structural analysis (statics & mechanics) -...

F2016abnAnalysis 1

Lecture 2

Applied Architectural Structures

ARCH 631

two

structural analysis

(statics & mechanics)

lecture

APPLIED ACHITECTURAL STRUCTURES:

STRUCTURAL ANALYSIS AND SYSTEMS

ARCH 631

DR. ANNE NICHOLS

FALL 2016

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Lecture 2


ARCH 631

Structural Requirements

• strength

• serviceability

– deflections

• efficiency

– economy of materials

• construction

• cost

• other

www.pbs.org/wgbh/buildingbig/

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Structure Requirements

• strength &

equilibrium

– safety

– stresses

not greater

than

strength

– adequate

foundation

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• stability &

stiffness

– stability of

components

– minimum

deflection and

vibration

– adequate

foundation

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• economy and

construction

– minimum material

– standard sized

members

– simple connections

and details

– maintenance

– fabrication/ erection


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Relation to Architecture

“The geometry and arrangement of the load-bearing members, the use of materials, and the crafting of joints all represent opportunities for buildings to express themselves. The best buildings are not designed by architects who after resolving the formal and spatial issues, simply ask the structural engineer to make sure it doesn’t fall down.” - Onouy & Kane

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• dead load

– static, fixed, includes

material weights, fixed

equipment

• live load

– transient and moving

loads (including

occupants)

• snow load

Structural Loads - STATIC

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W d hF C q A

• wind loads

– dynamic, wind pressures

treated as lateral static

loads on walls, pressure

or suction

– pressure determined from

wind velocity, qh

– dynamic effects include

motion from buffeting or

“vortex shedding”

Analysis 8

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Structural Loads – STATIC & DYNAMIC

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W

ZICWVR

• earthquake loads

– seismic, movement of

ground (3D)

– building mass responds

– static models often used,

V is static shear

• impact loads

– rapid, energy loads

Analysis 9

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Structural Loads - DYNAMIC

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Dynamic Response

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Dynamic Response

• period of vibration or

frequency

– wave

– sway/time period

• damping

– reduction in sway

• resonance

– amplification of sway

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Statics & Mechanics Review

• how loads affect our structures

– statics: things don’t move

• forces

• supports & connections

• equilibrium

– mechanics: things can change shape

• stress & strain

• deflections

• buckling

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Structural Math

• quantify environmental loads

– how big is it?

• evaluate geometry and angles

– where is it?

– what is the scale?

– what is the size in a particular direction?

• quantify what happens in the structure

– how big are the internal forces?

– how big should the beam be?

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Applied Architectural StructuresARCH 631

Physical Math

• physics takes observable phenomena

and relates the measurement with rules:

mathematical relationships

• need

– reference frame

– measure of length, mass, time, direction, velocity, acceleration, work, heat, electricity, light

– calculations & geometry

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Lecture 2


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Units

• measures

– US customary & SI

Units US SI

Length in, ft, mi mm, cm, m

Volume gallon liter

Mass lb mass g, kg

Force lb force N, kN

Temperature F C

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Vectors

• scalars – any quantity

• vectors - quantities with direction

– like displacements

– summation results in

the “straight line path”

from start to end

– normal vector is perpendicular to

something

y

xz

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Forces & Reactions

• Newton’s 3rd law:

– for every force of action there is an equal and opposite reaction alongthe same line

• external forces act on bodies

– can cause moments

• internal forces are

– in bodies

– between bodies (connections)

http://www.physics.umd.edu

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• convenient to resolve into 2 vectors

• at right angles

• in a “nice” coordinate system

• is between Fx and F from Fx

tany

x

F

F

2 2

x yF F F

sinyF F

cosxF F

Analysis 18

Lecture 2


ARCH 631

Force Components

Fy

Fx

F

x

y

Fy

Fx

F

Fy

Fx

F

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• weight (F = ma)

• concentrated

• distributed

– uniform

– linear

• friction

– F =N

W tA

Analysis 19

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Load Types

x

0

w

𝑤𝑥

2=𝑊

2

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( )load

w tributary widtharea

= ×

• tributary load

– think of water flow

– “concentrates” load of area into center

Analysis 20Lecture 2


Load Tracing

width

../animations/unesco_load.mov

../animations/unesco_load.mov

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• defined by magnitude and direction

