Engineers Edge

Beam Deflection and Analysis

  • This calculator contains numerous “comment boxes” which contain a wide variety of information including explanations of input or output items, equations used, data tables, etc. (Note: presence of a “comment box” is denoted by a “red triangle” in the upper right-hand corner of a cell. Merely move the mouse pointer to the desired cell to view the contents of that particular "comment box".)
  • This calculator follows the procedures and guidelines of the AISC 9th Edition Allowable Stress (ASD) Manual (1989).
  • In this calculator for members subjected to known loadings consisting of axial load (compression or tension) and/or uniaxial or biaxial bending, both the actual and allowable stress are computed, with the final result being a computed "stress ratio" of actual stress/allowable stress.
  • This calculator utilizes an "Allowable Stress Increase Factor" (ASIF) which is a multiplier of any of the calculated allowable stresses Fa, Fbx, and Fby and also the Euler column buckling stresses F'ex and F'ey. It is used and appears ONLY in the stress ratio calculation. Typically a value of 1.0 may be used. However, a value of 1.333 may be used for load combinations which include wind or seismic loads.
  • If an axially loaded compression member has a value of the maximum slenderness ratio K*L*12/r >200, then a message will appear. However, this calculator DOES NOT consider or deem a particular member as"inadequate" based on the slenderness ratio of 200 being exceeded.
  • For the case of combined axial compression with bending, if the calculated value of fa >=F'e (which is not allowed) then a warning (error!) message will appear.
  • When the values of either 'Lx', 'Ly', or 'Lb' are input = 0' (or actually <= 1.0'), this calculator will use a value = 1.0'. 11. When a stiffened element (web) of a member subjected to axial compression is classified as a "slender" element (exceeding non-compact limits) based on local buckling criteria, then the calculator complies with AISC Appendix B.
  • The values of 'Cb', 'Cmx', 'Cmy', 'Kx, and 'Ky' may be calculated (if applicable) by accessing the additional input data to the right of the main page in each of the calculation worksheets. Then, these calculated values can be input under the member design parameters on the main page. (Note: there are equations which very closely approximate the solutions for 'Kx' and 'Ky' obtained using the AISC Code Alignment Charts.)
  • This calculator does not calculate or check shear or deflection in member
  • This calculator does not consider torsion on member.
  • This calculator does not consider deduction for holes in members subjected to tension.

W Flange


Input Data: Yellow
Member Size:
Member Properties:
Select:
A = in.^2
d = in.
Member Loadings:
tw = in.
P =
'P' is the applied axial load on the member, which may be either a compression or tension load.
Sign convention: + = compression, - = tension
 kips bf = in.
Mx =
'Mx' is the applied flexural bending moment about the X-axis (major axis) of the member.
Note: the value input MUST BE positive (+).
 ft-kips tf = in.
My =
'My' is the applied flexural bending moment about the Y-axis (minor axis) of the member.
Note: the value input MUST BE positive (+).
 ft-kips rt = in.
d/Af =
Design Parameters:
Ix = in.^4
Fy = ksi Sx = in.^3
Note: See section of this worksheet (to
the right) for input data which
may be used to determine the
'Kx' to be used for input here.
Kx =
'Kx' is the effective length factor about the X-axis (major axis) for an axially loaded compression member. Typical values are as follows:

Column End Conditions "Kx" Value
(bottom-top) (Recommended)
Fixed-Fixed 0.65
Fixed-Pinned 0.80
Fixed-Slider 1.2
Pinned-Pinned 1.0
Fixed-Free 2.1
Pinned-Slider 2.0

where:
Fixed end denotes rotation fixed and translation fixed.
Pinned end denotes rotation free and translation fixed.
Slider end denotes rotation fixed and translation free.
Free end denotes rotation free and translation free.
rx = in.
Note: See section of this worksheet (to
the right) for input data which
may be used to determine the
'Ky' to be used for input here.
Ky =
'Ky' is the effective length factor about the Y-axis (minor axis) for an axially loaded compression member. Typical values are as follows:

