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This thesis primarily focused on the development of novel composite tubes by braiding. The objective was to use hierarchical scale technique, i.e., micro, meso and macro scales, with the transfer of information from one scale to another to develop novel braided composite tubes. This research was conducted and reported in three journal papers. The aim of the first paper was to predict plane elastic properties for E-glass/epoxy braided composite structures at different braid orientations, by analytical and finite element techniques. The lenticular shape has been used to describe the geometry of the tow. Modified lenticular geometric model was developed to improve an existing geometric model, in terms of tow parameters, thereafter, plane elastic properties from Chamis micromechanical model for E-glass fibre and epoxy matrix without any knockdown effects were used as benchmark to develop predictive models, namely; Lekhnitskii's methodology and braided unit cell meso-scale finite element model to account for the effects of tow geometry, undulations/crimp, cross-over and braid orientations on the plane elastic properties of E-glass/epoxy composite. The results showed agreement in trend between the predictive models, Chamis micromechanical model, and a similar existing model. However, the plane elastic properties were knocked down in predictive models by 30% in the E11 direction and 32% in the E22 direction, when compared with Chamis micro-mechanical model for largest ±65° braid angle, among the braid angles, considered. The aim of the second paper was to manufacture E-glass/epoxy braided tubes at different braid orientations by vacuum bag infusion technique, conduct internal pressure tests, and determine the hoop and axial moduli of the infused tubes. Lekhnitskii's methodology was also used to develop plane elastic moduli by experiment using microscopy results, and by calculation. The experimental elastic moduli of the infused tubes and the experimental elastic moduli from Lekhnitskii's methodology were used to compare the predictive elastic moduli for E-glass/epoxy braided structures by Chamis micro-mechanical model, and the braided unit cell meso-scale finite element model. The two were from another paper. Results showed a perfect agreement in trend between the experimental results and the predictive results. However, the values of the experimental results were close but lower than the predicted results. Optical microscopy was performed on braided tube cross-section to evaluate the level of crimp or undulation. This was done by the determination of tow centreline crimp angle and aspect ratio. Results show that when compared with the predicted crimp, there was an agreement in trend, although the experimental results were lower than the predicted. Also, the knockdown factor was evaluated and used to quantify the reduction in experimental elastic moduli when compared with the predicted. Results showed that the absences of crimp in the Chamis model caused a tremendous difference between it, other predicted models and the experiment results. The elastic moduli of Chamis were by far higher than all others, including other predictive models. The purpose of the third paper was to manufacture E-glass/epoxy braided tube at ±31°, ±45°, ±55°, ±65° braid orientations using vacuum bagging and resin infusion technique, to design and manufacture a rig for tube internal pressures experiment, to determine the hoop and axial stress performances of the tubes by internal pressure experiment, to compare experimental results with laminate analysis predictions to evaluate the effect of crimp on the internal pressure performance of the braided tubes. To use E-glass braided tow meso-scale unit cell finite element model to predict the tow critical stresses, and the optimum braided tube architecture, using tube hoop and axial failure stresses or strains. The tubes were manufactured and subjected to internal pressure test (2:1), to failure. Failure mode was by weeping and bursting. Hoop stress was twice the axial stress. The highest value of hoop stress was at the ±65° braid angle, higher than the hoop stresses at the ±31°, ±45°, and ±55 ° braid angles by 50%, 39%, and 28% respectively. Hoop stress increased with increase in braid angle. The experimental results were validated by laminate analysis predictions by Chamis micro-mechanical model and Lekhnitskii's methodology, and the trend of the laminate analysis prediction matched that of the experimental results. However, the predicted values were higher than the experimental results by 21%, 14%, 11%, 10% for the ±31°, ±45°, ±55°, ±65° braid angles for the Chamis micro-mechanical model and 5%, 7%, 7%, 5% for the ±31°, ±45°, ±55°, ±65 braid angles respectively for the Lekhnitskii's model, showing the severe effect of crimp in the experimental tube, mostly when compared with Chamis micro-mechanical model. Braided tow unit cell finite element model prediction, showed that tow axial stresses increased with increase in braid angle, while the tow transverse stresses decreased with increase in braid angle. The predictions showed that the tow critical stresses and the tube optimum braided architecture lie between the ±65° and 90° braid angles. The tow critical stresses are the stresses at which the tow decreasing transverse stress and the tow increasing axial stress causes the tube to fail.