• units: Nm, kft

• direction:

+ ccw (right hand rule)

- cw

• value found from Fand distance

• d also called “lever” or “moment” arm

M F d

F

A

C

B

d

Analysis 21

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Moments

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• analytically

• free body diagrams

0x xR F 0y yR F 0M M

http://www.physics.umd.edu

Analysis 22

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Equilibrium

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Free Body Diagram

• FBD (sketch)

• tool to see all forces on a body or a

point including

– external forces

– weights

– force reactions

– external moments

– moment reactions

– internal forces

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Supports and Connections

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Supports and Connections

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( )x Ax

∑ ∆=

A

( )y Ay

∑ ∆=

A

• “average” x & y of an area

• for a volume of constant thickness

– where γ is weight/volume

– center of gravity = centroid of area

W t Aγ∆ = ∆



Centroid

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2

x cx yI I Ad

• 2nd moment area

–math concept

– area x (distance)2

• need for behavior of

– beams

– columns

Analysis 27

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Moments of Inertia

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Internal and Pin Forces

– 3 equations per three-force body

– two-force body forces in line

– 2 reactions per pin + support forces

A

B

C

D

G

E F

Ax

Ay D

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Internal Beam V & M (+P)

• maximums needed for design

• Mmax at V = 0

V

L

+

M+

-

L/2

L

+V

+M

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Deflected Shape

• positive bending moment

– tension in bottom, compression in top

• negative bending moment

– tension in top, compression in bottom

• zero bending moment

– inflection point

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Pstress

A

• stress is a term for the intensity of a

force, like a pressure

• internal or applied

• force per unit area

Analysis 31

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Stress

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• normal stress is normal

to the cross section

• shear stress parallel

to a surface

t or c

Pf

A

v

P Pf

A td

Analysis 32

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Stress Types

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p

P Pf

A td= =

v

Tf

J

ρ=

• bearing stress on a

surface by contact in

compression

• torsional stress by shear

from twisting



Stress Types

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• single shear

• double shear

• bearing

2

4

vd

P Pf

A

2

4

2 2

2v

d

P PP

fA A

p

projected

P Pf

A td

Analysis 34

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Bolt Stresses

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• tension and compressive stress caused

by bending

• shear stress from bending

b

Mc Mf

I S

v ave

VQf

Ib

Analysis 35

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Bending Stresses

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longitudinalV VQ

p I

connected area

connector

VQnF p

I

• plates with

– nails

– rivets

– bolts

• splices

Analysis 36

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ARCH 631

Connectors Resisting Shear

x

y

ya

4”

2”

2”

12”

8” p

p p

4.43”

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Lstrain

L

Analysis 37

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Strain

• materials deform

• axially loaded materials change

length

• bending materials deflect

• STRAIN:

– change in length over length

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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

obtained from testing

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• important to us in - diagrams:

– straight section

– LINEAR-ELASTIC

– recovers shape

(no permanent

deformation)

f E

PL

AE

Analysis 39

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Stress to Strain

f

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Behavior Types

• brittle

• semi-brittle

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Plastic Behavior

• ductile

at yield stress

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• if we need to knowwhere max f and fv

happen:

Analysis 42

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Maximum Stresses

F

1c o s0 oA

Pf max

5.0sincos45

22

maxmax

f

A

Pf

o

v

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• - the rate of strain per degree

• UNITS : ,

• length change:

• thermal strain:

– no stress when movement allowed

Analysis 43

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Thermal Deformation

F C

LTT

TT

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• curvature, R

Analysis 44

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ARCH 631

Beam Deflections

EI

M

R

1

EI

xMcurvature

)(

dxEI

xMslope

)(

dxEI

xMdeflection

)(

a l lo wa b lea ctu a lxy )(max

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• short columns

• slenderness ratio = Le/r (L/d)

• radius of gyration =

Analysis 45

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ARCH 631

Column Stability

weak axis

aactual

critical FA

Pf

A

Ir

2

2

2

22

rL

E

LA

EAr

A

Pf

ee

criticalcritical

2

e

2

critical

rL

EAP

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• when a column gets stubby, Fy will limit the

load

• real world has loads

with eccentricity

• end conditions

Analysis 46

Lecture 2


ARCH 631

Column Stresses

LKLe

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