Column End Conditions "Ky" Value
(bottom-top) (Recommended)
Fixed-Fixed 0.65
Fixed-Pinned 0.80
Fixed-Slider 1.2
Pinned-Pinned 1.0
Fixed-Free 2.1
Pinned-Slider 2.0

where:
Fixed end denotes rotation fixed and translation fixed.
Pinned end denotes rotation free and translation fixed.
Slider end denotes rotation fixed and translation free.
Free end denotes rotation free and translation free.
Iy = in.^4
Lx =
'Lx' is the actual unbraced length of member for column-type (axial compression) buckling about X-axis (major axis).
Note: for input values of Lx <=1.0', this
program will use a value =1.0'.
ft. Sy = in.^3
Ly =
'Ly' is the actual unbraced length of member for column-type (axial compression) buckling about Y-axis (minor axis).
Note: for input values of Ly <=1.0', this
program will use a value =1.0'.
ft. ry = in.
Lb =
'Lb' is the actual unbraced length of the compression flange of the member for X-axis
(major axis) bending. The "unbraced length" can be more specifically defined as the distance between cross sections braced against twist or lateral displacement of the compression flange.
Notes:
1. For most cases, 'Lb' is equal to 'Ly'.
2. For cantilevers braced against twist only
at the support, 'Lb' may conservatively
be taken as the actual length.
3. For input values of Lb <=1.0', this
program will use a value =1.0'.
ft. J = in.^4
Note: See section of this worksheet (to
the right) for input data which
may be used to determine the
'Cb' to be used for input here.
Cb =
'Cb' is the allowable stress bending coefficient dependent on the moment gradient, for bending about the X-axis (major axis). 'Cb' is determined as follows:
Cb = 1.75+1.05*(Mx1/Mx2)+0.3*(Mx1/Mx2)^2 <= 2.3
where:
Mx1 = smaller X-axis (major axis) bending moment at either of the ends of the unbraced length
Mx2 = larger X-axis (major axis) bending moment at either of the ends of the unbraced length
Mx1/Mx2 = positive for reverse curvature bending (both have same signs)
= negative for single curvature bending (both have opposite signs)
Notes:
1. When the bending moment at any point within an unbraced length is larger than that at both
ends of this length, then use 'Cb' = 1.0.
2. When computing 'Fbx' to be used in AISC Eqn. H1-1:
a. For frames with sidesway (joint translation), then compute 'Cb' using above equation.
b. For frames without sidesway (braced against joint translation), then use 'Cb' = 1.0.
3. For cantilever beams, 'Cb' may be conservatively assumed = 1.0.
Cw = in.^6
Note: See section of this worksheet (to
the right) for input data which
may be used to determine the
'Cmx' to be used for input here.
Cmx =
'Cmx' is the coefficient applied to the X-axis (major axis) bending term in the interaction equation (H1-1) and is dependent upon column curvature caused by applied moments. The 'Cmx' coefficient value is determined as follows:
Category A:
For compression members in frames subject to joint translation (sidesway), Cmx = 0.85.
Category B:
For rotationally restrained compression members in frames braced against joint
translation (no sidesway) and not subject to transverse loading between their
supports in the plane of bending, Cmx =0.6-0.4*(Mx1/Mx2)
where:
Mx1 = smaller X-axis (major axis) bending moment at either of the ends of the unbraced length
Mx2 = larger X-axis (major axis) bending moment at either of the ends of the unbraced length
Mx1/Mx2 = positive for reverse curvature bending (both have same signs)
= negative for single curvature bending (both have opposite signs)
Category C:
For rotationally restrained compression members in frames braced against joint
translation (no sidesway) and subject to transverse loading between their
supports in the plane of bending, the following values of 'Cmx' are permitted :
1. For members whose ends are restrained against rotation in the plane of bending, Cmx = 0.85.
2. For members whose ends are unrestrained against rotation in the plane of bending, Cmx = 1.0.
Note: See section of this worksheet (to
the right) for input data which
may be used to determine the
'Cmy' to be used for input here.
Cmy =
'Cmy' is the coefficient applied to the Y-axis (minor axis) bending term in the interaction equation (H1-1) and is dependent upon column curvature caused by applied moments. The 'Cmy' coefficient value is determined as follows:
Category A:
For compression members in frames subject to joint translation (sidesway), Cmy = 0.85.
Category B:
For rotationally restrained compression members in frames braced against joint
translation (no sidesway) and not subject to transverse loading between their
supports in the plane of bending, Cmy =0.6-0.4*(My1/My2)
where:
My1 = smaller Y-axis (minor axis) bending moment at either of the ends of the unbraced length
My2 = larger Y-axis (minor axis) bending moment at either of the ends of the unbraced length
My1/My2 = positive for reverse curvature bending (both have same signs)
= negative for single curvature bending (both have opposite signs)
Category C:
For rotationally restrained compression members in frames braced against joint
translation (no sidesway) and subject to transverse loading between their
supports in the plane of bending, the following values of 'Cmy' are permitted :
1. For members whose ends are restrained against rotation in the plane of bending, Cmy = 0.85.
2. For members whose ends are unrestrained against rotation in the plane of bending, Cmy = 1.0.
ASIF =
'ASIF' is the Allowable Stress Increase Factor which is applied to all the allowable stresses and the Euler column buckling stresses used in the stress ratio calculation.
Note: for example, a value of 1.333 can be used for the
'ASIF' for load combinations which include wind or seismic.
Otherwise, use 1.0.
Qs =
'Qs' is the allowable stress reduction factor for an unstiffened compression element (flange) of member determined from AISC Appendix B and is calculated as follows:
when 95/SQRT(Fy/kc) < bf/(2*tf) < 195/SQRT(Fy/kc)
Qs = 1.293-0.00309*bf/(2*tf)*SQRT(Fy/kc) Eqn. A-B5-3
when bf/(2*tf) > 195/SQRT(Fy/kc)
Qs = 26,200*kc/(Fy*bf/(2*tf)) Eqn. A-B5-4
Note: Qs = 1.0 for all W, S, and M shapes for Fy = 36 or 50 ksi.
However, Qs < 1.0 for HP14X73, HP13X60, and HP12X53



