chapter 12: computing continuity diaphragm moments using … · which computes moments in...

Computing Continuity Diaphragm Moments Using DCALC

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Chapter 12: Computing Continuity Diaphragm Moments Using DCALC By Karl Hanson, S.E., P.E.*

August 2013 12.1 Introduction: This paper explains the methodology used in the DCALC program, “CONDIAPH” , which computes moments in continuity diaphragms used in prestressed concrete bridges. Multi-span precast prestressed concrete beam bridges are commonly designed to act as continuous structures by pouring the concrete deck without expansion joints. It has been common practice to design the beams using the following assumptions:

Dead load of beam and slab are carried by simply supported beams Superimposed dead loads and live loads are carried by the continuous structure

Although this design methodology has been a long standing practice, in recent years a concern has been raised about the structural integrity of the concrete diaphragms. To understand the issue in question, we first need to describe the stages of construction involved with this type of bridge.

The fabrication process begins at the precasting yard. It is common practice to cast a very long line of beams end-to-end between forms. The prestressing strands are anchored at the ends of the line, and are pre-tensioned a few hours before pouring the concrete.. The concrete used in the beams is typically high-early-strength concrete which is heat cured. It is not unusual to achieve the necessary “28 day” concrete strength in only 1 day. After the beams have reached the proper strength, the forms are stripped. The beams and strands are then sawed into individual beams, causing the force to be transferred to the beam concrete. The beams are then lifted from the stressing bed and are stored in the yard before being transported to the project site.


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With time, as the beams mature, they tend to increase in upward camber. The weight of the deck will cause the beams to deflect downward. Typically, there is a net upward increase in camber. If unrestrained, the beam ends will rotate as the beam/slab increases in camber. However, due to the continuity of the deck and diaphragm system, beam end rotation is resisted by moments at the continuity diaphragms. Referring to the sketch, a moment is “positive”, if the beams are tending to camber upward. The moment that develops in the continuity diaphragm depends on how old the beams are when the deck is poured. In general, the younger the beam is, the larger will be the continuity diaphragm moment. A concern has been raised that, if the moment in the continuity diaphragm is positive, then cracks may develop at the bottom of the diaphragm, if the diaphragm is under-reinforced. If the diaphragm does crack at the bottom, then the beam-slab system may not act as a continuous structure – that is, not until the beam deflects enough for the crack to close. AASHTO LRFD Specification Article 5.14.1.2.7, describes requirements for diaphragms which the designer must check:

For an extremely thorough discussion of this topic, refer to NCHRP Report 519, “Connection of Simple Span Precast Concrete Girders for Continuity” (Ref. 5)


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12.2 Time Dependent Material Properties To predict how these materially depend properties vary with time, this paper will use equations presented in Chapter 2 of the “PCI BRIDGE DESIGN MANUAL” (Ref. 1). 12.2.1 Strength of Concrete:

12.2.2 Modulus of Elasticity of Concrete:

For concrete strength greater than 8,000 psi,


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12.2.3 Shrinkage of Concrete:


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12.2.4 Creep of Concrete:


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12.4 A Simple Example of Time Loss Calculation For plain concrete, it is an easy matter computing the strains in a structural member.

Referring to above sketch showing a column, the total strain is,

εtotal = εelastic + εcreep + εshrinkage = εelastic+ εelastic * C(t,t0) + εshrinkage = εelastic (1 + C(t,t0)) + εshrinkage = s/(Ec)t * (1 + C(t,t0)) + εshrinkage

The total shortening is, Δ = εtotal * H Similarly, for a plain concrete beam, it is an easy matter computing flexural strains:

Stress at top of beam, stop = M/St

Stress at bottom of beam, sbot = M/Sb

Elastic strain at top of beam, εelastic, top = stop /(Ec)t

Elastic strain at bottom of beam, εelastic, bot = sbot /(Ec)t

Total strain at top of beam, εtotal, top = stop/(Ec)t * (1 + C(t,t0)) + εshrinkage

Total strain at bottom of beam, εtotal,bot = sbot/(Ec)t * (1 + C(t,t0)) + εshrinkage


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With reinforced concrete members, the problem of how to compute strains becomes less straight forward. Whereas concrete is subject to shrinkage and creep, steel is not. The steel will tend to resist strain changes in the concrete, due to strain compatibility.

Referring to the above sketch of a reinforced concrete column, the tendency of the concrete to shorten is resisted by the steel reinforcing bars.

