1

S2009abnBeam Stresses 1

Lecture 9

Elements of Architectural Structures

ARCH 614

ELEMENTS OF ARCHITECTURAL STRUCTURES:

FORM, BEHAVIOR, AND DESIGN

ARCH 614

DR. ANNE NICHOLS

SPRING 2014

nine

beams:

bending and shear stress

lecture

Beam Stresses 7

Lecture 18

Elements of Architectural Structures

ARCH 614

S2005abn

Beam Bending

• Galileo

– relationship between

stress and depth2

• can see

– top squishing

– bottom stretching

• what are the stress across the section?

Beam Stresses 8

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Pure Bending

• bending only

• no shear

• axial normal stresses

from bending can be

found in

– homogeneous materials

– plane of symmetry

– follow Hooke’s law x

y

Beam Stresses 9

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Bending Moments

• sign convention:

• size of maximum internal moment will

govern our design of the section

V

M + -

2

Beam Stresses 10

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Normal Stresses

• geometric fit

– plane sections

remain plane

– stress varies linearly

Beam Stresses 11

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Neutral Axis

• stresses vary linearly

• zero stress occurs at

the centroid

• neutral axis is line of

centroids (n.a.)

Beam Stresses 12

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Derivation of Stress from Strain

• pure bending =

arc shape

RL

R

L y c

½ ½

)( yRLoutside

R

y

R

RyR

L

LL

L

outside

Beam Stresses 13

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Derivation of Stress

R

EyEf

• zero stress at n.a.

maxfc

yf

R

Ecf max

R

L y c

½ ½

3

Beam Stresses 14

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Bending Moment

• resultant moment

from stresses =

bending moment!

AfyM

SfIc

fAy

c

fAy

c

yfmax

maxmaxmax 2

Beam Stresses 15

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Bending Stress Relations

c

IS

section modulus

I

Myfb

general bending stress

EI

M

R

1

curvature

S

Mfb

maximum bending stress

b

requiredF

MS

required section

modulus for design

Beam Stresses 16

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Transverse Loading and Shear

• perpendicular loading

• internal shear

• along with bending moment

Beam Stresses 17

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Bending vs. Shear in Design

• bending stresses

dominate

• shear stresses exist

horizontally with shear

• no shear stresses

with pure bending

4

Beam Stresses 18

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Shear Stresses

• horizontal & vertical

Beam Stresses 19

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Shear Stresses

• horizontal & vertical

Beam Stresses 20

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Beam Stresses

• horizontal with bending

Beam Stresses 21

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Equilibrium

• horizontalforce Vneeded

• Q is a moment area

xI

QVV T

allongitudin

5

Beam Stresses 22

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Moment of Area

• Q is a moment area with respect to the n.a.

of area above or below the horizontal

• Qmax at y=0

(neutral axis)

• q is shear flow:

I

QV

x

Vq Tallongitudin

Beam Stresses 23

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Shearing Stresses

• = 0 on the top/bottom

• b min may not be with Q max

• with h/4 b, fv-max 1.008 fv-ave

xb

V

A

Vfv

Ib

VQf avev

avevf

Beam Stresses 24

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

• fv-max occurs at n.a.

Rectangular Sections

12

3bhI 8

2bhyAQ

A

V

Ib

VQfv

2

3

Beam Stresses 25

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

• W and S sections

– b varies

– stress in flange negligible

– presume constant

stress in web

Steel Beam Webs

web

vA

V

A

Vf

2

3max

tweb

d

6

Beam Stresses 26

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Shear Flow

• loads applied in plane of symmetry

• cut made perpendicular

I

VQq

fv

fv

fvfv

fvfv

Beam Stresses 27

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Shear Flow Quantity

• sketch from Q

I

VQq

Beam Stresses 28

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

• plates with

– nails

– rivets

– bolts

• splices

Connectors Resisting Shear

x

y

ya

4”

2”

2”

12”

8” p

p p

4.43”

I

VQ

p

V allongitudin

pI

VQnF

areaconnected

connector

Beam Stresses 29

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Vertical Connectors

• isolate an area with vertical interfaces

pI

VQnF

areaconnected

connector

p p

p

7

Beam Stresses 30

Lecture 17

Elements of Architectural Structures

ARCH 614

S2004abn

Unsymmetrical Shear or Section

• member can bend and twist

– not symmetric

– shear not in that plane

• shear center

– moments balance