ENG1001 Lecture Notes - Lecture 7: Pure Bending, Neutral Axis
30 May 2018
School
Department
Course
Professor
Simple'beam'theory
Aka$Euler-Bernoulli$Beam$Theory
Developed$in$1750
•
Used$to$design$Eiffel$Tower
•
Real-life$structures$never$meet$assumptions$exactly,$but$usually$approximate$
them$well$enough$for$theory$to$be$fairly$accurate.
Although$many$engineers$forget$these$assumptions$and$often$apply$theory$
inappropriately,$conservative$nature$of$structural$design$(eg.$load$factors)$
compensates$for$this,$thus$design$error$is$rarely$a$cause$for$structural$failure.
Involves$consideration$of:
Type$of$material (Young's$mod)
=$measure$of$stiffness$of$material
○
=$slope$of$stress-strain$diagram$(σ/ε)
○
•
Way$beam$deforms (curves)
Curvature (K)$=$rate$of$change$of$beam$slope$(a$value$of$dy/dx$$at$a$
point)
○
K$=$1/R$=$dq/dS
○
○
○
•
Geometry of$beam$(eg.$cross-sect$area)•
Internal$equilibrium•
Assumptions$abt$beam
Long$relative to$its$depth$and$width.
Stresses$developed$perpendicular$to$beam$length$<<<$those$parallel,$
can$be$ignored.
○
•
Cross-section$constant along$length•
Symmetrical abt$YY$axis
No$twisting/torsion$occurring
○
•
Deflection/deformations$are$small•
Material$is$isotropic,$obey's$Hooke's$Law (i.e.$linear$elastic$/$E$=$σ/e)•
Plane$sections$remain$plane,$even$when$beam$is$subject$to$pure$bending
Experiences$0$shear$deformation
○
•
For$any$section,$integrating$the$area$above/below$neutral$axis$gives$total$
comp/tensile$force
When$no$axis$forces$applied,$C+T$=$0$(sum$of$x-forces$is$0)•
However,$the$2$forces$create$ a$couple/moment,$as$they$are$//,$not$in-line
This$moment$is$INTERNAL$BENDING$MOMENT!
○
•
Week$7:$Simple$Beam$Theory,$Deriv$of$Euler$Bernoulli$&$
bending$stress
Saturday,$2$September$ 2017
17:51
Document Summary
Week 7: simple beam theory, deriv of euler bernoul bending stress. Real-life structures never meet assumptions exactly, but usually approximate them well enough for theory to be fairly accurate. Although many engineers forget these assumptions and often apply theory inappropriately, conservative nature of structural design (eg. load factors) compensates for this, thus design error is rarely a cause for structural failure. Curvature (k) = rate of change of beam slope (a value of dy/dx at a point) Stresses developed perpendicular to beam length <<< those parallel, can be ignored. Material is isotropic, obey"s hooke"s law (i. e. linear elastic / e = /e) Plane sections remain plane, even when beam is subject to pure bending. For any section, integrating the area above/below neutral axis gives total comp/tensile force. When no axis forces applied, c+t = 0 (sum of x-forces is 0) However, the 2 forces create a couple/moment, as they are //, not in-line.
