Feasible Domain Notes

4.7 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- deflection or moment-curvature curves are assumed linear for the loadings considered. Typical stress diagrams for the two extreme initial and final loadings have been described in Fig. 4.7. Note that the highest stresses in the section occur at the extreme top and bottom fibers.

Two Extreme Two Allowable Stresses Two Extreme Loadings (Tension, Fibers ("minl Mmax) Compression) (Top, Bottom)

1 I Loadings

Eight Stress Inequality

( When all external moments are of the same sign, only ) I four conditions are binding I

Lel membt than 01 "more

where (4.2) cc suitablc

As I "stress are simi conditic membe~ checked against loading. support against initial a consider of them particula

Figure 4.13 Basis for the stress inequality conditions. The time in 1

Since two extreme loadings are generally critical and since for each, two I' to IV' prestress allowable stresses must be specified, at least four allowable stresses must be considered in the analysis. Since under flexural loading maximum stresses occur on the two extreme fibers (top and bottom), eight inequality equations comparing actual to IV ne stresses with allowable stresses can be derived (Fig. 4.13). They are of the form: inequalit

moments

Chapter 4 - FLEXURE: WORKING STRESS ANALYSIS AND DESIGN 155

(Actual stress) or (allowable stress) 1'1 Let us develop one of these equations for a pretensioned simply supported

member. The actual stress on the top fiber 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

where Mmin represents the dead load moment at the section considered. Equation (4.2) 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:

F;: I (Mmin - ZtiZ,) /(e, - kb) eo 5 kb + (1 1 & )(Mmin - ctiZt ) 11 F;: 2 (eo - kb) l(Mmin - FtiZt)

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. (4.2). 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 fiber for the initial loading is of concern and is checked against allowable limit (as for Eq. (4.2)), there is certainly no need to check, against allowable limit, the compressive stress on the same fiber and for the same loading. If for the same loading we were checking the section at the intermediate support of a two-span continuous beam, we would check the stress on the top fiber 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, i.e., 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 4.2 and 4.3. They have been numbered in roman notation I to IV and I' to IV'. The coefficient 77 was defined in Section 3.13 and is the ratio of the final 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 4.2 Useful ways of writing the four stress inequality conditions.

Table 4.3 1

Way co

Inequality equation

(4 / A,) [l - (e, / kb )] + Mmin / Zt 2 Oti (F, /,!&)[I-(eo /kt)]-Mmi, / z b I Oci [(Forq4)/&l[l-(eo /kb)l+Mmax IZt Ices [ ( F o r ~ F ; : ) f 4 I C 1 - ( e o / ~ t ) I - ~ m a x IZb >ot,

eo I kb + (1 /F,)(Mmin - OtiZt) eo I kt + (1 I F, )(Mmin + OciZb) eo 2 kb + [l/(F~rVF;;)l(Mmax -8csZt eo kt +[l/(For~q)l(Mrnax +5tsZb)

F, I (Mmin - StiZt ) l(eo - kb ) F, I (Mmin + OciZb)l(e0 -kt) F =vF;: 2 (Mmax -8csZ,)l(eo -kb) F =vF;: 2 (Mmax +OtsZb)l(eo -kt)

1 16 2 (e, - kb) l(Mmin - OtiZt) 1 1 4 2(e0 -kt)l(Mmin +OciZb) l / F = l l ~ F ; :


Table 4.3 Useful ways of writing the four complementary stress ineaualitv conditions.

Stress Way condition Inequality equation

1 I' ( F , / A , ) [ I - ( ~ ~ / ~ ~ ) ] + M , ~ ~ / & ISci 11' (4 /&)[ I - (eo l k t ) ] - ~ m i n IZb 2 qi III' [ (F or ' ~ & ) l & l [ l - ( e o ~ ~ b ) l + Mmax I Z r >Sts IV' [ ( F o ~ ~ & ) / ~ ] [ ~ - ( ~ ~ / ~ ~ ) ] - M ~ ~ ~ / Z ~ I C ~ ~

2 I' eo 2 kb + ( 1 I F;:)(Mmin - OciZt ) 11' eo 2 kt + ( 1 / &)(Mmin + otiZb) 111' eo I kb + [1/(F or '~&)I(Mmax -8tsZt ) IV' eo 5 kt + [1 / (F or 'IF;: )I(Mmax + ocs Zb)

3 I' F, 2 (Mmin - OciZt ) l(eo - kb) 11' F, 2 (Mmin +OtiZb)l(eo -k t ) HI' F=vF;: I (Mmax -OtsZt)l(eo -kb) IV' F=q& I (Mmax + ~ a Z b ) I ( e o -k t )

