Geotechnical Engineering Circular No. 9 Design, Analysis, and Testing of Laterally Loaded Deep Foundations that Support Transportation Facilities
Figure 11-2: Nominal and factored interaction diagrams
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- Figure 11- 3: Variation of φ with net tensile strain, t and dt/c for grade 60 reinforcement.
- 11.3.9 Axial Compression and Biaxial Bending for Non-Circular Members
| Figure 11- 2: Nominal and factored interaction diagrams. The interaction diagram uses a resistance factor (φ) that is variable and is determined by the strain conditions in the structural cross-section, at nominal strength. Therefore, resistance factors are different for compression-controlled and tension-controlled sections. Sections are considered tension controlled if the tensile strain (in the extreme tensile steel) at nominal strength is greater than 0.005. A value of 0.9 is used as φ fora tension-controlled section. A “compression-controlled” section uses a φ of 0.75 and is defined as a cross-section for which the net tensile strain (tin the extreme tensile steel at nominal strength is less than or equal to the compression controlled strain limit of 0.003 (refer to AASHTO Articles 5.7.2.1 and C 2014). Linear interpolation is used to determine ϕ for sections that transition between tension-controlled and compression-controlled (see plot in Figure 11-3). The transition formula for φ can also be given by Equation 11-12: 0.75 ≤ 𝜑𝜑 = 0.65 + 0.15 � 𝑑𝑑 𝑡𝑡 𝑐𝑐 − 1� ≤ Equation 11-12) Where c = Distance from extreme compression fiber to the neutral axis (ind t Distance from extreme compression fiber to the centroid of the extreme tensile element (in. 155 Figure 11- 3: Variation of φ with net tensile strain, t and dt/c for grade 60 reinforcement. Cases involving combined axial tension and bending are analyzed by applying the same concepts described above for combined axial compression and bending. A notable difference would be that the Strength Limit State is always tension-controlled; therefore, the resistance factor is φ = 0.90. 11.3.9 Axial Compression and Biaxial Bending for Non-Circular Members For axial compression loading, the factored Structural Limit State is taken. indicated in Equations 11-10 and 11-11. To determine the nominal compressive resistance, a straightforward calculation is performed considering either spiral or tie reinforcement. As mentioned in the AASHTO (2014) commentary, reduction factors are placed on the respective equations to account for unintended eccentricity. Further details on axial resistance of concrete piles can be found in Article 5.7.4.4 of AASHTO (2014). For members with spiral reinforcement 𝑃𝑃 𝑚𝑚 = 0.85 �0.85𝑓𝑓 𝑐𝑐 ′ �𝐴𝐴 𝑔𝑔 − 𝐴𝐴 𝑠𝑠𝑡𝑡𝑐𝑐 − 𝐴𝐴 𝑝𝑝𝑠𝑠 � + 𝐹𝐹 𝑦𝑦𝑐𝑐 𝐴𝐴 𝑠𝑠𝑡𝑡𝑐𝑐 − 𝐴𝐴 𝑝𝑝𝑠𝑠 �𝑓𝑓 𝑝𝑝𝑒𝑒 − 𝐸𝐸 𝑠𝑠𝑡𝑡 𝜀𝜀 𝑐𝑐𝑢𝑢 �� Equation 11-13) For members with tie reinforcement 𝑃𝑃 𝑚𝑚 = 0.80 �0.85𝑓𝑓 𝑐𝑐 ′ �𝐴𝐴 𝑔𝑔 − 𝐴𝐴 𝑠𝑠𝑡𝑡𝑐𝑐 − 𝐴𝐴 𝑝𝑝𝑠𝑠 � + 𝐹𝐹 𝑦𝑦𝑐𝑐 𝐴𝐴 𝑠𝑠𝑡𝑡𝑐𝑐 − 𝐴𝐴 𝑝𝑝𝑠𝑠 �𝑓𝑓 𝑝𝑝𝑒𝑒 − 𝐸𝐸 𝑠𝑠𝑡𝑡 𝜀𝜀 𝑐𝑐𝑢𝑢 �� Equation 11-14) 156 Where n Nominal compressive resistance (kips. f c ′ = Concrete compressive strength at 28 days, unless otherwise specified (ksi). f pe = Effective stress in the prestressing steel after losses (ksi). F yr = Yield stress of reinforcing steel (ksi). Ag Gross cross-sectional area (in. A str = Cross sectional area of longitudinal reinforcement (in. A ps = Cross sectional area of prestressing steel (in. Est Elastic modulus of prestressing steel (in. cu Failure