了解梁中的应力，高效的工程师（三）

了解梁中的应力，高效的工程师（三）

由于本文篇幅较长，分为四部分发布，这是第三部分。

Whena load is applied to a beam it will deform by bending, which generates internalstresses within the beam. These internal stresses can be represented by a shearforce and a bending moment acting on any cross-section of the beam. The shearforce is the resultant of vertical shear stresses, which act parallel to thecross-section, and the bending moment is the resultant of normal stresses,called bending stresses, which act perpendicular to the cross-section.

当载荷施加到梁上时，它会因弯曲而变形，从而在梁内产生内应力。这些内应力可以用作用在梁的任何横截面上的剪力和弯矩来表示。剪切力是垂直剪切应力的合力，垂直剪切应力平行于横截面作用，弯矩是垂直于横截面作用的法向应力（称为弯曲应力）的合力。

Thevideo below covers these bending and shear stresses that develop in beams inmore detail.

下面的视频更详细地介绍了梁中产生的这些弯曲和剪切应力。

Shear stresses for several common cross sections are discussedin the sections below.

以下各节将讨论几种常见截面的剪切应力。

Shear Stresses in Rectangular Sections

矩形截面中的剪切应力

The distribution of shear stress along the height of arectangularcross section is shown in the figurebelow:

沿矩形横截面高度的剪应力分布如下图所示：

The first moment of area at any given pointy1alongthe height of the cross section is calculated by:

沿横截面高度的任何给定点 y1处的第一弯矩面积计算公式为：

The maximum value ofQ occurs at the neutralaxis of the beam (where y1= 0):

Q 的最大值出现在梁的中性轴处（其中 y1= 0）：

The shear stress at any given pointy1alongthe height of the cross section is calculated by:

沿横截面高度的任何给定点 y1处的剪切应力计算公式为：

whereIc=b·h3/12 is the centroidal moment of inertiaof the cross section. The maximum shear stress occurs at the neutral axis ofthe beam and is calculated by:

其中 Ic= b·h3/12是横截面的质心惯性矩。最大剪应力发生在梁的中性轴上，计算公式为：

whereA= b·h is the area of the cross section.

其中 A=b·h是横截面的面积。

We can see from the previous equation that the maximum shearstress in the cross section is 50% higher than the average stressV/A.

从前面的方程中可以看出，截面中的最大剪应力比平均应力V/A高50%。

Shear Stresses in Circular Sections

圆形截面中的剪切应力

Acircularcross section is shown in the figurebelow:

圆形横截面如下图所示：

The equations for shear stress in a beam were derived using theassumption that the shear stress along the width of the beam is constant. Thisassumption is valid at the centroid of a circular cross section, although it isnot valid anywhere else. Therefore, while the distribution of shear stressalong the height of the cross section cannot be readily determined, the maximumshear stress in the section (occurring at the centroid) can still becalculated. The maximum value of first moment,Q, occurring at the centroid,is given by:

梁中剪应力方程的推导是假设沿梁宽度的剪应力是恒定的。这个假设在圆形横截面的质心上是有效的，尽管它在其他任何地方都无效。因此，虽然不能轻易确定沿截面高度的剪应力分布，但仍可以计算截面中的最大剪应力（发生在质心处）。发生在质心处的第一个矩 Q 的最大值由下式给出：

The maximum shear stress is then calculated by:

最大剪应力的计算公式如下：

whereb= 2r is the diameter (width) of the cross section, Ic= πr4/4 is the centroidal moment of inertia, and A = πr2isthe area of the cross section.

其中 b=2r 是横截面的直径（宽度），Ic=πr4/4 是质心惯性矩，A =πr2是横截面的面积。

Shear Stresses in Circular Tube Sections

圆管截面的剪切应力

Acirculartube cross section is shown in the figurebelow:

圆管横截面如下图所示：

The maximum value of first moment,Q, occurring at the centroid,is given by:

发生在质心处的第一个矩 Q 的最大值由下式给出：

The maximum shear stress is then calculated by:

最大剪应力的计算公式如下：

whereb= 2 (ro− ri) isthe effective width of the cross section, Ic= π (ro4−ri4) / 4 is the centroidal momentof inertia, and A= π (ro2− ri2) is the area of the cross section.

其中 b=2（ro−ri）是横截面的有效宽度，Ic=π（ro4−ri4）/4 是质心惯性矩，A =π（ro2− ri2）是横截面的面积。

Shear Stresses in I-Beams

工字钢中的剪应力

The distribution of shear stress along the web of anI-Beam is shown in the figure below:

沿工字梁腹板的剪应力分布如下图所示：

The equations for shear stress in a beam were derived using theassumption that the shear stress along the width of the beam is constant. Thisassumption is valid over the web of an I-Beam, but it is invalid for theflanges (specifically where the web intersects the flanges). However, the webof an I-Beam takes the vast majority of the shear force (approximately 90% -98%, according to Gere), and so it can be conservatively assumed that the web carriesall of the shear force.

梁中剪应力方程的推导是假设沿梁宽度的剪应力是恒定的。此假设在工字梁的腹板上有效，但对于法兰（特别是腹板与法兰相交的地方）无效。然而，工字梁的腹板承受了绝大部分的剪切力（根据Gere的说法，大约为90%-98%），因此可以保守地假设腹板承载了所有的剪力。

The first moment of the area of thewebof anI-Beam is given by:

工字梁腹板面积的第一矩由下式给出：

The shear stress along the web of the I-Beam is given by:

工字梁腹板上的剪应力由下式给出：

wheretwis the web thickness and Icisthe centroidal moment of inertia of the I-Beam:

其中 tw是腹板厚度，Ic是工字梁的质心惯性矩：

The maximum value of shear stress occurs at the neutral axis(y1= 0 ), and the minimum value of shear stress in the web occursat the outer fibers of the web where it intersects the flanges y1= ±hw/2 ):

剪切应力的最大值出现在中性轴（y1=0），腹板中剪切应力的最小值出现在腹板的外纤维处，它与法兰相交 y1= ±hw/2）：