Cross Sectional Properties Calculated by HyperBeam

The beam cross section is always defined in a y,z plane.

The x-axis is defined along the beam axis. The coordinate system you define is called the local coordinate system; the system parallel to the local coordinate system with the origin in the centroid is called the centroidal coordinate system; the system referring to the principal bending axes is called the principal coordinate system.

For shell sections, only the theory of thin walled bars is used. This means that for the calculation of the moments and product of inertia, terms of higher order of the shell thickness t are neglected. Thickness warping is also neglected.
Area
A = d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9maapeaabaGaamizaiaadgeaaSqabeqaniabgUIiYdaaaa@3B6C@
Area Moments of Inertia
I y y = z 2 d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG5bGaamyEaaqabaGccqGH9aqpdaWdbaqaaiaadQhadaah aaWcbeqaaiaaikdaaaGccaWGKbGaamyqaaWcbeqab0Gaey4kIipaaa a@3F98@
I z z = y 2 d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG6bGaamOEaaqabaGccqGH9aqpdaWdbaqaaiaadMhadaah aaWcbeqaaiaaikdaaaGccaWGKbGaamyqaaWcbeqab0Gaey4kIipaaa a@3F99@
Area Products of Inertia
I yz = yzdA MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG5bGaamOEaaqabaGccqGH9aqpdaWdbaqaaiaadMhacaWG 6bGaamizaiaadgeaaSqabeqaniabgUIiYdaaaa@3FA1@
Radius of Gyration
R g = I min A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGNbaabeaakiabg2da9maakaaabaWaaSaaaeaacaWGjbWa aSbaaSqaaiGac2gacaGGPbGaaiOBaaqabaaakeaacaWGbbaaaaWcbe aaaaa@3DBC@
Elastic Section Modulus
E y = I y y z max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWG5baabeaakiabg2da9maalaaabaGaamysamaaBaaaleaa caWG5bGaamyEaaqabaaakeaacaWG6bWaaSbaaSqaaiGac2gacaGGHb GaaiiEaaqabaaaaaaa@4009@
E z = I z z y max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWG6baabeaakiabg2da9maalaaabaGaamysamaaBaaaleaa caWG6bGaamOEaaqabaaakeaacaWG5bWaaSbaaSqaaiGac2gacaGGHb GaaiiEaaqabaaaaaaa@400B@
Max Coordinate Extension
y max = max | y | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0JaciyBaiaacgga caGG4bWaaqWaaeaacaWG5baacaGLhWUaayjcSdaaaa@41F8@
z max = max | z | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0JaciyBaiaacgga caGG4bWaaqWaaeaacaWG6baacaGLhWUaayjcSdaaaa@41FA@
Plastic Section Modulus
P y | z | d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWG5baabeaakmaapeaabaWaaqWaaeaacaWG6baacaGLhWUa ayjcSdGaamizaiaadgeaaSqabeqaniabgUIiYdaaaa@3FCA@
P z | y | d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWG6baabeaakmaapeaabaWaaqWaaeaacaWG5baacaGLhWUa ayjcSdGaamizaiaadgeaaSqabeqaniabgUIiYdaaaa@3FCA@
Torsional Constant
Solid
I t = I y y + I z z + ( z ω y y ω z ) d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG0baabeaakiabg2da9iaadMeadaWgaaWcbaGaamyEaiaa dMhaaeqaaOGaey4kaSIaamysamaaBaaaleaacaWG6bGaamOEaaqaba GccqGHRaWkdaWdbaqaamaabmaabaGaamOEamaalaaabaGaeyOaIyRa eqyYdChabaGaeyOaIyRaamyEaaaacqGHsislcaWG5bWaaSaaaeaacq GHciITcqaHjpWDaeaacqGHciITcaWG6baaaaGaayjkaiaawMcaaaWc beqab0Gaey4kIipakiaadsgacaWGbbaaaa@5435@
ω - Warping function
(see below for warping function)
Shell open
I t = 1 3 t 3 d s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG0baabeaakiabg2da9maalaaabaGaaGymaaqaaiaaioda aaWaa8qaaeaacaWG0bWaaWbaaSqabeaacaaIZaaaaOGaamizaiaado haaSqabeqaniabgUIiYdaaaa@404A@
t - Shell thickness
Shell closed
I t = 2 A m i F s i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaWG0baabeaakiabg2da9iaaikdadaaeabqaaiaadgeadaWg aaWcbaGaamyBaiaadMgaaeqaaOGaamOramaaBaaaleaacaWGZbGaam yAaaqabaaabeqab0GaeyyeIuoaaaa@4177@
