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z`PV%uSVOVxH!WYnL)yj-(Mhr-(NdqS3cU)s_<2vGB~{=$>afy_e58qed2xWE>798qF5UKHG(xPK=NXj? zLX=iQa7PJ=_$yNS71%3=$J%IKFuF3;Y&Jw2D6^-IEgH`3hg{y}P<#AgA{SP;4#R${ zP{Y0<`{t%NX?;Ho)0RsAok5!=6{Fd{SDh7mf>X`l!vq2CFIydL_8Je5=@;%;-i;%! zf@dEClL~!o-{1LPY#rv||NC~rpVa`%;eTQWW$(_N>0y?^b^cH2TS`58`Y_}E^LPIT D%+GVG literal 0 HcmV?d00001 diff --git a/src/pages/sol/shear_&_moment_diagrams.astro b/src/pages/sol/shear_&_moment_diagrams.astro index 5aefd3a43..7faf132f0 100644 --- a/src/pages/sol/shear_&_moment_diagrams.astro +++ b/src/pages/sol/shear_&_moment_diagrams.astro @@ -37,10 +37,10 @@ To determine how the internal shear force and bending moment change throughout t Sign conventions for internal forces and moments Sign conventions for internal forces and moments in 3D - + Sign conventions for internal forces and moments + diff --git a/src/pages/sol/transverse_shear.astro b/src/pages/sol/transverse_shear.astro index b64863ec4..3c7098509 100644 --- a/src/pages/sol/transverse_shear.astro +++ b/src/pages/sol/transverse_shear.astro @@ -118,10 +118,10 @@ Shear stress can also be evaluated at any given point ( - : shear force + : Shear force : First area moment of inertia of cut section ( ) : Second area moment of inertia of whole cross-sectional area - : beam thickness + : Width of cross section at given point From 13daa22002201c5f17f74f6b9b0f32c5e9b4145f Mon Sep 17 00:00:00 2001 From: mihikad2 Date: Tue, 3 Feb 2026 11:02:11 -0600 Subject: [PATCH 2/7] Page updates --- src/pages/sol/axial_loading.astro | 2 +- src/pages/sol/bending.astro | 2 +- src/pages/sol/design_considerations.astro | 117 +++++++++++++++++++- src/pages/sol/geometric_properties.astro | 6 +- src/pages/sol/shear_&_moment_diagrams.astro | 3 - src/pages/sol/sign_conventions.astro | 5 + src/pages/sol/torsion.astro | 2 +- src/pages/sol/transverse_shear.astro | 2 +- 8 files changed, 129 insertions(+), 10 deletions(-) diff --git a/src/pages/sol/axial_loading.astro b/src/pages/sol/axial_loading.astro index fdccf1183..44db63bc9 100644 --- a/src/pages/sol/axial_loading.astro +++ b/src/pages/sol/axial_loading.astro @@ -29,7 +29,7 @@ import DisplayEquation from "../../components/DisplayEquation.astro"
- +

For a scenario with only axial loading conditions, the stress tensor can be simplified as shown below.

diff --git a/src/pages/sol/bending.astro b/src/pages/sol/bending.astro index 09d97e3e3..5079772d0 100644 --- a/src/pages/sol/bending.astro +++ b/src/pages/sol/bending.astro @@ -49,7 +49,7 @@ A beam in bending will develop normal (tensile and compressive) bending stresses - +

For pure bending loading conditions, the stress tensor can be simplified as shown below, depending on whether bending occurs around the y or z axis.

diff --git a/src/pages/sol/design_considerations.astro b/src/pages/sol/design_considerations.astro index 101f36dfb..3a16f54f1 100644 --- a/src/pages/sol/design_considerations.astro +++ b/src/pages/sol/design_considerations.astro @@ -9,6 +9,7 @@ import Item from "../../components/Item.astro" import InlineEquation from "../../components/InlineEquation.astro" import DisplayEquation from "../../components/DisplayEquation.astro" import BlueText from "../../components/BlueText.astro" +import Example from "../../components/Example.astro" --- @@ -24,7 +25,121 @@ import BlueText from "../../components/BlueText.astro"
- + +

+Broadly speaking, the goal of a design is to prevent unwanted behaviors (or +failure modes) from occurring. There can be, and usually are, many different +failure modes which need to be considered simultaneously. +

