From 76fcb343758c14322b78613831b16e143ca6b21e Mon Sep 17 00:00:00 2001 From: mihikad2 Date: Wed, 10 Sep 2025 20:45:13 -0500 Subject: [PATCH 1/3] Rearranging 251 pages --- src/pages/sol.astro | 157 +++--- src/pages/sol/appendix.astro | 112 +++++ ...ling.astro => design_considerations.astro} | 0 ...ro => failure_engineering_materials.astro} | 0 src/pages/sol/intro.astro | 48 ++ src/pages/sol/material_properties.astro | 447 +++++++++--------- ...ial_loading.astro => simple_loading.astro} | 0 ...> statically_indeterminate_problems.astro} | 0 src/pages/sol/strain.astro | 143 +++--- ...ansformation.astro => superposition.astro} | 4 +- src/pages/sol/thermal_loading.astro | 48 ++ 11 files changed, 566 insertions(+), 393 deletions(-) create mode 100644 src/pages/sol/appendix.astro rename src/pages/sol/{buckling.astro => design_considerations.astro} (100%) rename src/pages/sol/{beam_deflection.astro => failure_engineering_materials.astro} (100%) create mode 100644 src/pages/sol/intro.astro rename src/pages/sol/{axial_loading.astro => simple_loading.astro} (100%) rename src/pages/sol/{failure_theories.astro => statically_indeterminate_problems.astro} (100%) rename src/pages/sol/{stress_transformation.astro => superposition.astro} (99%) create mode 100644 src/pages/sol/thermal_loading.astro diff --git a/src/pages/sol.astro b/src/pages/sol.astro index 538d343d9..b7549f39e 100644 --- a/src/pages/sol.astro +++ b/src/pages/sol.astro @@ -14,21 +14,12 @@ import "../../public/static/css/course_home_pages.css"
- + + +
  • Units
    • @@ -55,6 +46,23 @@ import "../../public/static/css/course_home_pages.css"
+ + + + + - + + +
  • Units
    • @@ -111,6 +109,7 @@ import "../../public/static/css/course_home_pages.css"
+ +
  • Sign Conventions
    • @@ -137,6 +137,7 @@ import "../../public/static/css/course_home_pages.css"
+
  • Built-up Members/Beams: Shear Flow
    • +     + + + - + + - - + + + - + + - + + - + + - + + -
  • Stress
    • +
  • Intro
  • Strain
    • +
  • Stress
  • Material properties
    • -
  • Axial loading
    • -
  • Torsion
    • -
  • Shear moment diagrams
    • -
  • Bending
    • -
  • Transverse shear
    • -
  • Pressure vessels
    • +
  • Simple loading
  • Combined loading
    • -
  • Stress transformation
    • -
  • Failure theories
    • -
  • Beam deflection
    • -
  • Buckling
    • +
  • Principle of Superposition
    • +
  • Statically Indeterminate Problems
    • +
  • Failure of Engineering Materials
    • +
  • Design Considerations
    • +
  • Appendix
    •
    diff --git a/src/pages/sol/appendix.astro b/src/pages/sol/appendix.astro new file mode 100644 index 000000000..8c0020805 --- /dev/null +++ b/src/pages/sol/appendix.astro @@ -0,0 +1,112 @@ +--- +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 Row from "../../components/Row.astro" +import Col from "../../components/Col.astro" +import CalloutCard from "../../components/CalloutCard.astro" +import CalloutContainer from "../../components/CalloutContainer.astro" +import Warning from "../../components/Warning.astro" +--- + + + + + +
    + +
    + +
    + + + + +Dimensionless [mm/mm, in/in, or %] + + + + +The units of stress can be found in the table below. + + + + + + + + + + + + + + + + + + + + + + + + + + +
    MeasureSI unitsImperial units
    Force[\(\rm N\)][\(\rm lb\)]
    Area[\(\rm m^2\)][\(\rm in^2\)]
    Stresspascal = [\(\rm Pa\)] = [\(\rm N/m^2\)]pound per square inch = [\(\rm psi\)] = [\(\rm lb/in^2\)]
    + + It is also important to recall some of the most common prefixes used to denote quantity. + + + + + + + + + + + + + + + + +
    kilopascal [\(\rm kPa\)]\(10^3\) [\(\rm Pa\)]
    megapascal [\(\rm MPa\)]\(10^6\) [\(\rm Pa\)]
    gigapascal [\(\rm GPa\)]\(10^9\) [\(\rm Pa\)]
    + +Typical sign convention for axial (normal) stress: + + Tension: + Compression: + + +     + +     diff --git a/src/pages/sol/buckling.astro b/src/pages/sol/design_considerations.astro similarity index 100% rename from src/pages/sol/buckling.astro rename to src/pages/sol/design_considerations.astro diff --git a/src/pages/sol/beam_deflection.astro b/src/pages/sol/failure_engineering_materials.astro similarity index 100% rename from src/pages/sol/beam_deflection.astro rename to src/pages/sol/failure_engineering_materials.astro diff --git a/src/pages/sol/intro.astro b/src/pages/sol/intro.astro new file mode 100644 index 000000000..077662b49 --- /dev/null +++ b/src/pages/sol/intro.astro @@ -0,0 +1,48 @@ +--- +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 Row from "../../components/Row.astro" +import Col from "../../components/Col.astro" +import CalloutCard from "../../components/CalloutCard.astro" +import CalloutContainer from "../../components/CalloutContainer.astro" +import Warning from "../../components/Warning.astro" +--- + + + + + +
    + +
    + +
    + +     diff --git a/src/pages/sol/material_properties.astro b/src/pages/sol/material_properties.astro index 612c6ce6e..7ac2fcd65 100644 --- a/src/pages/sol/material_properties.astro +++ b/src/pages/sol/material_properties.astro @@ -28,13 +28,12 @@ import Warning from "../../components/Warning.astro"
    -
    - - - - - -

