60-60-36.83-152.4-228.6-287.782614.4146875217.867890625-287.78-9.0847685298141.2203326939-186.53683765

2.50-0.009-0.018144.50-60-60-12.954-152.4-228.6-249.4542908.81375242.4011458333-249.454-11.6607220169165.2515877464-170.0597141485

2.750.0002727273-0.0079090909-0.0160909091158.957.9090909091-60-6017.1796363636-152.4-228.6-204.87036363643247.0744602273270.5895383523-204.8703636364-15.8494103422195.0242795384-147.6579446133

30.0005-0.007-0.0145173.414.5-60-6042.291-152.4-228.6-165.3093544.5975295.383125-165.309-21.4422535978223.1190096157-124.8669177041

3.250.0006923077-0.0062307692-0.0131538462187.8520.0769230769-60-6063.5390769231-152.4-228.6-129.61092307693810.0752644231317.5062720353-129.6109230769-29.3962512879249.7473989557-101.950713311

3.50.0008571429-0.0055714286-0.012202.324.8571428571-60-6081.7517142857-152.4-228.6-96.94828571434049.7166071429337.4763839286-96.9482857143-41.7719259016275.1007082562-79.0293583028

3.750.001-0.005-0.011216.7529-60-6097.536-152.4-228.6-66.7144268.0746875355.672890625-66.714-63.9756975672299.3432854905-56.1481869285

40.001125-0.0045-0.010125231.232.625-60-60111.34725-152.4-228.6-38.452754468.564375372.3803645833-38.45275-116.2092275585322.6131601938-33.3136877653

4.250.0012352941-0.0040588235-0.0093529412245.6535.8235294118-60-60123.5336470588-152.4-228.6-11.81635294124653.7970404412387.8164200368-11.8163529412-393.8437742685345.0250242995-10.5125447248

0-6100-549

4.50.0013333333-0.0036666667-0.0086666667260.138.6666666667-60-60134.366-152.4-228.613.4664825.80375402.150312513.466358.3695046785357.196844655411.9607334885

4.750.0014210526-0.0033157895-0.0080526316274.5541.2105263158-60-60144.0581052632-152.4-228.637.60810526324986.1879769737415.515664747837.6081052632132.582802087360.29068602232.609721353

50.0015-0.003-0.007528943.5-60-60152.781-152.4-228.660.7815136.2325428.01937560.78184.5039156973362.453922572151.4703612841

5.250.0015714286-0.0027142857-0.007303.4545.5714285714-60-60160.6731428571-152.4-228.683.12314285715276.9757589286439.747979910783.123142857163.4838334734363.789102033868.7650538028

5.50.0016363636-0.0024545455-0.0065454545317.947.4545454545-60-60167.8478181818-152.4-228.6104.74781818185409.2673863636450.772282197104.747818181851.6408597359364.379817552484.6724441265

5.750.0016956522-0.0022173913-0.0061304348332.3549.1739130435-60-60174.3986086957-152.4-228.6125.74860869575533.8092527174461.1507710598125.748608695744.0069223041364.295259576699.3376243108

60.00175-0.002-0.00575346.850.75-58-60180.4035-147.32-228.6151.28355651.18625470.9321875151.283537.3549412196361.5002328608116.1292897993

6.250.0018-0.0018-0.0054361.2552.2-52.2-60185.928-132.588-228.6185.995761.8896875480.1574739583185.9930.9795671138354.0002990537137.1227549125

6.50.0018461538-0.0016153846-0.0050769231375.753.5384615385-46.8461538462-60191.0275384615-118.9892307692-228.6219.13830769235866.3352884615488.8612740385219.138307692326.7700127387346.2379433817155.2055787621

6.750.0018888889-0.0014444444-0.0047777778390.1554.7777777778-41.8888888889-60195.7493333333-106.3977777778-228.6250.90155555565964.8771875497.0730989583250.901555555623.7737752334347.9511692708175.6310888889

70.0019285714-0.0012857143-0.0045404.655.9285714286-37.2857142857-60200.1338571429-94.7057142857-228.6281.42814285716057.8189285714504.8182440476281.428142857121.5252777035353.3727708333196.9997

7.250.0019655172-0.001137931-0.0042413793419.0557-33-60204.216-83.82-228.6310.8466145.4221875512.118515625310.84619.769989601358.4829609375217.5922

7.50.002-0.001-0.004433.558-29-60208.026-73.66-228.6339.2666227.91375518.9928125339.26618.3570229554363.29496875237.4862

7.750.0020322581-0.0008709677-0.0037741935447.9558.935483871-25.2580645161-60211.5901935484-64.155483871-228.6366.78470967746305.4911391129525.4575949261366.784709677417.1912595393367.8203164483256.7492967742

80.0020625-0.00075-0.0035625462.459.8125-21.75-60214.931625-55.245-228.6393.4866256378.3271875531.527265625393.48662516.2097687247372.0690859375275.4406375

8.250.0020909091-0.0006363636-0.0033636364476.8560-18.4545454545-60215.646-46.8745454545-228.6417.02145454556428.3896875535.699140625417.021454545515.4150095095374.9893984375291.9150181818

