beam introduction & internal...

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