4 I' 114 I (eo - kb)/(Mmin -CciZt) 11' l /F;: I (eo -kt )l(Mmin + OtiZb) 111' 1 /(F or 7 4 ) 2 (eo - kb)/(Mmax - OtsZt) IV' 1 l(F or 7 4 ) 2 (eo -kt ) l(M,,, + OcsZb)

All V' le0 1 < l(eo jrnP 1 = yt - (d , lmin = maximum practical eccentricity towards top fiber

Note: for condition IV', Mmax and i?,, may also be replaced by Msubined and G,,, 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 4.3 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 (i.e., 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

I 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 4.2 and 4.3, a fifth 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 (dC),in. Thus the design eccentricity e, must be less than or equal to a maximum practical value (eo),p = yb - (dc),,. Although in an analysis or investigation problem, condition V is obviously satisfied, 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 finally 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 e,, and/or 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 e, that would be acceptable for the problem at hand. This is clarified in the next sections.

4.8 GEOMETRIC INTERPRETATION OF THE STRESS INEQUALITY CONDITIONS

The geometric interpretation of the stress inequality conditions has been first explored by Magnel [Ref. 1 .lo]. 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 e, and Fi (or F = 77 Fi).

1 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 satisfied and the other where it is not. If e, is plotted versus Fi, the curves will be hyperbola. However, if e, is plotted versus l/Fi, then straight lines are obtained and the geometric representation is much simplified. For this reason, it is better to use the second way of writing the equations in Table 4.2 because they are written in the form: e, = a(llFi) + b where b is the intercept and a the slope of the line. When plotted as shown in Fig. 4.14, 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 Fi and e,, which satisfy the four stress inequality conditions I to IV. The practicality condition V can also be represented at

equality intersecl feasibili would h the line I cases (a) point ins (a) a nel case (c), leading represent by solvin (b) the sn correspon by (eo),

Figure 4.14 Fe


equality on the same graph by a horizontal line parallel to the l/Fi axis. If this line intersects the quadragon A, B, C, D, such as case (b) of Fig. 4.14, then a new reduced feasibility domain is defined such as EBCDG. Any point inside this new domain would have satisfactory (stress wise) and practically feasible values of Fi and e,. If the line representing condition V does not intersect the domain A, B, C, D, such as in cases (a) or (c) of Fig. 4.14, then either there is no practical solution (case (a)) or any point inside A, B, C, D represents a practically feasible solution (case (c)). 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, i.e., 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, Fi and e,. 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 Fi, after replacing e, by (eo)mp = ~ b - (dclrnin.

Figure 4.14 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 e, 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 satisfied and devise a corrective action. Other types of practical questions that can be best answered by using the above geometric representation are as follow:

1. Given an eccentricity e,, what are the minimum and maximum feasible values of the prestressing force? The answer to this question could lead to finding 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 first 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.

4.9 EXAMPLE: ANALYSIS AND DESIGN OF A PRESTRESSED BEAM

4.9.1 Simply Supported T Beam

This example is also continued in Sections4.12,4.15, 5.5, 6.10, 6.18,7.7, and 7.8.

Consider the pretensioned simply supported member shown in Fig. 4.15 with a span length of 70 feet. It is assumed that f,' = 5000 psi, fii = 4000 psi, = -1 89 psi, Cci = 2400 psi, = -424 psi, Fcs, = 2250 psi for sustained load, and = 3000 psi for the maximum service load. Normal weight concrete is used, i.e., yc= 150 pcf, live load = 100 psf and superimposed dead load = 10 psf. Assume: fpe = 150 ksi; 7 = f / f . = F IF; . = 0.83; fpi = 180.723 ksi; and (e,),? = yt, - 4 = 23.1 in. Pe P'

In order to calculate the stresses, the geometric properties of the section (g~ven in Fig. 4.15) and the applied bending moments are needed.

Minimum moment: Mmh = MG = 0.573(702/8) = 350.962 kips-ft Moment due to superimposed dead load: Mm = 0.04(702/8) = 24.5 kips-ft Moment due to live load: ML = 0.4(702/8) = 245 kips-ft Additional moment due to superimposed dead load and live load: AM= 0.44(702/8) = 269.5 kips-ft Maximum moment: M,, = Mmin + AM= 620.462 kips-ft Sustained moment: Ms,, = MG + MsD = 375.462 kips-ft

Figure 4

(a) Inve dian

Referring of momer

The rl C

(b) Plot tl stresst

The equatic 1/Fi on Fig


48 in SECTION PROPERTIES , I

(a) Investigate flexural stresses at midspan given: F = 229.5 kips (corresponding to ten %-in diameter strands), Fi = F/I~ = 276.5 kips, and eo = 23.1 in. !