8.50.0021176471-0.0005294118-0.0031764706491.360-15.3529411765-60215.646-38.9964705882-228.6439.34952941186470.02375539.1686458333439.349529411814.7263700468377.4180520833307.5446705882

8.750.0021428571-0.0004285714-0.003505.7560-12.4285714286-60215.646-31.5685714286-228.6461.22742857146508.5871875542.382265625461.227428571414.1114486787379.6675859375322.8592

90.0021666667-0.0003333333-0.0028333333520.260-9.6666666667-60215.646-24.5533333333-228.6482.69266666676544.08545.34482.692666666713.5574464911381.738337.8848666667

9.250.0021891892-0.0002432432-0.0026756757534.6560-7.0540540541-60215.646-17.9172972973-228.6503.77870270276576.5021875548.0418489583503.778702702713.0543473795383.6292942708352.6450918919

9.50.0022105263-0.0001578947-0.0025263158549.160-4.5789473684-60215.646-11.6305263158-228.6524.51547368426605.85375550.4878125524.515473684212.5942018519385.34146875367.1608315789

9.750.0022307692-0.0000769231-0.0023846154563.5560-2.2307692308-60215.646-5.6661538462-228.6544.92984615386632.1346875552.677890625544.929846153812.1706211071386.8745234375381.4508923077

100.002250-0.00225578600-60215.646-8.636-228.6556.416655.345554.6120833333556.4111.9612246365388.2284583333389.487

10.250.00226829270.0000731707-0.0021219512592.45602.1219512195-60215.646-3.2462439024-228.6576.24975609766675.4846875556.290390625576.249756097611.5843601092389.4032734375403.3748292683

10.360.00227606180.0001042471-0.0020675676598.808603.0231660232-59.9594594595215.646-0.9571583012-228.4455405405585.05130115836682.2149300541556.8512441712585.051301158311.421588016389.7958709198409.5359108108

10.50.00228571430.0001428571-0.002606.9604.1428571429-58215.6461.8868571429-220.98603.45285714296635.40375552.9503125603.452857142910.9957284508387.06521875422.417

10.750.00230232560.0002093023-0.0018837209621.35606.0697674419-54.6279069767215.6466.7812093023-208.1323255814635.64488372096553.0446293605546.0870524467635.644883720910.3092855731382.2609367127444.9514186047

110.00231818180.0002727273-0.0017727273635.8607.9090909091-51.4090909091215.64611.4530909091-195.8686363636667.03045454556471.9947727273539.3328977273667.03045454559.7026975734377.5330284091466.9213181818

11.250.00233333330.0003333333-0.0016666667650.25609.6666666667-48.3333333333215.64615.9173333333-184.15697.66333333336391.9621875532.663515625697.66333333339.1619580421372.8644609375488.3643333333

11.50.00234782610.0003913043-0.0015652174664.76011.347826087-45.3913043478215.64620.1874782609-172.9408695652727.59260869576312.6802717391526.0566893116727.59260869578.6761192957368.2396825181509.314826087

11.750.00236170210.0004468085-0.0014680851679.156012.9574468085-42.5744680851215.64624.2759148936-162.2087234043756.86319148946233.9051130319519.4920927527756.86319148948.2365018977363.6444649269529.8042340426

120.0023750.0005-0.001375693.66014.5-39.875215.64628.194-151.92375785.516256155.413125512.95109375785.516257.8361372219359.065765625549.861375

12.250.00238775510.0005510204-0.0012857143708.056015.9795918367-37.2857142857215.64631.9521632653-142.0585714286813.58959183676076.9989732143506.4165811012813.58959183677.4693666613354.4916067708569.5127142857

12.50.00240.0006-0.0012722.56017.4-34.8215.64635.56-132.588841.1185998.47375499.8728125841.1187.1315484272349.91096875588.7826

12.750.00241176470.0006470588-0.0011176471736.956018.7647058824-32.4117647059215.64639.0263529412-123.4888235294868.13352941185919.6633639706493.3052803309868.13352941186.818839687345.3136962316607.6934705882

130.00242307690.0006923077-0.0010384615751.46020.0769230769-30.1153846154215.64642.3593846154-114.7396153846894.66576923085840.4071153846486.7005929487894.66576923086.5280323851340.6904150641626.2660384615

13.250.00243396230.0007358491-0.0009622642765.856021.3396226415-27.9056603774215.64645.5666415094-106.3205660377920.74207547175760.556432783480.0463693986920.74207547176.2564279251336.032458579644.5194528302

13.50.00244444440.0007777778-0.0008888889780.36022.5555555556-25.7777777778215.64648.6551111111-98.2133333333946.38777777785679.97375473.3311458333946.38777777786.0017403895331.3318020833662.4714444444

13.750.00245454550.0008181818-0.0008181818794.756023.7272727273-23.7272727273215.64651.6312727273-90.4009090909971.62636363645598.5315056818466.5442921402971.62636363645.7620209941326.5810044981680.1384545455

140.00246428570.0008571429-0.00075809.26024.8571428571-21.75215.64654.5011428571-82.8675996.47964285715516.11125459.6759375996.47964285715.5355985338321.77315625697.53575

