Torsion and Thermal Effects in Architectural Structures: Lecture 17, Slides of Structural Design and Architecture

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Torsion & Temp 1Lecture 17

A

A

RCHITECTURAL

S

TRUCTURES

I:

S

TATICS AND

S

TRENGTH OF

M

ATERIALS

seventeen torsion& thermal effects

lecture

Torsion & Temp 2Lecture 17

A

Torsional Stress & Strain •^

can see torsionalstresses & twistingof axi-symmetricalcross sections^ –

torque

remain plane

undistorted

rotates

-^

not true for squaresections....

T T

Shearing Strain •^ Torsion & Temp 4Lecture 17

related to

is the radial distance from the centroid to the pointunder strain

-^

shear strain varies linearlyalong the radius:

γmax

is at

outer diameter

ρφ L

γ =

Torsion & Temp 5Lecture 17

A

Torsional Stress - Strain •^

know

and

•^

so

-^

where G is the Shear Modulus

L

G

ρφ

τ

ρφ^ L

γ =

γ

τ

G

f^ v

Shear Stress • Torsion & Temp 7Lecture 17

max

happens at outer diameter

•^

combined shear andaxial stresses^ –

maximum shearstress at 45° “twisted”plane

Shear strain •^ Torsion & Temp 8Lecture 17

knowing

and

•^

solve:

-^

composite shafts:

L

G

ρφ

τ

J

T

ρ

τ =

TL JG

φ

i i

i i

i^

G

J

L

T

Σ

φ =

Torsion & Temp 10Lecture 17

A

Open Thin-Walled Sections •^

with very large a/b ratios:

2

13

max

T ab

τ

G

TL ab

3

13

φ

Torsion & Temp 11Lecture 17

A

Shear Flow in Closed Sections •^

q is the internal shear force/unit length

-^

is the area bounded by the centerline

•^

s

i^

is the length segment, t

is the thicknessi

a t

T

τ

i^

i ti

s

TL t

2

a

φ

a

Torsion & Temp 13Lecture 17

A

Deformation Relationships •^

physical movement^ –

axially (same or zero)

rotations from axial changes

-^

relates

to P

steel

20 kN aluminum

aluminum

PL AE

δ

Torsion & Temp 14Lecture 17

A

Deformations from Temperature •^

atomic chemistry reactsto changes in energy

-^

solid materials

-^

can contract with decrease in temperature

-^

can expand with increase in temperature

•^

linear change canbe measured perdegree

Coefficients of Thermal ExpansionTorsion & Temp 16Lecture 17

12.8 x 10

Aluminum

10.4 x 10

Brass

10.1 x 10

Bronze

9.3 x 10

Copper

6.7 x 10

Wrought Iron

6.5 x 10

Steel

5.9 x 10

Cast Iron

5.5 x 10

Concrete

4.4 x 10

Glass

3.0 x 10

Wood

Coefficients (

α

) [in./in./

°F]

Material

Stresses and Thermal Strains •^ Torsion & Temp 17Lecture 17

if thermal movement is restrainedstresses are induced

bar pushes on supports

support pushes back

reaction causes internalstress

E L

P A

f

δ

=

Torsion & Temp 19Lecture 17

A

Superposition Method –^

total length change restrained tozero

0

T

P

δ

δ (^

)^

0

Δ

−

L T

PL AE

α

(^

) E T

P A

f^

Δ

−

−

α

(^

) L T

T^

Δ

=

α

δ

PL AE

p^

−

δ

constraint:

sub: