EXAMPLE 9.2 – Part IV PCI Bridge Design Manual 2011/12 Edition EXAMPLE 9.2 – Part IV PCI Bridge Design Manual BULB “T” (BT-72) THREE SPANS, COMPOSITE DECK LRFD SPECIFICATIONS Materials copyrighted by Precast/Prestressed Concrete Institute, 2011. All rights reserved. Unauthorized duplication of the material or presentation prohibited.

UNFACTORED SHEARS AND MOMENTS LIVE LOAD ENVELOPE INCLUDES IM FACTOR Midspan Values, symmetrical

STRENGTH LIMIT STATE Positive Moment Zones Since the dead loads and live loads produce stresses of the same sign, use maximum load factors. Find Mu at midspan. Mu = 1.25 DC + 1.5 DW + 1.75 (LL+IM) = 1.25(1391 + 2127 + 73) + 1.5 (128) + 1.75 (2115) = 8382 k-ft. This is the applied FACTORED load at midspan of a center span interior beam.

STRENGTH LIMIT STATE Positive Moment Zones Consider the section at strength limit state. Assume the stress block is entirely within the deck slab (a< 7.5”). If this is true, treat as a rectangular section.

STRENGTH LIMIT STATE Positive Moment Zones Because the stress block is assumed to be in the flange (slab), the properties of the SLAB concrete are used. Therefore, there is no need to “transform” the slab concrete to beam concrete. The actual effective width of 144” is used, not the transformed width.

STRENGTH LIMIT STATE Positive Moment Zones If the stress block had fallen into the precast beam: The section would be treated as a “T” beam. The section would be assumed to be made of “beam” concrete, so the beam concrete properties would be used. For this reason, the transformed slab and haunch widths would also be used.

STRENGTH LIMIT STATE Positive Moment Zones Equilibrium: Compression = Tension 0.85 fc’ b a = Aps fps

The value of fps can be found from: STRENGTH LIMIT STATE Positive Moment Zones The value of fps can be found from: (Eq’n 5.7.3.1.1-1) Then:

STRENGTH LIMIT STATE Positive Moment Zones c = depth of neutral axis b = width of compression block (flange in this case) Aps = area of TENSILE prestressing steel dp = depth to centroid of tensile prestressing steel k = a constant for the prestressing steel k = 0.28 for low relaxation steel Reminder: When a < ts , the stress block is in the slab. Use the effective width of the slab NOT the transformed width and use 1 of the slab concrete.

STRENGTH LIMIT STATE Positive Moment Zones If there is mild (nonprestressed) tensile steel, As, and mild compression steel, As’, and both yield (strength of fy ), the equation for c becomes: Rectangular section assumed. (Eq’n 5.7.3.1.1-4)

STRENGTH LIMIT STATE Positive Moment Zones Reminder of previously calculated values: Aps = 44 strand(0.153 in2) = 6.72 in2 fpu = 270 ksi dp = beam depth+haunch+slab-ybs = 72” + 0.5” + 7.5” – 5.82” = 74.18” ybs = distance from bottom of beam to centroid of prestressing steel. b = 144” fc’ = 4.0 ksi 1 = 0.85 (for 4 ksi concrete) k = 0.28 for low relaxation strand

STRENGTH LIMIT STATE Positive Moment Zones OK – Stress Block in Flange – Rect. Section

STRENGTH LIMIT STATE Positive Moment Zones Find the approximate stress in the prestressing steel: This equation gives the approximate stress in the prestressing steel. A more accurate value can be found from strain compatibility (see the PCI Design Handbook)

STRENGTH LIMIT STATE Positive Moment Zones From Equilibrium:

STRENGTH LIMIT STATE Positive Moment Zones The complete moment equation, assuming prestressed and nonprestressed tensile steel, compression steel and a flanged section (T beam) is given by Eq’n 5.7.3.2.2-1 : ds is the depth from the compression fiber to the tensile mild steel and ds’ is the depth from the compression fiber to the compression mild steel, bw is web width and hf is flange thickness. In this design example: As = As’= 0 and b = bw , so the equation simplifies to Mn = Apsfps(dp – a/2)

STRENGTH LIMIT STATE Positive Moment Zones OK for Strength Limit State in (+) Moment Zones IF the section is tension controlled.