+ +

+For example, consider just 2 plates bolted together and loaded to failure: + + + Example 1 + Example 2 + Example 3 + This one simple connection shows three different modes of failure: tension, shear & bolt failures. + +

+ +

+It is the job of the engineer to anticipate and design for all relevant failure modes.For a given design, successful operation may require: + + Stresses stay below material limits (see Von Mises failure criteria) + Stress from cyclic loading stays below the fatigue limit (see ME370 page) + Buckling is prevented (see buckling page) + Deformation stays below a prescribed limit (see beam deflection page) + Vibration deformation from dynamic loads stays below a prescribed limit + And more +

+ +

+Any way a product or component can fail to meet the desired performance can be referred to as a failure mode. These are occasionally categorized as + + Strength: exceeding the limit state results in material failure + Stability: exceeding the limit state results in a stability failure (like + buckling or + overturning) + Serviceability: exceeding the limit state results in compromised performance (excessive deflection) + +

+ +

+It can be very difficult to anticipate every possible failure mode. This broad view +of component performance is discussed in ME270 (Design) as “product design +specifications” https://www.mechref.org/design/pds/?origin=coursemenu) In solid +mechanics, we’ll focus on the strength failure modes. + +

+Note that the nomenclature can vary between industries, but the intent is the +same, to identify potential failures and design to prevent them, or reduce the risk +(probability x severity) to an acceptably low level. +

+ + +
+

+ + Automotive +

+ + Design Failure Mode and Effects Analysis: + Design FMEA is a type of FMEA that analyzes the product design, focusing on potential design-related + deficiencies, with emphasis on improving the design and ensuring product operation is safe and reliable + during useful life. +
+ + SAE J1739_202101: + https://saemobilus-sae-org.proxy2.library.illinois.edu/standards/j1739_202101 + -potential-failure-mode-effects-analysis-fmea-including-design-fmea- + supplemental-fmea-msr-process-fmea#view +
+ + Aerospace +
+ + Failure Mode and Effects Analysis (FMEA): + A procedure by which each potential failure mode or fault of a system is analyzed to determine the + consequences or effects thereof on the system, to classify each potential failure mode according to its + severity, and to recommend actions to eliminate, or compensate for, unacceptable effects. +
+ + SAE ARP5580: + https://saemobilus-sae-org.proxy2.library.illinois.edu/standards/arp5580- + recommended-failure-modes-effects-analysis-fmea-practices-non-automobile- + applications#view +
+ + Civil +
+ + Limit State: + A condition beyond which a structure or member becomes unfit for service and is judged either to be no + longer useful for its intended function (serviceability limit state) or to be unsafe (strength limit state). +
+ + ASCE7: + https://ascelibrary-org.proxy2.library.illinois.edu/doi/epdf/10.1061/9780784415 + 788 +
+ + US Government +
+ + Failure mode and effects analysis (FMEA): + A procedure by which each potential failure mode in a system is analyzed to determine the results or + effects thereof on the system and to classify each potential failure mode according to its severity. +
+ + MIL-STD-1629A: + https://www.dsiintl.com/wp-content/uploads/2017/04/mil_std_1629a.pdf + +

+
+
+ +
diff --git a/src/pages/sol/geometric_properties.astro b/src/pages/sol/geometric_properties.astro index 67f1170ed..09cea01cf 100644 --- a/src/pages/sol/geometric_properties.astro +++ b/src/pages/sol/geometric_properties.astro @@ -23,6 +23,7 @@ import DisplayTable from "../../components/DisplayTable.astro"
  • Centroid
  • Area Moment of Inertia
  • Polar Moment of Inertia
  • +
  • Simple Shapes Moment of Inertia
  • Summary: Moment of Inertia
  • @@ -97,7 +98,7 @@ The moment of inertia of the area with respect t
    - +
    Common shapes about the origin: and
    Common shapes about the centroid: and
    @@ -226,8 +227,9 @@ The moment of inertia of the area with respect t - +
    + diff --git a/src/pages/sol/shear_&_moment_diagrams.astro b/src/pages/sol/shear_&_moment_diagrams.astro index 7faf132f0..aa6ceaef1 100644 --- a/src/pages/sol/shear_&_moment_diagrams.astro +++ b/src/pages/sol/shear_&_moment_diagrams.astro @@ -38,9 +38,6 @@ To determine how the internal shear force and bending moment change throughout t Sign conventions for internal forces and moments Sign conventions for internal forces and moments in 3D -For a profile view depiction of a 3D force and moment diagram, the following convention is used for notating arrows going into or out of the page. - Sign conventions for internal forces and moments - diff --git a/src/pages/sol/sign_conventions.astro b/src/pages/sol/sign_conventions.astro index 706ce0788..9f0bededd 100644 --- a/src/pages/sol/sign_conventions.astro +++ b/src/pages/sol/sign_conventions.astro @@ -34,6 +34,7 @@ import BlueText from "../../components/BlueText.astro" Positive shear stress points towards the negative axis direction in a negative face. +
    @@ -87,6 +88,10 @@ These conventions assume the analysis is conducted on a horizontal beam with pos +For a profile view depiction of a 3D force and moment diagram, the following convention is used for notating arrows going into or out of the page. + Sign conventions for internal forces and moments + + \ No newline at end of file diff --git a/src/pages/sol/torsion.astro b/src/pages/sol/torsion.astro index d80ef33d2..43db3ff10 100644 --- a/src/pages/sol/torsion.astro +++ b/src/pages/sol/torsion.astro @@ -47,7 +47,7 @@ import Enumerate from "../../components/Enumerate.astro" - +