Similarly, for a beam subject to flexure, the reinforcing bars will resist any tendency of the concrete to shorten (or lengthen). Referring to the above sketch, a reinforcing bar located above the neutral axis is shown subject to shortening, which will cause a compressive stress in the bars. A step-by-step time analysis is an approach for solving these types of problems, Using relatively short time intervals, we can iterate the solution as follows:

1. Compute concrete strains (ignoring steel) for all days considered 2. Starting with t=0, continuing for each day to be considered,

compute the increment in total strain: Δε = εtotal (current) - εtotal (last) 3. Compute the incremental increase in steel force: ΔP = Δε * Es * As 4. Apply “-ΔP” as a negative (tension) force to the section 5. Compute incremental strains due to “-ΔP” from current day to all future days 6. Add strains to concrete strains computed in Step 1


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The below shows a simple spreadsheet calculation (adapted from Example 8.13.2.2.2 in then PCI Bridge Design Handbook): Column is 12"x12" w/(4)‐#9 bars, loaded with 216 k.

fc'= 3500 psi

Ec = 3372.165 ksi Cu= 1.65

n=Es/Ec= 8.599815 Su= 0.0004

Aeff = 174.3993 sq. in. (transformed area)

P0= 216 k

s = 1.238537 ksi

Reinf. Conc.

T Elastic Creep Creep Shrinkage Total Reinf. Force Force

(days) strain Coef. C Strain Strain Strain Stress DP P0‐DP

0 0.000367 0 0 0 0.000367 10.65119 42.60477 173.3952

1 0.000367 0.15 5.51E‐05 1.11E‐05 0.000433 12.57109 50.28437 165.7156

2 0.000367 0.217176 7.98E‐05 2.16E‐05 0.000469 13.5914 54.36559 161.6344

3 0.000367 0.267301 9.82E‐05 3.16E‐05 0.000497 14.41406 57.65622 158.3438

4 0.000367 0.308253 0.000113 4.1E‐05 0.000522 15.12419 60.49678 155.5032

5 0.000367 0.343227 0.000126 0.00005 0.000543 15.75697 63.0279 152.9721

6 0.000367 0.373913 0.000137 5.85E‐05 0.000563 16.33138 65.3255 150.6745

7 0.000367 0.401333 0.000147 6.67E‐05 0.000581 16.85921 67.43682 148.5632

8 0.000367 0.426164 0.000157 7.44E‐05 0.000598 17.34849 69.39396 146.606

9 0.000367 0.448881 0.000165 8.18E‐05 0.000614 17.80504 71.22016 144.7798

10 0.000367 0.469833 0.000173 8.89E‐05 0.000629 18.23325 72.93301 143.067

11 0.000367 0.489284 0.00018 9.57E‐05 0.000643 18.63657 74.54627 141.4537

12 0.000367 0.507442 0.000186 0.000102 0.000656 19.01776 76.07105 139.929

13 0.000367 0.524472 0.000193 0.000108 0.000668 19.37911 77.51645 138.4836

14 0.000367 0.540507 0.000199 0.000114 0.00068 19.72252 78.89009 137.1099

15 0.000367 0.555658 0.000204 0.00012 0.000691 20.04961 80.19846 135.8015

16 0.000367 0.570018 0.000209 0.000125 0.000702 20.36178 81.44712 134.5529

17 0.000367 0.583664 0.000214 0.000131 0.000712 20.66022 82.64089 133.3591

18 0.000367 0.596665 0.000219 0.000136 0.000722 20.946 83.78401 132.216

19 0.000367 0.609076 0.000224 0.000141 0.000732 21.22006 84.88023 131.1198

20 0.000367 0.620948 0.000228 0.000145 0.000741 21.48321 85.93285 130.0672

21 0.000367 0.632325 0.000232 0.00015 0.00075 21.73621 86.94484 129.0552

22 0.000367 0.643246 0.000236 0.000154 0.000758 21.97972 87.91888 128.0811

23 0.000367 0.653744 0.00024 0.000159 0.000766 22.21434 88.85738 127.1426

24 0.000367 0.66385 0.000244 0.000163 0.000774 22.44063 89.76251 126.2375

25 0.000367 0.673591 0.000247 0.000167 0.000781 22.65907 90.63627 125.3637

26 0.000367 0.682991 0.000251 0.00017 0.000789 22.87012 91.48048 124.5195

27 0.000367 0.692072 0.000254 0.000174 0.000796 23.0742 92.29678 123.7032

28 0.000367 0.700854 0.000257 0.000178 0.000802 23.27168 93.08673 122.9133


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Note that, with time, the portions of force carried by the concrete versus steel changes considerably.