14.250.00247368420.0008947368-0.0006842105823.656025.9473684211-19.8421052632215.64657.2703157895-75.59842105261020.96789473685432.6028453947452.71690378291020.96789473685.3210320064316.901832648714.6775263158

14.50.00248275860.0009310345-0.0006206897838.16027-18215.64659.944-68.581045.115347.90375445.65864583331045.115.1170726048311.9610520833731.577

14.750.00249152540.0009661017-0.000559322852.556028.0169491525-16.2203389831215.64662.5270508475-61.79949152541068.9235593225261.9183739407438.49319782841068.9235593224.9226329872306.9452384799748.2464915254

150.00250.001-0.00058676029-14.5215.64665.024-55.2451092.4255174.5575431.2131251092.4254.7367622491301.8491875764.6975

15.250.00250819670.0010327869-0.000442623881.456029.9508196721-12.8360655738215.64667.4390819672-48.90540983611115.62967213125085.7377612705423.81148010591115.62967213124.5586254008296.6680360741780.9407704918

15.50.0025161290.0010645161-0.0003870968895.96030.8709677419-11.2258064516215.64669.7762580645-42.77032258061138.55193548394995.3811693548416.28176411291138.55193548394.3874864323291.397234879796.9863548387

15.750.00252380950.0010952381-0.0003333333910.356031.7619047619-9.6666666667215.64672.0392380952-36.831161.20523809524903.4146875408.6178906251161.20523809524.2226942548286.0325234375812.8436666667

160.002531250.001125-0.00028125924.86032.625-8.15625215.64674.2315-31.07531251183.60218754809.76984375400.81415364581183.60218754.0636709652280.5699075521828.52153125

16.250.00253846150.0011538462-0.0002307692939.256033.4615384615-6.6923076923215.64676.3563076923-25.49769230771205.75461538464714.3823798077392.86519831731205.75461538463.909901998275.0056388221844.0282307692

16.50.00254545450.0011818182-0.0001818182953.76034.2727272727-5.2727272727215.64678.4167272727-20.08909090911227.67363636364617.1919318182384.76599431821227.67363636363.7609278191269.3361960227859.3715454545

16.750.00255223880.0012089552-0.0001343284968.156035.0597014925-3.8955223881215.64680.415641791-14.84194029851249.36970149254518.1417397388376.51181164491249.36970149253.6163368892263.5582681514874.5587910448

170.00255882350.0012352941-0.0000882353982.66035.8235294118-2.5588235294215.64682.3557647059-9.74911764711270.85264705884417.1783823529368.09819852941270.85264705883.4757596741257.6687389706889.5968529412

17.250.00256521740.0012608696-0.0000434783997.056036.5652173913-1.2608695652215.64684.2396521739-4.80391304351292.13173913044314.2515353261359.52096127721292.13173913043.3388635266251.664672894904.4922173913

17.50.00257142860.001285714301011.56037.28571428570215.64686.069714285701313.21571428574209.31375350.77614583331313.21571428573.2053482944245.5433020833919.251

17.750.00257746480.00130985920.00004225351025.956037.9859154931.2253521127215.64687.8482253521-8.28540845071321.15881690144199.4752508803349.95627090671321.15881690143.1786301519244.9693896347924.811171831

180.00258333330.00133333330.00008333331040.46038.66666666672.4166666667215.64689.5773333333-3.74651341.87683333334090.38375340.86531251341.87683333333.0482557329238.60571875939.3137833333

18.250.00258904110.00135616440.00012328771054.856039.32876712333.5753424658215.64691.25906849320.66805479451362.42312328773979.1542765411331.59618971181362.42312328772.9206449953232.1173327982953.6961863014

18.50.00259459460.00137837840.00016216221069.36039.9729729734.7027027027215.64692.89535135144.96329729731382.80464864863865.7490202703322.14575168921382.80464864862.7955857858225.5020261824967.9632540541

18.750.00260.00140.00021083.756040.65.8215.64694.4889.1441403.0283750.1321875312.5110156251403.0282.6728847803218.7577109375982.1196

190.00260526320.00142105260.00023684211098.26041.21052631586.8684210526215.64696.038736842113.21468421051423.09942105263632.269868421302.68915570181423.09942105262.5523655021211.8824089912996.1695947368

19.250.00261038960.00144155840.00027272731112.656041.80519480527.9090909091215.64697.549194805217.17963636361443.02483116883512.1299147727292.67749289771443.02483116882.4338665828204.87424502841010.1173818182

19.50.00261538460.00146153850.00030769231127.16042.38461538468.9230769231215.64699.020923076921.04292307691462.80984615383389.6818269231282.47348557691462.80984615382.3172402318197.73143990381023.9668923077

19.750.00262025320.00148101270.00034177221141.556042.94936708869.9113924051215.646100.455392405124.80840506331482.45979746843264.8966495253272.07472079381482.45979746842.2023508867190.45230455561037.7218582278

200.0026250.00150.00037511566043.510.875215.646101.85428.479751501.979753137.746875261.478906251501.979752.0890740205183.0352343751051.385825

