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*Internal Beam Forces 1
Lecture 13
*

*A
*

*ARCHITECTURAL STRUCTURES I:
STATICS AND STRENGTH OF MATERIALS
8
*

*thirteen
*

*beam forces –
internal
*

*lecture
*

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*Internal Beam Forces 2
Lecture 13
*

*n
*

*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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*Internal Beam Forces 3
Lecture 13
*

*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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*Internal Beam Forces 4
Lecture 13
*

*A
*

*Beam Loading
*• *concentrated force
*• *concentrated moment
*

– *spandrel beams
*

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*Internal Beam Forces 5
Lecture 13
*

*A
*

*Beam Loading
*• *uniformly distributed load (line load)
*• *non-uniformly distributed load
*

– *hydrostatic pressure
*– *wind loads
*

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*Internal Beam Forces 6
Lecture 13
*

*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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*Internal Beam Forces 7
Lecture 13
*

*A
*

*Beam Supports
*• *in the real world, modeled type
*

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*Internal Beam Forces 8
Lecture 13
*

*A
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*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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*Internal Beam Forces 9
Lecture 13
*

*V & M Diagrams
*• *tool to locate Vmax and Mmax
*• *necessary for designing
*• *have a different sign convention than
*

*external forces, moments, and reactions
*

R

(+)V (+)M

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*Internal Beam Forces 10
Lecture 13
*

*A
*

*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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*Internal Beam Forces 11
Lecture 13
*

*A
*

*Shear Sign Convention
*

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*Internal Beam Forces 12
Lecture 13
*

*A
*

*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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*Internal Beam Forces 13
Lecture 13
*

*Bending Moment Sign Convention
*

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*Internal Beam Forces 14
Lecture 13
*

*A
*

*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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*Internal Beam Forces 15
Lecture 13
*

*A
*

*Constructing V & M Diagrams
*• *along the beam length, plot V, plot M
*

*V
*

*L
*

*+
*

*M
+
*

*-
*

*L
*

*load diagram
*

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*Internal Beam Forces 16
Lecture 13
*

*A
*

*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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*Internal Beam Forces 17
Lecture 13
*

*A
*

*Method 1: Equilibrium
*• *cut sections at important places
*• *plot V & M
*

*V
*

*L
*

*+
*

*M
+
*

*-
*

*L/2
*

*L
*

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*Internal Beam Forces 18
Lecture 13
*

*A
abn
*

• *important places
*– *supports
*– *concentrated loads
*– *start and end of distributed loads
*– *concentrated moments
*

• *free ends
*– *zero forces
*

*Method 1: Equilibrium
*

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*Internal Beam Forces 19
Lecture 13
*

*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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*Internal Beam Forces 20
Lecture 13
*

*Architectural Structures I
ENDS 231
*

*S2008abn
*

*Method 2
*• *relationships
*

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*Internal Beam Forces 21
Lecture 13
*

*Method 2: Semigraphical
*• *Mmax occurs where V = 0 (calculus)
*

*V
*

*L
*

*+
*

*M
+
*

*-
*

*L no area
*

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*Internal Beam Forces 22
Lecture 13
*

*A
*

• *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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*Internal Beam Forces 23
Lecture 13
*

*A
*

• *line with slope, integrates to parabola
*

• *ex: load to shear, shear to moment
*

*Curve Relationships
*

*x
*

*y
*

*x
*

*y
*

⇒

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*Internal Beam Forces 24
Lecture 13
*

*A
*

• *parabola, integrates to 3rd order curve
*

• *ex: load to shear, shear to moment
*

*Curve Relationships
*

*x
*

*y
*

*x
*

*y
*

⇒

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*Internal Beam Forces 25
Lecture 13
*

*A
*

*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 jumps with concentrated load
5. Shear changes with area under load
*

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