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Analytic solutions for tapered column buckling
Compufers It Slrucfures Vol. 28, No. 5, pp. 677481, Printed in Great Britain.
004%7949/88 53.00 + 0.00 Pwgamon Press plc
1988
TECXNICAL
NOTE
ANALYTIC SOLUTIONS FOR TAPERED COLUMN BUCKLING W. GARTH SMITH Mosaic Software, Inc., 1972 Massachusetts
Avenue, Cambridge, MA 02140, U.S.A.
(Receiwd 28 January 1987) Ahstraet-The energy method and symbolic aided analysis provide analytical expressions for the critical buckling load of tapered columns. The theory developed applies to any column whose cross sectional variation can be written as a function of the axial coordinate. Examples and percentage errata are included.
NOTA~ON
amplitude of assumed deflected shape dimension of base of rectangular column cross ml section Young’s modulus for homogeneous material E H(z) dimension of height of rectangular column cross section height dimension at z = 0 height diiension at z = L moment of inertia about the centroid moment of inertia at 2 = 0 moment of inertia at 2 = L constant coefficient pertaining to moment of inertia model length of the column before deflection L M(z) bending moment dist~b~tion applied tip load P critical buckling load P RT) radius dimension of circular column cross section radius at z = 0 RI radiusatz=l R2 work done on column by tip load strain energy due to bending z assumed deflected shape Y”(z) vertical deflection of column i
STANDARD ~0~ FOR PR~I~NG CRlTICAL BUCKLlNG LOALtS WHERE THE CROSS SEclloN OF THE COLUMN VARIES
B
Predicting the load which will cause the column to buckle is traditionally done by numerical or finite element techniques. An exact method is to solve the differential equation d2 Y(z)
.wz) -+LuMfz)=O, dz2
with appropriate boundary conditions for the ends. Dinnik [ 1) shows that when the moment of inertia varies as a power, eqn (I) can be solved with Bessel functions where
I(z)=Z,
z ”
0 -
a
.
NON~I~RM COLUMN ANALYSIS AS APPLIED TO STRINGER DESIGN FOR WING BAYS
Given a ~~dirnen~ona~ moment ~s~bution corresponding to a wing loading (Fig. I). the need arises to design stringers/columns for individual wing bays (Fig. 2) that can resist the variable moment at each cross section (i.e. the column should not buckle). A traditional approach is to design the column to resist the highest bending moment (e.g. hf,) and have constant cross sectional area members for the bay. The weight penalty for such an approach can be prohibitive since at any location (other than where the largest moment occurs) the column is overdesigned. Alte~atively, an optimal stringer design is one where the functional moment of inertia variation of the member differs from that of the moment distribution by a constant (i.e. column strength-and consequently the factor of safety-are both constant). A practical analysis and manufacturing compromise is to design the cohnnn such that tlte dimensions taper linearly between ribs.
Fig. 1. Moment distribution as a function of wing length with wing bay positions indicated.
ringar
Fig 2. Wing bay two corres~nding dist~bution. 617
to Fig. 1 moment
Technical Note
678
Aerospace structures such as T-stringers and I-beams are columus whose moment of inertia is not simpIy described by eqn (2) and require a more general expr~sion associated with composite shapes of
where the &s are constant coefficients prudent on the coiumns’ cross section and by what order the dimensions taper.
A fixed-free column (Fig. 3) is used to demonstrate the present method. Wowever, the t~h~que is equally a~li~ble for other Sunday condit~ns. Traditional expressions are ~~b~~~ for work done by a ~mpr~ion and the resulting strain energy due to bendingI3, pp. SZ-SSJ where
Example: Using the energy method, establish the critical buckling load for a fixed-free square pyramid or a truncated cone colnmn (Fig. 4) whose moment of inertia varies about the x axis as n = 4 in eqn (3). This is a speciat case where, due to s~rne~, the Form of eqn (3) can be expressed in the form of eqn (2) and thus allows comparison between exact and energy method results. Conside~ng linear tapering ~mensio~s for the square pyramid and the moment of inertia for a square cross section (B = H in Fig. 4(a)),
~uatio~ 112) is then exuressed in the form of eqn (3) (i.e. n = 4), . I(z)=k,z°Ck,z’+E;,z2Ck,zJfk4z4, where
An assumed de&c&d shape that accounts for firsi mode conditions for a fixed-free cohzmn is then
~uckli~[3, p, 911and s&Mes the madam
into eqn (5) and using eqn (6) yields
The critical buckling condition ooaurs when V, = V, and substituting eqn (3) for l(z) in eqn (8) gives
Similarly, for a pinny-pinned
column
- 4H:H, + Hfjz’) cos2 E dz. ( > substituting nurne~~~ constants for the moment of inertia ratios allows expression of H, in terms of Hr, or
I
P
Y
Fig. 3. A fixed-free column subjected to a tip load.
