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S2009abn Mechanics of Materials 1 Lecture 4 Elements of Architectural Structures ARCH 614

ELEMENTS OF ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN

ARCH 614

• DR. ANNE NICHOLS

SPRING 2019

four

mechanics

• f materials

lecture

www.carttalk.com Mechanics of Materials 7 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

Mechanics of Materials

• MECHANICS
• MATERIALS

Mechanics of Materials 8 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

Mechanics of Materials

• external loads and their

effect on deformable bodies

• use it to answer question if structure

meets requirements of

– stability and equilibrium – strength and stiffness

• other principle building requirements
• economy, functionality and aesthetics

Mechanics of Materials 9 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

Knowledge Required

• material properties
• member cross sections
• ability of a material to resist breaking
• structural elements that resist excessive

– deflection – deformation

2

Mechanics of Materials 10 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

Problem Solving

• 1. STATICS:

equilibrium of external forces, internal forces, stresses

• 2. GEOMETRY:

cross section properties, deformations and conditions of geometric fit, strains

• 3. MATERIAL PROPERTIES:

stress-strain relationship for each material

• btained from testing

Mechanics of Materials 11 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

Stress

• stress is a term for the intensity of a

force, like a pressure

• internal or applied
• force per unit area

A P f stress  

Mechanics of Materials 12 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

• materials have a critical stress value

where they could break or yield

– ultimate stress – yield stress – compressive stress – fatigue strength – (creep & temperature)

Design

acceptance

• vs. failure

Mechanics of Materials 13 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

allowable actual

F f 

• we’d like
• stress distribution may

vary: average

• uniform distribution

exists IF the member is loaded axially (concentric)

Design (cont)

3

Mechanics of Materials 9 Lecture 4 Elements of Architectural Structures ARCH 614 S2009abn

Scale Effect

• model scale

– material weights by volume, small section areas

• structural scale

– much more material weight, bigger section areas

• scale for strength is not

proportional: L L L   

2 3

Mechanics of Materials 16 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

Normal Stress

• normal stress is normal

to the cross section

– stressed area is perpendicular to the load

A P f

c

• r

t



 



Mechanics of Materials 17 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

• stress parallel to a surface

Shear Stress

td P A P fv  

 

ave



Mechanics of Materials 18 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

• stress on a surface by

contact in compression

Bearing Stress

td P A P f p  

 



4

Mechanics of Materials 19 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

• normal stress caused by bending

Bending Stress

S M I Mc fb  

 



Mechanics of Materials 20 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

• shear stress caused by twisting

Torsional Stress

J T fv  

 



Mechanics of Materials 21 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

• what structural elements see shear?

– beams – bolts – splices – slabs – footings – walls

• wind
• seismic loads

Structures and Shear

connections



V

S2011abn

• connected members in tension cause

shear stress

• connected members in

compression cause bearing stress

Bolts

Mechanics of Materials 16 Lecture 4 Elements of Architectural Structures ARCH 614

5

S2011abn

• seen when 2 members are connected

Single Shear

Mechanics of Materials 17 Lecture 4 Elements of Architectural Structures ARCH 614

4

2

d v

P A P f   

S2011abn

Double Shear

Mechanics of Materials 18 Lecture 4 Elements of Architectural Structures ARCH 614

F=

• seen when 3 members are connected
• two areas

4 d v

2

2 P A 2 P A 2 P f    

S2011abn

td P A P f

projected p

 

• compression & contact
• projected area

Bolt Bearing Stress

Mechanics of Materials 19 Lecture 4 Elements of Architectural Structures ARCH 614

F=

Mechanics of Materials 15 Lecture 14 Elements of Architectural Structures ARCH 614 S2004abn

Strain

• materials deform
• axially loaded materials change

length

• bending materials deflect
• STRAIN:

– change in length

• ver length

L L strain    

(S)

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Stress & Strain 10 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Shearing Strain

• deformations

with shear

• parallelogram
• change in angles
• stress:
• strain:

– unitless (radians)

s

 L





        tan L

s

Stress & Strain 11 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Shearing Strain

• deformations

with torsion

• twist
• change in angle of line
• stress:
• strain:

– unitless (radians)

  L   

Stress & Strain 12 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Load and Deformation

• for stress, need P & A
• for strain, need  & L

– how? – TEST with load and measure – plot P/A vs. 

Stress & Strain 13 Lecture 15 Elements of Architectural Structures ARCH 614 S2005abn

Material Behavior

• every material has its own response

– 10,000 psi – L = 10 in – Douglas fir vs. steel?

7

Stress & Strain 14 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Behavior Types

• ductile - “necking”
• true stress
• engineering stress

– (simplified)

A P f 

• A

P f 

Stress & Strain 15 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Behavior Types

• brittle
• semi-brittle

Stress & Strain 16 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Stress to Strain

• important to us in f- diagrams:

– straight section – LINEAR-ELASTIC – recovers shape (no permanent deformation)

Stress & Strain 17 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Hooke’s Law

• straight line has constant slope
• Hooke’s Law
• E

– Modulus of elasticity – Young’s modulus – units just like stress

f

 E 1

   E f

8

Stress & Strain 18 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Stiffness

• ability to resist strain
• steels

– same E – different yield points – different ultimate strength

u

f

Stress & Strain 19 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Isotropy & Anisotropy

• ISOTROPIC

– materials with E same at any direction of loading – ex. steel

• ANISOTROPIC

– materials with different E at any direction of loading – ex. wood is orthotropic

Stress & Strain 20 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Elastic, Plastic, Fatigue

• elastic springs back
• plastic has permanent

deformation

• fatigue caused by

reversed loading cycles

Stress & Strain 21 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Plastic Behavior

• ductile

at yield stress

9

Stress & Strain 15 Lecture 15 Elements of Architectural Structures ARCH 614 S2006abn

Lateral Strain

• or “what happens to the cross section

with axial stress”

• strain in lateral direction

– negative – equal for isometric materials

z y

   E f x

x 

  

z y

f f

Stress & Strain 23 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Poisson’s Ratio

• constant relationship between

longitudinal strain and lateral strain

• sign!

x z x y

strain axial strain lateral            E f x

z y

      5 .   

