Hexcrete tower flexural analysis:

For design and analysis of the Hexcrete tower subject to flexure, two different

methodologies were utilized for linear and non-linear tower response. In both methodologies, it is assumed that the tower panels do not participate as compression or tension areas since the tower panels are not structurally connected vertically along the height of the tower. For linear response, an uncracked section analysis was applied based on basic stress equations where the stress in the test unit columns, σc, was found by using Equation 5-10. The stress in the vertical

post-tensioning tendons was assumed to be the effective stress resulting from jacking of the tendons during erection. This approach was used until decompression of the vertical tendons occurred. Decompression was assumed to occur when Equation 5-11 was true of the tower section. Strain in the columns and vertical tendons was then calculated based on basic mechanics properties for converting from stress to strain.

�(𝐹𝐹∗𝑁𝑁)+𝑃𝑃_{(𝐴𝐴∗𝑁𝑁)} � +𝑀𝑀_𝑆𝑆 = 𝜙𝜙𝑐𝑐 (Eq. 5-10)

where:

𝐹𝐹 = single column vertical PT force after jacking and long term losses N = number of columns

P = applied axial load of tower section

A = single column net area (accounts for voids due to PT ducts) M = overturning moment applied to tower section

𝑆𝑆 = section modulus of tower cross section

-1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 Op 1 Op 3 Op 5 Ex 1 Ex 3 Ex 5 St re ss ( ksi ) Predicted Measured

𝐹𝐹 ≤(𝑀𝑀∗𝐴𝐴)_(𝑆𝑆) −_𝑁𝑁𝑃𝑃 (Eq. 5-11)

For the non-linear tower analysis, an iterative displacement method first developed by Lewin was implemented assuming that decompression of the tower vertical tendons had already occurred and that a gap would open at a critical load location due to tendon decompression (Lewin & Sritharan, 2010). The first step in the analysis process was to define a critical tower rotation value corresponding to the magnitude of gap opening at the critical tower section. The tower rotation value depends on an assumed neutral axis depth at the critical section (Figure 5.15) which should be less than the critical section tower diameter (the neutral axis is an iterative value). The assumed neutral axis value and corresponding rotation were then input into Equation 5-12 to solve for the maximum column compression strain. Equation 5-12 treated the entire Hexcrete tower as a single hollow column and equated the critical section rotation to a constant plastic curvature over the plastic hinge length, Lp (Thomas & Sritharan, 2004). This method allows approximation of a maximum column strain for specific load and gap values without the need for calculating the ultimate tower capacity.

𝜖𝜖𝑐𝑐𝑐𝑐𝑎𝑎𝑥𝑥 = 𝑒𝑒𝑁𝑁𝐴𝐴(𝜃𝜃_𝑃𝑃𝑐𝑐𝑟𝑟 𝑝𝑝 +

𝑀𝑀𝑐𝑐𝑟𝑟

0.6𝐸𝐸𝐼𝐼𝑐𝑐𝑟𝑟) (Eq. 5-12)

where:

єcmax = maximum column strain for given rotation and neutral axis values

cNA = assumed neutral axis depth

θcr = rotation at critical section resulting from gap opening magnitude and assumed neutral axis

Lp = plastic hinge length, assumed to be 0.06L where L is equal to the distance of the critical section from the top of the tower

Mcr = moment at critical section E = elastic modulus of tower columns

Icr = moment of inertia of tower columns at critical section

Based on the assumptions that plane sections remained plane within the Hexcrete tower, that the tower concrete experienced a linear strain distribution, and that the maximum column strain occurred at the edge of the outermost column, average strain values for each column were calculated using a linear strain profile and similar triangles (Figure 5.16). If the outermost column experienced a strain higher than 0.003 it was assumed that the ultimate load condition was reached for the tower structure. The calculated column strains were then converted to column stresses and column forces.

Next, the elongation of the tower tendons due to rotation at the critical tower section were calculated based on the assumed neutral axis depth, corresponding critical rotation, and tendon location (Figure 5.17). The rotational elongations were then divided by the original tendon length, LT, to calculate the tendon. Strain was then converted to stress and added to the effective

stress due to tensioning of the tendons. If the resulting stress value exceeded 230 ksi, the strand would no longer act in a linear manner and the tower was considered to have reached its ultimate capacity. Tendon stresses were then converted to tendon forces.