Fig. 4. Cross sections of edumns for (a) square pyramid . I.\ . *, ana (a) an axrauy symmetnc truncated cone.
Technical Note
679
a2sef
Then, combining eqn (20) and the solution to (19) (which is referenced as Pm5 in Appendix A) allows comparison (Fig. 5) between Dinnik’s results and the energy method when the critical buckling load is expressed as
B = BI + ZB,L - ZBJL H = constant
mEI,
PC? = --p-
p,, = EH3(4( - B, i- B,) + x2(Bz f ~,))/(96~2) PC,2 = n
2
EH
3 (II2+
B,)/(24L*).
CONCLUSIONS The example included is a special case where the form of eqn (3) can be expressed in the form of eqn (2). However, this technique is intended when form (2) does not apply as in the case of composite shapes (Appendix B). Notice the low percentage error even when only the first term is used in the series approximation for the deflected shape for the example. In the past, the computations necessary to generate the forms in Appendices A and B would have tended to be prohibitive owing to their size. Symbolic analysis packages such as MuMath~ and MACSYMAIM allow for the generation and storage of such expressions in a more mana~ble manner. With the energy method and symbolic analysis it is only a matter of providing a functional description of the moment of inertia and the number of modes in order to generate the critical buckling load. Notice that by using effective lengths, the range of applicability of the fixed-free and pinned-pinned cases can be extended to cover boundary conditions not discussed here. Finally, it can be seen that for the sacrifice of expression size, an analytical solution can supersede the numerical.
case 2 3 = instant
p,s = ~EWW,W~W,
-
3H,) + 3Hj) - H;)
+ n2(12(Hz(H2(-Hz + H,) - H:) + Hi) + ~*(H~(H~(~~+ H,) + H:) + H~)))/(192~2~2) PC&= S&3( -(H:
+ H:) + H&H2
+ H,))
+ ~*(H*(~~(H* + H,f + H:> + H~))/{~~*). c@&?3 B-H H=H,+Zl&/L-ZH,/L P ws= E(120(Hz(H:(H, - 2H,) + 2H:) - H;) + ~~2qH~~H~-H~ + H,) -H:,
~e~~o~~e~geme~~-Dr Nithiam T. Sivaneri is thanked for his critical comments and guidance.
+ n2Ufz&Vf,W,
-I-H:)
+ H,) + H:) f H:)
f H~~))/(2~z~2) REFERENCES
Pcr6= E(IS(H,(H,(H,(H, - 4H,) + 6H:) - 4H;) + Hi)
1, A. N. Dinnik,
+ 2x2(5(-W:
Design of columns of varying cross sections. Tram. ASME 51, APM31-11 (1929) and 54, APM-54-16 (1932). 2. F. Bleich, Theorie and ~erechmmg der E&ertzett Bracken, pp. 136-142. Springer, Berlin (1924). 3. S. P. Timoshenko and J. M. Gere, Theory ofEkstic &se 4 Stability. McGraw-Hill, New York (l%i).
buckling formulas for the moment of inertia about the x-axis are given for five geometries whose dimensions taper linearly corresponding to Fig. 4. The odd number solutions are for the fixed-free case and the even number solutions am for the pinny-pinny case.
0.0
~2~~2~~~~~2~H2~~,~+
+
ff~~))/(l2~2~2).