An axial flux permanent magnet (AFPM) motor can be considered one of the optimal motor designs due to its higher torque density and compact sizing. To date, most AFPM models have adopted surface mounted and spoke type rotor designs for various applications. However, these designs provide poor protection for the magnet. Furthermore, the low saliency ratio of the surface mounted AFPM motors limits their performance in the flux weakening region. On the other hand, magnets in an AFPM can be placed inside well-designed flux barriers to guard against external forces. The advantages of the proposed design can be described in the following ways. First, the flux barriers provide protection for magnets in harsh operating conditions. Second, this rotor structure generates a satisfactory level of reluctance torque which extends the operating region of the motor. Third, the simpler flux barrier simplifies the manufacturing process of the motor and reduces the manufacturing cost. This paper describes the design, optimization and prototype development of a double stator single rotor (DSSR) axial flux interior permanent magnet (AFIPM) motor with a novel “H” shaped flux barrier.Once the design parameters of the rotor and stator have been pointed out, an initial design of the proposed AFIPM has been developed in finite element analysis (FEA) based on the machining requirements and design specifications. However, the initial AFIPM design exhibited lower average torque and higher ripple. As a result, a novel multistage optimization method has been developed to achieve the desired electromagnetic performance. This optimization method, which includes Taguchi orthogonal array, multivariate regression analysis and a genetic algorithm, calculates the optimal design parameters of the motor. Detailed electromagnetic finite element analysis has been executed to compare the performances of the optimized model with the benchmark design.An AFIPM prototype has been developed to verify the proposed design and optimization method. A detailed manufacturing process and mechanical assembly of the motor has been included. Magtrol 2WB65-HS dynamometer and a six-phase inverter utilizing SiC modules have been used to run the experimental tests. A detailed analysis of experimental results at different operating points shows good correlations with simulation results.

The development of residual stress in metal material additive manufacturing brings a significant obstacle to achieving parts with good quality. The residual stress can be destructive as the structural integrity and mechanical properties are seriously influenced during the manufacturing process. In this work, a finite element method based thermomechanical model is created to simulate the process of direct energy deposition for a copper alloy, GRCop-42. The thermal history of the deposition process is captured, and the development of residual stress & strain is calculated. The result of residual strain development inside the part is compared with the experimental measurement by energy dispersive x-ray diffraction. From the comparison, the result from thermomechanical model shows a high level of consistency with the EDD measurement result in the middle section of the part. However, the EDD result shows a higher residual strain at the left end of the specimen. Non-uniform geometry of printed specimen due to the bulging corners is measured and proved to be the reason for the discrepancy between simulation result and experimental result. Different laser parameters and material properties were also tested in the thermomechanical model to study their influence on the residual stress in additive manufacturing process. The work of this paper provides an approach to understand the development of residual stress, which will be crucial in enhancing the performance of metal additive manufacturing in a wide range of industrial applications.

This paper presents an evaluation of the shear lag factor for HSS tension members connected with two side plate gussets with longitudinal welds as given in Case 6 of Table D3.1 of AISC 360-16. The current AISC Specification for Case 6 does not permit weld lengths less than the perpendicular distance between the welds, and has the potential of producing negative shear lag factors. Similar issues previously existed for members given in Case 4 of Table 3.1. However, the AISC 360-16 Specification has adopted a mathematical model proposed by Fortney and Thornton for Case 4 of Table D3.1. The work presented in this paper offers: (1) a mathematical model for calculating the shear lag factor for Case 6 derived by repurposing the model adopted by AISC for Case 4 of Table D3.1; (2) the results of a parametric study comparing the results of the new mathematical model to the results using the current AISC method, and; (3) discusses the protocols developed for use in finite element analysis to evaluate the effectiveness of the proposed mathematical model. The proposed new mathematical model will permit longitudinal weld lengths less than the perpendicular distance between the welds, and removes the possibility of calculating a negative shear lag factor, while better representing the redistribution of cross-sectional stress near the connection region.

