ten top squishing bottom stretching beams: what are the stress - - PowerPoint PPT Presentation

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F2009abn

ten

beams: bending and shear stress

Beam Stresses 1 Lecture 10 Architectural Structures ARCH 331

lecture ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN

ARCH 331

• DR. ANNE NICHOLS

SPRING 2019

Beam Stresses 2 Lecture 10 Foundations Structures ARCH 331 F2008abn

Beam Bending

• Galileo

– relationship between stress and depth2

• can see

– top squishing – bottom stretching

• what are the stress across the section?

Beam Stresses 3 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 Hooke’s law

x y

Beam Stresses 4 Lecture 10 Foundations Structures ARCH 331 F2008abn

Bending Moments

• sign convention:
• size of maximum internal moment will

govern our design of the section

V M

+

2

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Normal Stresses

• geometric fit

– plane sections remain plane – stress varies linearly

Beam Stresses 6 Lecture 10 Foundations Structures ARCH 331 F2008abn

Neutral Axis

• stresses vary linearly
• zero stress occurs at

the centroid

• neutral axis is line of

centroids (n.a.)

Beam Stresses 7 Lecture 10 Foundations Structures ARCH 331 F2008abn

Derivation of Stress from Strain

• pure bending =

arc shape

 R L 

R  L y c ½ ½

 ) ( y R Loutside  

 

R y R R y R L L L L

• utside

           

Beam Stresses 8 Lecture 10 Foundations Structures ARCH 331 F2008abn

Derivation of Stress R Ey E f   

• zero stress at n.a.

max

f c y f  R Ec f 

max R  L y c ½ ½

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Beam Stresses 9 Lecture 10 Foundations Structures ARCH 331 F2008abn

Bending Moment

• resultant moment

from stresses = bending moment!

A fy M    S f I c f A y c f A y c yf

max max max max

       

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Beam Stresses 10 Lecture 10 Foundations Structures ARCH 331 F2008abn

Bending Stress Relations

c I S 

section modulus

I My fb 

general bending stress

EI M R  1

curvature

S M fb 

maximum bending stress b required

F M S 

required section modulus for design

Beam Stresses 11 Lecture 10 Foundations Structures ARCH 331 F2008abn

Transverse Loading and Shear

• perpendicular loading
• internal shear
• along with bending moment

Beam Stresses 12 Lecture 10 Foundations Structures ARCH 331 F2008abn

Bending vs. Shear in Design

• bending stresses

dominate

• shear stresses exist

horizontally with shear

• no shear stresses

with pure bending

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Beam Stresses 13 Lecture 10 Foundations Structures ARCH 331 F2008abn

Shear Stresses

• horizontal & vertical

Beam Stresses 14 Lecture 10 Foundations Structures ARCH 331 F2008abn

Shear Stresses

• horizontal & vertical

Beam Stresses 15 Lecture 10 Foundations Structures ARCH 331 F2008abn

Beam Stresses

• horizontal with bending

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• horizontal

force V needed

• Q is a moment area

T longitudinal

V Q V x I  

Beam Stresses 16 Lecture 10 Foundations Structures ARCH 331

Equilibrium

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• Q is a moment area with respect to the n.a.
• f area above or below the horizontal
• Qmax at y=0

(neutral axis)

• q is shear flow:

longitudinal T

V V Q q x I   

Beam Stresses 17 Lecture 10 Foundations Structures ARCH 331

Moment of Area

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• = 0 on the top/bottom
• b min may not be with Q max
• with h/4  b, fv-max  1.008 fv-ave

Beam Stresses 18 Lecture 10 Foundations Structures ARCH 331

Shearing Stresses x b V A V fv      Ib VQ f

ave v



 ave v

f 

Beam Stresses 19 Lecture 10 Foundations Structures ARCH 331 F2008abn

• fv-max occurs at n.a.

Rectangular Sections 12

3

bh I  8

2

bh y A Q   A V Ib VQ fv 2 3  

Beam Stresses 20 Lecture 10 Foundations Structures ARCH 331 F2008abn

• W and S sections

– b varies – stress in flange negligible – presume constant stress in web

Steel Beam Webs

web v

A V A V f  



2 3

max

tweb d

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Beam Stresses 21 Lecture 10 Foundations Structures ARCH 331 F2008abn

Shear Flow

• loads applied in plane of symmetry
• cut made perpendicular

I VQ q 

fv fv fv fv fv fv

Beam Stresses 22 Lecture 10 Foundations Structures ARCH 331 F2008abn

Shear Flow Quantity

• sketch from Q

I VQ q 

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• plates with

– nails – rivets – bolts

• splices

longitudinal

V VQ p I 

Beam Stresses 23 Lecture 10 Foundations Structures ARCH 331

Connectors Resisting Shear

x y ya 4” 2” 2” 12” 8” p p p 4.43”

p I VQ nF

area connected connector

 

Beam Stresses 24 Lecture 10 Foundations Structures ARCH 331 F2008abn

Vertical Connectors

• isolate an area with vertical interfaces

p I VQ nF

area connected connector

 

p p p

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Beam Stresses 25 Lecture 10 Foundations Structures ARCH 331 F2008abn

Unsymmetrical Shear or Section

• member can bend and twist

– not symmetric – shear not in that plane

• shear center

– moments balance