LINQ Min Example

void Main()
{
    var bronze = new ElementType { Name = "Bronze" };
    var steel = new ElementType { Name = "Steel" };

    var weakBronzeBeam = new Beam { Ix = 50, ElementType = bronze };
    var strongBronzeBeam = new Beam { Ix = 100, ElementType = bronze };

    var weakSteelBeam = new Beam { Ix = 80, ElementType = steel };
    var strongSteelBeam = new Beam { Ix = 200, ElementType = steel };

    var data = new [] { weakBronzeBeam, strongBronzeBeam, weakSteelBeam, strongSteelBeam };

    var iRequired = 70;

    var smallestBeamForTypes =
        from anyBeam in data
        where anyBeam.Ix >= iRequired
        group anyBeam by anyBeam.ElementType into beamTypeGroups
        let minIx = beamTypeGroups.Min(beam => beam.Ix)
        select new {
            ElementType = beamTypeGroups.Key,
            SmallestBeam = beamTypeGroups.First(beam => beam.Ix == minIx)
        };

    foreach(var smallestBeamForType in smallestBeamForTypes)
    {
        String.Format("For the element type {0} the smallest beam is {1}",
            smallestBeamForType.ElementType , smallestBeamForType.SmallestBeam).Dump();
    }
}

class ElementType
{
    public string Name { get; set; }

    public override string ToString()
    {
        return Name;
    }
}

class Beam
{
    public int Ix { get; set; }
    public ElementType ElementType { get; set; }

    public override string ToString()
    {
        return Ix.ToString();
    }
}