Prestress Conc.Des CES 5715
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154 Naaman PRESTRESSED CONCRETE ANALYSIS AND DESIGN 47 MATHEMATICAL BASIS FOR FLEXURAL ANALYSIS For the analysis it is assumed that all materials behave elastically in the working range of stresses applied The usual hypothesis of Hooke Navier Bresse and Bernouilli are assumed valid namely 1 The materials steel and concrete are elastic and there is a proportional relationship between stresses and strains Hooke s Law 2 Plane sections remain plane after bending Bernouilli and 3 There is a perfect bond between steel and concrete This is equivalent to saying that both the stress and strain diagrams along the section of concrete under bending are linear and that the changes in strains in the steel and in the concrete at the level of the steel are identical Also the load de ection or moment curvature curves are assumed linear for the loadings considered Typical stress diagrams for the two extreme initial and nal loadings have been described in Fig 47 Note that the highest stresses in the section occur at the extreme top and bottom bers Two Extreme ngg39g gme Two Extreme Loadings lt gt Tension lt gt Fibers Mmm Mmax Compression Top Bottom Eight Stress Inequality Conditions When all external moments are of the same sign only four conditions are binding Figure 413 Basis for the stress inequality conditions Since two extreme loadings are generally critical and since for each two allowable stresses must be speci ed at least four allowable stresses must be considered in the analysis Since under exural loading maximum stresses occur on the two extreme bers top and bottom eight inequality equations comparing actual stresses with allowable stresses can be derived Fig 413 They are of the form Chapter 4 FLEXURE WORKING STRESS ANALYSIS AND DESIGN 155 gt Actual stress or allowable stress 41 lt Let us develop one of these equations for a pretensioned simply supported member The actual stress on the top ber under initial conditions must be more than or equal to the allowable initial tensile stress since tension is negative we use more than or equal Therefore F Fe M GtilL AY1ZO 42 Ac Zr Zr where Mmin represents the dead load moment at the section considered Equation 42 could also be rewritten in several different ways one of which may be more suitable if a particular variable is to be emphasized such as for example Fz S Mmin EtiZteo kb 60 S kl 1Fi39Mmin TEN39ZI 4393 NF 2 ea kbMmin 6tiZt As mentioned above eight inequality equations which will also be described as stress conditions stress constraints or stress limit states can be derived and are similar in form to Eq 42 However in actual design problems out of the eight conditions four are generally non binding For example if for a simply supported member the tensile stress on the top ber for the initial loading is of concern and is checked against allowable limit as for Eq 42 there is certainly no need to check against allowable limit the compressive stress on the same ber and for the same loading If for the same loading we were checking the section at the intermediate support of a twospan continuous beam we would check the stress on the top ber against initial allowable compression and that eliminates the need to check against initial allowable tension Thus the number of inequality equations that must be considered in the analysis at a given section is essentially reduced to four ie four of them are binding while the four others are not The four that are binding in a particular design depend on the sign of the applied moments The eight stress inequality equations written in various ways are shown four at a time in Tables 42 and 43 They have been numbered in roman notation l to IV and I to IV The coef cient 77 was de ned in Section 313 and is the ratio of the nal prestressing force after all losses to the initial prestressing force For a cross section where all applied moments Mmin and Mmax are positive only stress inequalities I to IV need to be considered similarly if all applied moments are negative stress inequalities I to IV become binding When a particular section is subject to moments of different signs it is possible by inspection to select out of the eight 156 Naaman PRESTRESSED CONCRETE ANALYSIS AND DESIGN inequalities the four that would be binding on the other hand one can also check systematically the eight inequalities against allowable stresses and select the four that are binding This is typically what should be done if a computer program or a spreadsheet is written and the applied bending moment at any section can be of any sign Table 42 Useful ways of writing the four stress inequality conditions Stress Way condition Inequality equation 1 1 FiAcl eOkbMminZ 26 H FiAc1eoktMminZbSaci m WormACmeokbMmaxzt sacs 1v ltFornEAc1 eomoi MmZb2615 2 I ea S kl 1FiMmin 6tiZt 11 ea 3 kt 1Fi39Mmin 639ciZb III 90 2k 1F0r77Fi39Mmax 539csZt 1V 60 Z kt 1F OYUEMmax 539tst 3 I Fz39 3Mmin EtiZteo kb H SMmin639ciZbeo kt 111 F 77Fi39 2 Mmax 60sZteo kb 1V F77FIZMmax639tSZbeokt 4 I 1EZeo kbMmin 52izt II lFizeo kzMmin5cizb III lFl77ESeo kbMmax5cszt IV 1F177F se0 kMmax5stb All V e0 S e0 mp yb dc min 2 maximum practical eccentricity towards bottom ber Note for condition 111 Mmax and Em may also be replaced