5-2-1 Flexural Behavior
The flexural behavior of a structural element such as a beam, column, or a pile subjected to bending is dependent upon its flexural rigidity, EI, where E is the modulus of elasticity of the material of which it is made and I is the moment of inertia of the cross section about the axis of bending. In some instances, the values of E and I remain constant for all ranges of stresses to which the member is subjected, but there are situations where both E and I vary with changes in stress conditions because the materials are nonlinear or crack.
The variation in bending stiffness is significant in reinforced concrete members because concrete is weak in tension and cracks and because of the nonlinearity in stress-strain relationships. As a result, the value of E varies; and because the concrete in the tensile zone below the neutral axis becomes ineffective due to cracking, the value of I is also reduced. When a member is made up of a composite cross section, there is no way to calculate directly the value of E for the member as a whole.
The following is a description of the theory used to evaluate the nonlinear moment-curvature relationships in LPile.
Consider an element from a beam with an initial unloaded shape of abcd as shown by the dashed lines in Figure 5-3. This beam is subjected to pure bending and the element changes in shape as shown by the solid lines. The relative rotation of the sides of the element is given by the small angle dθ and the radius of curvature of the elastic element is signified by the length ρ measured from the center of curvature to the neutral axis of the beam. The bending strain εx in the beam is given by
x dx
ε = Δ ...(5-8)
where:
Δ = deformation at any distance from the neutral axis, and dx = length of the element along the neutral axis.
a
b
c
d dx
Δ ρ
η
M M
dθ a
b
c
d dx
Δ ρ
η
M M
dθ
Figure 5-3 Element of Beam Subjected to Pure Bending
The following equality is derived from the geometry of similar triangles
Δ η ρ =
dx ...(5-9)
where:
η = distance from the neutral axis, and ρ = radius of curvature.
Equation 5-10 is obtained from Equations 5-8 and 5-9, as follows:
ρ η ρ
η
ε = Δ = =
dx dx
x dx
1 ...(5-10)
From Hooke’s Law
x
x Eε
σ = ...(5-11)
where:
σx = unit stress along the length of the beam, and E = Young’s modulus.
Substituting Equation 5-10 into Equation 5-11, we obtain
ρ
σx = Eη ...(5-12)
From beam theory
I M
x
σ = η ...(5-13)
where:
M = applied moment, and
I = moment of inertia of the section.
Equating the right sides of Equations 5-12 and 5-13, we obtain
ρ η
η E
M =I ...(5-14) Cancelling η and rearranging Equation 5-14
ρ
= 1 EI
M ...(5-15)
Continuing with the derivation, it can be seen that dx = ρ dθ and
θ φ ρ = dx =
d
1 ...(5-16)
For convenience here, the symbol φ is substituted for the curvature 1/ρ. The following equation is developed from this substitution and Equations 5-15 and 5-16
φ
EI = M ...(5-17)
and because Δ = η dθ and εx = Δ/dx, we may express the bending strain as
εx = φ η ...(5-18)
The computation for a reinforced-concrete section, or a section consisting partly or entirely of a pile, proceeds by selecting a value of φ and estimating the position of the neutral axis. The strain at points along the depth of the beam can be computed by use of Equation 5-18, which in turn will lead to the forces in the concrete and steel. In this step, the assumption is made that the stress-strain curves for concrete and steel are those shown in Section 5-1-3.
With the magnitude of the forces, both tension and compression, the equilibrium of the section can be checked, taking into account the external compressive loading. If the section is not in equilibrium, a revised position of the neutral axis is selected and iterations proceed until the neutral axis is found.
Bending moment in the section is computed by integrating the moments of forces in the slices times the distances of the slices from the centroid. The value of EI is computed using Equation 5-17. The maximum compressive strain in the section is computed and saved. The computations are repeated by incrementing the value of curvature until a failure strain in the concrete or steel pipe, is reached or exceeded. The nominal (unfactored) moment capacity of the section is found by interpolation using the values of maximum compressive strain.
5-2-2 Axial Structural Capacity
The axial structural capacity, or squash load capacity, is the load at which a short column would fail. Usually, this capacity is so large that it exceeds the axial bearing capacity of the soil, except in the case of rock that is stronger than concrete. Several design equations are used to compute the axial structural capacity, depending on the type of section being analyzed. For reinforced concrete sections (not including prestressed concrete piles) the nominal (unfactored) axial structural capacity, Pn, is
y s s g c
n f A A A f
P =0.85 ′( − )+ ...(5-19)
where Ag is the gross cross-sectional area of the section, As is the cross-sectional area of the longitudinal steel, f′c is the specified compressive strength of concrete and fy is the specified yield strength of the longitudinal reinforcing steel.
Common design practice in North America and Europe is to restrict the steel reinforcement to be between 1 and 8 percent of the gross cross-sectional area for drilled shafts without permanent casing. Usually, reinforcement percentages higher than 3.5 to 4 percent are
attainable only by a combination of bundling of bars and by reducing the maximum aggregate size to be small enough to pass through the reinforcement cage. LPile has features that help the user to identify the combinations of reinforcement details that satisfy requirement for constructability.
For prestressed concrete piles, the equations for the nominal axial structural capacity differ depending on the cross-sectional shape and the level of prestressing. As for uncased reinforced concrete sections, the concrete stress at failure is assumed to be 0.85 f′c. With axial loading, the effective prestress in the section is lowered. At a compressive strain of 0.003, only about 60 percent of the prestressing remains in the member. Thus, the nominal strength can be computed as
(
c ps)
gn f f A
P = 0.85 ′−0.60 ...(5-20)
where fpc is the effective prestress.
The service load capacity for short column piles established by the Portland Cement Association is based on a factor of safety between 2 and 3 is
(
fc fpc)
AgN = 0.33 ′−0.27 ...(5-21)
Conventional construction practice in North American is to use effective prestressing of 600 to 1,200 psi (4.15 to 8.3 MPa) for driven piling. The level of prestressed used varies with the overall length of the pile and local practice. Usually, the designing engineer obtains the value of prestress and fraction of losses from the pile supplier.