    - If you want more details on this topic, consider taking a course in continuum mechanics. -

    -     - - -

    - Bold symbols, such as and , - represent tensors rather than scalar quantities. -

    - -

    - Additionally, some quantities might be represented with indicial notation. When indices on the right hand side of the equation are repeated, - this represents a summation where the index values range from 1 to 3. - For example, - represents - - This is a shorthand to make writing long summations easier. -

    -     - -     - -

    -Generally, the state of stress a 3D material can be represented as a second-rank tensor (i.e. a matrix) with components. -

    - Infinitesimal Stress element - - -

    -Using indicial notation, component of this tensor is denoted as . The same can be said for a material strain tensor. -

    - - Infinitesimal Strain Element - - -

    - -Using indicial notation, component of this tensor is denoted as . Note -that both the stress and strain tensors are symmetric, that is and -. - -

    - -

    - -For our purposes, lets bound the discussion of material stress-strain behavior to only the elastic range, that is, deformation is linear proportional applied load. -This is Hooke’s law. The relationship between stress and strain is referred to as the constitutive relationship of a material. In the most general case, this relationship is defined as a fourth-rank tensor with components. -This elasticity tensor, C, relating the stress and strain tensors is shown in indicial notation (rather than a matrix operation). +

    -

    +ADD CONTENT ABOUT ASSUMPTIONS - - This is short form representation of a summation, known as indicial notation. Whenever there are repeated indices on the right hand side of an equation, - as with k and l above (repeated in and ), these indices represent - a summation over three dimensions. So the example can be rewritten as shown below, where and . - - - +
    -Expanding over these two summations yields: - -

    +

    -What this means is there can be as many as 9 components of strain and 9 material coefficients that contribute to each stress component. -(You can see the benefit of indicial notation here.) + -

    +Loading in this region results in elastic behavior, meaning the material returns to its original shape when unloaded. This region of the diagram is mostly a straight line, limited by the proportional limit. This region ends at (yielding). For , the diagram is linear, and the behavior is elastic. For , the diagram is nonlinear, but the behavior is still elastic. -

    -However, not all 81 components of this tensor are independent, because of symmetry of stress and strain tensors there are only 21 independent components. If -the material is assumed to be isotropic (having the same behavior in all directions) this is further reduced to just 2 independent components (for instance, - Young’s modulus and , Poisson’s ratio). With this simplification, the 3D isotropic elastic material constitutive relation can be simplified. +Hooke's law is used for small deformations in the elastic region. The Young's modulus (also known as the elastic modulus) is the slope . - - -

    - -Here the engineering shear strains are introduced. -This, along with identification of shear stresses -as -allows for the expression for pure axial stress & strain in one direction to be represented as . -Rearranging the terms, relation can be expressed in a more general form. - - - -Similarly, the expression for pure shear stress & strain can be expressed as . - -Again, this can be expressed in a more general form. - -