S2004abn Stress & Strain 24 Lecture 15 Elements of Architectural Structures ARCH 614

Calculating Strain

• from Hooke’s law
• substitute
• get 

   E f L E A P    AE PL  

(L)

Stress & Strain 25 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Orthotropic Materials

• non-isometric
• directional values of

E and 

• ex:

– plywood – laminates – polymer composites

10

Stress & Strain 26 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

• why we use fave
• increase in stress at

changes in geometry

– sharp notches – holes – corners –

Stress Concentrations

Stress & Strain 27 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

2 2

max

• max

v

f A P f  



• if we need to know

where max f and fv happen:

Maximum Stresses

F

• max

A P f  1       cos 5 45 . sin cos        

Stress & Strain 28 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Maximum Stresses

Torsion & Temp 18 Lecture 16 Elements of Architectural Structures ARCH 614 S2004abn

Deformation Relationships

• physical movement

– axially (same or zero) – rotations from axial changes

• relates  to P

steel 20 kN  aluminum

AE PL  

11

Torsion & Temp 19 Lecture 16 Elements of Architectural Structures ARCH 614 S2004abn

Deformations from Temperature

• atomic chemistry reacts

to changes in energy

• solid materials
• can contract with decrease in temperature
• can expand with increase in temperature
• linear change can

be measured per degree

Torsion & Temp 20 Lecture 16 Elements of Architectural Structures ARCH 614 S2004abn

Thermal Deformation

•  - the rate of strain per degree
• UNITS : ,
• length change:
• thermal strain:

– no stress when movement allowed

 L

T

T

  

 

T

T

   F  C 

Torsion & Temp 21 Lecture 16 Elements of Architectural Structures ARCH 614 S2004abn

Coefficients of Thermal Expansion

12.8 x 10-6 Aluminum 10.4 x 10-6 Brass 10.1 x 10-6 Bronze 9.3 x 10-6 Copper 6.7 x 10-6 Wrought Iron 6.5 x 10-6 Steel 5.9 x 10-6 Cast Iron 5.5 x 10-6 Concrete 4.4 x 10-6 Glass 3.0 x 10-6 Wood Coefficients () [in./in./F] Material

Torsion & Temp 22 Lecture 16 Elements of Architectural Structures ARCH 614 S2004abn

E L A P f   

Stresses and Thermal Strains

• if thermal movement is restrained

stresses are induced

• 1. bar pushes on supports
• 2. support pushes back
• 3. reaction causes internal

stress

12

Torsion & Temp 23 Lecture 16 Elements of Architectural Structures ARCH 614 S2004abn

Superposition Method

– can remove a support to make it look determinant – replace the support with a reaction – enforce the geometry constraint

Torsion & Temp 24 Lecture 16 Elements of Architectural Structures ARCH 614 S2004abn

Superposition Method

– total length change restrained to zero

 

T P

 

 

    L T AE PL 

 E

T A P f      

 L

T

T

    AE PL

p

  

constraint: sub:

Mechanics of Materials 43 Lecture 4 Elements of Architectural Structures ARCH 614 S2007abn

Dynamics

• kinematics

– time, velocity, acceleration – linear motion – angular rotation

• kinetics

– forces causing motion W = m·g – work – conservation of energy

2

2 1 ) ( ) ( at t v t s  

Mechanics of Materials 44 Lecture 4 Elements of Architectural Structures ARCH 614 S2007abn

Dynamic Response

13

Mechanics of Materials 45 Lecture 4 Elements of Architectural Structures ARCH 614 S2007abn

Dynamic Response

• period of vibration or

frequency

– wave – sway/time period

• damping

– reduction in sway

• resonance

– amplification of sway

Mechanics of Materials 46 Lecture 4 Elements of Architectural Structures ARCH 614 S2007abn

Frequency and Period

• natural period of vibration

– avoid resonance – hard to predict seismic period – affected by soil – short period

• high stiffness

– long period

• low stiffness

“To ring the bell, the sexton must pull

• n the downswing of the bell in time

with the natural frequency of the bell.”

Stress & Strain 29 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Design of Members

• beyond allowable stress...
• materials aren’t uniform 100% of the

time

– ultimate strength or capacity to failure may be different and some strengths hard to test for

• RISK & UNCERTAINTY

A P f

u u 

Stress & Strain 30 Lecture 15 Elements of Architectural Structures ARCH 614 S2004abn

Factor of Safety

• accommodate uncertainty with a safety

factor:

• with linear relation between load and

stress:

S F load ultimate load allowable . 

stress allowable stress ultimate load allowable load ultimate S F   .

14

Stress & Strain 31 Lecture 15 Elements of Architectural Structures ARCH 614 S2005abn

Load and Resistance Factor Design

• loads on structures are

– not constant – can be more influential on failure – happen more or less often – UNCERTAINTY  - resistance factor  - load factor for (D)ead & (L)ive load

n L L D D u

R R R R      