yy)---------

θ2sin– )

I 1 3⁄ )I 2 I 3+ ] 2A3( )⁄

γyz z∂∂v

y∂∂w+= γzx x∂

∂wz∂

∂u+=

σxx

E 1 ν–( )εxx νε yy+[ ]1 2ν–( ) 1 ν+( )

---------------------------------------------------=

σ2–

2----------

σ2 σ3–

2------------------

σ3 σ1–

2------------------, ,

ax

M

Vy

w

x

y

Mz wx a–⟨ ⟩ 2

2--------------------–=

py w x a–⟨ ⟩ 0–=

FORMULA SHEET

Axial (Rods) Torsion (Shafts) Bending (Beams)

σij

∆F j

∆Ai----------

∆ Ai 0→

lim= σnn σxx θ2cos σyy θ2

sin 2τ xy θ θcossin+ += τnt σxx θ θsincos– σyy θ θcossin τ xy θ2cos θ2

sin–( )+ += 2θptan2τ xy

σxx σ–(-------------------=

σ1 2,σxx σyy+( )

2----------------------------

σxx σyy–

2-----------------------

2

τ xy2+±= εnn εxx θ2

cos εyy θ2sin γxy θsin θcos+ += γnt 2εxx θsin θcos– 2εyy θsin θcos γxy θ2

cos(+ +=

σnn n{ } T σ[ ] n{ }= τnt t{ } T σ[ ] n{ }= σtt t{ } T σ[ ] t{ }= S{ } σ[ ] n{ }=

σp3 I 1σp

2– I 2σp I– 3+ 0= I 1 σxx σyy σzz+ += I 2σxx τ xy

τ yx σyy

σyy τ yz

τ zy σzz

σxx τ xz

τ zx σzz

+ += I 3

σxx τ xy τ xz

τ yx σyy τ yz

τ zx τ zy σzz

=

x3

I 1– x2

I 2x I 3–+ 0= x1 2A αcos I 1 3⁄+= x2 3, 2– A α 60o±( )cos I 1 3⁄+= A I 1 3⁄( )2

I 2 3⁄–= 3αcos 2 I 1 3⁄( )3 (–[=

σoct σ1 σ2 σ3+ +( ) 3⁄= τoct13--- σ1 σ2–( )2 σ2 σ3–( )2 σ3 σ1–( )2

+ += εxx x∂∂u= εyy y∂

∂v= εzz z∂∂w= γxy y∂

∂ux∂

∂v+=

εxx σxx ν σ yy σzz+( )–[ ] E⁄= α∆T+ γxy τ xy G⁄= G E 2 1 ν+( )[ ]⁄= σxx εxx νε yy+[ ] E

1 ν2–( )

-------------------= εzzν

1 ν–------------

– εxx εyy+( )=

εxx

σxx

Ex--------

ν yx

E y--------σyy–= γxy

τ xy

Gxy---------=

ν yx

E y--------

νxy

Ex--------= σvon

1

2------- σ1 σ2–( )2 σ2 σ3–( )2 σ3 σ1–( )2

+ +=σ2

σC-------

σ1

σT------– 1≤ τmax max

σ1--------=

K I σnom πa= K II τnom πa= Kequiv K I2

K II2

+=

σ

σyield

Eεσ– yield

=

ε εyield≥

εyield– ε εyield≤ ≤

ε ε– yield≤

σ

σyield E2 ε εyield–( )+

E1ε

σ– yield E2 ε εyield+( )+

=

ε εyield≥

εyield– ε εyield≤ ≤

ε εyield≥

σEεn

E ε–( )n–

=ε 0≥ε 0<

ax

NF

ax

TT

ax

Mz

Vy

Mx

y

a

P

x

Mz

Vy

x

y

N F x a–⟨ ⟩ 0–= px F x a–⟨ ⟩ 1–

= T T x a–⟨ ⟩ 0–= t T x a–⟨ ⟩ 1–

= Mz M x a–⟨ ⟩ 0–=

py M x a–⟨ ⟩ 2––=

Mz P x a–⟨ ⟩ 1–=

py P x a–⟨ ⟩ 1––=

U o12---σε= U o

12--- σxxεxx σyyεyy σzzεzz τ xyγxy τ yzγyz τ zxγzx+ + + + +[ ]=

F 1=( )v1 xP( )M 2 x( )M 1 x( )