    - -

    - -     - - - -A stress-strain diagram is the relationship of normal stress as a function of normal strain. One way to collect these measurement is a uniaxial tension test in which a specimen at a very slow, constant rate (quasi-static). A load and distance are measured at frequent intervals. - - - - - - -

    - Things that effect the material properties include imperfections, different composition, rate or loading, or temperature. -

    -     - - Stress-strain diagram showing key properties -     - - -Loading in this region results elastic behavior, meaning the material returns to its original shape when unloaded. This region of the diagram is mostly a straight line, limited by the proportional limit). This region ends at (yielding). For , the diagram is linear, and the behavior is elastic. For , the diagram is nonlinear, but the behavior is still elastic. - - - Young's Modulus - - - - Hooke's law is used for small deformations in the elastic region. The Young's modulus is the slope . @@ -226,8 +90,9 @@ Loading in this region results elastic behavior, meaning the ma
    +     -Shear Modulus + Shear stress strain diagram @@ -235,18 +100,78 @@ Loading in this region results elastic behavior, meaning the ma -In the above equation, \(\nu\) refers to Poisson's ratio, further explained in a subsequent section. Only two of the three material constants (ie; , , ) are independent in isotropic materials. +In the above equation, \(\nu\) refers to Poisson's ratio. Only two of the three material constants (ie; , , ) are independent in isotropic materials. - + - + -Stresses above the plastic limit () cause the material to permanently deform. +Poisson's ratio

    - Yield Strength + As illustrated in the figure above, when an object is being compressed it expands outward, and when stretched it will become thinner. Depending on the material, it may experience more or less lateral deformation for a given amount of axial deformation. The ratio that governs the amount of lateral strain per unit of axial strain is called Poisson's ratio.

    + + + + + +

    Consider a cylinder with volume . We know that . Since the diameter is dependant on the change in length \(dx\) due to Poisson's ratio, we get

    + +

    We use the chain rule to find the change in volume

    + + + +

    + By definition, +

    + + + +

    + Returning to our first equation, +

    + + + +

    + Since \(\partial V \geq 0\) and \(\nu > 0\) in nicely behaved materials, \( 0 < \nu \leq 0.5\). In practice \(\nu = 0.5\) only if the material is incompressible, thus in most materials \(\nu < 0.5\). +

    + +     + + + + +

    + A negative Poisson's ratio is possible when the material structure is non-trival. +

    +     + +     + +
    + + + +ADD CONTENT ABOUT DUCTILITY + + + + + +Stresses above the plastic limit () cause the material to permanently deform. +

    Perfect plastic or ideal plastic: well-defined , stress plateau up to failure. Some materials (e.g. mild steel) have two yield points (stress plateau at ). Most ductile metals do not have a stress plateau; yield strength is then defined by the 0.002 (0.2%) offset method.

    @@ -259,9 +184,9 @@ Stresses above the plastic limit (Atoms rearrange in plastic region of ductile materials when a higher stress is sustained. Plastic strain remains after unloading as permanent set, resulting in permanent deformation. Reloading is linear elastic up to the new, higher yield stress (at A') and a reduced ductility.

    - Ultimate Strength - +     +

    The ultimate strength () is the maximum stress the material can withstand.

    @@ -273,11 +198,9 @@ Stresses above the plastic limit (), the middle of the material elongates before failure.

    -     - - - + +

    Also called fracture or rupture stress () is the stress at the point of failure for the material. Brittle and Ductile materials fail differently.

    @@ -312,103 +235,195 @@ Stresses above the plastic limit (For this reason, concrete is almost always reinforced with steel bars or rods whenever it is designed to support tensile loads. +

    - Strain Energy builds on this content in Engineering Materials. + Fatigue builds on this content in Engineering Materials and Mechanical Design.