20.250.00262962960.00151851850.00040740741170.456044.03703703711.8148148148215.646103.218074074132.06044444441521.37451851853008.2063541667250.68386284721521.37451851851.9772950825175.47870399311064.962162963

20.50.00263414630.00153658540.00043902441184.96044.560975609812.7317073171215.646104.548878048835.5538048781540.64868292682876.2502134146239.68751778461540.64868292681.8669085595167.78126244921078.4540780488

20.750.00263855420.00155421690.00046987951199.356045.072289156613.6265060241215.646105.847614457838.96298795181559.80660240962741.8547778614228.48789815511559.80660240961.7578171381159.94152870861084

210.00264285710.00157142860.00051213.86045.571428571414.5215.646107.115428571442.2911578.85242857142604.9975217.0831251578.85242857141.649930958151.95818751084

21.250.00264705880.00158823530.00052941181228.256046.058823529415.3529411765215.646108.353411764745.54070588241597.79011764712465.6568933824205.47140778191597.79011764711.5431669442143.82998544731084

21.50.00265116280.00160465120.00055813951242.76046.534883720916.1860465116215.646109.562604651248.71483720931616.62344186052323.8124709302193.65103924421616.62344186051.4374482089135.55572747091084

21.750.00265517240.00162068970.00058620691257.15604717215.646110.74451.8161635.3562179.4446875181.6203906251635.3561.3327035138127.13427343751084

220.00265909090.00163636360.00061363641271.66047.454545454517.7954545455215.646111.898545454554.84668181821653.99122727272032.5348863636169.3779071971653.99122727271.2288667877118.56453503791084

22.250.00266292130.00165168540.00064044941286.056047.898876404518.5730337079215.646113.027146067457.8092584271672.53240449441883.0652492978156.92210410811672.53240449441.1258766911109.84547287571084

22.50.00266666670.00166666670.00066666671300.56048.333333333319.3333333333215.646114.130666666760.7061690.98266666671731.01875144.25156251690.98266666671.0236762234100.976093751084

22.750.00267032970.00168131870.00069230771314.956048.758241758220.0769230769215.646115.209934065963.53907692311709.3450109891576.3791105769131.36492588141709.3450109890.922212368291.9554481171084

230.0026739130.00169565220.00071739131329.46049.173913043520.8043478261215.646116.265739130466.31056521741727.62230434781419.1307608696118.26089673911727.62230434780.821435771882.78262771741084

23.250.00267741940.00170967740.00074193551343.856049.580645161321.5161290323215.646117.298838709769.02245161291745.81729032261259.2588004032104.93823336691745.81729032260.721300451973.45676335691084

23.50.00268085110.00172340430.00076595741358.36049.978723404322.2127659574215.646118.309957446871.67663829791763.93259574471096.74896276691.39574689721763.93259574470.62176353363.9770228281084

23.750.00268421050.00173684210.00078947371372.756050.368421052622.8947368421215.646119.299789473774.27494736841781.9707368421931.587582236877.63229851971781.97073684210.522785005954.34260896381084

240.00268750.001750.00081251387.26050.7523.5625215.646120.26976.8191251799.934125763.761562563.6467968751799.9341250.424327508444.55275781251084

24.250.00269072160.00176288660.00083505151401.656051.123711340224.2164948454215.646121.218226804179.31084536081817.825072165593.258347293849.43819560781817.8250721650.326356125434.60673692551084

24.50.00269387760.00177551020.00085714291416.16051.489795918424.8571428571215.646122.148081632781.75171428571835.6457959184420.065892857135.00549107141835.64579591840.228838207124.503843751084

24.750.00269696970.00178787880.00087878791430.556051.848484848525.4848484848215.646123.059151515284.14327272731853.3984242424244.172642045520.34772017051853.39842424240.131743201514.24340411931084

24.9550.0026994590.00179783610.00089621321442.3996052.137247044725.9901823282215.646123.792607493586.06859467041867.906202163997.9170458478.15975382061867.90620216390.0524207515.71182767441084

Sheet1

M (k-ft)

P (k)

Column Analysis

Sheet2

fMn (k-ft)

fPn (k)

Column Analysis

Sheet3

Factored

Unfactored

Mn (k-ft)

Pn (k)

Column Analysis

Example: Axial Load vs. Moment Interaction DiagramCb

Chart2

-549

11.9607334885

32.609721353

51.4703612841

68.7650538028

84.6724441265

99.3376243108

116.1292897993

137.1227549125

155.2055787621

175.6310888889

196.9997

217.5922

237.4862

256.7492967742

275.4406375

291.9150181818

307.5446705882

322.8592

337.8848666667

352.6450918919

367.1608315789

381.4508923077

389.487

403.3748292683

409.5359108108

422.417

444.9514186047

466.9213181818

488.3643333333

509.314826087

529.8042340426

549.861375

569.5127142857

588.7826

607.6934705882

626.2660384615

644.5194528302

662.4714444444

680.1384545455

697.53575

714.6775263158

731.577

748.2464915254

764.6975

780.9407704918

796.9863548387

812.8436666667

828.52153125

844.0282307692

859.3715454545

874.5587910448

889.5968529412

904.4922173913

919.251

924.811171831

939.3137833333

953.6961863014

967.9632540541

982.1196

996.1695947368

1010.1173818182

1023.9668923077

1037.7218582278

1051.385825

1064.962162963

1078.4540780488

1084

1084

1084

1084

1084

1084

1084

1084

1084

1084

1084

1084

1084

1084

1084

1084

1084

1084

fMn (k-ft)

fPn (k)

