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pile analysis concrete piles pile supports # Rocscience | Improved Analysis of Reinforced Concrete Piles in RSPile Featured

This article depicts how the pile analysis under lateral loading, or couples is improved in RSPile by updating the pile flexural stiffness during the solution in cases of piles with combined flexural and axial stresses. This is raised because of the bending (flexural) stiffness of the pile varies with the level of cracking and the applied axial load. In previous versions and in most of the software the bending stiffness-moment relation is a single curve constructed in the beginning of the analysis and never updated or varied along the pile length. Usually, that curve is based on the loading of the pile head while pile load may change at other depths as well, which may have significant effect on the flexural stiffness.

In the latest version of *RSPile* release 3.014, this issue is addressed and solved. The results of the pile analysis with updated bending stiffness clearly varies with depth during the analysis. An example is solved in this article to compare the results between single and variable moment-curvature curves (bending stiffness) with depth.

In piles with reinforced concrete sections, one of the parameters for the analysis of piles under lateral loading or bending moments is the bending (flexural) stiffness, effective *EI*. In ACI 318–2019, and similar codes around the world, the effective *EI* may be approximated by the concrete modulus of elasticity, *Ec* , multiplied by an effective moment of inertia * Ie* , a value falling between the cracked section moment of inertia,

(EI)i= Mi/φi

Derivation for this relation may be found in classical textbooks of strength of materials, see for example Higdon et al. (1978). The moment curvature curve is prepared from the beginning to pick the corresponding *EI *at any moment-curvature level. This helps getting better results for the response of the pile to the applied loads.

The bending stiffness is highly dependent on the applied axial load at the section and the axial load varies with depth due to the soil resistance around the pile, i.e., the skin friction. The skin friction decreases the axial load with depth (or increases it in the case of negative skin friction) and it may also vary due to additional external loads applied on the pile at different depths.

In traditional pile analysis the bending stiffness is taken constant along the depth which is true for elastic sections. Engineers in hand calculations or simple spread sheets assume the reinforced concrete pile as an elastic section. Meanwhile, most of the software programs the relation between bending stiffness and the bending moment in reinforced concrete sections is assumed unchanged and a single *M-φ** *curve is used for the whole length of the pile (just like a column in a structural frame). Reese and Van Impe (2011) stated that “the errors that are involved in using the approximation where there is a change in the bending stiffness along the length of a pile are thought to be small but must be investigated as necessary”. Nevertheless, the error increases with new external loads added to the pile through the depth and in several other cases such as changes in section dimensions or materials (e.g. reinforcement ratio). This error is overcome in the recent version of *RSPile* 3.014.

This is accomplished by generating a series of moment-curvature curves based on the range of internal axial force experienced by the pile. The bending stiffness of the reinforced concrete pile is then obtained as a function of both internal axial force and curvature. Fig.1 depicts the difference between the use of a single *M-φ *relation and multiple ones in the modeling.

A pile with a rectangular reinforced concrete section of 500 mm depth and 800 mm width with 34 MPa concrete strength and reinforced with 10 #10 bars placed peripherally at 100mm cover. The pile is analyzed using *RSPile* under a lateral load in the long side direction of 150 kN applied at the pile head investigating the response applying different vertical downward loads at the top of the pile, 100 kN, 500 kN and 1000 kN. The section details are shown in the section designer of *RSPile* in Fig.2.

Pile length is taken as 20 m fully embedded in soil. To control soil parameters effects, the soil is assumed elastic with a constant modulus of lateral subgrade reaction of 1000 kN/m3 and a skin frictional spring of 10000 kN/m3 while the end bearing stiffness is set to zero to govern the axial load distribution by the skin friction only.

With the introduction of *Interaction Diagrams* in *RSPile*, users can now estimate the structural capacity of their concrete section in a single click. Fig.3 shows interaction diagram for the short column (pile), the nominal axial load capacity, *Pn* plotted against the nominal bending moment capacity *Mn* for a load applied at an angle α of 0o taken counterclockwise from *x’* axis. At the right side the *Mnx’-Mny’* plot is presented for a case of *Pn*=0 though the user may choose any load level within the pile capacity. From the left graph the three axial loads 100 kN, 500 kN and 1000 kN chosen for the example can be found being safe enough with the correspond moments that will be obtained in the analysis part.

Fig.4 shows the distribution of axial force through the depth which should be varying from the load applied at the pile head to zero at the bottom as no force is transferred to base (resistance is side resistance only). The model is run with *RSPile* v.3.013 (single *M- φ*

The results for calculated lateral displacements, rotations and calculated bending moment are plotted against depth for both solutions in Fig.5, Fig.6 and Fig.7 for applied downward vertical loads of 100 kN, 500 kN and 1000 kN respectively.

The differences between the results using a single *M- φ*

As it is expected, the variation in the results between the case with single *M- φ*

The above finding means that the original single *M- φ*

The new improved bending stiffness calculations using updated *M- φ*

Higdon, A., Ohlsen, E., Stiles, W.B., Weese, J.A. and Riley, W.F. (1978): “Mechanics of Materials”. SI version, 3rd ed., John Wiley and Sons, New York, USA, 752pp.

Reese, L.C. and Van Impe, W. (2011): “Single Piles and Pile Groups Under Lateral Loading”. 2nd ed. Taylor & Francis Group LLC, Boca Raton, USA.

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