    For pure torsion loading conditions, the stress tensor can be simplified as shown below, depending on whether is in the y or z direction for torsion calculations.

    diff --git a/src/pages/sol/transverse_shear.astro b/src/pages/sol/transverse_shear.astro index 3c7098509..6971a7070 100644 --- a/src/pages/sol/transverse_shear.astro +++ b/src/pages/sol/transverse_shear.astro @@ -39,7 +39,7 @@ Transverse loading creates corresponding longitudinal shear stresses - +

    For pure transverse shear loading conditions, the stress tensor can be simplified as shown below, depending on whether shearing occurs along the y or z axis.

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    Introduction/Big Picture:

    - +
    +
      +
    1. Introduction
    2. +
    +

    Material Behavior:

    diff --git a/src/pages/sol/axial_bending.astro b/src/pages/sol/axial_bending.astro deleted file mode 100644 index fab81099f..000000000 --- a/src/pages/sol/axial_bending.astro +++ /dev/null @@ -1,33 +0,0 @@ ---- -import Layout from "../../layouts/Layout.astro" -import Image from "../../components/Image.astro" -import Section from "../../components/Section.astro" -import SubSection from "../../components/SubSection.astro" -import SubSubSection from "../../components/SubSubSection.astro" -import Itemize from "../../components/Itemize.astro" -import Item from "../../components/Item.astro" -import InlineEquation from "../../components/InlineEquation.astro" -import DisplayEquation from "../../components/DisplayEquation.astro" -import BlueText from "../../components/BlueText.astro" - - ---- - - - - -
    - - -
    \ No newline at end of file diff --git a/src/pages/sol/intro.astro b/src/pages/sol/intro.astro new file mode 100644 index 000000000..8b1f15e14 --- /dev/null +++ b/src/pages/sol/intro.astro @@ -0,0 +1,95 @@ +--- +import Layout from "../../layouts/Layout.astro" +import Image from "../../components/Image.astro" +import Section from "../../components/Section.astro" +import SubSection from "../../components/SubSection.astro" +import SubSubSection from "../../components/SubSubSection.astro" +import Itemize from "../../components/Itemize.astro" +import Item from "../../components/Item.astro" +import InlineEquation from "../../components/InlineEquation.astro" +import DisplayEquation from "../../components/DisplayEquation.astro" +import BlueText from "../../components/BlueText.astro" + + +--- + + + + +
    + + +There are many fields of mechanics, the intent of this reference page is not to cover every one of them. It is important however to be aware of the relationships between the sub-field “Solid Mechanics” and mechanics as a whole. The figure below attempts to organize some of the fields included in the broad category of “mechanics” + + + +

    +Many introductory courses with similar scope use the name “mechanics of materials”, since the learning objectives are based on understanding the material level behavior. The distinction made here for “introductory solid mechanics” is to include interpretation of material level behavior at the component scale. The idea being to connect the behavior of materials to the behavior of components. What is a ‘component’? Broadly speaking, it could be any “engineering potato” that needs to be designed or evaluated. +Frequently the components engineers design can be interpreted as “beams” in that they have one dimension significantly longer than the other two, and usually a constant cross sectional shape along this length. Introductory solid mechanics takes the material level behavior and ‘accumulates’ into component level using some assumptions specific to beams (and pressure vessels). +