Column Analysis

Sheet1

Concrete Column Analysis

fy60ksiEs29000ksi

fc4ksir1

400f0.7

Ast13.81b10.85

Ast22.54

Ast33.81

h20in

b20in

Strain of the Steel

c (in)S1S2S3CcCs1Cs2TPMMPe

2-0.00075-0.012-0.02325115.6-21.75-60-60-82.8675-152.4-228.6-348.26752150.73375179.2278125-348.2675-6.1755224073106.6882118339-207.3117798883

2.25-0.0003333333-0.0103333333-0.0203333333130.05-9.6666666667-60-60-36.83-152.4-228.6-287.782614.4146875217.867890625-287.78-9.0847685298141.2203326939-186.53683765

2.50-0.009-0.018144.50-60-60-12.954-152.4-228.6-249.4542908.81375242.4011458333-249.454-11.6607220169165.2515877464-170.0597141485

2.750.0002727273-0.0079090909-0.0160909091158.957.9090909091-60-6017.1796363636-152.4-228.6-204.87036363643247.0744602273270.5895383523-204.8703636364-15.8494103422195.0242795384-147.6579446133

30.0005-0.007-0.0145173.414.5-60-6042.291-152.4-228.6-165.3093544.5975295.383125-165.309-21.4422535978223.1190096157-124.8669177041

3.250.0006923077-0.0062307692-0.0131538462187.8520.0769230769-60-6063.5390769231-152.4-228.6-129.61092307693810.0752644231317.5062720353-129.6109230769-29.3962512879249.7473989557-101.950713311

3.50.0008571429-0.0055714286-0.012202.324.8571428571-60-6081.7517142857-152.4-228.6-96.94828571434049.7166071429337.4763839286-96.9482857143-41.7719259016275.1007082562-79.0293583028

3.750.001-0.005-0.011216.7529-60-6097.536-152.4-228.6-66.7144268.0746875355.672890625-66.714-63.9756975672299.3432854905-56.1481869285

40.001125-0.0045-0.010125231.232.625-60-60111.34725-152.4-228.6-38.452754468.564375372.3803645833-38.45275-116.2092275585322.6131601938-33.3136877653

4.250.0012352941-0.0040588235-0.0093529412245.6535.8235294118-60-60123.5336470588-152.4-228.6-11.81635294124653.7970404412387.8164200368-11.8163529412-393.8437742685345.0250242995-10.5125447248

0-6100-549

4.50.0013333333-0.0036666667-0.0086666667260.138.6666666667-60-60134.366-152.4-228.613.4664825.80375402.150312513.466358.3695046785357.196844655411.9607334885

4.750.0014210526-0.0033157895-0.0080526316274.5541.2105263158-60-60144.0581052632-152.4-228.637.60810526324986.1879769737415.515664747837.6081052632132.582802087360.29068602232.609721353

50.0015-0.003-0.007528943.5-60-60152.781-152.4-228.660.7815136.2325428.01937560.78184.5039156973362.453922572151.4703612841

5.250.0015714286-0.0027142857-0.007303.4545.5714285714-60-60160.6731428571-152.4-228.683.12314285715276.9757589286439.747979910783.123142857163.4838334734363.789102033868.7650538028

5.50.0016363636-0.0024545455-0.0065454545317.947.4545454545-60-60167.8478181818-152.4-228.6104.74781818185409.2673863636450.772282197104.747818181851.6408597359364.379817552484.6724441265

5.750.0016956522-0.0022173913-0.0061304348332.3549.1739130435-60-60174.3986086957-152.4-228.6125.74860869575533.8092527174461.1507710598125.748608695744.0069223041364.295259576699.3376243108

60.00175-0.002-0.00575346.850.75-58-60180.4035-147.32-228.6151.28355651.18625470.9321875151.283537.3549412196361.5002328608116.1292897993

6.250.0018-0.0018-0.0054361.2552.2-52.2-60185.928-132.588-228.6185.995761.8896875480.1574739583185.9930.9795671138354.0002990537137.1227549125

6.50.0018461538-0.0016153846-0.0050769231375.753.5384615385-46.8461538462-60191.0275384615-118.9892307692-228.6219.13830769235866.3352884615488.8612740385219.138307692326.7700127387346.2379433817155.2055787621

6.750.0018888889-0.0014444444-0.0047777778390.1554.7777777778-41.8888888889-60195.7493333333-106.3977777778-228.6250.90155555565964.8771875497.0730989583250.901555555623.7737752334347.9511692708175.6310888889

70.0019285714-0.0012857143-0.0045404.655.9285714286-37.2857142857-60200.1338571429-94.7057142857-228.6281.42814285716057.8189285714504.8182440476281.428142857121.5252777035353.3727708333196.9997