Qa =
'Qa' is the ratio of effective profile area of an axially loaded compression member to its total (gross) profile area from AISC Appendix B and is calculated as follows:
when h/tw > 253/SQRT(Fy)
be = 253*tw/SQRT(f)*(1-44.3/((h/tw)*SQRT(f)) <= h Eqn. A-B5-8
Ae = A-(h-be)*tw
Qa = (A-(h-be)*tw)/A
where: be = effective length of web of member
h = d-2*tf
f = computed compressive stress based on effective area
Ae = effective area of member
Results:
'Sx(eff)' is the effective X-axis (major axis) section modulus of an axially loaded compression member, based on a reduced effective width of web, 'be', and is calculated as follows for W, S, M, and HP shapes:
Sx(eff) = Sx-tw*(d-2*tf-be)^3/(6*d)
'Sy(eff)' is the effective Y-axis (minor axis) section modulus of an axially loaded compression member, based on a reduced effective width of web, 'be', and is calculated as follows for W, S, M, and HP shapes:
Sy(eff) = Sy-(d-2*tf-be)*tw^3/(6*bf)
For X-axis Bending:
Kx*Lx/rx =
The expression 'Kx*Lx/rx' is the effective slenderness ratio for members subjected to axial compression load.
Note: 'Lx' is converted from feet to inches in
the evaluation of the expression.
 Lc =
'Lc' is the maximum unbraced length of the compression flange at which the allowable X-axis (major axis) bending stress maybe taken at 0.66*Fy, or from AISC Code Eqn. F1-3 when applicable.
Lc = smaller of: 76*bf/SQRT(Fy) or 20000/((d/Af)*Fy)
 ft.
Ky*Ly/ry =
The expression 'Ky*Ly/ry' is the effective slenderness ratio for members subjected to axial compression load.
Note: 'Ly' is converted from feet to inches in
the evaluation of the expression.
 Lu =
'Lu' is the maximum unbraced length of the compression flange at which the allowable X-axis (major axis) bending stress maybe taken at 0.60*Fy when Cb = 1.
 ft.
Cc =
'Cc' is the column (compression) slenderness ratio separating elastic and inelastic buckling, and is calculated as follows:
Cc = SQRT(2*p^2*E/Fy)
where:
E = modulus of elasticity for steel = 29,000 ksi
Lb/rt =
Note: In the expression 'Lb/rt', the
value of 'Lb' is converted from
feet to inches in the evaluation.
'fa' is the actual compression stress for an axially loaded compression member and is calculated as follows:
fa = P/A

