beam introduction & internal...
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S2015abnInternal Beam Forces 1Lecture 6
Elements of Architectural StructuresARCH 614
ELEMENTS OF ARCHITECTURAL STRUCTURES:
FORM, BEHAVIOR, AND DESIGN
ARCH 614
DR. ANNE NICHOLS
SPRING 2015
six
beam introduction &
internal forces
lecture
http:// nisee.berkeley.edu/godden
S2015abnInternal Beam Forces 2Lecture 6
Elements of Architectural StructuresARCH 614
Beams
• span horizontally
– floors
– bridges
– roofs
• loaded transversely by gravity loads
• may have internal axial force
• will have internal shear force
• will have internal moment (bending)
R
V
M
S2015abnInternal Beam Forces 3Lecture 6
Elements of Architectural StructuresARCH 614
Beams
• transverse loading
• sees:
– bending
– shear
– deflection
– torsion
– bearing
• behavior depends on
cross section shape
S2015abnInternal Beam Forces 4Lecture 6
Elements of Architectural StructuresARCH 614
Beams
• bending
– bowing of beam with loads
– one edge surface stretches
– other edge surface squishes
S2015abnInternal Beam Forces 5Lecture 6
Elements of Architectural StructuresARCH 614
Beam Stresses
• stress = relative force over an area
– tensile
– compressive
– bending
• tension and compression + ...
S2015abnInternal Beam Forces 6Lecture 6
Elements of Architectural StructuresARCH 614
Beam Stresses
S2015abnInternal Beam Forces 7Lecture 6
Elements of Architectural StructuresARCH 614
Beam Stresses
• tension and compression
– causes moments
Copyright © 1996-2000 Kirk Martini.
S2015abnInternal Beam Forces 8Lecture 6
Elements of Architectural StructuresARCH 614
Beam Stresses
• prestress or post-tensioning
– put stresses in tension area to “pre-compress”
S2015abnInternal Beam Forces 9Lecture 6
Elements of Architectural StructuresARCH 614
Beam Stresses
• shear – horizontal & vertical
S2015abnInternal Beam Forces 10Lecture 6
Elements of Architectural StructuresARCH 614
Beam Stresses
• shear – horizontal & vertical
S2015abnInternal Beam Forces 11Lecture 6
Elements of Architectural StructuresARCH 614
Beam Stresses
• shear – horizontal
S2015abnInternal Beam Forces 12Lecture 6
Elements of Architectural StructuresARCH 614
Beam Deflections
• depends on
– load
– section
– material
S2015abnInternal Beam Forces 13Lecture 6
Elements of Architectural StructuresARCH 614
Beam Deflections
• “moment of inertia”
S2015abnInternal Beam Forces 14Lecture 6
Elements of Architectural StructuresARCH 614
Beam Styles
• vierendeel
• open web joists
• manufactured
http:// nisee.berkeley.edu/godden
S2015abnInternal Beam Forces 15Lecture 6
Elements of Architectural StructuresARCH 614
Internal Forces
• trusses
– axial only, (compression & tension)
• in general– axial force
– shear force, V
– bending moment, M
A
A
B
B
F
F′ F
F
F F′
T´
V
T´
T
S2015abnInternal Beam Forces 16Lecture 6
Elements of Architectural StructuresARCH 614
Beam Loading
• concentrated force
• concentrated moment
– spandrel beams
S2015abnInternal Beam Forces 17Lecture 6
Elements of Architectural StructuresARCH 614
Beam Loading
• uniformly distributed load (line load)
• non-uniformly distributed load
– hydrostatic pressure = γh
– wind loads
S2015abnInternal Beam Forces 18Lecture 6
Elements of Architectural StructuresARCH 614
Beam Supports
• statically determinate
• statically indeterminate
L L L
simply supported
(most common)
overhang cantilever
L
continuous
(most common case when L1=L2)
L
L L
Propped Restrained
S2015abnInternal Beam Forces 19Lecture 6
Elements of Architectural StructuresARCH 614
Beam Supports
• in the real world, modeled type
S2015abnInternal Beam Forces 20Lecture 6
Elements of Architectural StructuresARCH 614
Internal Forces in Beams