7.250.0019655172-0.001137931-0.0042413793419.0557-33-60204.216-83.82-228.6310.8466145.4221875512.118515625310.84619.769989601358.4829609375217.5922

7.50.002-0.001-0.004433.558-29-60208.026-73.66-228.6339.2666227.91375518.9928125339.26618.3570229554363.29496875237.4862

7.750.0020322581-0.0008709677-0.0037741935447.9558.935483871-25.2580645161-60211.5901935484-64.155483871-228.6366.78470967746305.4911391129525.4575949261366.784709677417.1912595393367.8203164483256.7492967742

80.0020625-0.00075-0.0035625462.459.8125-21.75-60214.931625-55.245-228.6393.4866256378.3271875531.527265625393.48662516.2097687247372.0690859375275.4406375

8.250.0020909091-0.0006363636-0.0033636364476.8560-18.4545454545-60215.646-46.8745454545-228.6417.02145454556428.3896875535.699140625417.021454545515.4150095095374.9893984375291.9150181818

8.50.0021176471-0.0005294118-0.0031764706491.360-15.3529411765-60215.646-38.9964705882-228.6439.34952941186470.02375539.1686458333439.349529411814.7263700468377.4180520833307.5446705882

8.750.0021428571-0.0004285714-0.003505.7560-12.4285714286-60215.646-31.5685714286-228.6461.22742857146508.5871875542.382265625461.227428571414.1114486787379.6675859375322.8592

90.0021666667-0.0003333333-0.0028333333520.260-9.6666666667-60215.646-24.5533333333-228.6482.69266666676544.08545.34482.692666666713.5574464911381.738337.8848666667

9.250.0021891892-0.0002432432-0.0026756757534.6560-7.0540540541-60215.646-17.9172972973-228.6503.77870270276576.5021875548.0418489583503.778702702713.0543473795383.6292942708352.6450918919

9.50.0022105263-0.0001578947-0.0025263158549.160-4.5789473684-60215.646-11.6305263158-228.6524.51547368426605.85375550.4878125524.515473684212.5942018519385.34146875367.1608315789

9.750.0022307692-0.0000769231-0.0023846154563.5560-2.2307692308-60215.646-5.6661538462-228.6544.92984615386632.1346875552.677890625544.929846153812.1706211071386.8745234375381.4508923077

100.002250-0.00225578600-60215.646-8.636-228.6556.416655.345554.6120833333556.4111.9612246365388.2284583333389.487

10.250.00226829270.0000731707-0.0021219512592.45602.1219512195-60215.646-3.2462439024-228.6576.24975609766675.4846875556.290390625576.249756097611.5843601092389.4032734375403.3748292683

10.360.00227606180.0001042471-0.0020675676598.808603.0231660232-59.9594594595215.646-0.9571583012-228.4455405405585.05130115836682.2149300541556.8512441712585.051301158311.421588016389.7958709198409.5359108108

10.50.00228571430.0001428571-0.002606.9604.1428571429-58215.6461.8868571429-220.98603.45285714296635.40375552.9503125603.452857142910.9957284508387.06521875422.417

10.750.00230232560.0002093023-0.0018837209621.35606.0697674419-54.6279069767215.6466.7812093023-208.1323255814635.64488372096553.0446293605546.0870524467635.644883720910.3092855731382.2609367127444.9514186047

110.00231818180.0002727273-0.0017727273635.8607.9090909091-51.4090909091215.64611.4530909091-195.8686363636667.03045454556471.9947727273539.3328977273667.03045454559.7026975734377.5330284091466.9213181818

11.250.00233333330.0003333333-0.0016666667650.25609.6666666667-48.3333333333215.64615.9173333333-184.15697.66333333336391.9621875532.663515625697.66333333339.1619580421372.8644609375488.3643333333

11.50.00234782610.0003913043-0.0015652174664.76011.347826087-45.3913043478215.64620.1874782609-172.9408695652727.59260869576312.6802717391526.0566893116727.59260869578.6761192957368.2396825181509.314826087

11.750.00236170210.0004468085-0.0014680851679.156012.9574468085-42.5744680851215.64624.2759148936-162.2087234043756.86319148946233.9051130319519.4920927527756.86319148948.2365018977363.6444649269529.8042340426

120.0023750.0005-0.001375693.66014.5-39.875215.64628.194-151.92375785.516256155.413125512.95109375785.516257.8361372219359.065765625549.861375

12.250.00238775510.0005510204-0.0012857143708.056015.9795918367-37.2857142857215.64631.9521632653-142.0585714286813.58959183676076.9989732143506.4165811012813.58959183677.4693666613354.4916067708569.5127142857

12.50.00240.0006-0.0012722.56017.4-34.8215.64635.56-132.588841.1185998.47375499.8728125841.1187.1315484272349.91096875588.7826

12.750.00241176470.0006470588-0.0011176471736.956018.7647058824-32.4117647059215.64639.0263529412-123.4888235294868.13352941185919.6633639706493.3052803309868.13352941186.818839687345.3136962316607.6934705882

130.00242307690.0006923077-0.0010384615751.46020.0769230769-30.1153846154215.64642.3593846154-114.7396153846894.66576923085840.4071153846486.7005929487894.66576923086.5280323851340.6904150641626.2660384615