A m i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacaWGTbGaamyAaaqabaaaaa@38C8@ - Area enclosed by cell i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@
F s i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGZbGaamyAaaqabaaaaa@38D3@ - Shear flow in cell i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@
Elastic Torsion Modulus
Solid
E t = I t max ( y 2 + z 2 + z ω y y ω z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWG0baabeaakiabg2da9maalaaabaGaamysamaaBaaaleaa caWG0baabeaaaOqaaiGac2gacaGGHbGaaiiEaaaadaqadaqaaiaadM hadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG6bWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaamOEamaalaaabaGaeyOaIyRaeqyYdChaba GaeyOaIyRaamyEaaaacqGHsislcaWG5bWaaSaaaeaacqGHciITcqaH jpWDaeaacqGHciITcaWG6baaaaGaayjkaiaawMcaaaaa@533F@
Shell open
E t = I t max t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWG0baabeaakiabg2da9maalaaabaGaamysamaaBaaaleaa caWG0baabeaaaOqaaiGac2gacaGGHbGaaiiEaiaaysW7caWG0baaaa aa@405C@
Shell closed
E t = I t max ( F s i t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWG0baabeaakiabg2da9maalaaabaGaamysamaaBaaaleaa caWG0baabeaaaOqaaiGac2gacaGGHbGaaiiEamaabmaabaWaaSaaae aacaWGgbWaaSbaaSqaaiaadohacaWGPbaabeaaaOqaaiaadshaaaaa caGLOaGaayzkaaaaaaaa@434F@
Shear Center
y s = I y z I y ω I z z I z ω I y y I z z I y z 2 I y ω = y ω d A , I z ω = z ω d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGZbaabeaakiabg2da9maalaaabaGaamysamaaBaaaleaa caWG5bGaamOEaaqabaGccaWGjbWaaSbaaSqaaiaadMhacqaHjpWDae qaaOGaeyOeI0IaamysamaaBaaaleaacaWG6bGaamOEaaqabaGccaWG jbWaaSbaaSqaaiaadQhacqaHjpWDaeqaaaGcbaGaamysamaaBaaale aacaWG5bGaamyEaaqabaGccaWGjbWaaSbaaSqaaiaadQhacaWG6baa beaakiabgkHiTiaadMeadaqhaaWcbaGaamyEaiaadQhaaeaacaaIYa aaaaaakiaaysW7caaMe8UaaGjbVlaaysW7caaMe8UaamysamaaBaaa leaacaWG5bGaeqyYdCNaeyypa0dabeaakmaapeaabaGaamyEaiabeM 8a3jaadsgacaWGbbGaaiilaiaadMeadaWgaaWcbaGaamOEaiabeM8a 3bqabaGccqGH9aqpdaWdbaqaaiaadQhacqaHjpWDcaWGKbGaamyqaa Wcbeqab0Gaey4kIipaaSqabeqaniabgUIiYdaaaa@717B@
z s = I y z I y ω I y z I z ω I y y I z z I y z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacaWGZbaabeaakiabg2da9maalaaabaGaamysamaaBaaaleaa caWG5bGaamOEaaqabaGccaWGjbWaaSbaaSqaaiaadMhacqaHjpWDae qaaOGaeyOeI0IaamysamaaBaaaleaacaWG5bGaamOEaaqabaGccaWG jbWaaSbaaSqaaiaadQhacqaHjpWDaeqaaaGcbaGaamysamaaBaaale aacaWG5bGaamyEaaqabaGccaWGjbWaaSbaaSqaaiaadQhacaWG6baa beaakiabgkHiTiaadMeadaqhaaWcbaGaamyEaiaadQhaaeaacaaIYa aaaaaakiaaysW7aaa@5401@
Warping Constant (normalized to the shear center)
I ω ω = ω 2 d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacqaHjpWDcqaHjpWDaeqaaOGaeyypa0Zaa8qaaeaacqaHjpWD daahaaWcbeqaaiaaikdaaaGccaWGKbGaamyqaaWcbeqab0Gaey4kIi paaaa@4204@
Shear deformation coefficients
α z z = 1 Q y 2 ( τ x y 2 | Q z = 0 + τ x z 2 | Q z = 0 ) d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadQhacaWG6baabeaakiabg2da9maalaaabaGaaGymaaqa aiaadgfadaqhaaWcbaGaamyEaaqaaiaaikdaaaaaaOWaa8qaaeaada qadaqaaiabes8a0naaDaaaleaacaWG4bGaamyEaaqaaiaaikdaaaGc daabbaqaamaaBaaaleaadaWgaaadbaGaamyuamaaBaaabaGaamOEaa qabaGaeyypa0JaaGimaaqabaaaleqaaaGccaGLhWoacqGHRaWkcqaH epaDdaqhaaWcbaGaamiEaiaadQhaaeaacaaIYaaaaOWaaqqaaeaada WgaaWcbaWaaSbaaWqaaiaadgfadaWgaaqaaiaadQhaaeqaaiabg2da 9iaaicdaaeqaaaWcbeaaaOGaay5bSdaacaGLOaGaayzkaaaaleqabe qdcqGHRiI8aOGaamizaiaadgeaaaa@5957@