Initially the concrete carries 173.4 k At 28 days, the concrete force diminishes to 123 k at 28 days. At 90 days (not shown), the concrete force is 98 k At 2005 days (not shown), the concrete force is 64 k.

12.5 Approach Used By DCALC Although the preceding simple example was easily computed using a spreadsheet, a similar analysis for a beam becomes unwieldy using a spreadsheet. Although similar in concept to the example, prestressed concrete bridge beams have many additional factors to consider:

Stresses and strains for a beam need to be computed at multiple sections in order to compute camber and continuity moments

Both axial force and moment are to be considered Prestress force loss must be considered An estimate of the time that the deck is constructed is a factor Construction of the deck changes the stiffness properties of the beam

Obviously the computation of continuity moments is not an easy task. A time-step analysis, written in a programming language, is ideally suited as the most efficient and modern approach for this type of problem. Almost any degree of complexity can be handled using a computerized approach, with the power to compute factors for each day considered. DCALC’s program “CONDIAPH” was written to compute the continuity moments in bridges of this type.

CONDIAPH was written to read the output from DCALC’s prestressed beam design program (PCBRIDGE).

The user will enter information about curing methods of the beam and the deck. The user will enter an assumed day of deck pour CONDIAPH computes the continuity moments at all piers CONDIAPH shows if the combined moments of continuity, dead loads and 50%

maximum positive live load satisfy the AASHTO LRFD Article 5.14.1.2.7 requirements


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12.4 Description of Time-Dependent Analysis Approach The total concrete strain at any time, t, can be separated into three components:

εf = the immediate strain due to applied stress, f εcr = the time-dependent creep strain εsh = free shrinkage strain

We will be computing the strains in the top and bottom fibers of the prestressed concrete beams, at every 10th point along each span, and storing these two strain values for each day to be considered.

Positive Sign Conventions Used

The above sketch shows positive sign convention that will be used for strains, moments and prestressing force eccentricity. The prestressing eccentricity, “ep”, is shown positive above the neutral axis. (Typically prestressing is located below the neutral axis; therefore in that case “ep” will be negative.) The principle of superposition will be used, where the above strains are computed for each separate loading case and are added. The following loading cases will be considered:

Step 1: Beam Weight and Initial Prestress Step 2: Weight of Deck Pour Step 3: Prestress Losses and Deck Forces After Deck Pour Step 4: Calculation of Continuity Moments

Strains will be computed at the top and bottom of the beam, in a very simple manner, using algorithms that are repeated from day to day.


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Step 1: Calculate Strains Due to Beam Weight and Initial Prestress: When the prestressing steel force is transferred to the beam, the beam is subject to a axial force (due to the prestress force, P) and a moment (due to P*ep and the beam dead load moment, Mbeam) . Initial losses in the prestress force are due to relaxation, Lr , and elastic shortenting, ES: Lr = log10(24*t)/Kr *(fi/fpy – 0.55)* fi (Eq. 8.6.5.3-1) ES = εprestress * 28500 where, for low relaxation steel, Kr = 45, fpy = 243 (Table 8.13.1.4-1) for stress-relieved steel, Kr = 10, fpy = 229.5 Prestress stress after release, fp0 = fpi – Lr – ES Prestress force at release, Pi = fp0 * AP * Nstrands The algorithm for this step is as follows:

1. Compute beam dead load moment at 10th points of each span 2. Assume an initial prestress, fi. 3. Compute Lr 4. Compute Pi 5. Compute initial stresses and strains:

Initial stresses are, ftop = Pi/A + (Mbeam + Pi*ep)/St

fbot = Pi/A - (Mbeam + Pi*ep)/Sb Initial strains are, εtop = ftop/Ec (at time of prestressing)

εbot = fbot/Ec

6. Compute εprestress at each prestressing level 7. Go to Step 3. Iterate this process several times 8. Compute creep strains, εcr,top = εtop * C(t,t0 ) (where “t0”=day of transfer )

εcr,bot = εbot * C(t,t0 ) 9. Compute beam shrinkage strains

Store the above strains in the following arrays:

ElasticStrain(N%, J%, T%).Top and ElasticStrain(N%, J%, T%).Bot CreepStrain(N%, J%, T%).Top and CreepStrain(N%, J%, T%).Bot ShrinkageStrain(N%, J%, T%).Top and StrainStrain(N%, J%, T%).Bot

