a structures orm, behavior, and design arch 331 r....
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![Page 1: A STRUCTURES ORM, BEHAVIOR, AND DESIGN ARCH 331 R. Nfaculty-legacy.arch.tamu.edu/.../courses/arch331/lect6.pdf · 2014-05-23 · 1. Find reaction forces & moments Plot axes, underneath](https://reader034.vdocument.in/reader034/viewer/2022050502/5f9463b361a3a570ef4022ac/html5/thumbnails/1.jpg)
Su2014abn Internal Beam Forces 1
Lecture 6
Architectural Structures
ARCH 331
ARCHITECTURAL STRUCTURES:
FORM, BEHAVIOR, AND DESIGN
ARCH 331
DR. ANNE NICHOLS
SUMMER 2014
six
beams –
internal forces & diagrams
lecture
http:// nisee.berkeley.edu/godden
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Su2014abn Internal Beam Forces 2
Lecture 6
Architectural Structures
ARCH 331
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
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Su2014abn Internal Beam Forces 3
Lecture 6
Architectural Structures
ARCH 331
Beams
• transverse loading
• sees:
– bending
– shear
– deflection
– torsion
– bearing
• behavior depends on
cross section shape
![Page 4: A STRUCTURES ORM, BEHAVIOR, AND DESIGN ARCH 331 R. Nfaculty-legacy.arch.tamu.edu/.../courses/arch331/lect6.pdf · 2014-05-23 · 1. Find reaction forces & moments Plot axes, underneath](https://reader034.vdocument.in/reader034/viewer/2022050502/5f9463b361a3a570ef4022ac/html5/thumbnails/4.jpg)
Su2014abn Internal Beam Forces 4
Lecture 6
Architectural Structures
ARCH 331
Beams
• bending
– bowing of beam with loads
– one edge surface stretches
– other edge surface squishes
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Su2014abn Internal Beam Forces 5
Lecture 6
Architectural Structures
ARCH 331
Beam Stresses
• stress = relative force over an area
– tensile
– compressive
– bending
• tension and compression + ...
![Page 6: A STRUCTURES ORM, BEHAVIOR, AND DESIGN ARCH 331 R. Nfaculty-legacy.arch.tamu.edu/.../courses/arch331/lect6.pdf · 2014-05-23 · 1. Find reaction forces & moments Plot axes, underneath](https://reader034.vdocument.in/reader034/viewer/2022050502/5f9463b361a3a570ef4022ac/html5/thumbnails/6.jpg)
Su2014abn Internal Beam Forces 6
Lecture 6
Architectural Structures
ARCH 331
Beam Stresses
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Su2014abn Internal Beam Forces 7
Lecture 6
Architectural Structures
ARCH 331
Beam Stresses
• tension and compression
– causes moments
Copyright © 1996-2000 Kirk Martini.
