Beam Stresses 1 S2009abn Lecture 18 ARCH 614 bending and...
Transcript of Beam Stresses 1 S2009abn Lecture 18 ARCH 614 bending and...
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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