The above uses VisualBasic’s nomenclature “%” for integers, where N% = Span number J% = Point in span (J%=1 is beginning of span, J%=11 is end of span) T% = Days since the beam concrete was pour (CONDIAPH uses a maximum of 1890 days)


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Step 2: Calculate Strains Due to Deck Weight: After the beams have been transported to the construction site and have been erected, the deck is formed and poured. The weight of the deck is initially carried by the non-composite beam section. The algorithm for this step is as follows:

1. Compute deck dead load moment at 10th points of each span 2. Compute increases in stresses and strains due to deck dead load:

Stress increases are,

Δftop = Mdeck /St Δfbot = - Mdeck /Sb

Strain increases are,

Δεtop = Δftop/Ec (using Ec at time of deck pour) Δεbot = Δfbot/Ec

3. Compute creep strains, Δεcr,top = Δεtop * χ* (C(t,t0 ) - C(t,tpour))

Δεcr,bot = Δεbot * χ *(C(t,t0 ) - C(t,tpour)) where “χ” = “aging factor” (see below) “tpour”=day of deck pour

4. Add the above strain increase to the current values of strains: εtop = εtop (previous) + Δεtop εbot = εbot (previous) + Δεbot

εcr,top = εcr,top (previous) + Δεcr,top εcr,bot = εcr,bot (previous) + Δεcr,bot

Concrete Aging Factor, “χ”: Referring back to the PCI Bridge Design Handbook, Equation 2.5.8.1-2a, the ultimate creep coefficient is defined as,

Cu = 1.88*kc = 1.88*kla * kh * ks Of these factors, “kla” is a correction for loading age. It is convenient to compute the one ultimate creep coefficient, Cu, using kla for the beam concrete at the time of prestress force transfer. Then, for loadings occurring after the transfer, to adjust this constant by an aging factor.

Cu = χ * Cu, at time of transfer

The value of χ can be determined by the ratio of kla constants,

χ = kla (at time of loading)/kla (computed at time of transfer)

(Note: A value of χ = 0.8 of has commonly been used.)


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Step 3:Calculate Prestress Losses and Deck Forces Increase After Deck Pour: After the deck is poured, the beam will be subjected the additional weight of the deck. As the beam cures, the stiffness properties of the beam change from non-composite beam properties to composite beam/deck properties.

The algorithm for this step is as follows:

1. For all days after the deck pour, compute the following: Beam concrete (f’c,beam)t and (Ebeam)t Deck concrete (f’c,deck)t and (Edeck)t Composite section properties, Ac,Yna, Sct, Scb based on n=Edeck/Ebeam Deck “free” shrinkage strain, εsh, deck

2. Using the strains, εtop and εbot, computed in Step 1, compute the strain, εp ,at the

level of prestressing strands.

3. For each level of strands, compute the incremental prestress force loss, ΔP = -εp * 28500 * Ap (tension force if εp > 0)

4. Using the strains, εtop and εbot, computed in Step 1, compute the strain at centroid

of the deck, εdeck.

5. For each day, compute the differential change in deck force, ΔN = -(εdeck – εsh, deck) * (Ebeam)t * Adeck (tension force if εdeck – εsh, deck > 0)

6. Using the section properties of the composite beam, the increase in stresses at the top and bottom of the beam are computed as, Δftop = (∑ΔP + ΔN)/Ac + (∑ΔP*ep + ΔN*ed)/Sct (“∑”=sum ΔP for all strands) Δfbot = (∑ΔP + ΔN)/Ac - (∑ ΔP*ep + ΔN*ed)/Scb

7. Elastic strain increases, Δεtop = Δftop/Ec (constant for all days in future)

Δεbot = Δfbot/Ec

Creep strain increases, Δεcr, top = Δεtop * χ*(C(t,t0) – C(t,tcurrent)) Δεcr,bot = Δεbot * χ*(C(t,t0) – C(t,tcurrent))

8. Add the above strains to strains computed in Step 1.


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Step 4: Calculate Continuity Moments: At this point of the calculation, the total strains have been computed at the top and bottom of the beam at tenth points, for every day:

εtotal, top = εtop + εcr, top + εsh, top

εtotal, bot = εbot + εcr, bot + εsh, bot

Recalling the sketch on p. 12-6, we are now in a position to compute the rate of curvature, beam rotations and deflections:

The rate of curvature per unit length is,

dϕ/dl = (εtotal, top - εtotal,bot )/ H (where H = beam depth) The algorithm for this step is as follows:

1. For each day, simple span beam end rotations are computed using the conjugate beam method, This involves computing a simple numerical integration of the rates of curvature.