0.1
0.2
0.3
W:)+N:)
- ZR,/L
Pcr?= EU2O(R,(R:(R, - 2RJ + 2R;) - R;)
The critical
Aa
+
R i= R, + ZRJL
APPENDIX A
+ H:) + H2H,(# + H:))
+ ~‘~2qR*~R~~-R~ + R,) - R:) + Rf) + “2VV.R,(R,tR, + R,) + R:) + R:) + R~)))l~80~L2)
oa4
0.5
11 -------
1.202 1,505 1.710
---
1111
1.870 2.002
-----0.886 1.500 1.684 1.824 1.942 2.047
m2
------ig;z:ntl I
-I
124.8 111.9 I
I
1 6.7 1 3.8 1 2.2 ( 1.2 1 0.6 I
I
I
I
I
Fig. 5. m, corresponds to Dinnik’s results [3, p. 130) and m2 corresponds to the energy method. The percent error is generated as the taper of the column becomes si~ifi~nt and this is due to the actual deflected shape no longer being a simple trigonometric shape as assumed.
680
Technical Note pcrs = ~(15(R~t~*(R~(R~ - 4RJ + 6R:) - 4R;) + R:) + 2x2(5( -(R$
+ R:) + R,R,(R$ + Rf))
f ~*(R~(R~(R~(R~f R,) + R:) + R:) + R~)))/(~~L~). Case 5
Combining I(z) and the dimensional descriptions H(z), T(z) and W(z) with eqn (9) yields
B = B, + ZB&
- Z&/L
H = H, i-Z&/L
- ZHJL
B(Z),
Pn,, = - (((32n4 - 64Orr* + 3840ET; + ((24~~ + 480x ’ - 576O)ET,
pcr9 = JV~~~(H;(H,(H,(~B,
-B,I
-3H&)
+ 3H:B,)
+ HXB, - 23,)) + ~2(~5(H~t~~(~~(-4~*
+ (-48x4
+ 3,)
+3H,B~~-3H~~~)~H~(-B*~4B,))
C(16x4ET:+(~-24n4-4SOn?
f ~~(H*(~*~~~(4B~ + B,) + fJs(3Bz t ZB,)) + H:(2B,
+ 5760fEHz - 16n4EH1)T, i- 38,))
+ (24d
+ H:(Bz + 4B,~)))/(9~~zLz)
+ (12~’ + 240x2 - 2SSO)EH, Hz + 4n4EHf)T2 + (Sn4 - 160d + 1920)ETf
+ 3H3 - 3H:) + ff;))
+ ((-
+ ~~(5(B,~~~(-4~~ + B&H,(H:
12x4 + 24On* - 2880~~~~ - SdEH,)Tf
+ ((6nd + 120x ’ - 1~0)~~~ + Sn4EH, H,
+ 3H,) + Hi)
+ (6n* - 12&c2 + MO)EHf)T,
+ 3H:) - 4H;))
f ~~(B,(~,(~~(4~,
- 4SOn* + 2880)~~~
- 3H,) + 3H:) - H;)
pw,o = EOO(B,(H,(ff,(H, + B*(H,(~,(-~,
+ 96&r2 - 576OEN,
f (- 127r’ - 240x2 + 288O)~~,)T~
+ (-4n4
+ 3H2j + 2H;) i- Hi)
+ S0a2 - 48O)EH;
+ ~~~H,~H~~~,+ 2H,) + 3H:)
+ (- 3n“ - 60x” -I-72O)EH, H;
+ 4H~))))/(240 x2L2).
-2n%HfH,+f-n4+20n’--24O)EH;)W, i- ((Sd f 160~ - 1920)ET; +(16x%2-,
~PPENR~ B The critical buckling formula for the moment of inertia about the x-axis is given for a fixed-free I-beam with linearly tapering Dimensions.