A torsional moment is experienced by single plate shear connections through lateral torsional buckling and by effect of lap eccentricity. To satisfy this torsional demand on the connection, a designer can change the dimensions of the connection or provide stabilizer plates. Fortney and Thornton (2016) have studied the analysis, design and detailing of the stabilizer plates and have recommended three different types of stabilizer plates, namely Type I, Type II, and Type III.This paper presents a study on the behavior of stabilizer plates as envisioned by Fortney and Thornton (2016) subjected to torsion. Calibrations of bolts, welds and plate buckling are also performed in ABAQUS as they constitute the most influential mechanisms that influence the behavior of the stabilizer plate in the connection. Finite element models of Type Ia, Type Ib, Type II and Type III stabilizer plates were generated in ABAQUS, as well as a benchmark model without any stabilizer plates based on the design examples given by Fortney and Thornton (2016). The models were subjected to a load at the midspan of the connected beam inducing a torsion in the connection. The simulated responses of the stabilizer plates are compared and contrasted. The results show that the Type II stabilizer plate has a better response in imparting torsional rigidity to the connection than the other stabilizer plates for a single plate shear connection. However, Type III stabilizer plate is recommended if using a moment connection. Also, a side study on pretensioning of a bolt in ABAQUS shows that the bolt load feature to pretension the bolt is best suited if using a Standard solver in ABAQUS.

Supercritical CO2 (sCO2) power cycles are superior to traditional water-based, air-breathing, direct-fired, open Brayton cycles; or indirect-fired, closed Rankine cycles in terms of efficiency and equipment footprint. They hold great potential in nuclear power production, fossil fuel power plants, concentrated solar power, geothermal power, and ship propulsion. To unlock the potential of sCO2 power cycles, technology readiness must be demonstrated on the scale of 10 – 600 MWe and at the sCO2 temperatures and pressures of 350 – 700 ºC and 15 – 30 MPa for nuclear industries. The lack of suitable shaft seals at sCO2 operating conditions is one of the main challenges at the component level. So far, conventional seals are incapable of handling sCO2 pressure and temperature in one way or another. This study proposes a patented novel Elasto-Hydrodynamic (EHD) high-pressure, high temperature, and scalable shaft seal for sCO2 turbomachinery that offers low leakage, minimal wear, low cost, and no stress concentration. The focus of this paper was to conduct a proof-of-concept study with the help of computer simulations. To this end, a fully coupled Fluid-Structure Interaction (FSI) modeling approach was adopted, and the simulations were carried out in COMSOL Multiphysics software. The modeling approach was presented thoroughly, the results were discussed, and the subsequent research steps were highlighted. Under the EHD mechanism: the higher the pressures are, the tighter the sealing becomes while sustaining a continuous sCO2 film. a 2" diameter shaft static test rig was designed at Georgia Southern University before the actual dynamic testing at Sandia National Laboratories for proof-of-concept purposes. For simplicity, the working fluid was chosen to be Nitrogen, and the tests were conducted at room temperature. The test rig consisted of a 16.5 MPA N2 tank, steel tubing with compression type fittings, and OMEGA – PX5500C0-2.5KA10E pressure sensor to measure the pressure of the N2 after the pressure regulator from the tank, a chamber housing the 2" steel shaft and EHD seal (two prototype steel seals with seal clearances of 25 μm and 50 μm), a CONAX – K-INC12-U-T3(5FT)-7.5" temperature sensor to collect temperature data in the chamber, and an OMEGA – FMA-1623AI mass flow meter with temperature and pressure measurement capability to evaluate conditions after the test chamber. The pressure, temperature, and mass flow rate data were collected via a National Instruments DAQ – Module NI-9205 and a LabVIEW program. The tests were conducted for an inlet pressure of 0.1 ~ 15 MPa and at two different clearances of 25 μm and 50 μm. The mass flow rate exhibited a quadratic trend with increased pressure, which proved the hypothesis.

This thesis concerns the simulation of locomotion in vertebrates. The state-of-the-art uses genetic algorithms together with solid body kinematics to generate possible solutions for stable gaits. In recent work, this methodology led to a hopping gait in a dinosaur and the researchers wondered if this was realistic. The purpose of the research carried out in this thesis was to examine whether quick and simple finite-element analyses could be added to the simulator, to evaluate a simple ―break or―not break failure criterion. A break would rule out gaits that might damage theowner‘s skeleton. Linear elastic analysis was considered as a possible approach as it would add little overhead to the simulations. The author used X-ray computed tomography and the finite element method to examine the axial loading of a barnacle goose femur. The study considered four levels of detail for a linear elastic simulation, finding that all the analyses carried out overestimated the strength of the bone, when considering safety factors. The conclusion is that to incorporate stress-strain analysis into the gait simulation requires more realistic models of bone behaviour that incorporate the nonlinear response of bone to applied loading. A new study focusing on the use of novel techniques such as model order reduction is recommended for future work. The outputs of this research include chapters written up as journal papers covering a 4D tomography experiment; a level of detail study; an analysis of a purported tendon avulsion injury in Tyrannosaurus rex and a review of the elastic properties of bone.

