家 » Principal and Maximum Shear Stresses (Mohr’s Circle)

Principal and Maximum Shear Stresses (Mohr’s Circle)

1882-01-01

Christian Otto Mohr

The principal stresses, [latex]\sigma_1[/latex] and [latex]\sigma_2[/latex], are the maximum and minimum normal stresses at a point, occurring on planes with zero shear stress. On Mohr’s circle, these correspond to the two points where the circle intersects the horizontal ([latex]\sigma_n[/latex]) axis. The maximum in-plane shear stress, [latex]\tau_{max}[/latex], is equal to the radius of the circle, [latex]R[/latex].

Identifying principal stresses and maximum shear 强调 is a primary application of Mohr’s circle. The principal stresses are the eigenvalues of the stress tensor and represent the extreme values of normal stress. They are found at the intersections of the circle with the [latex]\sigma_n[/latex] axis, calculated as [latex]\sigma_{1,2} = \sigma_{avg} \pm R[/latex], where [latex]\sigma_{avg}[/latex] is the center of the circle and [latex]R[/latex] is its radius. The planes on which these stresses act are called principal planes, and they are mutually orthogonal. On Mohr’s circle, the angle [latex]2\theta_p[/latex] from the reference state to the principal state can be found using trigonometry: [latex]\tan(2\theta_p) = \frac{2\tau_{xy}}{\sigma_x – \sigma_y}[/latex].

The maximum in-plane shear stress, [latex]\tau_{max}[/latex], corresponds to the highest and lowest points on the circle, with a magnitude equal to the circle’s radius, [latex]R[/latex]. The planes of maximum shear are oriented at 45 degrees to the principal planes. This is visually represented on the circle by a 90-degree rotation from the principal stress points. Understanding these maximum values is critical in engineering design, as material failure, particularly in ductile materials, is often initiated by shear stress. Failure theories, such as the Tresca (Maximum Shear Stress) criterion, directly use this value to predict the onset of yielding.

可靠性设计（DfR）, 故障分析, 故障模式和影响分析（FMEA）, 材料, 材料科学, 机械工业, 机械性能, 应力腐蚀, 结构工程

UNESCO Nomenclature: 3328

– Materials science and engineering

类型

抽象系统

中断

实质性

使用方法

广泛使用

前体

Rankine’s theory of earth 压力
Cauchy’s stress tensor
Navier’s equations of motion for elastic solids
The concept of eigenvalues and eigenvectors in linear algebra

应用

failure analysis of materials (e.g., tresca and von mises yield criteria)
design of pressure vessels and pipes
structural analysis of bridges and buildings
geotechnical engineering for slope stability analysis

专利：

潜在的创新想法

级别需要会员

您必须是！！等级！！会员才能访问此内容。

立即加入

已经是会员？在此登录

Related to: principal stress, maximum shear stress, Mohr’s circle, stress analysis, failure criteria, Tresca criterion, solid 力学, material science, structural design, normal stress.