    + +

    +This ‘accumulation’ of material level behaviors can be thought of in terms of force quantities or deformation quantities. (the two columns of the image below) +

    + + Big Picture of Solid Mechanics + +

    +Continuing from where a statics prerequisite left off, the idea of internal force in a beam describes the component level, but at the material level, this force is experienced as stress. Each point within the component experiences some material scale stress. Because materials are not infinitely stiff, they deform as a result of this stress. This deformation is described by the constitutive relations, i.e. the material scale relationship between force and displacement quantities (stress and strain). The material deformation, strain, everywhere within a component can then be accumulated into component level displacement, or deformation. The remaining connection between component level force and deformation, is the component “stiffness”. A quantity derived from three fundamental relationships. This relationship visualized in the tonti diagram above shows three fundamental relationships of introductory mechanics: + + Equilibrium (between force and stress) + Constitutive relations (between stress and strain) + Compatibility (accumulation of strain into displacement) + +

    + +

    +In a statics prerequisite course, analysis of internal forces was limited to ‘determinate’ structures. That is, only equilibrium was needed to determine component reactions and internal forces. (Determinate implies there is only one load path, so the forces carried in that load path are independent of any deformations, which is why statics does not consider deformation.) In many applications, system designs are indeterminate. That is, there are multiple load paths, or load carrying mechanisms within a design. In these systems, equilibrium alone is insufficient. What is necessary are the 3 items listed above, equilibrium, constitutive relations and compatibility. The internal forces in an indeterminate structure must satisfy all three. Introductory solid mechanics uses these three relationships as applied to beam behavior to solve indeterminate problems. +

    + +

    +Not all engineering problems can be idealized into a “beam-like” component. However, the same three relationships above can be used to approximate the responses of more complex systems. These problems can be characterized as ‘boundary value problems’ in that we typically know some conditions on the boundaries such as displacement constraints or applied loads, and typically want to solve for results within the structure, such as stress or strain. Evaluating the three relationships in a 1D, 2D or 3D domain results in a differential equation relating forces and displacements. (An ODE if quantities are only dependent on spatial derivatives, a PDE if they are also dependent on temporal derivatives, but that is dynamics, which is not part of the scope of introductory mechanics) The approximations made in idealizing components as ‘beams’ are exactly the same approximations made when defining “beam elements” in a finite element software. In some sense, the 1D or beam idealizations covered in introductory solid mechanics can be considered an introduction into finite element methods, which also includes 2D and 3D methods for solving boundary value problems with the same three fundamental relationships. +

    + +
    + + +

    +The scope of this class is intended to cover the fundamental relationships in a way that establishes an understanding of classical solid mechanics. There are many practical considerations outside the scope of this class that we introduce but do not fully investigate. (plasticity, nonlinearity, fatigue…. ) +The intent here is to make sure the assumptions made in idealization are made explicit/conscious/deliberate, so that when these conditions are no longer met in a design scenario, it is clear that relationships depending on those conditions are no longer valid. +

    + +

    +The intent here is to make sure the assumptions made in idealization are made explicit/conscious/deliberate, so that when these conditions are no longer met in a design scenario, it is clear that relationships depending on those conditions are no longer valid. +

    + +

    +Introductory solid mechanics begins with an introduction to stress and strain (2.1 & 2.2) (i.e. force and displacement quantities at the material scale). The concepts of stress and strain as tensor quantities is also introduced. Then the relationship between these is covered in section 2.3, along with a broad overview of other material behaviors beyond linear elasticity. Because many engineering tasks are aimed at understanding and designing to avoid failure, section 2.4 and 2.5 discuss material failure theories, and design philosophies respectively. +

    + +

    +Section 3 is focused on linear elastic relationships for different loading conditions as applied to 1D components (beams). The intent here is to develop the three fundamental relationships for these loading conditions as applied to beam behavior. Each loading condition is explored individually. In terms of how externally applied actions (like forces & moments) are balanced by internal stresses, and how material strains are accumulated into component deformations. +

    + +

    +Section 4 extends to the application of the concepts developed in Section 3 to multiple loading conditions occurring simultaneously. These include shear and moment diagrams in beams, beam buckling, pressure vessels etc. +