    - Deformation does work on the material: equal to internal strain energy (by energy conservation). + SN curve examples + + If stress does not exceed the elastic limit, the specimen returns to its original configuration. However, this is not the case if the loading is repeated thousands or millions of times. In such cases, rupture will happen at stress lower than the fracture stress - this phenomenon is known as fatigue. +

    +     +     + +
    + Strain Energy builds on this content in Engineering Materials. - + Deformation does work on the material: equal to internal strain energy (by energy conservation). - Energy density. + - Can be generalized to any deformation: areas under stress-strain curves + Energy density. - + Can be generalized to any deformation: areas under stress-strain curves + + +
    + +
    + + + + + Isotropic: material properties are independent of the direction + Anisotropic: material properties depend on the direction (ie; composites, wood, and tissues) + + + + + + +ADD CONTENT HERE + + + + + +ADD CONTENT HERE + + + + + + + +

    + If you want more details on this topic, consider taking a course in continuum mechanics.

    - +

    - Fatigue builds on this content in Engineering Materials and Mechanical Design. + Bold symbols, such as and , + represent tensors rather than scalar quantities.

    +

    - SN curve examples + Additionally, some quantities might be represented with indicial notation. When indices on the right hand side of the equation are repeated, + this represents a summation where the index values range from 1 to 3. + For example, + represents - If stress does not exceed the elastic limit, the specimen returns to its original configuration. However, this is not the case if the loading is repeated thousands or millions of times. In such cases, rupture will happen at stress lower than the fracture stress - this phenomenon is known as fatigue. + This is a shorthand to make writing long summations easier.

    +     -     +

    +Generally, the state of stress a 3D material can be represented as a second-rank tensor (i.e. a matrix) with components. +

    + Infinitesimal Stress element + - - - +

    +Using indicial notation, component of this tensor is denoted as . The same can be said for a material strain tensor. +

    - - Isotropic: material properties are independent of the direction - Anisotropic: material properties depend on the direction (ie; composites, wood, and tissues) - + Infinitesimal Strain Element + -     +

    - +Using indicial notation, component of this tensor is denoted as . Note +that both the stress and strain tensors are symmetric, that is and +. -Poisson's ratio +

    - As illustrated in the figure above, when an object is being compressed it expands outward, and when stretched it will become thinner. Depending on the material, it may experience more or less lateral deformation for a given amount of axial deformation. The ratio that governs the amount of lateral strain per unit of axial strain is called Poisson's ratio. + +For our purposes, lets bound the discussion of material stress-strain behavior to only the elastic range, that is, deformation is linear proportional applied load. +This is Hooke’s law. The relationship between stress and strain is referred to as the constitutive relationship of a material. In the most general case, this relationship is defined as a fourth-rank tensor with components. +This elasticity tensor, C, relating the stress and strain tensors is shown in indicial notation (rather than a matrix operation). +

    - + + This is short form representation of a summation, known as indicial notation. Whenever there are repeated indices on the right hand side of an equation, + as with k and l above (repeated in and ), these indices represent + a summation over three dimensions. So the example can be rewritten as shown below, where and . + + + - +Expanding over these two summations yields: + - -

    Consider a cylinder with volume . We know that . Since the diameter is dependant on the change in length \(dx\) due to Poisson's ratio, we get

    - -

    We use the chain rule to find the change in volume

    +

    - +What this means is there can be as many as 9 components of strain and 9 material coefficients that contribute to each stress component. +(You can see the benefit of indicial notation here.) -

    - By definition, -

    +

    - +

    +However, not all 81 components of this tensor are independent, because of symmetry of stress and strain tensors there are only 21 independent components. If +the material is assumed to be isotropic (having the same behavior in all directions) this is further reduced to just 2 independent components (for instance, + Young’s modulus and , Poisson’s ratio). With this simplification, the 3D isotropic elastic material constitutive relation can be simplified. -

    - Returning to our first equation, -

    + - +

    -

    - Since \(\partial V \geq 0\) and \(\nu > 0\) in nicely behaved materials, \( 0 < \nu \leq 0.5\). In practice \(\nu = 0.5\) only if the material is incompressible, thus in most materials \(\nu < 0.5\). -

    - -     +Here the engineering shear strains are introduced. +This, along with identification of shear stresses +as +allows for the expression for pure axial stress & strain in one direction to be represented as . +Rearranging the terms, relation can be expressed in a more general form. - + - +Similarly, the expression for pure shear stress & strain can be expressed as . + +Again, this can be expressed in a more general form. + +

    + +

    + + + + + +A stress-strain diagram is the relationship of normal stress as a function of normal strain. One way to collect these measurement is a uniaxial tension test in which a specimen at a very slow, constant rate (quasi-static). A load and distance are measured at frequent intervals. + + + + + +

    - A negative Poisson's ratio is possible when the material structure is non-trival. + Things that effect the material properties include imperfections, different composition, rate or loading, or temperature.

