This example is similar to Example 1, but in this case, the pile is replaced by a drilled shaft (bored pile). The soil properties and ground slope angle are the same as those used in Example 1. The design issue with a reinforced-concrete pile is to find the nominal bending moment capacity and an appropriate value of flexural stiffness (EI) to use in the computations.
As in Example 1, an axial thrust load of 88.8 kN (20 kips) is assumed. The pile head is assumed fixed against rotation in the first loading case and free to rotate in the second loading case. The problem is to find the lateral load for each case that will cause the shaft to fail. Both of these loading cases might be used in a practical problem to establish the bounds for the solution if the rotational restraint caused by embedment of the top of the pile causes the pile head to be between fixed and free.
A drilled shaft with an outside diameter of 760 mm (30 in.) and a length of 15.2 m (50 ft) is used in this example. The reinforcing steel consists of 12 bars with outside diameter of 25 mm (corresponding to No. 8 bars in US practice) and spaced equally around a 610 mm (24 in.) diameter circle as shown in Figure 6-7. The ultimate strengths of the reinforcing steel and the concrete are 414 MPa (60 ksi) and 27.6 MPa (4.0 ksi), respectively.
Bending Moment vs. Depth
LPile 2013.7.01, © 2013 by Ensoft, Inc.
Loading Case 1
number of axial thrust loads. After the first run, the user may read the estimated axial capacities of the pile section in compression and tension from the output report and use these values to set the upper and lower values of axial thrust for the second analysis.
Figure 6-7 Cross-section of Drilled Shaft for Example 2.
An excerpt from the output report for Example 2a for the axial structural capacities is shown below:
Axial Structural Capacities:
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Nom. Axial Structural Capacity = 0.85 Fc Ac + Fy As = 13031.123 kN Tensile Load for Cracking of Concrete = -1424.929 kN Nominal Axial Tensile Capacity = -2532.072 kN
Using these values, axial thrust values were entered ranging from -2,500 to 13,000 kN.
The resulting factored interaction diagram generated by the Presentation Graphics feature is shown in Figure 6-8.
The curves for moment versus curvature for multiple axial thrust forces are shown in Figure 6-9 and the curves for EI versus bending moment are shown in Figure 6-10.
Computations of nominal bending moment capacities are determined when the concrete compressive strain at failure equals 0.003. For the axial load of 88.8 kN, the nominal bending moment capacity, Mnom, was taken from the curve as 731.8 kN-m. For design, a resistance factor for moment capacity equal to 0.65 was assumed, which gives a factored (ultimate) moment capacity of 475.7 kN-m.
The computations for nominal moment capacity could have been done for only the one axial load level, however, the full interaction diagram was developed to demonstrate the influence of axial load for this particular problem. As seen in Figure 6-8, an increase in the axial load up to a point will increase the value of the moment capacity so the axial thrust load was not multiplied by the global factor of safety to get the moment capacity.
Chapter 6 – Example Problems
Figure 6-8 Factored Interaction Diagram for Example 2a.
Factored Interaction Diagram
LPile 2013.7.01, © 2013 by Ensoft, Inc.
Section 1, Rf = 1.00
Moment vs. Curvature - All Sections
Moment, kN-meters
Figure 6-10 Bending Stiffness versus Bending Moment for Example 2a.
In earlier versions of LPile, the user had to select a constant value of bending stiffness to use in an analysis. This is no longer needed, as LPile will automatically vary the value of bending stiffness in proportion to the bending curvature developed in the pile.
The load-deflection curves and moment versus shear force curves for free-head conditions are shown in Figure 6-11 and for fixed-head conditions are shown in Figure 6-12. The scales of the two figures have been set equal to aid comparing the two sets of graphs.
The free-head shaft reaches its nominal moment capacity at a shear load of approximately 530 kN and its factored moment capacity at a shear load of 346 kN at a deflection of 0.035 m.
The fixed-head shaft reaches it nominal moment capacity at a shear load of 550 kN and its factored moment capacity at a shear load of 352 kN at a deflection of 0.0076 m. By happenstance, the load-carrying capacity of the two pile-head conditions are nearly equal.
However, the load-deflection response of the fixed-head shaft is substantially stiffer.
To illustrate the differences in deflection and bending moment versus depth for the two pile-head fixity conditions, a fourth analysis was performed for pile-head shear loads equal to 346 for the free-head shaft and 352 kN for the fixed-head shaft. The results of this analysis are shown in Figure 6-13.
The length of the pile may be reduced if there are more than two points of zero deflection, which ensures that the pile acts as a stable pile. The LPile can perform a series of analyses with different lengths of piles, so the user can compare pile length versus deflection at the pile head. The curves of top deflection versus pile length for free and fixed-head conditions is shown in Figure 6-14.
EI vs. Moment - All Sections
LPile 2013.7.01, © 2013 by Ensoft, Inc.
Thrust = -2500.00 kN
Chapter 6 – Example Problems
Figure 6-11 Shear Force versus Top Deflection and Maximum Bending Moment versus Top Shear Load for Free-head Conditions in Example 2b.
Shear Force vs. Top Deflection
Figure 6-13 Results for Free-head and Fixed-head Loading Conditions for Example 2d
Figure 6-14 Top Deflection versus Pile Length for Example 2d
Perhaps it is of interest to note that the lateral loads that were computed for the steel pile and for the bored pile were of significant magnitude, indicating that different types of piles can be used economically to sustain lateral loads.
Free-head Shaft
Chapter 6 – Example Problems