by Msustagned and 6cm to satisfy the second provision on allowable compression given in the 2002 ACI Code Chapter 4 FLEXURE WORKING STRESS ANALYSIS AND DESIGN 157 Table 43 Useful ways of writing the four complementary stress inequality conditions Stress Way condition Inequality equation 1 I39 EAC1 eo kbMmm Zt 3 5C 1139 EACl e0k MminZb 25 11139 F or nEAC1 eOkbMmaxZt 265 IV F or 77EAc1 eokrlMmaxZbsacs 2 139 60 2 kb lFlMmir1 ECZ H 60 2 kt 1EMmin OZiZb HI39 60 g k 1F 0r 77ElMmax 5tsZt V39 ea 3 kt 1F 0r 775Mmax 6cst 3 l39 2111mmECI39Zleo kb II39 Fz39 Z Mmm 5n39Zbeo 49 11139 F 2 WE S Mmax 6ts39Zteo 1 IV F77Fi SIlIinax 5csteokt 4 139 HE S 60 kbMmir1 601Z II39 lFi 3eoklMmin6tizb 11139 F or UPDZ 30 kwMmax O lsZt IV F or 77Fi2eoktMmax6cszb All V e0sle0 mplz y dc mm maximum practical eccentricity towards top ber Note for condition IV Mmax and 50 may also be replaced by Msutained and Ecm to satisfy the second provision on allowable compression given in the 2002 ACI Code Note that the stress inequalities I to IV given in Table 43 are described here as complementary stress inequalities This is because often one does not need to use them It can be shown that if all applied moments are negative stress conditions I to IV can still be used provided the concrete section is assumed in its inverted position ie use properties of inverted section and the sign of the moments is changed from negative to positive the position of F within the cross section remains unchanged It is because stress conditions I to IV can cover the majority of practical problems that 158 Naaman PRESTRESSED CONCRETE ANALYSIS AND DESIGN they are often encountered alone in the technical literature and with no reference to the four others In Tables 42 and 43 a fth condition numbered V has been included and will be described as the practicality condition Essentially it states that the prestressing force must be inside the concrete section with an adequate cover dcmin Thus the design eccentricity eo must be less than or equal to a maximum practical value 60 p yb dcmin Although in an analysis or investigation problem condition V is obviously satis ed in a design problem condition V can be binding and can be used with advantage in optimizing or simplifying a solution This is why it has been included in the tables Note nally that in the case of external prestressing the maximum practical eccentricity is independent of the minimum concrete cover In an analysis or investigation problem the above stress inequality equations can be directly checked as all quantities are known Thus one can verify the allowable stress limit states In a design problem however these inequalities can be used to either determine exactly or put bounds on some of the unknown variables such as prestressing force F eccentricity e0 andor section properties For example if the concrete cross section is given the stress conditions can be used to determine bounds on all the possible values of F and eo that would be acceptable for the problem at hand This is clari ed in the next sections 48 GEOMETRIC INTERPRETATION OF THE STRESS INEQUALITY CONDITIONS The geometric interpretation of the stress inequality conditions has been rst explored by Magnel Ref 110 As emphasized throughout this text the geometric representation can be a very useful and powerful technique for the solution of many problems where the working stress design approach is used Let us assume that the geometric properties of the concrete cross section are given including the depth of the section which can be estimated a priori then only two unknown variables remain in equations I to IV namely ea and F or F 77 F i One can plot on a two dimensional scale the curves corresponding to the four equations at equality Each curve will separate the plane into two parts one where the inequality is satis ed and the other where it is not If 60 is plotted versus F i the curves will be hyperbola However if e0 is plotted versus lFi then straight lines are obtained and the geometric representation is much simpli ed For this reason it is better to use the second way of writing the equations in Table 42 because they are written in the form e0 a1F39 b where b is the intercept and a the slope of the line When plotted as shown in Fig 414 the inequality equations delineate a domain of feasibility limited by a quadragon A B C D Essentially any point inside this feasibility domain has coordinates F and e0 which satisfy the four stress inequality conditions I to IV The practicality condition V can also be represented at Chapter4 FLEXURE WORKING STRESS ANALYSIS AND DESIGN 159 equality on the same graph by a horizontal line parallel to the lFl axis If this line intersects the quadragon A B C D such as case b of Fig 414 then a new reduced feasibility domain is de ned such as EBCDG Any point inside this new domain would have satisfactory stress wise and practically feasible values of F and e0 If the line representing condition V does not intersect the domain A B C D such as in cases a or c of Fig 414 then either there is no practical solution case a or any point inside A B C D represents a practically feasible solution case 0 In case a a new concrete cross sectiOn must be used leading to higher section moduli In case c since any point of the domain A B C D