+(-i2n4-240~2+2S80~EH*
- 8z4EH,)T:
+ ((24x4 - 4807~~+ 5760)ETf
f (( -24~1’ +480x2 - i6~4~~*)~,(6~4
- 576O)EH, + 120~’
- l~O)~H~
f Sn4EHIH, + (6~“ - 120na f l~O~EH~)T~ f (32x4 +640x2 - 3840)EF; + (( - I 2n4 + 240x2 - 2SSO)EH~ +(-48x4
-960x2
+ 576O~~H,}T~
+ (4n4EH; +(12x4-240x2 -t. 288O)EH, Hz + (24~ f 4807~’ - 2880)EH:)T, -+ t--n4 - 20x*-t 2~)~~~
- 2~4~~,H~
+(-3rt4+60s2-720)EH:Hz +(-4x4-807?+480)EH;)W, + (( - 32x 4 + 6401~2 - 384O~B* -----B-
+(-&I-
160n’f
192O)B,)~T~
+ ((( - 24n4 - 4807~~i- 5?60)8, - 16n4B,)ET, + ((487~~- 960~’ + 57SO)B, + (127~~4 24Oz* - 2SSO)B,)~H~ *
+ ((12~~ + 240~’ - 2SSO)B, + Sn4B,)Eff,)T;
+ (((-24~~
+ 4SOn’
- .5’76O)B,- 16x%,)ETf + (((24~~ + 480n2 - 576O)B, + 16n4B,)EH, + (M?r4Bz + (24~~ - 480x2 + 576O)B,)EH,)T,
Fig. 31. An K-beam with B, H, T and W tapering finearly as a function of z, where z = 0 is the built-in end and z = L is the free end.
+ (( -24~~ + 480x* - ZSSO)B, +(-6x’-
120n2 i ~~O)B,)~H~
f (( - 12x4 - 24od
+ 2SSO)B2
Technical Note - 8n’B,)EH,H,
+ ((-6n4 + 120d - 144O)B,
- 4n4B,)EH;)T2 + (( - 8n4 + 160n2 - 192O)B,
28 I-,
+ (((-6~~ - 1207r2+ 144O)B, - 4n4B,)EH; + (( - 12x4 + 240~~ - 288O)B,
+ (-32x4 - 640~~ + 384O)B,)ET:
- 8n4B2)EH,H2 +((-6n4
+ ((8rr4B2+ (12~~ - 2407~~+ 288O)B,)EH,
+ (-24n’-
+ ((12~~ - 24011~+ 288O)B,
+ TJ(96OdP).
+(48x4 + 960~~ - 576O)B,)EH,)T:
C.A.S
681
+ 12h2-
480~~ + 288O)B,)EHf
144O)B,
004%7949/88 53.00 + 0.00 Pwgamon Press plc
1988
TECXNICAL
NOTE
ANALYTIC SOLUTIONS FOR TAPERED COLUMN BUCKLING W. GARTH SMITH Mosaic Software, Inc., 1972 Massachusetts
Avenue, Cambridge, MA 02140, U.S.A.
(Receiwd 28 January 1987) Ahstraet-The energy method and symbolic aided analysis provide analytical expressions for the critical buckling load of tapered columns. The theory developed applies to any column whose cross sectional variation can be written as a function of the axial coordinate. Examples and percentage errata are included.
NOTA~ON
amplitude of assumed deflected shape dimension of base of rectangular column cross ml section Young’s modulus for homogeneous material E H(z) dimension of height of rectangular column cross section height dimension at z = 0 height diiension at z = L moment of inertia about the centroid moment of inertia at 2 = 0 moment of inertia at 2 = L constant coefficient pertaining to moment of inertia model length of the column before deflection L M(z) bending moment dist~b~tion applied tip load P critical buckling load P RT) radius dimension of circular column cross section radius at z = 0 RI radiusatz=l R2 work done on column by tip load strain energy due to bending z assumed deflected shape Y”(z) vertical deflection of column i
STANDARD ~0~ FOR PR~I~NG CRlTICAL BUCKLlNG LOALtS WHERE THE CROSS SEclloN OF THE COLUMN VARIES
B
Predicting the load which will cause the column to buckle is traditionally done by numerical or finite element techniques. An exact method is to solve the differential equation d2 Y(z)
.wz) -+LuMfz)=O, dz2
with appropriate boundary conditions for the ends. Dinnik [ 1) shows that when the moment of inertia varies as a power, eqn (I) can be solved with Bessel functions where
I(z)=Z,
z ”
0 -
a
.
NON~I~RM COLUMN ANALYSIS AS APPLIED TO STRINGER DESIGN FOR WING BAYS
Given a ~~dirnen~ona~ moment ~s~bution corresponding to a wing loading (Fig. I). the need arises to design stringers/columns for individual wing bays (Fig. 2) that can resist the variable moment at each cross section (i.e. the column should not buckle). A traditional approach is to design the column to resist the highest bending moment (e.g. hf,) and have constant cross sectional area members for the bay. The weight penalty for such an approach can be prohibitive since at any location (other than where the largest moment occurs) the column is overdesigned. Alte~atively, an optimal stringer design is one where the functional moment of inertia variation of the member differs from that of the moment distribution by a constant (i.e. column strength-and consequently the factor of safety-are both constant). A practical analysis and manufacturing compromise is to design the cohnnn such that tlte dimensions taper linearly between ribs.
Fig. 1. Moment distribution as a function of wing length with wing bay positions indicated.
ringar
Fig 2. Wing bay two corres~nding dist~bution. 617
to Fig. 1 moment
Technical Note
678
Aerospace structures such as T-stringers and I-beams are columus whose moment of inertia is not simpIy described by eqn (2) and require a more general expr~sion associated with composite shapes of
where the &s are constant coefficients prudent on the coiumns’ cross section and by what order the dimensions taper.
A fixed-free column (Fig. 3) is used to demonstrate the present method. Wowever, the t~h~que is equally a~li~ble for other Sunday condit~ns. Traditional expressions are ~~b~~~ for work done by a ~mpr~ion and the resulting strain energy due to bendingI3, pp. SZ-SSJ where
Example: Using the energy method, establish the critical buckling load for a fixed-free square pyramid or a truncated cone colnmn (Fig. 4) whose moment of inertia varies about the x axis as n = 4 in eqn (3). This is a speciat case where, due to s~rne~, the Form of eqn (3) can be expressed in the form of eqn (2) and thus allows comparison between exact and energy method results. Conside~ng linear tapering ~mensio~s for the square pyramid and the moment of inertia for a square cross section (B = H in Fig. 4(a)),
~uatio~ 112) is then exuressed in the form of eqn (3) (i.e. n = 4), . I(z)=k,z°Ck,z’+E;,z2Ck,zJfk4z4, where
An assumed de&c&d shape that accounts for firsi mode conditions for a fixed-free cohzmn is then
~uckli~[3, p, 911and s&Mes the madam
into eqn (5) and using eqn (6) yields
The critical buckling condition ooaurs when V, = V, and substituting eqn (3) for l(z) in eqn (8) gives
Similarly, for a pinny-pinned
column
- 4H:H, + Hfjz’) cos2 E dz. ( > substituting nurne~~~ constants for the moment of inertia ratios allows expression of H, in terms of Hr, or
I
P
Y
Fig. 3. A fixed-free column subjected to a tip load.
Fig. 4. Cross sections of edumns for (a) square pyramid . I.\ . *, ana (a) an axrauy symmetnc truncated cone.
Technical Note
679
a2sef
Then, combining eqn (20) and the solution to (19) (which is referenced as Pm5 in Appendix A) allows comparison (Fig. 5) between Dinnik’s results and the energy method when the critical buckling load is expressed as
B = BI + ZB,L - ZBJL H = constant
mEI,
PC? = --p-
p,, = EH3(4( - B, i- B,) + x2(Bz f ~,))/(96~2) PC,2 = n
2
EH
3 (II2+
B,)/(24L*).
CONCLUSIONS The example included is a special case where the form of eqn (3) can be expressed in the form of eqn (2). However, this technique is intended when form (2) does not apply as in the case of composite shapes (Appendix B). Notice the low percentage error even when only the first term is used in the series approximation for the deflected shape for the example. In the past, the computations necessary to generate the forms in Appendices A and B would have tended to be prohibitive owing to their size. Symbolic analysis packages such as MuMath~ and MACSYMAIM allow for the generation and storage of such expressions in a more mana~ble manner. With the energy method and symbolic analysis it is only a matter of providing a functional description of the moment of inertia and the number of modes in order to generate the critical buckling load. Notice that by using effective lengths, the range of applicability of the fixed-free and pinned-pinned cases can be extended to cover boundary conditions not discussed here. Finally, it can be seen that for the sacrifice of expression size, an analytical solution can supersede the numerical.
