STRUCTURES: FOUNDATIONS FORM, BEHAVIOR, AND DESIGN
ARCH 331
Beam Bending
Galileo
relationship between stress and depth2
DR. ANNE NICHOLS SPRING 2009
ten
lecture
can see
top squishing bottom stretching
beams: bending and shear stress
Beam Stresses 1 Lecture 10 Foundations Structures ARCH 331 F2008abn
what are the stress across the section?
Beam Stresses 2 Lecture 10 Foundations Structures ARCH 331 F2008abn
Pure Bending
bending only no shear axial normal stresses from bending can be found in
homogeneous materials plane of symmetry follow Hookes law
Beam Stresses 3 Lecture 10 Foundations Structures ARCH 331
Bending Moments
sign convention: + V
M
size of maximum internal moment will govern our design of the section
y x
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Beam Stresses 4 Lecture 10
Foundations Structures ARCH 331
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1
Normal Stresses
geometric fit
plane sections remain plane stress varies linearly
Neutral Axis
stresses vary linearly zero stress occurs at the centroid neutral axis is line of centroids (n.a.)
Beam Stresses 5 Lecture 10
Foundations Structures ARCH 331
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Beam Stresses 6 Lecture 10
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Derivation of Stress from Strain
pure bending = arc shape
R
Derivation of Stress
zero stress at n.a.
L = R
f = E =
L y
Loutside = ( R + y )
c
=
L
=
Loutside L (R + y ) R y = = L R R
Foundations Structures ARCH 331 F2008abn Beam Stresses 8 Lecture 10
Ey R Ec f max = R y f = f max c
R
c
L
y
Beam Stresses 7 Lecture 10
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2
Bending Moment
resultant moment from stresses = bending moment!
Bending Stress Relations
1 M = R EI
curvature
My fb = I
general bending stress
I S= c
section modulus
M = fyA
=
f f yf max yA = max y 2 A = max I = f max S c c c
Foundations Structures ARCH 331 F2008abn
M fb = S
maximum bending stress
Beam Stresses 10 Lecture 10 Foundations Structures ARCH 331
S required
M Fb
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required section modulus for design
Beam Stresses 9 Lecture 10
Transverse Loading and Shear
Bending vs. Shear in Design
bending stresses dominate
perpendicular loading internal shear along with bending moment
shear stresses exist horizontally with shear no shear stresses with pure bending
F2008abn Beam Stresses 12 Lecture 10 Foundations Structures ARCH 331 F2008abn
Beam 11 Stresses Lecture 10
Foundations Structures ARCH 331
3
Shear Stresses
horizontal & vertical
Shear Stresses
horizontal & vertical
Beam Stresses 13 Lecture 10
Foundations Structures ARCH 331
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Beam Stresses 14 Lecture 10
Foundations Structures ARCH 331
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Beam Stresses
horizontal with bending
Equilibrium
horizontal force V needed
Vlongitudinal =
VT Q x I
Q is a moment area
Beam Stresses 15 Lecture 10 Foundations Structures ARCH 331 F2008abn
Beam Stresses 16 Lecture 10 Foundations Structures ARCH 331 F2008abn
4
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:
Shearing Stresses V V fv = = A b x VQ f v ave = Ib
f v ave 0 on the top/bottom = b min may not be with Q max with h/4 b, fv-max 1.008 fv-ave
q=
Beam Stresses 17 Lecture 10
Vlongitudinal x
=
VT Q I
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Foundations Structures ARCH 331
Beam Stresses 18 Lecture 10
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Rectangular Sections
bh 3 I= 12
Q = Ay = bh
2
Steel Beam Webs
W and S sections
8
b varies
d tweb
fv =
VQ 3V = Ib 2 A
fv-max occurs at n.a.
stress in flange negligible presume constant f v max stress in web
F2008abn Beam Stresses 20 Lecture 10 Foundations Structures ARCH 331
=
3V V 2 A Aweb
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Beam Stresses 19 Lecture 10
Foundations St...