STRENGTH LIMIT STATE Negative Moment Zones Since the dead loads and live loads produce stresses of the same sign, use maximum load factors. Also, only the loads carried as by the continuous span cause negative moment. Thus, the only DC is the barrier: Mu = 1.25 DC + 1.5 DW + 1.75 (LL+IM) = 1.25(-197) + 1.5 (-345) + 1.75 (-2328) = -4837 k-ft. This is the applied FACTORED load over the pier of a center span interior beam.

STRENGTH LIMIT STATE Negative Moment Zones In the negative moment area, the bottom of the bottom flange is the compression area (use BT72 concrete properties). The tensile steel is placed in the deck.

STRENGTH LIMIT STATE Negative Moment Zones Consider the section at strength limit state. Assume the stress block is entirely within the bottom flange (a< 6”). If this is true, treat as a rectangular section.

STRENGTH LIMIT STATE Negative Moment Zones The general equation for equilibrium is:

STRENGTH LIMIT STATE Negative Moment Zones a = As fy / 0.85 fc’ b This is the general equation with Ap = As’ = 0 and a= 1c.

STRENGTH LIMIT STATE Negative Moment Zones Once again, the complete moment equation, assuming prestressed and nonprestressed tensile steel, compression steel and a flanged section (T beam) is given by Eq’n 5.7.3.2.2-1: In the negative moment case, Ap = As‘= 0 and b = bw , so the equation simplifies to: Mn = Asfs(ds – a/2)

Mn = As fy (ds – a/2) In this equation, As and a are unknown. STRENGTH LIMIT STATE Negative Moment Zones Mn = As fy (ds – a/2) In this equation, As and a are unknown.

STRENGTH LIMIT STATE Negative Moment Zones Although As and “a” are unknown, it is possible to estimate “a”. Since a is usually an order of magnitude smaller than d, even a gross error in “a” leads to a small error in the moment arm, ds -a/2, and a reasonably accurate estimate of As.

STRENGTH LIMIT STATE Negative Moment Zones  = 0.9 for reinforced concrete in flexure IF tension controlled. The bottom flange is 6” high. We want to keep the assumption of a rectangular section and it is better to overestimate “a” as this will underestimate the moment arm. Assume a = 6”

If the steel centroid is in the center of the slab: STRENGTH LIMIT STATE Negative Moment Zones If the steel centroid is in the center of the slab: ds = 72” + 0.5” + 7.5”/2 = 76.25”

STRENGTH LIMIT STATE Negative Moment Zones The bars should be placed within the lesser of: The effective flange width = 144” 1/10 of the average length of adjacent spans= [(119+120)/2](12)/10 = 143” Note that if the 1/10 span controls, additional, nonstructural bar is required in the areas outside of 1/10 span width but within the effective width. The amount of bar required is 0.4% of the area outside of the width determined by 1/10 span. In this case, the two are practically equal. It would be impractical to supply additional steel in the extra 1/2 inch on side with the required amount of steel, which would be 0.03 in2.

STRENGTH LIMIT STATE Negative Moment Zones The bars are be placed within the effective width of 144”. The bridge design handbook suggests: #5 @ 12” Top mat = 9 bars X 0.31 in2 = 2.79 in2 #4 @ 12” Bottom mat = 9 bars X 0.22 in2 = 1.99 in2 #7 Split between top and bottom mat = 18 bars x 0.6 in2 = 10.8 in2 Total = 15.6 in2 #7 placed between each #4 on bottom mat and between each #5 on top mat. Spacing is 8 in c/c. Now ds = 75.6 in.

STRENGTH LIMIT STATE Negative Moment Zones As =15.6 in2: This is close enough to 6 inch that the section can be assumed a rectangle.