'ft' is the allowable tension stress for an axially loaded tension member and is calculated as follows:
ft = P/A
 ksi fbx =
'fbx' is the actual X-axis (major axis) bending stress and is calculated as follows:
fbx = Mx*12/Sx

 ksi
'Fa' is the allowable compression stress for an axially loaded compression member and is calculated as follows:
For: K*L*12/r <= Cc = SQRT(2*p^2*E/Fy)
use Eqn. E2-1:
Fa = (1-(K*L*12/r)^2/(2*Cc)^2)*Fy/(5/3+3*(K*L*12/r)/(8*Cc)-(K*L*12/r)^3/(8*Cc^3))
For: K*L*12/r > Cc = SQRT(2*p^2*E/Fy)
use Eqn. E2-2:
Fa = 12*p^2*E/(23*(K*L*12/r)^2)
Note: the larger value of either Kx*Lx*12/rx or Ky*Ly*12/ry is to be used in the equations above
to determine 'Fa'.

'Ft' is the allowable tension stress for an axially loaded tension member and is calculated as follows:
Ft = 0.60*Fy
 ksi Fbx =
'Fbx' is the allowable X-axis (major axis) bending stress and is calculated as follows:
For either compression or tension due to bending,
when bf/(2*tf) <= 65/SQRT(Fy), and d/tw compact criteria are met,
and Lb <= Lc:
Fbx = 0.66*Fy (Eqn. F1-1)
when 65/SQRT(Fy) < bf/(2*tf) <= 95/SQRT(Fy)
and Lb <= Lc:
Fbx = Fy*(0.79-0.002*bf/(2*tf)*SQRT(Fy)) (Eqn. F1-3)
when bf/(2*tf) > 95/SQRT(Fy) and Lb <= Lc:
Fbx = 0.60*Fy (Eqn. F1-5)
For tension due to bending,
when the compact criteria are not met,
Fbx = 0.60*Fy

For compression due to bending, and member is either compact or non-
compact and Lb > Lc:
when SQRT(102000*Cb/Fy) <= Lb*12/rt <= SQRT(510000*Cb/Fy):
Fbx = (2/3-Fy*(Lb*12/rt)^2/(1530000*Cb))*Fy <= 0.60*Fy (Eqn. F1-6)
when Lb*12/rt >= SQRT(510000*Cb/Fy):
Fbx = 170000*Cb/((Lb*12/rt)^2) <= 0.60*Fy (Eqn. F1-7)
and for ANY value of Lb*12/rt:
Fbx = 12000*Cb/(Lb*12*d/Af) <= 0.60*Fy (Eqn. F1-8)
Note: for 'Fbx' use larger value of either Eqn. F1-6 and Eqn. F1-8, or Eqn. F1-7
and F1-8, depending on the value of 'Lb*12/rt' as noted above. Also, note
that Eqn. F1-8 is applicable only to sections with a compression flange that
is solid and approximately rectangular.



 ksi
Pa =
'Pa' is the allowable axial load for compression (or tension if applicable), and is calculated as follows:
Pa = Fa*A
 kips Mrx =
'Mrx' is the allowable resisting moment for X-axis (major axis) bending, and is calculated as follows:
Mrx = Fbx*Sx/12
 ft-kips
For Y-axis Bending:
fby =
'fby' is the actual Y-axis (minor axis) bending stress and is calculated as follows:
fby = My*12/Sy