13.250.00243396230.0007358491-0.0009622642765.856021.3396226415-27.9056603774215.64645.5666415094-106.3205660377920.74207547175760.556432783480.0463693986920.74207547176.2564279251336.032458579644.5194528302

13.50.00244444440.0007777778-0.0008888889780.36022.5555555556-25.7777777778215.64648.6551111111-98.2133333333946.38777777785679.97375473.3311458333946.38777777786.0017403895331.3318020833662.4714444444

13.750.00245454550.0008181818-0.0008181818794.756023.7272727273-23.7272727273215.64651.6312727273-90.4009090909971.62636363645598.5315056818466.5442921402971.62636363645.7620209941326.5810044981680.1384545455

140.00246428570.0008571429-0.00075809.26024.8571428571-21.75215.64654.5011428571-82.8675996.47964285715516.11125459.6759375996.47964285715.5355985338321.77315625697.53575

14.250.00247368420.0008947368-0.0006842105823.656025.9473684211-19.8421052632215.64657.2703157895-75.59842105261020.96789473685432.6028453947452.71690378291020.96789473685.3210320064316.901832648714.6775263158

14.50.00248275860.0009310345-0.0006206897838.16027-18215.64659.944-68.581045.115347.90375445.65864583331045.115.1170726048311.9610520833731.577

14.750.00249152540.0009661017-0.000559322852.556028.0169491525-16.2203389831215.64662.5270508475-61.79949152541068.9235593225261.9183739407438.49319782841068.9235593224.9226329872306.9452384799748.2464915254

150.00250.001-0.00058676029-14.5215.64665.024-55.2451092.4255174.5575431.2131251092.4254.7367622491301.8491875764.6975

15.250.00250819670.0010327869-0.000442623881.456029.9508196721-12.8360655738215.64667.4390819672-48.90540983611115.62967213125085.7377612705423.81148010591115.62967213124.5586254008296.6680360741780.9407704918

15.50.0025161290.0010645161-0.0003870968895.96030.8709677419-11.2258064516215.64669.7762580645-42.77032258061138.55193548394995.3811693548416.28176411291138.55193548394.3874864323291.397234879796.9863548387

15.750.00252380950.0010952381-0.0003333333910.356031.7619047619-9.6666666667215.64672.0392380952-36.831161.20523809524903.4146875408.6178906251161.20523809524.2226942548286.0325234375812.8436666667

160.002531250.001125-0.00028125924.86032.625-8.15625215.64674.2315-31.07531251183.60218754809.76984375400.81415364581183.60218754.0636709652280.5699075521828.52153125

16.250.00253846150.0011538462-0.0002307692939.256033.4615384615-6.6923076923215.64676.3563076923-25.49769230771205.75461538464714.3823798077392.86519831731205.75461538463.909901998275.0056388221844.0282307692

16.50.00254545450.0011818182-0.0001818182953.76034.2727272727-5.2727272727215.64678.4167272727-20.08909090911227.67363636364617.1919318182384.76599431821227.67363636363.7609278191269.3361960227859.3715454545

16.750.00255223880.0012089552-0.0001343284968.156035.0597014925-3.8955223881215.64680.415641791-14.84194029851249.36970149254518.1417397388376.51181164491249.36970149253.6163368892263.5582681514874.5587910448

170.00255882350.0012352941-0.0000882353982.66035.8235294118-2.5588235294215.64682.3557647059-9.74911764711270.85264705884417.1783823529368.09819852941270.85264705883.4757596741257.6687389706889.5968529412

17.250.00256521740.0012608696-0.0000434783997.056036.5652173913-1.2608695652215.64684.2396521739-4.80391304351292.13173913044314.2515353261359.52096127721292.13173913043.3388635266251.664672894904.4922173913

17.50.00257142860.001285714301011.56037.28571428570215.64686.069714285701313.21571428574209.31375350.77614583331313.21571428573.2053482944245.5433020833919.251

17.750.00257746480.00130985920.00004225351025.956037.9859154931.2253521127215.64687.8482253521-8.28540845071321.15881690144199.4752508803349.95627090671321.15881690143.1786301519244.9693896347924.811171831

180.00258333330.00133333330.00008333331040.46038.66666666672.4166666667215.64689.5773333333-3.74651341.87683333334090.38375340.86531251341.87683333333.0482557329238.60571875939.3137833333

18.250.00258904110.00135616440.00012328771054.856039.32876712333.5753424658215.64691.25906849320.66805479451362.42312328773979.1542765411331.59618971181362.42312328772.9206449953232.1173327982953.6961863014

18.50.00259459460.00137837840.00016216221069.36039.9729729734.7027027027215.64692.89535135144.96329729731382.80464864863865.7490202703322.14575168921382.80464864862.7955857858225.5020261824967.9632540541

18.750.00260.00140.00021083.756040.65.8215.64694.4889.1441403.0283750.1321875312.5110156251403.0282.6728847803218.7577109375982.1196

190.00260526320.00142105260.00023684211098.26041.21052631586.8684210526215.64696.038736842113.21468421051423.09942105263632.269868421302.68915570181423.09942105262.5523655021211.8824089912996.1695947368