The present work has been produced as part of an on-going collaboration between the University of Manchester and Amec, with the primary aim of furthering mechanistic understanding of corrosion processes in zirconium alloys out-of-reactor. Zirconium alloys are used as cladding material for nuclear fuel pellets, and correct understanding of the corrosion process in autoclave is essential to predicting material behaviour in-reactor. This EngD thesis is composed of five proposed papers that investigate observations and hypotheses under the theme of mechanical degradation in oxides formed on zirconium alloys in autoclave. First investigations concern observed stress relaxation in zirconium oxide. Finite element analysis is used to capture mechanical aspects of the corrosion process and apply this to stress behaviour determined previously using synchrotron x-ray diffraction. The results indicate that a mechanism other than creep or hydrogen induced lattice strain must be present to account for the observed stress relaxation. One such potential mechanism is crack formation; statistical analysis of scanning electron microscopy images has been used to identify a link between the development of roughness at the metal-oxide interface, crack formation in the oxide and transition points or acceleration in the corrosion kinetics. Parameters such as the median radius of curvature and profile slope (Rdq) have been applied, as these parameters do not require the definition of a periodic wavelength or amplitude. These and other parameters are related to information in literature to indicate that for samples of Zircaloy-4 and ZIRLOTM, which go through transition, the interface roughness changes in a way that would increase localised stress concentrations. The third material is an experimental low tin alloy, which under the same oxidation conditions, and during the same time period, does not appear to go through transition and does not develop an interface roughness in the same way. A critical assessment of finite element analysis applied to oxidising non-planar interfaces shows the significant limitations in the existing mechanism for representing oxidation expansion and stress formation. Autoclave oxidation experiments of artificially roughened samples of Zircaloy-4 were carried out to further understand the impact of out-of-plane stress generation. The results indicate a divergence based on surface roughness after ~86 days oxidation. SEM examination of images in cross section highlighted accelerated oxidation above surface roughness peaks, and an increased crack area in rougher samples. Finally, finite element analysis of the tetragonal to monoclinic phase transformation showed that biaxial compressive stress relaxation, or the tri-axial tensile stress associated with an advancing crack tip, could reduce the transformation strain energy and destabilise the tetragonal phase. The volumetric expansion and shear strain associated with the phase transformation produces stress in the surrounding oxide sufficient to generate nano-scale cracks perpendicular to the metal-oxide interface. This would allow fast ingress routes for oxygen containing species, and therefore acceleration in the corrosion kinetics.

The development of lightweight concretes has made a contribution to advances in structural design. It would be useful to further improve the mechanical properties of lightweight concrete formulations whilst enhancing their resistance to fire degradation and reduced thermal conductivity. Improving the sustainability of any new proposed lightweight concrete formulation is desirable, for example by the inclusion of waste stream components into the formulation.This thesis describes an investigation of the mechanical, thermal and fire resistance properties of a new type of expanded clay lightweight concrete formulation in which varying quantities of sand are replaced by crushed glass aggregate, in conjunction with the addition of metakaolin (which may be available as a waste component from the manufacture of paper) as a partial replacement for the cement. The investigation involved short and long-term laboratory testing of a range of mechanical and thermal properties of individual concrete formulations and small scale structural elements consisting of masonry blocks made from these formulations (so called wallettes). An extensive programme of Finite Element Analysis using Abaqus was also performed.The results obtained show that it is possible to produce a structural expanded clay lightweight concrete that possesses good thermal properties by incorporating of ground glass and metakaolin. Compressive and splitting tensile strengths, as well as the modulus of elasticity, increased with an increase in the metakaolin content, while concrete density decreased. Reductions in thermal conductivity and improvements in fire resistance criteria were also observed in comparison with conventional lightweight concrete mixtures. For example, measured thermal conductivity values ranged from 0.092 W/m.K to 0.177 W/m.K, and the insulation criterion (an indicator of resistance to fire) reached up to 110 minutes for a concrete member with a thickness of 29 mm. The highest resistance to the effects of high temperatures was observed for concrete mixes containing either 15% or 30% recycled glass with 10% metakaolin.The maximum axial loads at failure were 474 kN and 558 kN for reference and modified wallettes respectively, implying corresponding bearing capacities of 7.1 MPa and 8.3 MPa. The critical path of the failure mode was similar for all of the wallettes tested and normally began underneath the load point, then passed through the concrete blocks and head joint to reach the toe of the wallette. The masonry wallettes formulated using reference lightweight concrete blocks exhibited failure due to explosive spalling at 400 oC with no applied mechanical load, whereas the second type of masonry wallettes (the modified wallettes) did not show such behaviour.The results of Finite Element Analysis showed that the coefficient of thermal convection had the most influence upon the insulation criterion. From a structural perspective, the key parameters were the value of penalty stiffness and imperfections in wallette construction. In general, a close agreement between the measured and simulated results was observed for both the thermal and structural finite element models at ambient and high temperatures.