    +
    + + +Relationship between internal stresses and deformations produced by external forces acting on deformable bodies, and design principles based on mechanics of solids: normal stresses, shear stresses, and deformations produced by tensile, compressive, torsional, and bending loading of members; beam deflections; elastic energy and impact; multi-dimensional stress states; buckling of columns. + + +
    \ No newline at end of file From 9bd218c7ce4a423b7f27f3d99128c0f3059d31c4 Mon Sep 17 00:00:00 2001 From: mihikad2 Date: Wed, 4 Feb 2026 10:08:18 -0600 Subject: [PATCH 4/7] Update design_considerations.astro --- src/pages/sol/design_considerations.astro | 11 ++++++----- 1 file changed, 6 insertions(+), 5 deletions(-) diff --git a/src/pages/sol/design_considerations.astro b/src/pages/sol/design_considerations.astro index 3a16f54f1..8d5194cd3 100644 --- a/src/pages/sol/design_considerations.astro +++ b/src/pages/sol/design_considerations.astro @@ -17,7 +17,7 @@ import Example from "../../components/Example.astro"
    - +

    Broadly speaking, the goal of a design is to prevent unwanted behaviors (or failure modes) from occurring. There can be, and usually are, many different @@ -162,9 +162,6 @@ A structure is safe if its strength exceeds the required strength. Similarly, - - -

    Depending on the industry or application, the acceptability of a design may be interpreted differently. For some applications showing a factor of safety of 4 may be sufficient. @@ -173,6 +170,10 @@ For example, a sling used for lifting, such as the one shown below, is specified Web sling + + + +

    In the civil/structural engineering industry, the variability of loading (wind, seismic, etc.) has been studied extensively. This is also true for various structural capacities in building materials like steel, concrete wood, etc. Given a loading and resistance that are both random variables, a simple “safety factor” does not result in a consistent level of probability of failure across all structures. From 0522df7e607fcf259f639e4f1c12cb93559c4a69 Mon Sep 17 00:00:00 2001 From: mihikad2 Date: Wed, 4 Feb 2026 10:20:00 -0600 Subject: [PATCH 5/7] Update intro.astro --- src/pages/sol/intro.astro | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/pages/sol/intro.astro b/src/pages/sol/intro.astro index 8b1f15e14..13e261db2 100644 --- a/src/pages/sol/intro.astro +++ b/src/pages/sol/intro.astro @@ -31,7 +31,7 @@ import BlueText from "../../components/BlueText.astro"

    -There are many fields of mechanics, the intent of this reference page is not to cover every one of them. It is important however to be aware of the relationships between the sub-field “Solid Mechanics” and mechanics as a whole. The figure below attempts to organize some of the fields included in the broad category of “mechanics” +There are many fields of mechanics, the intent of this reference page is not to cover every one of them. It is important however to be aware of the relationships between the sub-field “Solid Mechanics” and mechanics as a whole. The figure below attempts to organize some of the fields included in the broad category of “mechanics”. @@ -65,7 +65,7 @@ Not all engineering problems can be idealized into a “beam-like” component. - +

    The scope of this class is intended to cover the fundamental relationships in a way that establishes an understanding of classical solid mechanics. There are many practical considerations outside the scope of this class that we introduce but do not fully investigate. (plasticity, nonlinearity, fatigue…. ) The intent here is to make sure the assumptions made in idealization are made explicit/conscious/deliberate, so that when these conditions are no longer met in a design scenario, it is clear that relationships depending on those conditions are no longer valid. From 04328f87404ae778accc26d28b7aa35ae64c3ae0 Mon Sep 17 00:00:00 2001 From: mihikad2 Date: Wed, 18 Feb 2026 01:53:06 -0600 Subject: [PATCH 6/7] Updates --- src/components/Navbar.astro | 8 +-- src/pages/sol/design_considerations.astro | 70 +++++++++++------------ src/pages/sol/sign_conventions.astro | 13 +++-- 3 files changed, 45 insertions(+), 46 deletions(-) diff --git a/src/components/Navbar.astro b/src/components/Navbar.astro index 128fc8ef7..6f4181d96 100644 --- a/src/components/Navbar.astro +++ b/src/components/Navbar.astro @@ -221,7 +221,7 @@ const path = Astro.url.pathname; } .solids-menu{ - columns: 2; + columns: 1; } @media (width <= 578px){ @@ -426,7 +426,7 @@ const path = Astro.url.pathname;