σ = My Iz
(12.4)
物理条件：材料是线弹性材 料所以可以应用胡克定律， 即σ = Eε。线性变化的应变 必然引起线性变化的应力。 因此，如正应变的变化规律 一样，正应力从在中性轴处 的零应力线性地变化到离 中性轴最远处的最大值。
静力学条件：由横截面上正 应力的合力应为零可以确 定中性轴的位置。
ε = a′b′ − ab = (ρ + y)dθ − ρdθ = y
ab
ρdθ
ρ
(12.1)
注意到中性层上的任 意线不改变长度。根据定 义，沿 a′b′的正应变可表示 为：
47
Stresses in Beams
Physical Condition. The material behaves in a linear-elastic manner so that Hooke’s law applies, that is, σ = Eε. A linear variation of normal strain must then be the consequence of a linear variation in normal stress. Hence, like the normal strain variation, normal stress will vary from zero at the member’s neutral axis to a maximum value, a distance farthest from the neutral axis.
Equation 12.4 is often referred to as the flexure formula. It is used to determine the normal stress in a straight member, having a cross section that is symmetrical with respect to an axis, and the moment is applied perpendicular to this axis.
式 12.4 就是弯曲正应 力公式。该式用来确定有不 变的对称横截面的直梁在 受到垂直于梁轴线的力矩 作用下的正应力。
Stresses in Beams
CHAPTER 12 STRESSES IN BEAMS
12.1 The Flexure Formula
The beam is considered to be pure bending when it is subjected to a bending moment without either shearing or axial forces. Further, the beam has a uniform cross section, the cross section is symmetric about a vertical axis, the loads act in the plane of symmetry, and bending takes place in the same plane.
∫ ∫ yσ dA = E y 2dA = M
A
ρA

(12.3)
Here the integral represents the moment of inertia of the cross-sectional area, computed about the neutral axis and symbolized as Iz. Substituting equation 12.2 into equation 12.3,
几何条件：为了探讨材料是 如何变形的，一个离梁端距 离为 x 处，未变形宽度为 dx 的单元体被分离出来。该单 元体的未变形和变形后的 形状如图 12.2 所示。
dx
O1
O2
x
y
a
b
ρ
dθ
y O1 dx O2
a’
b’
Figure 12.2
Notice that any line segment located on the neutral surface does not change its length. By definition, the normal strain along a′b′ is determined as follow,
在弯矩作用下的可变 形杆件变形特性使得杆件 的下半部材料受拉，上半部 材料受压。因此，这两部分 材料的中间一定存在着一 个面，称为中性层，这层上 纵向的材料纤维的长度没 有变化。
由此可得关于应力使 材料产生变形的三个假设。 第一，在中性层上的纵向坐 标 x 长度保持不变。力矩使 梁产生弯曲变形，从而直线 变为在 x-y 对称平面内的曲 线。第二，变形时所有的横 截面保持平面并且与纵向 线垂直。第三，任何横截面 自身的变形将被忽略。
The beam has been loaded by couples to produce pure bending. After the bending moment is determined when the external couples were applied to the beam, the stress distribution on the cross section of a beam needs to be developed.
Figure 12.1 (a)
Figure 12.1 (b)
46
Stresses in Beams
The behaviour of any deformable bar subjected to a bending moment causes the material within the bottom portion of the bar to stretch and the material within the top portion to compress. Consequently, between these two regions there must be a surface, called the neutral surface, in which longitudinal fibres of the material will not undergo a change in length.
The stress in the beam can be determined from the requirement that the resultant internal moment M must be equal to the moment produced by the stress distribution about the neutral axis.
12.1 弯曲正应力公式
当梁上只有弯矩而没 有剪力和轴力作用时为纯 弯曲。并且梁的横截面不变 化，有竖直方向的对称轴， 加载在对称平面内，弯曲也 发生在同一平面内。
力偶加载在梁上产生 纯弯曲。加载后确定了梁内 的弯矩后，需要确定应力在 梁横截面的分布情况。
如图 12.1(a)所示的未 变形的杆，横截面为矩形， 杆上标示纵向线和横向线。 当加载一个弯矩后，标示线 如图 12.1(b)所示。图中可以 看出，纵向线变为曲线，横 向线保持为直线但有一定 的旋转。
From these observations the following three assumptions can be made regarding the way the stress deforms the material. First, the longitudinal axis x, which lies within the neutral surface, does not experience any change in length. Rather the moment will tend to deform the beam so that this line becomes a curve that lies in the x-y plane of symmetry. Second, all cross sections of the beam remain plane and perpendicular to the longitudinal axis during the deformation. And third, any deformation of the cross section within its own plane will be neglected.
σ = Ey ρ
(12.2)
Statics Condition. The position of the neutral axis on the cross section can be located by satisfying the condition that the resultant force produced by the stress distribution over the cross-section area must be equal to zero.
∫Aσ dA
=
∫A
Ey ρ
dA
=
E ρ
∫A
ydA
=
0
Since E/ρ is not equal to zero, then
∫A ydA = 0
This condition can only be satisfied if the neutral axis is also the horizontal centroidal axis for the cross section. Consequently, once the centroid for the member’s cross-sectional area is determined, the location of the neutral axis is known.