is feasible one must select the one leading to the smallest prestressing force ie point A intersection of lines representing conditions I and IV The corresponding analytical solution is obtained by solving two equations I and IV to determine two unknowns F and ea In case b the smallest value for the prestressing force is obtained by solving IV and V the corresponding analytical solution is obtained by solving IV for F i after replacing e0 by eomp yb 39 damm e i39 331 iiii39wa 39I3 39 0 I Beam Sectional I 39 39ETypic k c5 gtToward minimum Fl 39 I 55 53 i 0 l 1 E 3 Four F 7060 stress kb conditions 5 1 a I I B H 39 3 Practical condition 3quot Feasible 80 E Gymp domain D V 1 39 b s III C l V 6 0 M Figure 414 Feasibility domain defined by the stress inequality conditions 160 Naaman PRESTRESSED CONCRETE ANALYSIS AND DESIGN Note that the geometric interpretation of the stress inequality conditions gives a very clear picture of the state of a given problem or what should be done about a particular problem For example in a given analysis or design problem one can plot the feasibility domain and check if the proposed values of F and ea are represented by a point that falls inside the domain if it falls inside there is no need to check the stresses if it does not one can spot right away the condition or conditions that are not satis ed and devise a corrective action Other types of practical questions that can be best answered by using the abdve geometric representation are as follow 1 Given an eccentricity e0 what are the minimum and maximum feasible values of the prestressing force The answer to this question could lead to nding a range of live loads which can be carried by a particular beam 2 Given a prestressing force what is the range of feasible eccentricities at a given section This type of problem arises when in a pretensioning bed beams of different span lengths are prestressed simultaneously Two examples are treated next In the rst one the geometric representation of the stress inequality conditions and the feasibility domain are used in both an investigation problem and a design problem where the concrete section is given The second example illustrates the use of the feasibility domain at two critical sections midspan and support of a cantilever beam and the choice of an acceptable prestressing force for the two sections 49 EXAMPLE ANALYSIS AND DESIGN OF A PRESTRESSED BEAM 491 Simply Supported T Beam This example is also continued in Sections412 415 55 610 618 77 and 78 Consider the pretensioned simply supported member shown in Fig 415 with a span length of 70 feet It is assumed that fc39 5000 psi fc i 4000 psi 5 189 psi EC 2400 psi 55 424 psi Jesus 2250 psi for sustained load and E 3000 psi for the maximum service load Normal weight concrete is used Le 70 150 pcf live load 100 psf and superimposed dead load 10 psf Assume fpe 150 ksi 77 fpe fp FF 083fp 180723 ksi and eomp yb 4 231 in In order to calculate the stresses the geometric properties of the section given in Fig 415 and the applied bending moments are needed 0 Minimum moment Mmin MG 05737028 350962 kipsft o Moment due to superimposed dead load MSD 0047028 245 kips ft o Moment due to live load ML 047028 245 kips 0 Additional moment due to superimposed dead load and live load AM 0447028 2695 kips 0 Maximum moment Mmax Mmin AM 620462 kipsft 0 Sustained moment Mm MG MSD 375462 kipsft Chapter 4 FLEXURE WORKING STRESS ANALYSIS AND DESIGN 161 llt 48 i J SECTION PROPERTIES A 550 in2 82065 in4 yt 129 in yb 271 in 40 in 2 6362 in3 2b 3028 in3 kt 551 in b 1157 in wG 0573 klf m 7337 Figure 415 3 Investigate flexural stresses at midspan given F 2295 kips corresponding to ten zin diameter strands F Fn 2765 kips and e0 231 in Referring to the four stress inequality equations given in Table 42 way 1 and multiplying the values of moments by 12000 in order to have units of pounds per square inch leads to Condition I o l e quotJ M i 2 EH Ac k A 276500 231 350962 x 12000 in2 1611351 gta 189ps1 OK 550 l 157 6362 The results for the other conditions are given as follows Condition 11 0w 1219 psi lt EC 2400 psi OK 03965 2 754 psi lt E 3000 psi for Mmax Condition III or OK 10m 292 psi lt Em 2250 psi for MM Condition IV a 292 psi gt 65 424 psi OK Therefore the section is satisfactory with respect to exural stresses b Plot the feasibility domain for the above problem and check geometrically if allowable stresses are satis ed The equations at equality given in Table 42 way 2 are used to plot linear relationships of ea versus lF on Fig 416 They are reduced to the following convenient form the first ofwhich is detailed Condition 1 e0 3 kb 1FMmin EZ 11571B350962x12000189x6362 162 Naaman PRESTRESSED CONCRETE ANALYSIS AND DESIGN Beam Section IV V ltgt Figure 416 Feasibility domain for example 49 which can be put in the following convenient form for this as well as for the other conditions Condition 1 106 e0 11575410 F1 106 e0 551ll4787 F 106 e0 1157 14024 F lorMmm Condition 11 I Condition III S US 6 e0 21157 ll818 for M gtControls 139 106 e0 2 5517424 E Condition IV Chapter 4 FLEXURE WORKING STRESS ANALYSIS AND DESIGN 163 where e0 is in inches and F i is in pounds Also equation V showing eamp 231 in is plotted in Fig 416 The ve lines delineate a feasibility domain ABCD Let