case 2 3 = instant
p,s = ~EWW,W~W,
-
3H,) + 3Hj) - H;)
+ n2(12(Hz(H2(-Hz + H,) - H:) + Hi) + ~*(H~(H~(~~+ H,) + H:) + H~)))/(192~2~2) PC&= S&3( -(H:
+ H:) + H&H2
+ H,))
+ ~*(H*(~~(H* + H,f + H:> + H~))/{~~*). c@&?3 B-H H=H,+Zl&/L-ZH,/L P ws= E(120(Hz(H:(H, - 2H,) + 2H:) - H;) + ~~2qH~~H~-H~ + H,) -H:,
~e~~o~~e~geme~~-Dr Nithiam T. Sivaneri is thanked for his critical comments and guidance.
+ n2Ufz&Vf,W,
-I-H:)
+ H,) + H:) f H:)
f H~~))/(2~z~2) REFERENCES
Pcr6= E(IS(H,(H,(H,(H, - 4H,) + 6H:) - 4H;) + Hi)
1, A. N. Dinnik,
+ 2x2(5(-W:
Design of columns of varying cross sections. Tram. ASME 51, APM31-11 (1929) and 54, APM-54-16 (1932). 2. F. Bleich, Theorie and ~erechmmg der E&ertzett Bracken, pp. 136-142. Springer, Berlin (1924). 3. S. P. Timoshenko and J. M. Gere, Theory ofEkstic &se 4 Stability. McGraw-Hill, New York (l%i).
buckling formulas for the moment of inertia about the x-axis are given for five geometries whose dimensions taper linearly corresponding to Fig. 4. The odd number solutions are for the fixed-free case and the even number solutions am for the pinny-pinny case.
0.0
~2~~2~~~~~2~H2~~,~+
+
ff~~))/(l2~2~2).
0.1
0.2
0.3
W:)+N:)
- ZR,/L
Pcr?= EU2O(R,(R:(R, - 2RJ + 2R;) - R;)
The critical
Aa
+
R i= R, + ZRJL
APPENDIX A
+ H:) + H2H,(# + H:))
+ ~‘~2qR*~R~~-R~ + R,) - R:) + Rf) + “2VV.R,(R,tR, + R,) + R:) + R:) + R~)))l~80~L2)
oa4
0.5
11 -------
1.202 1,505 1.710
---
1111
1.870 2.002
-----0.886 1.500 1.684 1.824 1.942 2.047
m2
------ig;z:ntl I
-I
124.8 111.9 I
I
1 6.7 1 3.8 1 2.2 ( 1.2 1 0.6 I
I
I
I
I
Fig. 5. m, corresponds to Dinnik’s results [3, p. 130) and m2 corresponds to the energy method. The percent error is generated as the taper of the column becomes si~ifi~nt and this is due to the actual deflected shape no longer being a simple trigonometric shape as assumed.
680
Technical Note pcrs = ~(15(R~t~*(R~(R~ - 4RJ + 6R:) - 4R;) + R:) + 2x2(5( -(R$
+ R:) + R,R,(R$ + Rf))
f ~*(R~(R~(R~(R~f R,) + R:) + R:) + R~)))/(~~L~). Case 5
Combining I(z) and the dimensional descriptions H(z), T(z) and W(z) with eqn (9) yields
B = B, + ZB&
- Z&/L
H = H, i-Z&/L
- ZHJL
B(Z),
Pn,, = - (((32n4 - 64Orr* + 3840ET; + ((24~~ + 480x ’ - 576O)ET,
pcr9 = JV~~~(H;(H,(H,(~B,
-B,I
-3H&)
+ 3H:B,)
+ HXB, - 23,)) + ~2(~5(H~t~~(~~(-4~*
+ (-48x4
+ 3,)
+3H,B~~-3H~~~)~H~(-B*~4B,))
C(16x4ET:+(~-24n4-4SOn?