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Su2014abn Internal Beam Forces 8
Lecture 6
Architectural Structures
ARCH 331
Beam Stresses
• prestress or post-tensioning
– put stresses in tension area to
“pre-compress”
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Su2014abn Internal Beam Forces 9
Lecture 6
Architectural Structures
ARCH 331
Beam Stresses
• shear – horizontal & vertical
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Su2014abn Internal Beam Forces 10
Lecture 6
Architectural Structures
ARCH 331
Beam Stresses
• shear – horizontal & vertical
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Su2014abn Internal Beam Forces 11
Lecture 6
Architectural Structures
ARCH 331
Beam Stresses
• shear – horizontal
![Page 12: A STRUCTURES ORM, BEHAVIOR, AND DESIGN ARCH 331 R. Nfaculty-legacy.arch.tamu.edu/.../courses/arch331/lect6.pdf · 2014-05-23 · 1. Find reaction forces & moments Plot axes, underneath](https://reader034.vdocument.in/reader034/viewer/2022050502/5f9463b361a3a570ef4022ac/html5/thumbnails/12.jpg)
Su2014abn Internal Beam Forces 12
Lecture 6
Architectural Structures
ARCH 331
Beam Deflections
• depends on
– load
– section
– material
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Su2014abn Internal Beam Forces 13
Lecture 6
Architectural Structures
ARCH 331
Beam Deflections
• “moment of inertia”
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Su2014abn Internal Beam Forces 14
Lecture 6
Architectural Structures
ARCH 331
Beam Styles
• vierendeel
• open web joists
• manufactured
http:// nisee.berkeley.edu/godden
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Su2014abn Internal Beam Forces 15
Lecture 6
Architectural Structures
ARCH 331
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
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Su2014abn Internal Beam Forces 16
Lecture 6
Architectural Structures
ARCH 331
Beam Loading
• concentrated force
• concentrated moment
– spandrel beams
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Su2014abn Internal Beam Forces 17
Lecture 6
Architectural Structures
ARCH 331
Beam Loading
• uniformly distributed load (line load)
• non-uniformly distributed load
– hydrostatic pressure = h
– wind loads
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Su2014abn Internal Beam Forces 18
Lecture 6
Architectural Structures
ARCH 331
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
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Su2014abn Internal Beam Forces 19
Lecture 6
Architectural Structures
ARCH 331
Beam Supports
• in the real world, modeled type
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Su2014abn Internal Beam Forces 20
Lecture 6
Architectural Structures
ARCH 331
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
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Su2014abn Internal Beam Forces 21
Lecture 6
Architectural Structures
ARCH 331
V & M Diagrams
• tool to locate Vmax and Mmax (at V = 0)
• necessary for designing
• have a different sign convention than
external forces, moments, and reactions
R
(+)V
(+)M
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Su2014abn Internal Beam Forces 22
Lecture 6
Architectural Structures
ARCH 331
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
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Su2014abn Internal Beam Forces 23
Lecture 6
Architectural Structures
ARCH 331
Shear Sign Convention
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Su2014abn Internal Beam Forces 24
Lecture 6
Architectural Structures
ARCH 331
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
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Su2014abn Internal Beam Forces 25
Lecture 6
Architectural Structures
ARCH 331
Bending Moment Sign Convention
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Su2014abn Internal Beam Forces 26
Lecture 6
Architectural Structures
ARCH 331
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
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Su2014abn Internal Beam Forces 27
Lecture 6
Architectural Structures
ARCH 331
Constructing V & M Diagrams
• along the beam length, plot V, plot M
V
L
+
M +
-
L
load diagram
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Su2014abn Internal Beam Forces 28
Lecture 6
Architectural Structures
ARCH 331
Mathematical Method
• cut sections with x as width
• write functions of V(x) and M(x)
V
L
+
M +
-
x
L
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Su2014abn Internal Beam Forces 29
Lecture 6
Architectural Structures
ARCH 331
Method 1: Equilibrium
• cut sections at important places
• plot V & M
V
L
+
M +
-
L/2
L
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Su2014abn Internal Beam Forces 30
Lecture 6
Architectural Structures
ARCH 331
• important places
– supports
– concentrated loads
– start and end of distributed loads
– concentrated moments
• free ends
– zero forces
Method 1: Equilibrium