Figure 5.17. Tendon location in relation to critical rotation and neutral axis depth

Once column and tendon forces were calculated, force equilibrium was checked by summation of the column compression, tendon tension, and applied tower axial forces. If equilibrium was achieved, the assumed neutral axis depth was correct. If not, the neutral axis depth was iterated until equilibrium was reached.

The non-linear tower numerical method was then compared to the experimental test unit results. Before experimental testing, load cells were installed between the top of two test unit columns and the vertical tendon multi-strand anchors. Since the numerical method provided strain values for both test unit columns and tendons, the strain in a vertical tendon anchored to a load cell was examined. A test unit load case corresponding to tendon decompression was selected and the critical section was found to be located at the base of the test unit columns. Gap opening at the base of the selected column was measured by a Linear Variable Displacement Transducer (LVDT). Test unit properties were input into the non-linear model and the neutral

axis depth was iterated until force equilibrium was obtained. The resulting tendon strain was converted to force and compared to the load cell measurements (Figure 5.18). The graph shows that the numerical method values are typically within 2 to 3 kips of the measured data at high load values, which provides confidence in the accuracy of the numerical method.

Figure 5.18. Comparison of measured and non-linear anaylsis tendon forces 5.5 Conclusion:

In order to better understand the behavior of the Hexcrete tower system, finite element centerline models of the designed tower systems and Hexcrete experimental test unit were created in SAP2000. The same modeling technique was used for both the towers and test unit and the test unit was then validated using experimental data for multiple load cases. The Hexcrete test unit was more flexible than the created SAP model due to the post-tensioning of multiple precast concrete members. The SAP models were subsequently adjusted to match the test unit data and then applied to the full Hexcrete tower system. In examining the SAP tower models, it was found that two of the tower designs, the full concrete HT2 and HT3 towers, did not meet the necessary frequency requirements. Both towers experienced large displacements at the top of the towers under operational and extreme loads and will require changes in tower design. Simulations were also run in the verified SAP models to investigate the effects of increasing the vertical spacing of the circumferential post-tensioning and using HSC panels. It was found that the spacing of the tendons could be doubled without detrimental effects to the tower system.

Numerical methods were also created to simplify the initial tower design process. The derived equations calculated the deflection of the tower system, predicted panel stresses, and

-30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 1 122 243 364 485 606 727 848 969 1090 1211 1332 1453 1574 1695 1816 1937 2058 2179 Re sul ta nt T endo n Fo rc e ( ki ps ) Time Load Cell Numerical method

quantified the tower flexural behavior. Each numerical method was formulated and compared to Hexcrete test unit data for verification. The methods were found to be an effective alternative to finite element models for preliminary estimation of Hexcrete tower behavior. Opportunity for further refinement of these methods may be possible during future development of a Hexcrete prototype structure.

5.6 References:

ACI Committee 318. (2011). Building Code Requirements for Structural Concrete (ACI-318)

and Commentary. Farmington Hills: American Concrete Institute.

Computer and Structures, I. (2014). Structural and Earthquake Engineering Software, SAP2000 Version 17.

Hearn, E. (1997). Mechanics of Materials 2, 3rd Edition. Woburn, MA: Butterworth-Heinemann. Lewin, T., & Sritharan, S. (2010). Design of 328-ft (100-m) Tall Wind Turbine Towers Using

UHPC. Ames, IA: Department of Civil, Construction, and Enviromental Engineering

Report ERI-ERI-10336.

Sritharan, S., & Lewin, T. (2015). U.S. Patent No. 9,016,012.

Sritharan, S., Lewin, T., & Schmitz, G. M. (2014). U.S. Patent No. 8,881,485.

Thomas, D. J., & Sritharan, S. (2004). An Evaluation of Siesmic Design Guidelines Proposed for

Precast Jointed Wall Systems. Iowa State University.

Twigden, K., Sritharan, S., & Henry, R. (2017). Cyclic testing of unbonded post-tensioned concrete wall systems with and without supplemental damping. Engineering Structures, 406-420.

CHAPTER 6 – SURFACE PRESSURE ANALYSIS OF HEXCRETE WIND TURBINE