2. After the deck is poured, the incremental increase in beam end rotations are computed with respect to the slopes of the continuity diaphragms.

3. After computing the increase in simple span beam end rotations, the following fixed end moments are calculated: ML = 4*Ec*I/L*(-Δ∅L) + 2*Ec*I/L*(-Δ∅R) (Eq. 8.13.4.3.1-1)

MR = 2*Ec*I/L*(-Δ∅L) + 4*Ec*I/L*(-Δ∅R) (Eq. 8.13.4.3.1-2) (These FEMS will rotate the beam slopes back into alignment with the slope of the continuity diaphragms)

4. A moment distribution analysis is performed for the continuous structure using the above fixed end moments. This yields the incremental increase in continuity moments.

5. After solving for the incremental continuity moments for each day, the beam strains are adjusted for the effects of the incremental increase in continuity moments.

6. Go to Step 3, for next day considered 7. The final continuity moments are the sum of all the incremental increases in

continuity moments for all the days considered.


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12.5 A Design Example: The following example has been taken from NCHRP Report 519, “Connection of Simple-Span Precast Concrete Girders for Continuity” (Ref. 5), as “Design Example 1: AASHTO Type III Girder”. This bridge is a two span precast prestressed girder bridge. The bridge was analyzed using a program called “Restraint”. Details from this publication are shown below. Continuity restraint moments were computed at the pier for three cases of deck pours: 28 days, 60 days and 90 days.

(From Reference 5)


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(From Reference 5)

(From Reference 5) DCALC’s program “CONDIAPH” input: In order to make this comparison, some assumptions needed to be made to simplify the input. The design example is lengthy, describing strand pattern optimization. For this example, in the DCALC program “PCBRIDGE”, one strand pattern was assumed, using the one shown above in Figure D-4.1.2-1. Also, the design example uses different concrete strengths for each case. For this example, in the DCALC program “PCBRIDGE”, the following concrete strengths were assumed for all three cases:

At initial prestress, f’ci = 5.13 ksi At service, f’c=7.00 ksi


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(CONDIAPH input data)


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(CONDIAPH output for deck poured at 28 days)


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Comparison of restraint moments computed in NCHRP 591 and CONDIAPH:

(CONDIAPH results shown superimposed on results from Reference 5) As shown in the above comparison, the results appear to be reasonably close. As stated, the design example is somewhat lengthy, therefore some simplifications were necessary in the CONDIAPH input. The upper limit of days used by CONDIAPH has arbitrarily been set to 1890 days – approximately 5 years. We see that the NCHRP researchers have extended their analysis for a substantially longer period, up to 8000 days – approximately 22 years. We can see that in the range from 1890 days to 8000 days, there is a slight increase in moments. Summary: The current AASHTO LRFD Specifications require bridge designers to check for positive moments in continuity diaphragms, if the deck is poured before the beams have reached an age of 90 days. The computation of continuity moments typically requires an involved time step analysis. For this purpose, DCALC’s “CONDIAPH” program computes continuity diaphragm moments. The results shown in this paper appear to agree reasonably well with a published example. The CONDIAPH program was written to be used with the DCALC prestressed beam design program, “PCBRIDGE”, making this analysis a simple process.


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List of References: Reference 1: PCI “Precast Prestressed Concrete Bridge Design Manual”, Chapters 2 (dated October 97) and Chapter 8 (dated July 03) Reference 2: “Design Recommendations for the Optimized Continuity Diaphragm for Prestressed Concrete Bulb-T Beams”, by Stephanie Koch and Carin L. Roberts-Wollman, Department of Civil and Environmental Engiineering, Virginia Polytechnic Institute & State University, November 2008 Reference 3:”Prestressed Concrete Bridges”, by Christian Menn, originally pulished 1986 under the title “Stahlbetonbrücken” by Springer-Verlag, Wien Reference 4: “Creep Analysis of Prestressed Concrete Structures Using Creep-Transformed Section Properties”, by Walter H. Dilger, published in the PCI Journal, January-February 1982

Reference 5: NCHRP Report 519, “Connection of Simple-Span Precast Concrete Girders for Continuity”, by Miller, Castrodale, Mirmiran and Hastak, published 2004 by the Transportation Research Board (this publication can be downloaded at http://www.nap.edu)

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