In high seismic areas of the world, dissipation of energy created by earthquakes is vital to keep structures intact and more importantly the conservation of life. Steel structures use braced and moment frames to dissipate this energy through various mechanisms. This paper’s focus is on moment frames, and more specifically on Bolted Flange Plate connections with non-orthogonal geometry. With the complexity of designs increasing due to advancements in research and the want for large open spaces in many structures, moment frames are a popular option because no diagonal bracing members are required. Yet, in AISC 358-10 (AISC, 2010b), only orthogonal moment frame connections are discussed without any mention of how to design for a non-orthogonal connection. Using the finite element software ABAQUS, the effects of non-orthogonal geometry on Bolted Flange Plate connections, both skewed and sloped, were analyzed. Unlike welded moment connections, this connection requires bolts, and therefore two different bolt configurations were also studied, one having the bolt rows staying parallel to the column face and the other having the bolt rows perpendicular to the beam axis. The performance of these finite element models were compared to previous experimental results, based on the orthogonal connection, conducted at the University of California – San Diego. Using the moment-rotation capacity of the connection, distribution of stresses, and location of the plastic hinge, recommendations for the design of non-orthogonal Bolted Flange Plate connections were determined.

Viscoelastic coatings are currently being used in a variety of industries to provide shock absorption, energy absorption, noise reduction, and vibration isolation. They are commonly used through aerospace, automotive, and electronic industries. These materials have a complex behavior that causes their mechanical properties to vary with both temperature and frequency. Testing historically has been done on a composite material due to the inability of the coatings to support themselves at high temperatures. One common method is the vibrating beam technique, which examines two cantilever beam specimen, an uncoated beam and a coated beam. The change in modal properties is used to determine the properties of the viscoelastic material by itself. However, when the environment is harsh or space is restricted, measurements and excitations cannot be made in a traditional contacting manor. This paper examines methods of non-contact excitation and measurements and explores the effects from these methods.Specifically, magnetic excitation was examined as a source of non-contact excitation on a non-ferrous material. This requires that a magnetic material be attached to the beams in order for the magnetic excitation to be successful. The mass loading effects were examined for the two beam specimen to most accurately define the modal parameters of the beams. This excitation source was also used to test the specimen at a range of temperatures to determine the effect temperature had on the modal parameters of the specimen as well. Finite element models were subsequently created to see if the results from the experimental modal analysis could be confirmed with the FEA results.

This paper investigates the implementation of damage mechanics into finite element models of fillet-welded structural steel assemblies in order to predict connection fracture at realistic displacements. Utilizing a previously developed IDS (instability, ductile, and shear) material failure envelop for aluminum extrusions, the failure loci of the aluminum material were scaled to meet the point of fracture in welded steel lap splice assemblies, loaded in-plane at various angles, with a fillet weld leg size of 5 mm. The weld electrode under investigation was CSA E480xx, which is equivalent to AWS E70xx electrodes. A linear relationship for damage data scaling factors, as they relate to the loading angle of the fillet weld, was established. Three similar assemblies with fillet leg sizes of 9 mm were modeled employing this damage scaling relationship and deformations at fracture were found to coincide with experimental results and established relationships. This modeling methodology was also investigated for use in a T-stub connection tension scenario.