us check if the given values of F i and ea are represented by a point which belongs to the feasible region 1 36 X10396 1 276500 The representative point is shown in Fig 416 as point A Since it is on line AD it belongs to the feasible region and therefore all allowable stresses are satis ed Note that all stresses would still be satisfied if the eccentricity is reduced to approximately 21 in for the same force This is shown as point A on line AB and allows the designer to accept a reasonable tolerance on the value of e0 actually achieved during the construction phase c Assuming the prestressing force is not given determine its design value and corresponding eccentricity This is essentially a typical design problem where the concrete cross section is given It can be solved directly analytically or from the graphical representation of the feasibility domain In any case the graphical representation helps in the analytical solution It dictates the choice of point A of Fig 416 as the solution that minimizes the prestressing force Point A corresponds to the intersection of line V representing e0mp with that representing stress condition IV The corresponding value of F is obtained by replacing e0 by 60quotp in Eq IV way 3 of Table 42 that is Mmax ESZb 620462x12000 424x3028 e0p k 231551 2153681b and F F 259479 lb 2595 kips 083 Graphically the coordinates ofA can be read in Fig 416 as ea 231 in and 1F 39 X 10 6 which leads to F 257000 lb 257 kips It can be seen that the graphical solution gives essentially the same answer as the analytical one Note that the practical value of the prestressing force to use in the design should correspond to an integer number of tendons In this case exactly 938 strands each with a nal force of 2295 kips would be required The number is rounded off to 10 The resulting higher prestressing force allows for an acceptable tolerance on the value of e0 which can be varied now from 231 in to 2133 in see Table 45 d If the beam is to be used with different values of live loads what is the maximum value of live load it can sustain Referring to the stress inequality conditions it can be observed that conditions I and II which do not depend on the live load moment do not change and therefore lines I and II of Fig 416 are xed Increasing the value of the live load will increase the value of Mmax and thus will change the slopes of lines 111 and IV so as to reduce the size of the feasible domain Consequently point A of the feasible domain will move in the direction of AD and line BA tends to rotate about the intercept point k toward CD Similarly line 111 will rotate about the intercept kb towards line I The maximum value of live load correspond to the line that merges rst with the other one In this case it is the live load that will make lines II and IV coincide or have same slopes Therefore 164 Naaman PRESTRESSED CONCRETE ANALYSIS AND DESIGN Minux EI SZb 2114787x106 which leads to Mmax 10811193 lb in 90093 kips ft Subtracting from Mmax the values of moments due to dead load and superimposed dead load 375462 kipsft leads to a live load moment of 525468 kipsft from which the live load can be determined as 858 plf or 2145 psf The representative point in Fig 416 is D which shows the following coordinates e0 231 in and 10617 25 ie F 400000 lb 400 kips The reader is encouraged to check numerically in this case that the two allowable stresses 5 and 55 are attained exactly while the two others are satis ed as indicated by the geometric representation Note that such a design may have to be revised if the assumed value of e0 cannot be practically achieved Note also that while the limit capacity of this prestressed beam in now attained from an allowable stress point of view it can still be designed to carry a larger live load should partial prestressing be considered 492 Simply Supported T Beam with Single Cantilever on One Side Consider the same beam as in the previous example that is same section same loading same material properties and same main span of 70 it However assume that it has a cantilever on one side spanning 10 ft Fig 417a Also assume that in addition to the dead and live loads already considered a concentrated load of magnitude P 30 kips is applied at the free end of the cantilever Plot the feasibility domain for the two critical sections in span and at the right support on the same graph and determine an acceptable prestressing force and its eccentricity at the two sections The dead and live loads have to be placed in such a way as to produce minimum and maximum moments at each critical section In order to minimize the computations only the condition for maximum service compression under maximum load will be considered that is the corresponding stress condition for allowable compression under sustained load will be ignored In any case these stresses generally do not control the design 1343931133ff3j f G G L SD V llllllllllllllTiTllll llllllililllllllllll Maximum at midspan Maximum at support Minimum at support Minimum at midspan Figure 417a Loading arrangements leading to the maximum and minimum moments It can be shown that for the support section C the minimum moment is obtained when only the own weight of the beam is considered the maximum moment is obtained when in addition to the own