梁的应力的深入讨论.ppt

C
My z Mz y 0
Iy
Iz
The angle b between neutral
axis and the z axis is
n
tan b y M y Iz
z Mz Iy
And the maximum stresses occur at points located farthest from the neutral axis.
tan b y M y Iz Iz tan
z Mz Iy Iy (1)When the load lies in the xy
b M My
plane and z is the neutral axis;
z Mz C
(2)When the load lies in the xz plane and y is the neutral axis;
directions of rotation.
Bending Stresses and Neutral Axis
y
My n A(y,z)
z
Mz
C
The normal stress at point A is
x
' x
'' x
My Iy
z Mz Iz
y
The equation to determine the n neutral axis is
1 8
qzl 2
1 8
q sin
l2
x
M zmaxy Iz
M ymaxz Iy
1 ql2 cos
8 bh3
y
1 ql2 sin
8 hb3
z
12
12
3ql 2 2bh
(
c
os
h2
y
sin
b2
z)
q 3.0 kN/m
by
D
h
zo l
E
q26.6
Maximum stresses (pnts D and E)
oz l
q 30 y
Stresses under Symmetric bending
Longitudinal plane of symmetry, The plane of bending
F2
B
Symmetric axis
F1
FB
A
The deformed axis lies in the plane the loads act
Examle 6-4
Vocabulary Roof purlin, roof sheathing, top chords,
q 3.0 kN/m
by
h
zo l
q26.6
Example 6-4
q 3.0 kN/m
by
D
h
l
M zmax
1 8
qyl 2
1 8
q
cos
l2
zo
E
q26.6
M
y max
On a positive face, the
y
moments are positive
My
when their vectors
point in the positive
directions of the
x corresponding axes,
z
Mz
and the right-hand rule
for vectors gives the
(3)When the principal moments of inertia are equal (Iz =Iy). In case (3), all axes through the centroid are principal axes and all have the same moment of inertia. Whatever direction of the plane of loading, it is always a principal plane and perpendicular to the neutral axis.
FA In symmetric bending the deformed axis lies in the
same plane that all loads act, so it belongs to planar bending.
Sign Conventions for Bending Moments
axis and the z axis is
tan b y M y Iz Iz tan
z Mz Iy Iy So usually the neutral axis is not perpendicular to the loading plane.
Special Cases (b )
yP
Chapter 6 Stresses in Beams (Advanced Topics)
6.1 Introduction
Vocabulary • Composite beams , beams with inclined loads,
unsymmetric beams, shear stresses in thinwalled beams, shear centers
6.4 Doubly Symmetric Beams with Inclined Loads
Vocabulary • Inclined loads, a doubly symmetric cross
section, resolve, components, superpose,
b q 2 kN/m
My z Mz y 0
Iy
Iz
It is shown that the neutral axis is a straight line passing through the centroid C.
Bending Stresses and Neutral Axis
y
My n A(y,z)
b
z Mz
E
D
3ql 2 Βιβλιοθήκη Baidubh
( cos
h
sin
b
)
4.01MPa
Neutral axis:
x
3ql 2 2bh
( cos
h2
y
sin
b2
z)
cos sin
y
z0
h2
b2
tan b y h2 tan 1.125
z b2
b 48.4
Points where maximum stresses occur
Relationship Between the Neutral Axis and the Inclination of the Loads
y
P My(x)=Psin (l-x)
z
yP b M My
z Mz C
Mz(x)= Pcos (l-x)
x
The angle b between neutral