f ~~(H*(~*~~~(4B~ + B,) + fJs(3Bz t ZB,)) + H:(2B,
+ 5760fEHz - 16n4EH1)T, i- 38,))
+ (24d
+ H:(Bz + 4B,~)))/(9~~zLz)
+ (12~’ + 240x2 - 2SSO)EH, Hz + 4n4EHf)T2 + (Sn4 - 160d + 1920)ETf
+ 3H3 - 3H:) + ff;))
+ ((-
+ ~~(5(B,~~~(-4~~ + B&H,(H:
12x4 + 24On* - 2880~~~~ - SdEH,)Tf
+ ((6nd + 120x ’ - 1~0)~~~ + Sn4EH, H,
+ 3H,) + Hi)
+ (6n* - 12&c2 + MO)EHf)T,
+ 3H:) - 4H;))
f ~~(B,(~,(~~(4~,
- 4SOn* + 2880)~~~
- 3H,) + 3H:) - H;)
pw,o = EOO(B,(H,(ff,(H, + B*(H,(~,(-~,
+ 96&r2 - 576OEN,
f (- 127r’ - 240x2 + 288O)~~,)T~
+ (-4n4
+ 3H2j + 2H;) i- Hi)
+ S0a2 - 48O)EH;
+ ~~~H,~H~~~,+ 2H,) + 3H:)
+ (- 3n“ - 60x” -I-72O)EH, H;
+ 4H~))))/(240 x2L2).
-2n%HfH,+f-n4+20n’--24O)EH;)W, i- ((Sd f 160~ - 1920)ET; +(16x%2-,
~PPENR~ B The critical buckling formula for the moment of inertia about the x-axis is given for a fixed-free I-beam with linearly tapering Dimensions.
+(-i2n4-240~2+2S80~EH*
- 8z4EH,)T:
+ ((24x4 - 4807~~+ 5760)ETf
f (( -24~1’ +480x2 - i6~4~~*)~,(6~4
- 576O)EH, + 120~’
- l~O)~H~
f Sn4EHIH, + (6~“ - 120na f l~O~EH~)T~ f (32x4 +640x2 - 3840)EF; + (( - I 2n4 + 240x2 - 2SSO)EH~ +(-48x4
-960x2
+ 576O~~H,}T~
+ (4n4EH; +(12x4-240x2 -t. 288O)EH, Hz + (24~ f 4807~’ - 2880)EH:)T, -+ t--n4 - 20x*-t 2~)~~~
- 2~4~~,H~
+(-3rt4+60s2-720)EH:Hz +(-4x4-807?+480)EH;)W, + (( - 32x 4 + 6401~2 - 384O~B* -----B-
+(-&I-
160n’f
192O)B,)~T~
+ ((( - 24n4 - 4807~~i- 5?60)8, - 16n4B,)ET, + ((487~~- 960~’ + 57SO)B, + (127~~4 24Oz* - 2SSO)B,)~H~ *
+ ((12~~ + 240~’ - 2SSO)B, + Sn4B,)Eff,)T;
+ (((-24~~
+ 4SOn’
- .5’76O)B,- 16x%,)ETf + (((24~~ + 480n2 - 576O)B, + 16n4B,)EH, + (M?r4Bz + (24~~ - 480x2 + 576O)B,)EH,)T,
Fig. 31. An K-beam with B, H, T and W tapering finearly as a function of z, where z = 0 is the built-in end and z = L is the free end.
+ (( -24~~ + 480x* - ZSSO)B, +(-6x’-
120n2 i ~~O)B,)~H~
f (( - 12x4 - 24od
+ 2SSO)B2
Technical Note - 8n’B,)EH,H,
+ ((-6n4 + 120d - 144O)B,
- 4n4B,)EH;)T2 + (( - 8n4 + 160n2 - 192O)B,
28 I-,
+ (((-6~~ - 1207r2+ 144O)B, - 4n4B,)EH; + (( - 12x4 + 240~~ - 288O)B,
+ (-32x4 - 640~~ + 384O)B,)ET:
- 8n4B2)EH,H2 +((-6n4
+ ((8rr4B2+ (12~~ - 2407~~+ 288O)B,)EH,
+ (-24n’-
+ ((12~~ - 24011~+ 288O)B,
+ TJ(96OdP).
+(48x4 + 960~~ - 576O)B,)EH,)T:
C.A.S
681
+ 12h2-
480~~ + 288O)B,)EHf
144O)B,