//this calculates the 'initial' tension from the initial stress strain curve, assumption is that no plastic elongation has occured yet, but the wire does not experience linear elasticity.
//this tension is typically called the 1-hour creep tension,'short' term tension condition, or stringing tensions.
//vaguely this calculation goes like this: 1)estimate length at design case 2) calculate strain for that length 3) calculate stress for that strain 4) calculate the average tension for that stress
//5) find the horizontal tension that results in that average tension 6) calculate the wire length for that horizontal tension from the wire geometry and return to step 2) with the new wire length estimate
//in practice this is accomplished by plugging stress= h/rho and strain = (arclength- originallength)/originallength into the stress strain equation
//this results in an equation that has only one unknown, horizontal tension. this can be solved with a newton-raphson loop.
//our input stress strain equation is in % strain, so the engineering strain must be multiplied by 100
public static double CalculateInitialTensions(this Weather weather, Wire wire, Creep creep)
{
double originalLength = wire.CalculateOriginalLength(creep);
double originalLengthDesignTemp = originalLength + WireExtensions.CalculateWireThermalCoefficient(wire) * originalLength * (weather.Temperature - wire.StartingTemp);
double lengthEstimate = Math.Sqrt(Math.Pow(weather.FinalSpanLength, 2) + Math.Pow(weather.FinalElevation, 2));
double horizontalTension = CalculateFinalLinearForce(weather, wire) * (lengthEstimate * lengthEstimate) / (8 * Math.Sqrt(3 * lengthEstimate * (Math.Abs(originalLengthDesignTemp - lengthEstimate)) / 8));
//refactor this so wireStressStrains are not calculated both here and in the wireExtensions
double wireStressStrainK0 = wire.OuterStressStrainList[0] + wire.CoreStressStrainList[0];
double wireStressStrainK1 = wire.OuterStressStrainList[1] + wire.CoreStressStrainList[1];
double wireStressStrainK2 = wire.OuterStressStrainList[2] + wire.CoreStressStrainList[2];
double wireStressStrainK3 = wire.OuterStressStrainList[3] + wire.CoreStressStrainList[3];
double wireStressStrainK4 = wire.OuterStressStrainList[4] + wire.CoreStressStrainList[4];
double NewtonDiff = 1000;
while (Math.Abs(NewtonDiff) > 0.001d)
{
//standard newton raphson where f(x)=stress-strain equation with x (strain) substituted for (arclength-oglength)/oglength*100. newton raphson to solve for horizontal tension
double arcLength = CalculateArcLength(weather.FinalSpanLength, weather.FinalElevation, horizontalTension / CalculateFinalLinearForce(weather, wire));
double arcLengthPrime = CalculateArcLengthPrime(horizontalTension, weather.FinalSpanLength, CalculateFinalLinearForce(weather, wire), weather.FinalElevation);
double function = -horizontalTension / wire.TotalCrossSection + wireStressStrainK0 + wireStressStrainK1 * (arcLength / originalLengthDesignTemp - 1) * 100 +
wireStressStrainK2 * (10000 * Math.Pow(arcLength, 2) / Math.Pow(originalLengthDesignTemp, 2) - 20000 * arcLength / originalLengthDesignTemp + 10000) +
wireStressStrainK3 * (1000000 * Math.Pow(arcLength, 3) / Math.Pow(originalLengthDesignTemp, 3) - 3000000 * Math.Pow(arcLength, 2) / Math.Pow(originalLengthDesignTemp, 2) + 3000000 * arcLength / originalLengthDesignTemp - 1000000) +
wireStressStrainK4 * (100000000 * Math.Pow(arcLength, 4) / Math.Pow(originalLengthDesignTemp, 4) - 400000000 * Math.Pow(arcLength, 3) / Math.Pow(originalLengthDesignTemp, 3) + 600000000 * Math.Pow(arcLength, 2) / Math.Pow(originalLengthDesignTemp, 2) - 400000000 * arcLength / originalLengthDesignTemp + 100000000);
double functionPrime = -1 / wire.TotalCrossSection + wireStressStrainK1 * arcLengthPrime / originalLengthDesignTemp * 100 +
wireStressStrainK2 * (10000 * 2 * arcLength * arcLengthPrime / Math.Pow(originalLengthDesignTemp, 2) - 20000 * arcLengthPrime / originalLengthDesignTemp) +
wireStressStrainK3 * (1000000 * 3 * Math.Pow(arcLength, 2) * arcLengthPrime / Math.Pow(originalLengthDesignTemp, 3) - 3000000 * 2 * arcLength * arcLengthPrime / Math.Pow(originalLengthDesignTemp, 2) + 3000000 * arcLengthPrime / originalLengthDesignTemp) +
wireStressStrainK4 * (100000000 * 4 * Math.Pow(arcLength, 3) * arcLengthPrime / Math.Pow(originalLengthDesignTemp, 4) - 400000000 * 3 * Math.Pow(arcLength, 2) * arcLengthPrime / Math.Pow(originalLengthDesignTemp, 3) + 600000000 * 2 * arcLength * arcLengthPrime / Math.Pow(originalLengthDesignTemp, 2) - 400000000 * arcLengthPrime / originalLengthDesignTemp);
NewtonDiff = function / functionPrime;
horizontalTension = horizontalTension - NewtonDiff;
}
return(horizontalTension);
}
