Copyrighted Materials CapygtCrpUy Press fort wwo APPENDIX A Cross-Sectional Properties of Thin-Walled Composite Beams In the following tables au and du are the elements of the compliance matrix of symmetrical laminates (Egs.3.29 and 3.30)evaluated at the midsurface;aij, Bij,8ij are the elements of the compliance matrix of nonsymmetrical laminates (Eq.3.23).These properties are evaluated at each wall segment's"neutral"plane. In the main text these elements are identified by the superscript o.To simplify the notation,in this Appendix the superscript o is omitted. The subscripts fl,f2,and w refer to flange 1,flange 2 and the web,respectively. The modified material properties are (B16)2 11= 011 866 a11= 0111 822 E1=（B11- P16⊙16 a1= P12δ12 f11- 866 822 811= 811- (616)2 866 81 811一822 The location of the"neutral"plane with respect to the midplane is open section 0=- orthotropic unsymmetrical layup (A.1) 011 arbitrary cross section MP open section 0=- 骊 arbitrary layup (A.2) symmetrical cross section closed section 0=- orthotropic unsymmetrical layup A.3 arbitrary cross section. BMP and M are evaluated at the midplanes. 453
APPENDIX A Cross-Sectional Properties of Thin-Walled Composite Beams In the following tables a11 and d11 are the elements of the compliance matrix of symmetrical laminates (Eqs. 3.29 and 3.30) evaluated at the midsurface; αi j , βi j , δi j are the elements of the compliance matrix of nonsymmetrical laminates (Eq. 3.23). These properties are evaluated at each wall segment’s “neutral” plane. In the main text these elements are identified by the superscript . To simplify the notation, in this Appendix the superscript is omitted. The subscripts f1, f2, and w refer to flange 1, flange 2 and the web, respectively. The modified material properties are α11 = - α11 − (β16) 2 δ66 . α11 = α11 − β2 12 δ22 β11 = β11 − β16δ16 δ66 β11 = β11 − β12δ12 δ22 δ11 = - δ11 − (δ16) 2 δ66 . δ11 = δ11 − δ2 12 δ22 . The location of the “neutral” plane with respect to the midplane is = −βMP 11 δMP 11 open section orthotropic unsymmetrical layup arbitrary cross section (A.1) = −βMP 11 δ MP 11 open section arbitrary layup symmetrical cross section (A.2) = −βMP 11 δ MP 11 closed section orthotropic unsymmetrical layup arbitrary cross section. (A.3) βMP 11 and δMP 11 are evaluated at the midplanes. 453
454 CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS The stiffness is(Eq.6.196) c66=a66- 哈 866 (A.4) where a66,B66,and 866 are evaluated at the "neutral"plane. Table A.1.The tensile and bending stiffnesses and the coordinates of the centroid.The layup of each wall segment is orthotropic and symmetrical.The properties are evaluated at each wall segment's "neutral"plane,which is at the midplane. 个z EA=益+ bw 面，=品号+盖+品 2 bw 2 El z =(du).xan f EA= 2b bw (a11) +(an) 火= 1 E (益+六4) Ely= be d2 @+ 2b: (d1) ,+12ai x=品4-P+盒+品（等+） 个2 EA= d 2b线十anm 2b. (au) 2b2 2 面w=品号+斋+ 立ix=品之+ 2b. 260 ,十12(ai: R EA= w=x=π（然+品）