19.250.00261038960.00144155840.00027272731112.656041.80519480527.9090909091215.64697.549194805217.17963636361443.02483116883512.1299147727292.67749289771443.02483116882.4338665828204.87424502841010.1173818182

19.50.00261538460.00146153850.00030769231127.16042.38461538468.9230769231215.64699.020923076921.04292307691462.80984615383389.6818269231282.47348557691462.80984615382.3172402318197.73143990381023.9668923077

19.750.00262025320.00148101270.00034177221141.556042.94936708869.9113924051215.646100.455392405124.80840506331482.45979746843264.8966495253272.07472079381482.45979746842.2023508867190.45230455561037.7218582278

200.0026250.00150.00037511566043.510.875215.646101.85428.479751501.979753137.746875261.478906251501.979752.0890740205183.0352343751051.385825

20.250.00262962960.00151851850.00040740741170.456044.03703703711.8148148148215.646103.218074074132.06044444441521.37451851853008.2063541667250.68386284721521.37451851851.9772950825175.47870399311064.962162963

20.50.00263414630.00153658540.00043902441184.96044.560975609812.7317073171215.646104.548878048835.5538048781540.64868292682876.2502134146239.68751778461540.64868292681.8669085595167.78126244921078.4540780488

20.750.00263855420.00155421690.00046987951199.356045.072289156613.6265060241215.646105.847614457838.96298795181559.80660240962741.8547778614228.48789815511559.80660240961.7578171381159.94152870861084

210.00264285710.00157142860.00051213.86045.571428571414.5215.646107.115428571442.2911578.85242857142604.9975217.0831251578.85242857141.649930958151.95818751084

21.250.00264705880.00158823530.00052941181228.256046.058823529415.3529411765215.646108.353411764745.54070588241597.79011764712465.6568933824205.47140778191597.79011764711.5431669442143.82998544731084

21.50.00265116280.00160465120.00055813951242.76046.534883720916.1860465116215.646109.562604651248.71483720931616.62344186052323.8124709302193.65103924421616.62344186051.4374482089135.55572747091084

21.750.00265517240.00162068970.00058620691257.15604717215.646110.74451.8161635.3562179.4446875181.6203906251635.3561.3327035138127.13427343751084

220.00265909090.00163636360.00061363641271.66047.454545454517.7954545455215.646111.898545454554.84668181821653.99122727272032.5348863636169.3779071971653.99122727271.2288667877118.56453503791084

22.250.00266292130.00165168540.00064044941286.056047.898876404518.5730337079215.646113.027146067457.8092584271672.53240449441883.0652492978156.92210410811672.53240449441.1258766911109.84547287571084

22.50.00266666670.00166666670.00066666671300.56048.333333333319.3333333333215.646114.130666666760.7061690.98266666671731.01875144.25156251690.98266666671.0236762234100.976093751084

22.750.00267032970.00168131870.00069230771314.956048.758241758220.0769230769215.646115.209934065963.53907692311709.3450109891576.3791105769131.36492588141709.3450109890.922212368291.9554481171084

230.0026739130.00169565220.00071739131329.46049.173913043520.8043478261215.646116.265739130466.31056521741727.62230434781419.1307608696118.26089673911727.62230434780.821435771882.78262771741084

23.250.00267741940.00170967740.00074193551343.856049.580645161321.5161290323215.646117.298838709769.02245161291745.81729032261259.2588004032104.93823336691745.81729032260.721300451973.45676335691084

23.50.00268085110.00172340430.00076595741358.36049.978723404322.2127659574215.646118.309957446871.67663829791763.93259574471096.74896276691.39574689721763.93259574470.62176353363.9770228281084

23.750.00268421050.00173684210.00078947371372.756050.368421052622.8947368421215.646119.299789473774.27494736841781.9707368421931.587582236877.63229851971781.97073684210.522785005954.34260896381084

240.00268750.001750.00081251387.26050.7523.5625215.646120.26976.8191251799.934125763.761562563.6467968751799.9341250.424327508444.55275781251084

24.250.00269072160.00176288660.00083505151401.656051.123711340224.2164948454215.646121.218226804179.31084536081817.825072165593.258347293849.43819560781817.8250721650.326356125434.60673692551084

24.50.00269387760.00177551020.00085714291416.16051.489795918424.8571428571215.646122.148081632781.75171428571835.6457959184420.065892857135.00549107141835.64579591840.228838207124.503843751084

24.750.00269696970.00178787880.00087878791430.556051.848484848525.4848484848215.646123.059151515284.14327272731853.3984242424244.172642045520.34772017051853.39842424240.131743201514.24340411931084

24.9550.0026994590.00179783610.00089621321442.3996052.137247044725.9901823282215.646123.792607493586.06859467041867.906202163997.9170458478.15975382061867.90620216390.0524207515.71182767441084

Sheet1

M (k-ft)

P (k)

Column Analysis

Sheet2

fMn (k-ft)

fPn (k)

Column Analysis

Sheet3

Factored

Unfactored

Mn (k-ft)

Pn (k)

Column Analysis