재료역학(130527)

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재료역학(130527)
Dep. Of Materials Science and Engineering
Definition and applications of beams
 Beam?
: Long and slender structural elements that are called on
to support transverse as well as axial loads.
Definition and applications of beams
 Beam vs. Truss
: A truss is special beam element that can resist axial
deformation only
Shear Force & Bending Moment
1. Free-body diagram
2. Static (equilibrium) equations:
 F  ?,  F
M ?
x
y
?
continued.
V(x)
x
M(x)
x
continued.
3. Geometric considerations
V(x)
x
M(x)
x
continued.
Example : Statics analysis
P1
A
P2
C
m
n
B
Distributed loads
Free body diagram
 Accumulated load up to a certain point is of your interest.
 The shear force and moment at the point will be
counterbalanced with the accumulated load.
 An equivalent point load to the distributed load can be
considered for your better understanding.
Equivalent point load
Q

The distributed load q(x) is statically equivalent to a concentrated
load of magnitude Q placed at the centroid of the area under q(x)
diagram.
Successive integration method
Example 4.1
Example
Find RA and MA at the left end A
Shear Force Diagram(S.F.D.) & Bending Moment Diagram(B.M.D.)
- 1 concentrated load
S.F.D.
B.M.D.
Shear Force Diagram(S.F.D.) & Bending Moment Diagram(B.M.D.)
- uniformly distributed load
Shear Force Diagram(S.F.D.) & Bending Moment Diagram(B.M.D.)
- 2 concentrated load
P1
A
P2
C
D
B
Normal stresses in beams
- Stress distributions in bending beams
 Understanding of the stresses induced in beams by
bending loads took many years to develop. Galileo worked
on this problem, but the theory as we use it today is
usually credited principally to the great mathematician
Leonard Euler (1707~1783).
 Beams develop normal stresses in the lengthwise
direction that vary from a maximum in tension at one
surface, to zero at the beam's midplane, to a maximum in
compression at the opposite surface.
 Shear stresses are also induced, although these are often
negligible in comparision with the normal stresses when
the length-to-height ratio of the beam is large.
continued.
 Pure bending state
 Uniform bending moment
 V = dM / dx = 0
 M = constant
continued.
Assumptions in Pure Bending Theory
 The material of the beam is homogeneous and isotropic.
 The value of Young's Modulus of Elasticity is same in tension and compression.
 The transverse sections which were plane before bending, remain plane after bending
also.
 The beam is initially straight and all longitudinal filaments bend into circular arcs with a
common centre of curvature.
 The radius of curvature is large as compared to the dimensions of the cross-section.
 Each layer of the beam is free to expand or contract, independently of the layer, above or
below it.
continued.
Neutral axis
Compression
Tension
Neutral surface
continued.
 Shear strain, 


vB
vA
dy
v 
v

Δv  vB  v A   v A  dx   v A  dx
x 
x

δ v
tanγ  γ 

dx x
Calculation procedures of normal stresses
 Relating the magnitude of these axial normal stresses to
the bending moment.
1. Geometric statement:
u
v
 tanθ ~
y
x
u   yv,x
2. Kinematic equation:
du
ε xx 
  yv,xx
dx
Spatial rate of change of the slope of
v,xx
the beam deflection curve, the “slope
of the slope”  “curvature” of the
beam.
3. Constitutive equation:
d 2v
 2
dx
σ xx  Eε xx   yEv,xx
4. Calculating bending moment:
Moment of inertia with
respect to the centroidal
axis, I.
- For a rectangular cross section of height h and width b,
I is:
• The stress varies linearly from zero at the neutral axis to a maximum at the
outer surface.
• The stress varies inversely with the moment of inertia of the cross section,
and it is independent of the material's properties.

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