OK, IF the section is tension controlled. STRENGTH LIMIT STATE Negative Moment Zones Check Maximum Moment: OK, IF the section is tension controlled.

STRENGTH LIMIT STATE Tension controlled, compression controlled and transition sections are defined by the strain in the extreme tension steel at Mn. Strain in the extreme tensile steel is defined in the diagram. This definition applies to prestressed and non-prestressed steel.

TENSION CONTROLLED SECTION A section is TENSION CONTROLLED if the extreme steel strain > 0.005. This applies to prestressed and non-prestressed steel. If tension controlled  = 1 for prestressed  = 0.9 for non-prestressed  is interpolated for partially prestressed.

COMPRESSION CONTROLLED SECTION A section is COMPRESSION CONTROLLED if the extreme steel strain < balanced. For non-prestressed steel, the limit is fy/Es For prestressed steel, the limit is 0.002. If compression controlled,  = 0.75 for members with ties or spirals.

TRANSITION SECTION A transition section is has an extreme tensile steel strain between tension and compression controlled. For transition sections,  is interpolated based in extreme tensile steel strain.

SECTION  FACTORS Definition of  for prestressed steel and GR 60 non-prestressed steel.

SECTION  FACTORS Check for Tension Control: The tension control limit can be found as: So a section is tension controlled if:

SECTION  FACTORS In the prestressed girder (positive moment zone), it was found that: c = 4.30 inches dt = 72+7.5-2 = 77.5 in c/ dt = 4.30/77.5 = 0.055 < 0.375 Tension controlled  = 1.0

SECTION  FACTORS In the reinforced (negative moment) section c = a/1 = 6.04/.7 = 8.60” from the calculation of Mn Assume 2.5 inches clear cover to the #5 top mat steel: dt = 72 + 7.5 – 2.5 – (5/8)/2=76.7 in c/dt = 8.60 / 76.7 = 0.112 < 0.375 Tension Controlled

MINIMUM STEEL POSITIVE MOMENT SECTION Article 5.7.3.3.2 requires: Mn =Mr > lesser of 1.2 Mcr or 1.33Mu Mcr = Cracking Moment 1.33Mu = 1.33 (8381 k-ft) = 11150 k-ft

MINIMUM STEEL POSITIVE MOMENT SECTION Equation 5.7.3.3.2-1

MINIMUM STEEL POSITIVE MOMENT SECTION

MINIMUM STEEL POSITIVE MOMENT SECTION Since 1.2Mcr = 7837 k-ft < 1.33Mu = 11147 k-ft 1.2Mcr controls. Mn = 10649 k-ft > 1.2Mcr = 7837 k-ft OK

MINIMUM STEEL NEGATIVE MOMENT SECTION

DEVELOPMENT LENGTH OF PRESTRESSING STEEL Eq’n 5.11.4.2-1 The term  = 1.6. (5.11.4.2) The prestressing steel must be embedded 10.6 ft from the point of maximum stress. The point of maximum stress is at midspan and the embedment is 59.5 ft. OK

CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB According to Article 5.7.3.4 the spacing of the mild steel reinforcement in the layer closest to the tension face shall satisfy equation 5.7.3.4-1.

CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB Exposure factor = 1.00 for Class 1 exposure 0.75 for Class 2 exposure condition fs = Tensile stress in steel reinforcement at the service limit state, ksi dc= Thickness of concrete cover measured from extreme tension fiber to center of the flexural reinforcement located closest therto βs= h= Overall height of the section, in

CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB To find fs, the cracked moment of inertia is needed:

CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB Use bottom as reference point:

CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB

CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB Mserv = -197k-ft - 345 k-ft - 2328 k-ft = -2870 k-ft = - 34400 k-in If 2.5 in cover and Class 1 ( = 1) dc = 2.94 in = 2.5 + (7/16)

CONTROL OF CRACKING BY DISTRIBUTED REINFORCING IN THE SLAB So maximum spacing = 15. 3 inches for Class 1 ( = 1) 10.0 inches for Class 2 ( = 0.75) Actual spacing is 8 inches c/c. OK