The I-Beam Section:
Moment of Inertia

Van Warren
Warren Design Vision
Little Rock, Arkansas
September 2, 1995

Problem

An I-Beam Section.
Find the moment of inertia about the x, y, z, x1, y1, and z1 axes.

Variables

I  = moment of inertia
	K  = radius of gyration
	B  = width of I-Beam
	H  = height of I-Beam
	Tf = thickness of top and bottom flange.
	Tw = width of web

	A "subscript" of 'p' indicates a primed or shifted coordinate.
	A "subscript" of 'x', 'y', or 'z' indicates a centroidal axis.
	A "subscript" of '1' after an axis subscript' indicates a displaced axis.
	A "subscript" of '1' after an I or K indicates a section identifier.  For example, I2x is the moment of inertia of the web section about the 'x' axis.

Initializations

Ip =.
Ix =.
Iy =.
Iz =.
I1x =.
I2x =.
I3x =.
I1y =.
I2y =.
I3y =.
WebH =.
I1xp =.

strip2[u_] := First[First[u]] ;
ruleToEquality[u_] :=  u /. Rule -> Equal ;
equalityToRule[u_] :=  u /. Equal -> Rule ;
ruleToSet[u_] :=  u /. Rule -> Set ;
ruleToFunction[u_] :=  u /. Rule -> SetDelayed ;
rePresent[u_] := strip2[u] ;
enSconce[u_] := ruleToEquality[u] ;

srf[s_, l_]:=StringReplace[s,Join[{"["->"(", "]"->")"},l]];

numFix[s_]:=
StringReplace[s,
{
	"B"->"IBeamB",
	"H"->"IBeamH",
	"T"->"IBeamT",
	"1.*"->"",
	"0.3333333333333333333"->"(1/3)",
	"(-x^2"->"(-(x^2)",
	"(-a^2"->"(-(a^2)",
	"(-b^2"->"(-(b^2)",
	"(-x^4"->"(-(x^4)",
	"(-a^4"->"(-(a^4)",
	"(-b^4"->"(-(b^4)"
}
];

tsf[s_] := ToString[s, FormatType->InputForm];

fix[e_, l_] := srf[tsf[e], l];

outPut[e_] := Print[numFix[ToString[InputForm[e]]]];

Ix

The x axis passes through the center of the I-Beam.  There are three sections of the I-Beam.  The top flange, labeled '1', the bottom flange, labeled '3' and the web labeled section '2'..  The web is that portion of the beam from the top of the bottom flange to the bottom of the top flange.  The moment of inertia of composite areas is just the sum of the moments of inertia of the primitive pieces, appropriately displaced under the parallel axis thereom.  We will start with the moment of inertia for the web, section '2':

WebH = (H - 2 Tf);
I2x = Tw WebH^3 / 12;

The moment of inertia for the top web, undisplaced is:

I1x = B Tf^3 / 12;

We can place the top web in its proper position by use of the parallel axis thereom which states that:

Ip = I + A d^2;

Where A is the area of the section being moved and d is the distance the section was displaced.

I1xp = I1x + A1 d1^2 /. A1->B Tf /. d1->(WebH+Tf)/2

Where A is the area of the section being moved and d1 is the distance that section '1' was displaced to put it at the top of the I-Beam.  Now the moment of inertia of the composite area is just:

Ix = Simplify[I2x + 2 I1xp];

The following expressions can be output to a spreadsheet or program for repetitive evaluation under varying conditions:

outPut[Ix]

Iy

The y axis passes through the center of the I-Beam.  Again, the moment of inertia of composite areas is just the sum of the moments of inertia of the primitive pieces, in this case no vertical displacement is necessary.

I2y = WebH Tw^3 / 12;

The moment of inertia for the top web is:

I1y = Tf B^3 / 12;

Iy = Simplify[12 (I2y + 2 I1y)]/12;

The following expressions can be output to a spreadsheet or program for repetitive evaluation under varying conditions:

outPut[Iy]

Iz

The z axis passes through the center of the I-Beam.  The moment of inertia about the 'z' axis is the sum of Ix and Iy.

Iz = Ix + Iy;

Ix1

The I-Beam sits on the x1 axis.  To find the moment of inertia about the x1 axis we again employ the parallel axis thereom:

A1 = B    Tf;
A2 = WebH Tw;
A3 = A1;
A = A1 + A2 + A3;
d = H/2;

Ix1 = Ix + A d^2;

outPut[Simplify[Ix1]]

outPut[Simplify[A]]

Iy1

The I-Beam sits to the right of the y1 axis.  We again employ the parallel axis thereom:

d = B/2;

Iy1 = Iy + A d^2;

outPut[Simplify[Iy1]]

Iz1

The z1 axis passes through the center of the I-Beam.  The moment of inertia about the 'z' axis is the sum of Ix and Iy.

Iz1 = Ix1 + Iy1;

Testing

Input parameters for a realistic example.

inputs =
{
	B->1, H->1, Tf -> 1/4, Tw -> 1/4
};


Ix /. inputs

Icheck = B Tf^3/6 + (B Tf/2) (H-Tf)^2 + (Tw/12)(H-2 Tf)^3

Icheck /. inputs