strain of concrete in compression (in/in). Biaxial flexural resistance must satisfy the following checks. Additional information maybe found in Article 5.7.4.5 of the AASHTO (2014) specifications. If : 𝑃𝑃 𝑢𝑢 ≥ 0.10ϕ𝑓𝑓 𝑐𝑐 ′ 𝐴𝐴 𝑔𝑔 1 𝑃𝑃 𝑐𝑐𝑥𝑥𝑦𝑦 = 1 𝑃𝑃 𝑐𝑐𝑥𝑥 + 1 𝑃𝑃 𝑐𝑐𝑦𝑦 − 1 ϕ 𝑃𝑃 0 ≤ 1.0 Equation 11-15) In which 𝑃𝑃 0 = 0.85𝑓𝑓 𝑐𝑐 ′ �𝐴𝐴 𝑔𝑔 − 𝐴𝐴 𝑠𝑠𝑡𝑡𝑐𝑐 − 𝐴𝐴 𝑝𝑝𝑠𝑠 � + 𝐹𝐹 𝑦𝑦𝑐𝑐 𝐴𝐴 𝑠𝑠𝑡𝑡𝑐𝑐 − 𝐴𝐴 𝑝𝑝𝑠𝑠 (𝑓𝑓 𝑝𝑝𝑒𝑒 − 𝐸𝐸 𝑝𝑝 𝜀𝜀 𝑐𝑐𝑢𝑢 ) If 𝑃𝑃 𝑢𝑢 < 0.10ϕ𝑓𝑓 𝑐𝑐 ′ 𝐴𝐴 𝑔𝑔 : � 𝑀𝑀 𝑢𝑢𝑥𝑥 𝑀𝑀 𝑐𝑐𝑥𝑥 + 𝑀𝑀 𝑢𝑢𝑦𝑦 𝑀𝑀 𝑐𝑐𝑦𝑦 � ≤ 1.0 (Equation 11-16) Where: P u = Factored axial load. P rx = Factored axial resistance determined on basis that only eccentricity, ey, is present (kips. P ry = Factored axial resistance determined on basis that only eccentricity, ex, is present (kips. P rxy = Factored axial resistance in biaxial flexure (kips. M ux = Factored flexural moment about x-axis (kip-in). M rx = Factored flexural resistance about x-axis (kip-in) (AASHTO (2014) Section 8.5.2.3). M uy = Factored flexural moment about y-axis (kip-in). M ry = Factored flexural resistance about y-axis (kip-in) (AASHTO (2014) Section 8.5.2.3). 𝜑𝜑 𝑐𝑐 = Resistance factor for axial compression (AASHTO (2014) Table 8-6). 𝑓𝑓 𝑐𝑐 ′ = Concrete compressive strength at 28 days, unless otherwise specified (ksi). f pe = Effective stress in the prestressing steel after losses (ksi). F y r = Yield stress of reinforcing steel (ksi). Ag Gross cross-sectional area (in. 157 t Cross sectional area of longitudinal reinforcing steel ins Cross sectional area of prestressing steel (in. 𝐸𝐸 𝑝𝑝 = Elastic modulus of prestressing steel (in. u = Failure strain of concrete in compression (in. P 0 = Nominal axial resistance of a section at 0.0 eccentricity. The analysis of the prestressed concrete section’s response to a combination of an axial load and two orthogonal moments is complex. The concrete and the prestressing steel stress-strain relationships are assumed. For concrete, assume a maximum concrete strength of 0.85f’ c to include loading time effects on the concrete strength and all points on the stress strain curve are reduced to 85 percent of the short time values. Bi-axial interaction diagrams are determined for each of an increasing set of axial loads up to the maximum axial strength condition. An illustration of one of these interaction diagrams fora particular axial load is shown in Figure 11-4. These diagrams are determined for the entire range of axial loads up to the axial failure case. With increasing axial load the maximum moment strength becomes smaller. A three- dimensional interaction diagram can then be constructed with the axial load on the vertical axis and a particular interaction diagram at each level of axial load. Imagine a stack of these interaction diagrams. Thus, a three-dimensional failure surface is defined. The equation of the failure surface can be generated by fitting a surface through the interaction diagrams at each level. When the necessary failure surfaces are available, the analysis at a load level can be checked by examining whether the vector of the forces on the section (axial, Mx and My) falls within or outside the failure envelope. The deformations associated with the three applied forces make it possible to determine the displacements associated with the various load levels. This elegant and powerful analysis algorithm produces excellent results. Well-designed graphics make it possible for the foundation specialist to easily evaluate the results. |