α z y = 1 Q y Q z ( τ x y | Q y = 0 τ x y | Q z = 0 + τ x z | Q y = 0 τ x y | Q z = 0 ) d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadQhacaWG5baabeaakiabg2da9maalaaabaGaaGymaaqa aiaadgfadaWgaaWcbaGaamyEaaqabaGccaWGrbWaaSbaaSqaaiaadQ haaeqaaaaakmaapeaabaWaaeWaaeaacqaHepaDdaqhaaWcbaGaamiE aiaadMhaaeaaaaGcdaabbaqaamaaBaaaleaadaWgaaadbaGaamyuam aaBaaabaGaamyEaaqabaGaeyypa0JaaGimaaqabaaaleqaaaGccaGL hWoacaaMe8UaeqiXdq3aa0baaSqaaiaadIhacaWG5baabaaaaOWaaq qaaeaadaWgaaWcbaWaaSbaaWqaaiaadgfadaWgaaqaaiaadQhaaeqa aiabg2da9iaaicdaaeqaaaWcbeaakiabgUcaRiabes8a0naaDaaale aacaWG4bGaamOEaaqaaaaakmaaeeaabaWaaSbaaSqaamaaBaaameaa caWGrbWaaSbaaeaacaWG5baabeaacqGH9aqpcaaIWaaabeaaaSqaba aakiaawEa7aiaaysW7cqaHepaDdaqhaaWcbaGaamiEaiaadMhaaeaa aaGcdaabbaqaamaaBaaaleaadaWgaaadbaGaamyuaiaadQhacqGH9a qpcaaIWaaabeaaaSqabaaakiaawEa7aaGaay5bSdaacaGLOaGaayzk aaaaleqabeqdcqGHRiI8aOGaamizaiaadgeaaaa@6F81@
α z z = 1 Q z 2 ( τ x y 2 | Q y = 0 + τ x z 2 | Q y = 0 ) d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadQhacaWG6baabeaakiabg2da9maalaaabaGaaGymaaqa aiaadgfadaqhaaWcbaGaamOEaaqaaiaaikdaaaaaaOWaa8qaaeaada qadaqaaiabes8a0naaDaaaleaacaWG4bGaamyEaaqaaiaaikdaaaGc daabbaqaamaaBaaaleaadaWgaaadbaGaamyuamaaBaaabaGaamyEaa qabaGaeyypa0JaaGimaaqabaaaleqaaaGccaGLhWoacqGHRaWkcqaH epaDdaqhaaWcbaGaamiEaiaadQhaaeaacaaIYaaaaOWaaqqaaeaada WgaaWcbaWaaSbaaWqaaiaadgfadaWgaaqaaiaadMhaaeqaaiabg2da 9iaaicdaaeqaaaWcbeaaaOGaay5bSdaacaGLOaGaayzkaaaaleqabe qdcqGHRiI8aOGaamizaiaadgeaaaa@5956@
Shear stiffness factors
k y y = 1 α z z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa aaleaacaWG5bGaamyEaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaa cqaHXoqydaWgaaWcbaGaamOEaiaadQhaaeqaaaaaaaa@3EB2@
k y z = 1 α y z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa aaleaacaWG5bGaamOEaaqabaGccqGH9aqpdaWcaaqaaiabgkHiTiaa igdaaeaacqaHXoqydaWgaaWcbaGaamyEaiaadQhaaeqaaaaaaaa@3F9F@
k z z = 1 α y y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa aaleaacaWG6bGaamOEaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaa cqaHXoqydaWgaaWcbaGaamyEaiaadMhaaeqaaaaaaaa@3EB2@
Shear stiffness
S i i = k i i G A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbGaamyAaaqabaGccqGH9aqpcaWGRbWaaSbaaSqaaiaa dMgacaWGPbaabeaakiaadEeacaWGbbaaaa@3E7A@
Warping Function
2 ω = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaW baaSqabeaacaaIYaaaaOGaeqyYdCNaeyypa0JaaGimaaaa@3BFC@
( ω y z ) n y + ( ω z + y ) n z = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiabgkGi2kabeM8a3bqaaiabgkGi2kaadMhaaaGaeyOeI0Ia amOEaaGaayjkaiaawMcaaiaad6gadaWgaaWcbaGaamyEaaqabaGccq GHRaWkdaqadaqaamaalaaabaGaeyOaIyRaeqyYdChabaGaeyOaIyRa amOEaaaacqGHRaWkcaWG5baacaGLOaGaayzkaaGaamOBamaaBaaale aacaWG6baabeaakiabg2da9iaaicdaaaa@4F14@
For solid sections, the warping function is computed using a finite element formulation. This may lead to un-physically high stresses in geometric singularities (sharp corners) that get worse with mesh refinement. This may cause problems computing the elastic torsion modulus.

Nastran Type Notation

/1= I zz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4laiaaig dacqGH9aqpcaWGjbWaaSbaaSqaaiaadQhacaWG6baabeaaaaa@3B62@

/2= I yy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4laiaaik dacqGH9aqpcaWGjbWaaSbaaSqaaiaadMhacaWG5baabeaaaaa@3B61@

/12= I yz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4laiaaig dacaaIYaGaeyypa0JaamysamaaBaaaleaacaWG5bGaamOEaaqabaaa aa@3C1D@

K1= K yy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaaig dacqGH9aqpcaWGlbWaaSbaaSqaaiaadMhacaWG5baabeaaaaa@3B7F@

K2= K zz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaaik dacqGH9aqpcaWGlbWaaSbaaSqaaiaadQhacaWG6baabeaaaaa@3B82@