454 CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS The stiffness αν 66 is (Eq. 6.196) αν 66 = α66 − β2 66 δ66 , (A.4) where α66, β66, and δ66 are evaluated at the “neutral” plane. Table A.1. The tensile and bending stiffnesses and the coordinates of the centroid. The layup of each wall segment is orthotropic and symmetrical. The properties are evaluated at each wall segment’s “neutral” plane, which is at the midplane. bf z y b d w 2 d f w ?EA= 2bf (a11)f + bw (a11)w EI yy = bf (a11)f d2 2 + 2bf (d11)f + b3 w 12(a11)w EI zz = bw (d11)w + 2b3 f 12(a11)f bf bw d 2 d yc df z y f w ?EA= 2bf (a11)f + bw (a11)w yc = 1 EA 2bf (a11)f bf 2 + bw (a11)w df EI yy = bf (a11)f d2 2 + 2bf (d11)f + b3 w 12(a11)w EI zz = bw (a11)w (df − yc) 2 + bw (d11)w + 2 (a11)f y3 c 3 + (bf−yc) 3 3 bf z y b d w 2 d f w df ?EA= 2bf (a11)f + 2bw (a11)w EI yy = bf (a11)f d2 2 + 2bf (d11)f + 2b3 w 12(a11)w EI zz = bw (a11)w d2 f 2 + 2bw (d11)w + 2b3 f 12(a11)f R ?EA= 2Rπ a11 EI yy = EI zz = π R3 a11 + R d11
Table A.2.The tensile and bending stiffnesses and the coordinates of the centroid.The layup of each wall segment is orthotropic and unsymmetrical.The properties and 61 are evaluated at each wall segment's "neutral"plane, which is at =品+品。+品 点(d+盒) ZcA (u)a 面x=品+品倍+告 1 面w=部d-}+品。+部。+品。+武 1+2 3 EA=+部。+品 =(d+益) 1 ⊙ 北=言（品+部婴+盒） 立n=品d-P+品天 br +品+品+成（色地）〉 面=盒-P+盒+器+器 1品 +盒e-aP+品(- 面x=a-)d-)-部（學-）& +盒-)（伶-） Table A.3.The tensile and bending stiffnesses.The layup of each wall segment is orthotropic and unsymmetrical.Doubly symmmetrical cross section.The property is evaluated at each wall segment's "neutral"plane,which is at 2.The properties and 81 are evaluated at each wall segment's"neutral"plane,which is at 2. A=論+盖 d b2 2 面w=品号+流+器 长 可x=品等+斋+ b d br R EA=2Rm品 w=面x=π(R品+R品)
Table A.2. The tensile and bending stiffnesses and the coordinates of the centroid. The layup of each wall segment is orthotropic and unsymmetrical. The properties α11 and δ11 are evaluated at each wall segment’s “neutral” plane, which is at . z y b d w f1 zc f2 bf1 bf2 bw1 bw2 w zw ?EA= bf1 (α11)f1 + bf2 (α11)f2 + bw (a11)w zc = 1 ?EA bf1 (α11)f1 d + bw (a11)w zw EI zz = bw (d11)w + 1 (α11)f1 b3 f1 12 + 1 (α11)f2 b3 f2 12 EI yy = bf1 (α11)f1 (d − zc) 2 + bf2 (α11)f2 z2 c + bf1 (δ11)f1 + bf2 (δ11)f2 + 1 (a11)w b3 w1+b3 w2 3 z y d zc bf1 bf2 bw1 b zw w2 yf1 yw yc f1 f2 w ?EA= bf1 (α11)f1 + bf2 (α11)f2 + bw (α11)w zc = 1 ?EA bf1 (α11)f1 d + bw (α11)w zw yc = 1 ?EA bf1 (α11)f1 yf1 + bf2 (α11)f2 bf2 2 + bw (α11)w yw EI yy = bf1 (α11)f1 (d − zc) 2 + bf2 (α11)f2 z2 c + bf1 (δ11)f1 + bf2 (δ11)f2 + 1 (α11)w b3 w1+b3 w2 3 EI zz = bw (α11)w (yw − yc) 