 ksi
Fby =
'Fby' is the allowable Y-axis (minor axis) bending stress and is calculated as follows:
For either compression or tension due to bending,
when bf/(2*tf) <= 65/SQRT(Fy):
Fby = 0.75*Fy (Eqn. F2-1)
when 65/SQRT(Fy) < bf/(2*tf) <= 95/SQRT(Fy):
Fby = Fy*(1.075-0.005*bf/(2*tf)*SQRT(Fy)) (Eqn. F2-2)
when bf/(2*tf) > 95/SQRT(Fy):
Fby = 0.60*Fy (Eqn. F2-3)
 ksi
Mry =
'Mry' is the allowable resisting moment for Y-axis (minor axis) bending, and is calculated as follows:
Mry = Fby*Sy/12
 ft-kips
X-axis Euler Stress:
Y-axis Euler Stress:
F'ex =
F'ex is the Euler compressive buckling stress divided by factor of safety for the X-axis (major axis), and is calculated as follows:
F'ex = 12*p^2*E/(23*(Kx*Lx*12/rx)^2)
 ksi F'ey =
F'ey is the Euler compressive buckling stress divided by factor of safety for the Y-axis (minor axis), and is calculated as follows:
F'ey = 12*p^2*E/(23*(Ky*Ly*12/ry)^2)
 ksi
Stress Ratio:
S.R. =
"S.R." is the Stress Ratio for the member which is calculated as follows:
For members with combined axial compression and bending when fa/Fa > 0.15 per Eqn. H1-1:
S.R. = fa/(ASIF*Fa) + Cmx*fbx/((1-fa/(ASIF*F'ex))*(ASIF*Fbx)) + Cmy*fby/((1-fa/(ASIF*F'ey))*(ASIF*Fby)) <= 1.0
and per Eqn. H1-2:
S.R. = fa/(ASIF*0.60*Fy) + fbx/(ASIF*Fbx) + fby/(ASIF*Fby) <= 1.0
Note: program will display the results of the larger value obtained from either Eqn. H1-1 or Eqn. H1-2
For members with combined axial compression and bending when fa/Fa <= 0.15 per Eqn. H1-3:
S.R. = fa/(ASIF*Fa) + fbx/(ASIF*Fbx) + fby/(ASIF*Fby) <= 1.0

