Strength of Materials I EGCE201 กำลังวัสดุ 1. Instructor: ดร.วรรณสิริ พันธ์อุไร ( อ . ปู ) ห้องทำงาน : 6391 ภาควิชาวิศวกรรมโยธา E-mail: email@example.com โทรศัพท์ : 66(0) 2889-2138 ต่อ 6391. Symmetric Bending of Beams.
Instructor: ดร.วรรณสิริ พันธ์อุไร (อ.ปู)
โทรศัพท์: 66(0) 2889-2138ต่อ6391
Under action of M and M’, the member will bend but will remain symmetric with respect to the plane containing the couples.
There exists a surface // to the upper and lower faces of the member
(see as a line on the cross-section) where no elongation and the bending
normal stress is zero. This surface is called the neutral axis.
L = rq
Consider an arc some distance y above the neutral surface
The length of arc JK can be expressed as
L’ = (r- y)q
d = L’-L = (r- y)q - rq = - yq
The normal strain is max when y is the largest
This equation can be satisfied only if
The first area moment of the cross section about its NA = 0
Taking the moment about the z axis = 0
is the 2nd area moment of the cross section w.r.t. z axis
Top Surface (+y)
Bottom Surface (-y)
stress in x direction but also bending about y axis.
If the bending moment is about the y axis, a similar
The resistance to bending would be
the same if each section were made
of the same material,where the 2nd
material was multiplied by n
Try it for yourself at home
Transform section to all steel
In designing a beam, it is critical to determine the internal
shear force (V) and bending moment distribution (M). This
is accomplished by constructing shear and bending moment
In general, the load distribution across the width of the
beam is assumed to be applied uniformly. Therefore,
a beam can be analyzed in 2 dimensions rather than 3.
1. Determine the reactions at each support.
Neither half is in equilibrium
so that static equilibrium is maintained. This is done
Through internal forces and moments.
a segment of beam a distance x from the left end whose
width is dx.
eliminate last term = 0
V = constant
M = f(x)
V = f(x)
M = f(x2)
V = no effect
M = spike
Dr. Wonsiri Punurai (Bo)