EI--------------------------------- xd

0

L

∫= M 1=( )xd

dv1 xP( )M 2 x( )M 1 x( )

EI--------------------------------- xd

0

L

∫= v1 xP( )F∂

∂U B=xd

dv1 xP( )M∂

∂U B=

Unsymmetric Bending

zxd

dw– v v x( )= w w x( )=

x2

2w

Ezx

2

2

d

d w τ xy 0 σxx«≠ τ xz 0 σxx«≠

M z yσxx AdA∫–= M y zσxx Ad

A∫–=

z τ xz AdA∫=

I yzM y

I yz2

–-----------------

yI zzM y I yzMz–

I yyI zz I yz2

–------------------------------------

z–

Qz I yzQy–

yI zz I yz2

–------------------------------

V y

I zzQy I yzQz–

I yyI zz I yz2

–--------------------------------------

– V z

I yzM y

z I yz2

–---------------------

x2

2

d

d w 1E---

I zzM y I yzMz–

I yyI zz I yz2

–-----------------------------------------

=

x

Mz V– y=xd

dV z pz x( )–=xd

dM y V– z=

Axial (Rods) Torsion (Shafts) Symmetric Bending (Beams)Displace-ments

Strains

Stresses

InternalForces &Moments

StrainEnergy

C. StrainEnergy

u x y z, ,( ) u x( )= φ x y z, ,( ) φ x( )=u x y z, ,( ) y

xddv

–= v v x( )= w 0= u x y z, ,( ) yxd

dv–=

εxx xddu= γxθ ρ

xddφ

= εxx yx

2

2

d

d v–= εxx y

x2

2

d

d v– z

d

d–=

σxx Eεxx Exd

du= = τ xθ Gγxθ Gρ

xddφ

= = σxx Eεxx Eyx

2

2

d

d v–= = τ xy 0 σxx«≠ σxx Ey

x2

2

d

d v– –=

N σxx AdA∫= T ρτxθ Ad

A∫= N σxx Ad

A∫ 0= =

M z yσxx AdA∫–= V y τ xy Ad

A∫=

N σxx AdA∫ 0= =

V y τ xy AdA∫= V

σxxNA-----=

τ xθTρJ

-------= σxx

M z y

I zz-----------

–=

q τ xstV yQz

I zz--------------

–= =

σxx

I yyMz –

I yyI zz

--------------------

–=

q τ xstI yy

I y

---------

–= =

xddu N

EA-------= u2 u1–

N x2 x1–( )EA

---------------------------=xd

dφ TGJ-------= φ2 φ1–

T x2 x1–( )GJ

--------------------------=

x2

2

d

d v M z

EI zz-----------= v

M z

EI-------- xd∫ dx C1x C2+ +∫=

x2

2

d

d v 1E---

I yyMz –

I yyI z

---------------------

=

σxx( )i

NEi

E j A jj 1=

n

∑-----------------------= τ xθ( )

i

GiρT

G jJ jj 1=

n

∑-----------------------------=

σxx( )i

Ei yM z

E j I zz( )j

j 1=

n

∑-------------------------------–= q τ xst

QcompV y

E j I zz( )j

j 1=

n

∑-------------------------------------

–= =

u2 u1–N x2 x1–( )

E j A j∑---------------------------= φ2 φ1–

T x2 x1–( )G jJ j∑[ ]

--------------------------= vM z

E j I zz( )j∑

---------------------------- xd∫ dx C1x C2+ +∫=

xddN px x( )–=

xddT t x( )–=

xd

dV y py x( )–=xd

dMz V– y=xd

dV y py x( )–=d

d

xdd

EAxd

duo px x( )–= xd

dGJ

xddφ

t x( )–=

x2

2

d

dEI zz

x2

2

d

d v

py x( )=

U a EAxd

du

22⁄= U t GJ

xddφ

22⁄= U b EI zz

x2

2

d

d v

2

2⁄=

U a N2

2EA( )⁄= U t T2

2GJ( )⁄= U b M z2

2EI zz( )⁄=