For members with combined axial tension and bending:
S.R. = ft/(ASIF*Ft) + fbx/(ASIF*Fbx) + fby/(ASIF*Fby) <= 1.0
Note: in this case the Stress Ratio computed from just the compressive bending stress(s) must also be checked.
S.R. =
If S.R. > 1.0, then either increase the member size or if possible add/revise framing so that the unbraced length values (Lx, Ly, and Lb) can be reduced, thereby increasing the individual allowable stresses and decreasing the resulting stress ratio.
10 Lightest W Shapes 5 Lightest S Shapes
1.000 1.000
2.000 2.000
3.000 3.000
4.000 4.000
5.000 5.000
6.000
7.000
8.000
9.000
10.000
Optional Input Used to Determine Cb, Cm, & K Values:
Determine Cb by equation:
Single or Reverse curvature?
Mx1 =
'Mx1' is the smaller X-axis (major axis) bending moment at either of the ends of the unbraced length.
 ft-kips
Mx2 =
'Mx2' is the larger X-axis (major axis) bending moment at either of the ends of the unbraced length
Note: if 'Mx2' is input = 0, then 'Cb' will be = 1.0
 ft-kips
Cb =
'Cb' is the allowable stress bending coefficient dependent on the moment gradient, for bending about the X-axis (major axis). 'Cb' is determined as follows:
Cb = 1.75+1.05*(Mx1/Mx2)+0.3*(Mx1/Mx2)^2 <= 2.3
where:
Mx1 = smaller X-axis (major axis) bending moment at either of the ends of the unbraced length
Mx2 = larger X-axis (major axis) bending moment at either of the ends of the unbraced length
Mx1/Mx2 = positive for reverse curvature bending (both have same signs)
= negative for single curvature bending (both have opposite signs)
Notes:
1. When the bending moment at any point within an unbraced length is larger than that at both
ends of this length, then use 'Cb' = 1.0.
2. When computing 'Fbx' to be used in AISC Eqn. H1-1:
a. For frames with sidesway (joint translation), then compute 'Cb' using above equation.
b. For frames without sidesway (braced against joint translation), then use 'Cb' = 1.0.
3. For cantilever beams, 'Cb' may be conservatively assumed = 1.0.
Determine Cmx by equation:
Single or Reverse curvature?
Mx1 =
'Mx1' is the smaller X-axis (major axis) bending moment at either of the ends of the unbraced length.
 ft-kips
Mx2 =
'Mx2' is the larger X-axis (major axis) bending moment at either of the ends of the unbraced length
Note: if 'Mx2' is input = 0, then 'Cmx' will be = 1.0
 ft-kips
Cmx =
'Cmx' is the coefficient applied to the X-axis (major axis) bending term in the interaction equation (H1-1) and is dependent upon column curvature caused by applied moments. The 'Cmx' coefficient value is determined as follows:
Category A:
For compression members in frames subject to joint translation (sidesway), Cmx = 0.85.
Category B:
For rotationally restrained compression members in frames braced against joint
translation (no sidesway) and not subject to transverse loading between their
supports in the plane of bending, Cmx =0.6-0.4*(Mx1/Mx2)
where:
Mx1 = smaller X-axis (major axis) bending moment at either of the ends of the unbraced length
Mx2 = larger Xaxis (major axis) bending moment at either of the ends of the unbraced length
Mx1/Mx2 = positive for reverse curvature bending (both have same signs)
= negative for single curvature bending (both have opposite signs)
Category C:
For rotationally restrained compression members in frames braced against joint
translation (no sidesway) and subject to transverse loading between their
supports in the plane of bending, the following values of 'Cmx' are permitted :
1. For members whose ends are restrained against rotation in the plane of bending, Cmx = 0.85.
2. For members whose ends are unrestrained against rotation in the plane of bending, Cmx = 1.0.
Determine Cmy by equation:
Single or Reverse curvature?
My1 =
'My1' is the smaller Y-axis (minor axis) bending moment at either of the ends of the unbraced length.
 ft-kips
My2 =
'My2' is the larger Y-axis (minor axis) bending moment at either of the ends of the unbraced length.
Note: if 'My2' is input = 0, then 'Cmy' will be = 1.0
 ft-kips
Cmy =
'Cmy' is the coefficient applied to the Y-axis (minor axis) bending term in the interaction equation (H1-1) and is dependent upon column curvature caused by applied moments. The 'Cmy' coefficient value is determined as follows:
Category A:
For compression members in frames subject to joint translation (sidesway), Cmy = 0.85.
Category B:
For rotationally restrained compression members in frames braced against joint
translation (no sidesway) and not subject to transverse loading between their
supports in the plane of bending, Cmy =0.6-0.4*(My1/My2)
where:
My1 = smaller Y-axis (minor axis) bending moment at either of the ends of the unbraced length
My2 = larger Y-axis (minor axis) bending moment at either of the ends of the unbraced length
My1/My2 = positive for reverse curvature bending (both have same signs)
= negative for single curvature bending (both have opposite signs)
Category C:
For rotationally restrained compression members in frames braced against joint
translation (no sidesway) and subject to transverse loading between their
supports in the plane of bending, the following values of 'Cmy' are permitted :
1. For members whose ends are restrained against rotation in the plane of bending, Cmy = 0.85.
2. For members whose ends are unrestrained against rotation in the plane of bending, Cmy = 1.0.
Determine Kx by alignment charts:
Braced or Unbraced Frame?
Ga =
'Ga' represents the "a" joint (end) of the two ends of the column section being considered in the determination of effective length using the AISC Code Alignment Charts or equations representing the Alignment Charts.
'G' is determined as follows:
G = S (Ic/Lc)/S (Ig/Lg)
where:
S = indicates a summation of all members rigidly connected to that join
and within the plane of buckling of the column
Ic = moment of inertia of column taken about axis perpendicular to plane
of buckling
Lc = unsupported length of column section
Ig = moment of inertia of girder or other restaining member taken about
axis perpendicular to plane of buckling
Lg = unsupported length of girder or other restraining member
Note:
For column ends supported by but not rigidly connected to a foundation, 'G'
may be asumed equal to 10 for practical designs. If the column end is rigidly
attached to the foundation, 'G' may be assumed equal to 1.0.
Gb =
'Gb' represents the "b" joint (end) of the two ends of the column section being considered in the determination of effective length using the AISC Code Alignment Charts or equations representing the Alignment Charts.
'G' is determined as follows:
G = S (Ic/Lc)/S (Ig/Lg)
where:
S = indicates a summation of all members rigidly connected to that join
and within the plane of buckling of the column
Ic = moment of inertia of column taken about axis perpendicular to plane
of buckling
Lc = unsupported length of column section
Ig = moment of inertia of girder or other restaining member taken about
axis perpendicular to plane of buckling
Lg = unsupported length of girder or other restraining member
Note:
For column ends supported by but not rigidly connected to a foundation, 'G'
may be asumed equal to 10 for practical designs. If the column end is rigidly
attached to the foundation, 'G' may be assumed equal to 1.0.
Kx =
'Kx' is the effective length factor about the X-axis (major axis) for an axially loaded compression member as determined by the AISC Code Alignment Charts or equations representing Alignment Charts.
For Braced Frames:
The theoretical equation representing the Nomograph is as follows:
Ga*Gb/4*(p/K)^2+((Ga+Gb)/2)*(1-p/K/(TAN(p/K)))+2*TAN(p/(2*K)/(p/K) = 1
For which the following approximate equation may be used to directly solve for 'K':
K = SQRT((Ga+0.41)*(Gb+0.41)/((Ga+0.82)*(Gb+0.82)))
Note: above approximate equation underestimates 'K' by not more than 0.1%
and overestimates it by less than 1.5%.
For Unbraced Frames:
The theoretical equation representing the Nomograph is as follows:
Ga*Gb*(p/K)^2-36/(6*(Ga+Gb)) = (p/K)/TAN(p/K)
For which the following approximate equation may be used to directly solve for 'K':
K = SQRT((1.6*Ga*Gb+4.0*(Ga+Gb)+7.5)/(Ga+Gb+7.5))
Note: above approximate equation estimates 'K' within 2% error.