• like method of sections / joints
– no axial forces
• section must be in equilibrium
• want to know where biggest internal
forces and moments are for designing
R
V
M
S2015abnInternal Beam Forces 21Lecture 6
Elements of Architectural StructuresARCH 614
V & M Diagrams
• tool to locate Vmax and Mmax
• necessary for designing
• Mmax occurs when V = 0
R
(+)V
(+)M
S2015abnInternal Beam Forces 22Lecture 6
Elements of Architectural StructuresARCH 614
Sign Convention
• shear force, V:
– cut section to LEFT
– if ∑Fy is positive by statics, V acts down and is POSITIVE
– beam has to resist shearing apart by V
R
(+)V
(+)M
S2015abnInternal Beam Forces 23Lecture 6
Elements of Architectural StructuresARCH 614
Shear Sign Convention
S2015abnInternal Beam Forces 24Lecture 6
Elements of Architectural StructuresARCH 614
Sign Convention
• bending moment, M:
– cut section to LEFT
– if ∑Mcut is clockwise, M acts ccw and is POSITIVE – flexes into a “smiley” beam has to resist bending apart by M
R
(+)V
(+)M
S2015abnInternal Beam Forces 25Lecture 6
Elements of Architectural StructuresARCH 614
Bending Moment Sign Convention
S2015abnInternal Beam Forces 26Lecture 6
Elements of Architectural StructuresARCH 614
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
S2015abnInternal Beam Forces 27Lecture 6
Elements of Architectural StructuresARCH 614
Constructing V & M Diagrams
• along the beam length, plot V, plot M
V
L
+
M+
-
L
load diagram
S2015abnInternal Beam Forces 28Lecture 6
Elements of Architectural StructuresARCH 614
Mathematical Method
• cut sections with x as width
• write functions of V(x) and M(x)
V
L
+
M+
-
x
L
S2015abnInternal Beam Forces 29Lecture 6
Elements of Architectural StructuresARCH 614
Equilibrium Method
• cut sections at important places
• plot V & M
V
L
+
M+
-
L/2
L
S2015abnInternal Beam Forces 30Lecture 6
Elements of Architectural StructuresARCH 614
• important places
– supports
– concentrated loads
– start and end of distributed loads
– concentrated moments
• free ends
– zero forces
Equilibrium Method
S2015abnInternal Beam Forces 31Lecture 6
Elements of Architectural StructuresARCH 614
Equilibrium Method
• relationships
S2015abnInternal Beam Forces 32Lecture 6
Elements of Architectural StructuresARCH 614
Basic Procedure
1. Find reaction forces & moments
Plot axes, underneath beam load
diagram
V:
2. Starting at left
3. Shear is 0 at free ends
4. Shear has 2 values at point loads
5. Sum vertical forces at each section
S2015abnInternal Beam Forces 33Lecture 6
Elements of Architectural StructuresARCH 614
Basic Procedure
M:
6. Starting at left
7. Moment is 0 at free ends
8. Moment has 2 values at moments
9. Sum moments at each section
10.Maximum moment is where shear = 0!
S2015abnInternal Beam Forces 34Lecture 6
Elements of Architectural StructuresARCH 614
Shear Through Zero
• slope of V is w (-w:1)
shear
load
height = VA
w (force/length)
width = x
w
VxVwx A
A=⇒=⋅
A
S2015abnInternal Beam Forces 35Lecture 6
Elements of Architectural StructuresARCH 614
Tools
• software & spreadsheets help
• http://www.rekenwonder.com/atlas.htm
S2015abnInternal Beam Forces 36Lecture 6
Elements of Architectural StructuresARCH 614
Tools – Multiframe
• in computer lab
S2015abnInternal Beam Forces 37Lecture 6
Elements of Architectural StructuresARCH 614
Tools – Multiframe
• frame window
– define beam members
– select points, assign supports
– select members,assign section
• load window
– select point or member, add point or distributedloads
S2015abnInternal Beam Forces 38Lecture 6
Elements of Architectural StructuresARCH 614
Tools – Multiframe
• to run analysis choose
– Analyze menu
• Linear
• plot
– choose options
– double click (all)
• results
– choose options