Geotechnical Engineering Circular No. 9 Design, Analysis, and Testing of Laterally Loaded Deep Foundations that Support Transportation Facilities

Figure 11-4: Moment interaction diagram

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Figure 11-
4: Moment interaction diagram.
11.3.10 Cases with Axial and Bending Moment (Non-Linear Behavior)
As the bending moment on a reinforced concrete section increases to the point at which it produces tensile stresses on one side of the shaft exceeding the tensile strength of the concrete, the section cracks, and a significant reduction in the E
p
I
p of the section at that location will occur. The axial load (if compressive), when it has no eccentricity, produces a uniform distribution of compressive stresses in the section that superimposes with the bending- induced stress distribution. As a result, the behavior of the section is a function of the relative magnitude of both axial loads and moments. The stress (σ) versus strain (ε) curves for concrete and reinforcing steel that are used by some of computer programs for design of laterally loaded deep foundations are shown in Figures 11-5 and 11-6, respectively (O’Neill and Reese 1999). The curve for concrete exhibits an initial almost linear response followed by a nonlinear curve up to the peak stress (defined as f’
c
), and a decreasing linear segment that ends at the maximum strain of concrete.

159 Referring to Figure 11-5:
f"
c

= 0.85 f’
c
(Equation 11-17)
E
c
(initial slope of the stress-strain curve) = 4,730 (f’
c
)
0.5 Equation 11-18)
(ε < ε
o
)
𝒇𝒇
𝑨𝑨
= 𝒇𝒇
𝑨𝑨
"
�𝟐𝟐�𝜺𝜺 𝜺𝜺
𝟎𝟎
� � − �𝜺𝜺 𝜺𝜺
𝟎𝟎
� Equation 11-19)
𝜀𝜀
0
= 1.7 𝑓𝑓
𝑐𝑐
′
/𝐸𝐸
𝑐𝑐
𝑓𝑓
𝑐𝑐
= Equation 11-20) Equation 11-21) Where f’
c
and E
c
are the concrete compressive strength and elastic modulus, respectively. In these equations, the units of E
c
, f’
c
, f”
c
and f
r
are in MPa. Note that approximately 6.9 Mpa = 1 ksi. In Figure 11-6:
𝜀𝜀
𝑦𝑦
= 𝑓𝑓
𝑦𝑦
𝐸𝐸
� and E = 200,000 MPa (or 29,000 ksi)
(Equation 11-22) Where f
y
and E are the steel yield strength and elastic modulus, respectively. Most reinforcing steel used currently in piles shafts is Grade 420 (US. Grade 60), which has a nominal yield stress f
y
= 420 MPa (60 ksi).

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