Reference: "Historical Note on K-Factor Equations", by Pierre Dumonteil
AISC Engineering Journal - 2nd Quarter - 1999
Determine Ky by alignment charts:
Braced or Unbraced Frame?
Ga =
'Ga' represents the "a" joint (end) of the two ends of the column section being considered in the determination of effective length using the AISC Code Alignment Charts or equations representing the Alignment Charts.
'G' is determined as follows:
G = S (Ic/Lc)/S (Ig/Lg)
where:
S = indicates a summation of all members rigidly connected to that join
and within the plane of buckling of the column
Ic = moment of inertia of column taken about axis perpendicular to plane
of buckling
Lc = unsupported length of column section
Ig = moment of inertia of girder or other restaining member taken about
axis perpendicular to plane of buckling
Lg = unsupported length of girder or other restraining member
Note:
For column ends supported by but not rigidly connected to a foundation, 'G'
may be asumed equal to 10 for practical designs. If the column end is rigidly
attached to the foundation, 'G' may be assumed equal to 1.0.
Gb =
'Gb' represents the "b" joint (end) of the two ends of the column section being considered in the determination of effective length using the AISC Code Alignment Charts or equations representing the Alignment Charts.
'G' is determined as follows:
G = S (Ic/Lc)/S (Ig/Lg)
where:
S = indicates a summation of all members rigidly connected to that join
and within the plane of buckling of the column
Ic = moment of inertia of column taken about axis perpendicular to plane
of buckling
Lc = unsupported length of column section
Ig = moment of inertia of girder or other restaining member taken about
axis perpendicular to plane of buckling
Lg = unsupported length of girder or other restraining member
Note:
For column ends supported by but not rigidly connected to a foundation, 'G'
may be asumed equal to 10 for practical designs. If the column end is rigidly
attached to the foundation, 'G' may be assumed equal to 1.0.
Ky =
'Ky' is the effective length factor about the Y-axis (minor axis) for an axially loaded compression member as determined by the AISC Code Alignment Charts or equations representing Alignment Charts.
For Braced Frames:
The theoretical equation representing the Nomograph is as follows:
Ga*Gb/4*(p/K)^2+((Ga+Gb)/2)*(1-p/K/(TAN(p/K)))+2*TAN(p/(2*K)/(p/K) = 1
For which the following approximate equation may be used to directly solve for 'K':
K = SQRT((Ga+0.41)*(Gb+0.41)/((Ga+0.82)*(Gb+0.82)))
Note: above approximate equation underestimates 'K' by not more than 0.1%
and overestimates it by less than 1.5%.
For Unbraced Frames:
The theoretical equation representing the Nomograph is as follows:
Ga*Gb*(p/K)^2-36/(6*(Ga+Gb)) = (p/K)/TAN(p/K)
For which the following approximate equation may be used to directly solve for 'K':
K = SQRT((1.6*Ga*Gb+4.0*(Ga+Gb)+7.5)/(Ga+Gb+7.5))
Note: above approximate equation estimates 'K' within 2% error.

Reference: "Historical Note on K-Factor Equations", by Pierre Dumonteil
AISC Engineering Journal - 2nd Quarter - 1999

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