I The results for the other conditions are given as follows: Condition 11: a,; = 12 19 psi < FCi = 2400 psi OK

a,, = 754 psi < = 3000 psi for M,,,, Condition 111: OK

a,,, = 292 psi < G,, = 2250 psi for M,,, lor Condition IV: a, = -292 psi > = -424 psi OK

Therefore the section is satisfactory with respect to flexural stresses. I (b) Plot the feasibility domain for the above problem and check geometrically if allowable

stresses are satisfied.

The equations at equality given in Table 4.2 (way 2 ) are used to plot linear relationships of e, versus 1IFi on Fig. 4.16. They are reduced to the following convenient form, the first ofwhich is detailed:

I Condition I: e, 5 kb + (l lF;:)(Mmi, -i?,;Z,) = 11.57 +(l/F,)(350.962 x 12000+ 189 x 6362)

c-- Naaman - PRESTRESSED CONCRETE ANALYSIS AND DESIGN

I Figure 4.16 Feasibility domain for example 4.9.1. which can be put in the following convenient form, for this as well as for the other conditions: Condition I: eo = 11.57 +5.410[$)

where e, 4.16. TI

Let L feasible

The the feasil be satisfi point A ' actually 2

(c) Assu eccei

This is es directly a graphical as the so11 representi obtained 1

and

Graph which leac the same a the design with a f i n a higher pre! now from :

(d) If the live lo

Condition 111: Referring t depend on Increasing of lines 111 feasible do: kt) toward value of liv load that w


where e, is in inches and Fi is in pounds. Also equation V showing (eo),p = 23.1 in is plotted in Fig. 4.16. The five lines delineate a feasibility domain ABCD.

Let us check if the given values of Fi and e, are represented by a point which belongs to the feasible region:

1 - -

- -- I - 3.6 x Fi 276,500

The representative point is shown in Fig. 4.16 as point A'. Since it is on line AD, it belongs to the feasible region and therefore all allowable stresses are satisfied. 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 eo actually achieved during the construction phase.

Assuming the eccentricity.

prestressing force is not given, determine I its design value and corresponding

This is essentially a typical design problem where the concrete cross section is given. It can be solved directly analytically or Erom 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. 4.16 as the solution that minimizes the prestressing force. Point A corresponds to the intersection of line V representing (e& with that representing stress condition IV. The corresponding value of F is obtained by replacing e, by (eo),p in Eq. IV (way 3) of Table 4.2; that is:

(eo )mp - 4 23.1+5.51 and

9.5 kips

Graphically the coordinates of A can be read in Fig. 4.16 as e, = 23.1 in and l/Fi = 3.9 X which leads to Fi = 257,000 Ib = 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 wit

design should c :h a final force c

.orrespond to an inte ,f 22.95 kips would

:ger number o be required.

i' tendons. The numb1

In this case, exactly 9.38 strands each er is rounded off to 10. The resulting

higher prestressing force allows for an acceptable tolerance on the value of e,, which can be varied now from 23.1 in to 2 1.33 in (see Table 4.5).

(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? I

Referring to the stress inequality conditions, it can be observed that conditions I and I1 (which do not depend on the live load moment) do not change and therefore lines I and I1 of Fig. 4.16 are fixed. 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 kt) toward CD. Similarly line 111 will rotate about the intercept, kt,, towards line I. The maximum value of live load correspond to the line that merges first with the other one. In this case, it is the live load that will make lines I1 and IV coincide or have same slopes. Therefore:

Naaman - PRESTRESSED CONCRETE ANALYSIS AND DESIGN -

weig M,,+O z

=11.4787x106 unifc 7

which leads to M,, = 10,8 1 1,193 Ib-in = 900.93 kips-ft. Subtracting from M,,, the values of moments due to dead load and superimposed dead load (375.462 kips-ft), leads to a live load moment of 525.468 kips-ft, from which the live load can be determined as 858 plf or 214.5 psf. The I representative point in Fig. 4.16 is D which shows the following coordinates: eo = 23.1 in and 1 0 ~ 1 ~ ~ very ; = 2.5, i.e., Fj= 400,000 lb = 400 kips. The reader is encouraged to check numerically in this case ft fro] that the two allowable stresses and % are attained exactly while the two others are satisfied, as the m indicated by the geometric representation: Note that such a design may have to be revised if the (Fig. a assumed value of eo 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.

4.9.2 Simply Supported T Beam with Single Cantilever on One Side w

Consider the same beam as in the previous example, that is, same section, same loading, same build t material properties, and same main span of 70 ft. However, assume that it has a cantilever on one shown side spanning 10 ft (Fig. 4.17a). 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.

L + SD

Maximum at midspan s

Maximum at support Minimum at support Minimum at midspan

Figure 4.17a Loading arrangements leading to the maximum and minimum moments. No allowab It can be shown that for the support section, C, the minimum moment is obtained when only the lines le:

own weight of the beam is considered; the maximum moment is obtained when in addition to the own

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Feasible Domain Notes