Currently, guidelines for safely using shipping containers for building applications do not exist. The shipping container’s structural integrity, modification properties, foundation limits, building code regulations, and reinforcing limits are mostly unknown. As a result, this research begins the investigation of shipping containers’ structural limitations thus aiding the development of container building construction and design requirements. The main research objective is to develop structural guidelines for International Organization for Standardization (ISO) shipping containers used for non-shipping applications. This paper provides insight into the ISO shipping container’s structural strength which is further investigated using finite element computer modeling. The finite element analysis shows how both modified and unmodified container models respond under given loading scenarios. The loading scenarios incorporate the effect of gravity and lateral loading, and the computer simulations demonstrate the effectiveness of the container walls and roof to resist the loads. Structural engineering considerations, foundations, and connections for shipping containers used in building applications are presented.

The goal of this project was to develop a sterile surgical device for gaining peritoneal cavity access without external incisions via the body's major natural orifices. Such Natural Orifice Translumenal Endoscopic Surgery (NOTES) interventions have recently received innovative attention as the next step in minimally-invasive surgery, however, the safety and efficacy depends greatly on sterility. The research for this thesis focused primarily on gastric interventions is divided into four main foci: tensile testing and determination of the most accurate material model for human gastric tissue, FEA analysis of the balloon dilation, experimentally validating these results using a silicone tissue phantom, and the paper design of a theoretical prototype. Tensile testing the human gastric tissue provided the only material properties for the entire stress-strain curve known to the literature to be accurate. A series of tests were conducted on several different freshly donated organs. Statistical analyses were performed comparing the inner and outer layers, and the 0° and 90° orientation. These results showed that while visually different, the elastic portion of the stress-strain curve showed no statistical differences between layers or orientation. A FE model was created in 2D axisymmetric and 3D to determine the minimum size incision needed to dilate large enough to allow passage for the device and endoscope without inducing irreversible damage to the tissue. Conclusions from tensile testing led to the material model being hyperelastic, homogeneous, and isotropic. ABAQUS Explicit was used to model the quasi-steady state problem and to more effectively manage contact definitions in the 3D simulation. The models were also simulated with a silicone material for experimental validation. Using the same assumptions from the ABAQUS model, a physical experiment was performed with the silicone tissue phantom. From a circular incision of 10mm, several final diameters were tested. Rigid objects were used to dilate rather than balloons for ease of visualization. The surface had nodal coordinates drawn on the material and digital images taken before and after dilation. Coordinates were extracted using xyExtract.exe, and strains calculated with a user defined MATLAB program. Taking the above work into consideration, three different prototype designs were proposed. All three incorporate dual-donut balloons as the primary means of dilating the initial incision, holding the device in place, and providing a mechanical means for ensuring sterility while maintaining insufflation. The first embodiment simply incorporates dual balloons on the end of a PTFE sheath. The second utilizes a corkscrew-shaped singular balloon crimped in the middle to form the dual expansion zones. Finally, the last prototype design uses a second outer sheath to encapsulate both balloons which provides the dilatory force to the tissue, and acts as a long cylindrical balloon to stabilize the length of the sheath.

Cleavage fracture has been a very important subject for engineers for a long time because of the catastrophic result it may cause. The experimental results of cleavage fracture exhibit a large amount of scatter and show significant constraint effect, which motivated the development of statistical and micromechanics based methods in order to deal with such problem. The Weibull stress model, which is based on the weakest link statistics, uses two parameters, m and σu, to describe the inherent distribution of the micro-scale cracks once the plastic deformation has occurred and to define the relationship between the macro and micro-scale driving forces for cleavage fracture. In this paper we examine constraint effects on cleavage fracture toughness numerically using a constraint function g(M) derived from the Weibull stress model. The non-dimensional function g(M) describes the evolution of constraint loss effects on fracture toughness relative to reference plane-strain small scale yielding (SSY) condition (T-stress = 0). We performed detailed finite element analyses of single-edge notched bending specimens and compute g(M) functions for them. The g-function varies with parameters of the Weibull stress model, material flow properties and specimen geometry but not with absolute specimen size. Knowing the g-function one can construct fracture driving force curves for each absolute size of interest.