2 + bw (δ11)w + 1 (α11)f1 b3 f1 12 + 1 (α11)f2 b3 f2 12 + bf1 (α11)f1 (yc − yf1) 2 + bf2 (α11)f2 yc − bf2 2 2 EI yz = bf1 (α11)f1 (yf1 − yc) (d − zc) − bf2 (α11)f2 bf2 2 − yc zc + bw (α11)w (yw − yc) bw 2 − zc Table A.3. The tensile and bending stiffnesses. The layup of each wall segment is orthotropic and unsymmetrical. Doubly symmmetrical cross section. The property α11 is evaluated at each wall segment’s “neutral” plane, which is at . The properties α11 and δ11 are evaluated at each wall segment’s “neutral” plane, which is at . bf z y b d w 2 d f w df ?EA= 2bf (α11)f + 2bw (α11)w EI yy = bf (α11)f d2 2 + 2bf (δ11)f + 2b3 w 12(α11)w EI zz = bw (α11)w d2 f 2 + 2bw (δ11)w + 2b3 f 12(α11)f R ?EA= 2Rπ 1 α11 EI yy = EI zz = π R3 1 α11 + R 1 δ11
456 CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS Table A.4.The tensile and bending stiffnesses and the coordinates of the centroid.The cross section is symmetrical about the z-axis.The layup of each wall segment is arbitrary.The properties and 1 are evaluated at each wall segment's "neutral"plane,which is at p. bw d EA=品+品。+品 w2 le= (部d+品) EA（(a1a 品d-+品+高+品。+（“） Elyy=(u) d; EA=斋+品 2e= d n=品+高h(保-刘'+盒d-+盒 2 个2 f EA心篇+品。+流 2bw ⊙ b 云≈（盒d+盒 业 iy≈ma 、be 2b 十12@1 +品d-P+品+盖（传-
456 CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS Table A.4. The tensile and bending stiffnesses and the coordinates of the centroid. The cross section is symmetrical about the z-axis. The layup of each wall segment is arbitrary. The properties α11 and δ11 are evaluated at each wall segment’s “neutral” plane, which is at . z y b d w f1 zc f2 bf1 bf 2 bw1 bw2 w zw ?EA= bf1 (α11)f1 + bf2 (α11)f2 + bw (a11)w zc = 1 ?EA bf1 (α11)f1 d + bw (a11)w zw EI yy = bf1 (α11)f1 (d − zc) 2 + bf2 (α11)f2 z2 c + bf1 (δ11)f1 + bf2 (δ11)f2 + 1 (a11)w b3 w1+b3 w2 3 bf bw d zc df z w f ?EA= 2bf (α11)f + bw (α11)w zc = 1 ?EA 2bf (α11)f bf 2 + bw (α11)w df EI yy = 2 (α11)f b3 f 12 + 2 (α11)f bf bf 2 − zc 2 + bw (α11)w (df − zc) 2 + bw (δ11)w bf z b y d w f1 w f2 zc ?EA≈ bf1 (α11)f1 + bf2 (α11)f2 + 2bw (α11)w zc ≈ 1 ?EA bf (α11)f1 d + 2bw (α11)w d 2 EI yy ≈ bf1 (δ11)f1 + bf2 (δ11)f2 + 2b3 w 12(α11)w + bf1 (α11)f1 (d − zc) 2 + bf2 (α11)f2 z2 c + 2bw (α11)w d 2 − zc 2
CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS 457 Table A.5.The warping and torsional stiffnesses and the location of the shear center.The layup of each wall segment is orthotropic and unsymmetrical.The properties 866 and are evaluated at each wall segment's"neutral"plane,which is at a,=4(意+） bw El。=24 个2 a=4(高。+。+高） 品 品 ed (on)a 12 e=d- (1h(a11)e 蝴 a=4(0+念） e= (11)h d (n). d b21 3b 2d Ei。= a1m)f(o11)1(e1i)w 12 + a=4(0+念)〉 Elo= bd 2 候+bsd+d2) 12(2h+d02 (a11) (a11)w :=4(高+念） Ei。=0
CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS 457 Table A.5. The warping and torsional stiffnesses and the location of the shear center. The layup of each wall segment is orthotropic and unsymmetrical. The properties δ66 and δ11 are evaluated at each wall segment’s “neutral” plane, which is at . bf z y d SC 2 d f w f bw GI t = 4 2bf (δ66)f + bw (δ66)w EI ω = 1 (α11)f d2b3 f 24 bf1 z y d SC f1 w f2 e bf2 bw GI t = 4 bf1 (δ66)f1 + bf2 (δ66)f2 + bw (δ66)w EI ω = b3 f2 (α11)f2 12 ed e = d b3 f1 (α11)f1 b3 f1 (α11)f1 + b3 f2 (α11)f2 bf d SC e z y f w bw GI t = 4 2bf (δ66)f + bw (δ66)w e = 3b2 f (α11)f 6bf (α11)f + d (α11)w EI ω = b3 f d2 1 (α11)f 12 3bf (α11)f + 2d (α11)w 6bf (α11)f + d (α11)w bf d SC z y bf w f f bw GI t = 4 2bf (δ66)f + bw (δ66)w EI ω = b3 f d2 12(2bf+d) 2 2(b2 f+bfd+d2 ) (α11)f + 3 bfd (α11)w bf z y d SC bw f w GI t = 4 bf (δ66)f + bw (δ66)w EI ω = 0
458 CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS Table A.6.The warping and torsional stiffnesses of closed-section beams.The layup of each wall segment is orthotropic and unsymmetrical.is given by Eq.(A.4. 个2 2dd d 业 2 G=4+w El.≈0 d R a=要 Ei。=0 Table A.7.Shear compliances of closed-section beams.The layup of each wall segment is orthotropic and unsymmetrical.The properties 666 and 61 are evaluated at each wall segment's"neutral"plane,which is at. is given by Eq.(A.4). 个2 不 =+ 2d 6Py经 d 业 2 -会+器 Syz=0 ⅓=1+欲号 为=1+号欲含 R 红=w=赠 S:=0
458 CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS Table A.6. The warping and torsional stiffnesses of closed-section beams. The layup of each wall segment is orthotropic and unsymmetrical. αν 66 is given by Eq. (A.4). bf z y b d w 2 d f w df GI t = 2d2 f d2 (αν 66)f df+(αν 66)wd EI ω ≈ 0 R GI t = 2R3π αν 66 EI ω = 0 Table A.7. Shear compliances of closed-section beams. The layup of each wall segment is orthotropic and unsymmetrical. The properties δ66 and δ11 are evaluated at each wall segment’s “neutral” plane, which is at . αν 66 is given by Eq. (A.4). bf z y b d w 2 d f w df szz = (αν 66)w 2d + 1 6 (αν 66)f df d2 γ 2 z syy = (αν 66)f 2df + 1 6 (αν 66)wd d2 f γ 2 y syz = 0 γz = 1 + 1 3 (α11)f (α11)w d df γy = 1 + 1 3 (α11)w (α11)f df d R szz =syy = αν 66 Rπ syz = 0
CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS 459 Table A.8.Shear compliances.The layup of each wall segment is orthotropic and unsymmetrical.The coordinates of the centroid ye,ze are given in Tables A.1 through A.4,and the coordinate of the shear center e in Table A.5.The properties are evaluated at each wall segment's "neutral"plane,which is at.is given by Eq.(A.4). w=1.2 2bg .-g+ =4 $:=Sw=Sw=0 where=1+台 w=1.2 （+） 如(+安)》 a-學+吉+ d w=是 ()s b (i6)a (6)e Sy:=33=0 where =竖dc= n=1+欲4 h=1+号0a n=号g+ a-學+3 文=2n粤+d52ln =装 (6):25-38 $:=Sw=0 where=1+x号 6,=-达 1 8如=4 38-96,+368 py=32-,F ≈1.2 13-38y Qyu 2d 2-8y 13-38_ qoou=ddi 2-8 38-96+38≈1.2 Po=32-
CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS 459 Table A.8. Shear compliances. The layup of each wall segment is orthotropic and unsymmetrical. The coordinates of the centroid yc, zc are given in Tables A.1 through A.4, and the coordinate of the shear center e in Table A.5. The properties are evaluated at each wall segment’s “neutral” plane, which is at . αν 66 is given by Eq. (A.4). bf z d bw y 2 d f f w syy = 1.2 (αν 66)f 2bf szz = (αν 66)w d + 1 6 (αν 66)f bf d2 γ 2 sωω = 2.4 d2 (αν 66)f bf syz =syω =szω = 0 where γ = 1 + 1 6 (α11)f (α11)w d bf bf2 zc zsc z y SC bw d bf1 e w f2 f1 syy = 1.2 (αν 66)f1 bf1(1+δsc) 2 + (αν 66)f2 bf2(1+ 1 δsc ) 2 szz = (αν 66)w d + 1 12 (αν 66)f1bf1 d2 γ 2 1 + 1 12 (αν 66)f2bf2 d2 γ 2 2 sωω = 1.2 d2 (αν 66)f1 bf1 + (αν 66)f2 bf2 syω = 1.2 d − (αν 66)f1 bf1(1+δsc) + (αν 66)f2 bf2(1+ 1 δsc ) syz =szω = 0 where δc = d−zc zc δsc = d−e e γ1 = 1 + 1 3 (α11)f1 (α11)w d bf1(1+ 1 δc ) γ2 = 1 + 1 3 (α11)f2 (α11)w d bf2(1+δc) yc ysc z y SC d df e w f syy = ρy 2 (αν 66)f df + d 1 3q2 yu αν 66 w szz = (αν 66)w d + 2 3 (αν 66)f df d2 γ 2 sωω = 2ρω 1 d2 (αν 66)f df + d 1 5q2 ωu αν 66 w szω = (αν 66)f d2 2(5−3δω) γ (8−4δω) syz =syω = 0 where γ = 1 + 1 6 (α11)f (α11)w d df δy = df−yc yc δω = 1 df e −1 ρy = 3 5 8−9δy+3δ2 y (2−δy) 2 ≈ 1.2 qyu = 1 2df 3−3δ y 2−δy qωu = 1 ddf 3−3δω 2−δω ρω = 3 5 8−9δω+3δ2 ω (2−δω) 2 ≈ 1.2
460 CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS Table A.9.Shear compliances.The layup of each wall segment is orthotropic and unsymmetrical.The coordinate of the centroid z is given in Table A.2.The properties are evaluated at each wall segment's "neutral"plane,which is at 2.is given by Eq.(A.4). n=0.6+d5.6 d -+ d 三=24学+d52.(o W =-能+(1-)g如 d620-105 30=S=0 d where y=1+ B=4y-3 :=à(1-) 8u= 1 i上4 1+m.面 g=a受 3-4qyzc 8y=3-24w4 w=1.2 SC p+吉(arb呢 S= bw ⊙ 米 So =0 d Syz=Syo=S20=0 W 业 where 13-38 qa d2-8: 6=- 38-96:+3≈12 p=32-}
460 CROSS-SECTIONAL PROPERTIES OF THIN-WALLED COMPOSITE BEAMS Table A.9. Shear compliances. The layup of each wall segment is orthotropic and unsymmetrical. The coordinate of the centroid zc is given in Table A.2. The properties are evaluated at each wall segment’s “neutral” plane, which is at . αν 66 is given by Eq. (A.4). d df df f w syy = 0.6 (αν 66)f df + d 1 5q2 yz αν 66 w szz = (αν 66)w d + 4 15 (αν 66)f df d2 β2 sωω = 2.4 1 d2 (αν 66)f df + d 1 3q2 ωz αν 66 w syz = −(αν 66)f dβ 2−3δ y 20−10δy + (αν 66)w 5 1 − 1 β qyz syω =szω = 0 where γ = 1 + 1 6 (α11)f (α11)w d df β = 4γ − 3 qyz = 3 4df 1 − 1 β δω = 1 1+ (α11)f (α11)w d df qωz = 1 ddf 3−3δω 2−δω δy = 3−4qyzdf 3−2qyzdf zc d bf f w SC zsc syy = 1.2 (αν 66)f bf szz = ρ (αν 66)w bw + 1 12 αν 66 f bfq2 zt sωω = ∞ syz =syω =szω = 0 where qzt = 1 d 3−3δz 2−δz δz = d−zc zc ρ = 3 5 8−9δz+3δ2 z (2−δz) 2 ≈ 1.2