Chapter 3 Prestressing with Post-Tensioning
7.3 Strength Limit Verification—Flexure
7.3.1 Factored Loads for Longitudinal Flexure
Loads for bridge design specified by the LRFD Specifications are described in Article 3.3.2.
Those loads that act to produce bending moments in the superstructure of a concrete box girder bridge are:
• DC = dead load of structural components and nonstructural components
• DW = dead load of wearing surface and utilities
• LL = vehicular live load
• IM = vehicular dynamic allowance
• PS = secondary moments from post-tensioning
• CR = bending moments caused by concrete creep
• SH = bending moments caused by concrete shrinkage
Chapter 7 – Longitudinal Analysis & Design 147 of 369
• TU = bending moment caused by uniform temperature rise or fall
• TG = bending moments caused by thermal gradient (positive or negative) Dynamic load allowances are defined in AASHTO LRFD Article 3.6.2. Load effects caused by the application of truck and tandem loads are to be increased by a dynamic load allowance equal to:
(Eqn. 7.20) 33
1 1.33
Dynamic Load Allowance= + 100=
Moments PS, CR, SH, and TU are defined in Article 3.12 as force effects due to superimposed deformations. The secondary post-tensioning moments (PS) are those caused by the restraining effect of adjacent spans and integral substructures (see section 6.6). Considering construction of complete continuous units in a single phase, the moments caused by concrete creep, concrete shrinkage, and uniform thermal changes are induced only when the substructure is integral with the superstructure. The shortening/extension of the bridge deck displaces the piers, which in turn, induces bending moments in the superstructure (see sections 6.8, 6.9, and 6.10). For typical bridges on bearings, moments produced by concrete creep, concrete shrinkage and uniform temperature change are equal to zero.
The LRFD Specifications define five strength limit state load combinations (Strength I through Strength V). Of these five load combinations, those with wind (Strength III and Strength V) are not likely to govern with regard to maximum superstructure bending moment. The three remaining strength load combinations can be expressed as:
Strength I: (Eqn. 7.21)
Strength II: (Eqn. 7.22)
Strength IV: (Eqn. 7.23)
For bridges on bearings, these simplify to:
Strength I: (Eqn. 7.24)
Chapter 7 – Longitudinal Analysis & Design 148 of 369 Strength II: (Eqn. 7.25)
Strength IV: (Eqn. 7.26)
The Strength II limit state is intended for special owner-specified design vehicles and/or permit vehicles use to evaluate bridges. These vehicles vary with transportation agencies. The engineer should consult individual agency requirements and ensure that the newly designed bridge adequately carries these vehicles. For this manual, the information presented uses the LRFD design notional loading in the Strength I and Strength IV limit states.
Example: Using the three-span bridge in example 1 found in appendix C, whose analysis model is shown in figure 7.7, compute factored bending moments at selected locations within the bridge.
The bending moment results from analysis by the BD2 computer program in units of foot-kips are:
Table 7.1 – Example Bridge 1 Bending Moments (ft-kips)
Using the load factors of equation 7.21 and 7.23 the factored moment combinations for Strength Limit States I and IV are:
Table 7.2 – Example Bridge 1 Bending Moments (ft-kips) Loading 0.42L Span 1
Node 8
1.0L Span 1 Left of Node 16
0.0L Span 2 Right of Node 16
0.5L Span 1 Node 25
DC 15,125 -26,585 -29,178 17,761
DW 1,339 -2,353 -2,583 1,572
LL+I 12,306 -11,348 -13,778 11,795
TU 2,165 -5,067 -2,622 2,622
PS 1,964 4,594 9,834 9,799
CR 1,161 2,717 -1,543 -1,543
SH -2,642 -6,181 3,618 3,618
Strength Case
0.42L Span 1 Node 8
1.0L Span 1 Left of Node 16
0.0L Span 2 Right of Node 16
0.5L Span 1 Node 25
I 44,016 -58,023 -53,861 58,386
IV 26,262 -44,811 -37,044 42,185
Chapter 7 – Longitudinal Analysis & Design 149 of 369 7.3.2.1 Strain Compatibility
LRFD Article 5.7.3 provides guidance and requirements for the design of prestressed concrete flexural members at strength limit states. Equations leading to the computation of flexural resistance for both rectangular sections and t-beam sections are presented in LRFD Article 5.7.3. This manual presents the more general approach presented in Article 5.7.3.2.5, in which strain compatibility throughout the depth of the member is considered for determining internal forces under ultimate conditions.
Consider the rectangular prestressed concrete beam shown in figure 7.20. The beam, with height of h and width of b, is reinforced with a single layer of prestressing steel with a total area of steel equal to Aps, located a distance dp from the extreme compression fiber. The bending moment Mn is the nominal resistance of the cross section in flexure. The nominal flexural resistance is produced by the internal couple of compression in the concrete and tension in the prestressing steel, acting along their lines of action.
Figure 7.20 – Flexural Resistance by Strain Compatibility
Figure 7.20 shows a strain diagram over the depth of the member. LRFD Article 5.7.2 provides guidelines for establishing the strain diagram:
• The diagram is linear over the depth of the beam.
• The diagram passes through a neutral axis at depth (c) from the extreme compression fiber.
• The ultimate strain in the concrete (εcu) at the extreme compression fiber is equal to 0.003 in/in.
It is important to note that the prestressing steel only provides resistance in the internal equilibrium shown in figure 7.20. This is different from considering the prestressing force as a load effect in the summation of stresses at service limit states. The difference is seen in the strain in the prestressing steel at nominal flexural resistance. The strain on which the stress in the prestressing steel is computed is the sum of the strain at the effective level of stress in the prestressing plus the change in strain resulting from strain compatibility. This can be expressed as:
(Eqn. 7.27)