![Page 31: A STRUCTURES ORM, BEHAVIOR, AND DESIGN ARCH 331 R. Nfaculty-legacy.arch.tamu.edu/.../courses/arch331/lect6.pdf · 2014-05-23 · 1. Find reaction forces & moments Plot axes, underneath](https://reader034.vdocument.in/reader034/viewer/2022050502/5f9463b361a3a570ef4022ac/html5/thumbnails/31.jpg)
Su2014abn Internal Beam Forces 31
Lecture 6
Architectural Structures
ARCH 331
Method 2: Semigraphical
• by knowing
– area under loading curve = change in V
– area under shear curve = change in M
– concentrated forces cause “jump” in V
– concentrated moments cause “jump” in M
D
C
CD
x
x
wdxVV D
C
CD
x
x
VdxMM
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Su2014abn Internal Beam Forces 32
Lecture 6
Architectural Structures
ARCH 331
Method 2
• relationships
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Su2014abn Internal Beam Forces 33
Lecture 6
Architectural Structures
ARCH 331
Method 2: Semigraphical
• Mmax occurs where V = 0 (calculus)
V
L
+
M +
-
L no area
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Su2014abn Internal Beam Forces 34
Lecture 6
Architectural Structures
ARCH 331
• integration of functions
• line with 0 slope, integrates to sloped
• ex: load to shear, shear to moment
Curve Relationships
x
y
x
y
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Su2014abn Internal Beam Forces 35
Lecture 6
Architectural Structures
ARCH 331
• line with slope, integrates to parabola
• ex: load to shear, shear to moment
Curve Relationships
x
y
x
y
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Su2014abn Internal Beam Forces 36
Lecture 6
Architectural Structures
ARCH 331
• parabola, integrates to 3rd order curve
• ex: load to shear, shear to moment
Curve Relationships
x
y
x
y
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Su2014abn Internal Beam Forces 37
Lecture 6
Architectural Structures
ARCH 331
Basic Procedure with Sections
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
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Su2014abn Internal Beam Forces 38
Lecture 6
Architectural Structures
ARCH 331
Basic Procedure with Sections
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!
(locate where V = 0)
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Su2014abn Internal Beam Forces 39
Lecture 6
Architectural Structures
ARCH 331
Basic Procedure by Curves
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 jumps with concentrated load
5. Shear changes with area under load
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Su2014abn Internal Beam Forces 40
Lecture 6
Architectural Structures
ARCH 331
Basic Procedure by Curves
M:
6. Starting at left
7. Moment is 0 at free ends
8. Moment jumps with moment
9. Moment changes with area under V
10.Maximum moment is where shear = 0!
(locate where V = 0)
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Su2014abn Internal Beam Forces 41
Lecture 6
Architectural Structures
ARCH 331
Shear Through Zero
• slope of V is w (-w:1)
shear
load
height = VA
w (force/length)
width = x
wV
xVwx AA
A
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Su2014abn Internal Beam Forces 42
Lecture 6
Architectural Structures
ARCH 331
Parabolic Shapes
• cases
+
up fast,
then slow
+
up slow,
then fast
-
down fast,
then slow
down slow,
then fast
- -
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Su2014abn Internal Beam Forces 43
Lecture 6
Architectural Structures
ARCH 331
Deflected Shape & M(x)
• -M(x) gives shape indication
• boundary conditions must be met
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Su2014abn Internal Beam Forces 44
Lecture 6
Architectural Structures
ARCH 331
Boundary Conditions
• at pins, rollers,
fixed supports: y = 0
• at fixed supports: = 0
• at inflection points
from symmetry: = 0
• ymax at 0dx
dy
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Su2014abn Internal Beam Forces 45
Lecture 6
Architectural Structures
ARCH 331
Tabulated Beam Formulas
• how to read charts
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Su2014abn Internal Beam Forces 46
Lecture 6
Architectural Structures
ARCH 331
Tools
• software & spreadsheets help
• http://www.rekenwonder.com/atlas.htm
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Su2014abn Internal Beam Forces 47
Lecture 6
Architectural Structures
ARCH 331
Tools – Multiframe
• in computer lab
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Su2014abn Internal Beam Forces 48
Lecture 6
Architectural Structures
ARCH 331
Tools – Multiframe
• frame window
– define beam members
– select points, assign supports
– select members,
assign section
• load window
– select point or member,
add point or distributed
loads
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Su2014abn Internal Beam Forces 49
Lecture 6
Architectural Structures
ARCH 331
Tools – Multiframe
• to run analysis choose
– Analyze menu
• Linear
• plot
– choose options
– double click (all)
• results
– choose
options