    - + Stress-strain diagram showing key properties +     + diff --git a/src/pages/sol/axial_loading.astro b/src/pages/sol/simple_loading.astro similarity index 100% rename from src/pages/sol/axial_loading.astro rename to src/pages/sol/simple_loading.astro diff --git a/src/pages/sol/failure_theories.astro b/src/pages/sol/statically_indeterminate_problems.astro similarity index 100% rename from src/pages/sol/failure_theories.astro rename to src/pages/sol/statically_indeterminate_problems.astro diff --git a/src/pages/sol/strain.astro b/src/pages/sol/strain.astro index 6a3c29270..245432247 100644 --- a/src/pages/sol/strain.astro +++ b/src/pages/sol/strain.astro @@ -16,7 +16,15 @@ import CalloutContainer from "../../components/CalloutContainer.astro" @@ -49,11 +48,59 @@ import CalloutContainer from "../../components/CalloutContainer.astro" - + -Dimensionless [mm/mm, in/in, or %] + + +

    + Initial and final lengths of some section of the specimen are measured, perhaps by some handheld device such as a ruler. Axial strain computed directly by following formula: +

    + + + + Accurate measurements of strain in this way may require a fairly large initial length. + +     + + + +

    + A clip-on device that can measure very small deformations. Two clips attach to a specimen before testing. The clips are attached to a transducer body. The transducer outputs a voltage. Changes in voltage output are converted to strain. +

    + + A tensile test in the Materials Testing Instructional Laboratory, Talbot Lab, UIUC + +     + + +

    +Small electrical resistors whose resistance charges with strain. Change in resistance can be converted to strain measurement. Often sold as "rosettes", which can measure normal strain in two or more directions. Can be bonded to test specimen. +

    + Rosette strain gauge arrangement and example + +     + + +

    +A calibrated wire is set into vibration and its frequency is measured. Small changes in the length of the wire as a result of strain produce a measurable change in frequency, allowing for accurate strain measurements over relatively long gauge lengths. +

    + Vibrating wire strain gauge attached to the side of a bridge + + +     + + + +

    +Image placed on surface of test specimen. Image may consist of speckles or some regular pattern. Deformation of image tracked by digital camera. Image analysis used to determine multiple strain component. +

    + Experiment set up. The diffuse light source consists of two fluorescent tube lights that produce white light, behind a translucent plastic sheet. + + strain calculated through DIC of straight-curved specimen with an applied load of 114 N from TAM 456, UIUC. + + +     -     @@ -117,16 +164,6 @@ Dimensionless [mm/mm, in/in, or %] - - Average shear strain - - - - - Engineering (shear) strain: Compute angle from length changes and original (undeformed) total length. - True (shear) strain: Integrate infinitesimal angle changes. - - @@ -167,57 +204,15 @@ The tensor contains:

    - - - - -

    - Initial and final lengths of some section of the specimen are measured, perhaps by some handheld device such as a ruler. Axial strain computed directly by following formula: -

    - - - - Accurate measurements of strain in this way may require a fairly large initial length. - -     - - - -

    - A clip-on device that can measure very small deformations. Two clips attach to a specimen before testing. The clips are attached to a transducer body. The transducer outputs a voltage. Changes in voltage output are converted to strain. -

    - - A tensile test in the Materials Testing Instructional Laboratory, Talbot Lab, UIUC - -     - - -

    -Small electrical resistors whose resistance charges with strain. Change in resistance can be converted to strain measurement. Often sold as "rosettes", which can measure normal strain in two or more directions. Can be bonded to test specimen. -

    - Rosette strain gauge arrangement and example - -     - - -

    -A calibrated wire is set into vibration and its frequency is measured. Small changes in the length of the wire as a result of strain produce a measurable change in frequency, allowing for accurate strain measurements over relatively long gauge lengths. -

    - Vibrating wire strain gauge attached to the side of a bridge - - -     - - - -

    -Image placed on surface of test specimen. Image may consist of speckles or some regular pattern. Deformation of image tracked by digital camera. Image analysis used to determine multiple strain component. -

    - Experiment set up. The diffuse light source consists of two fluorescent tube lights that produce white light, behind a translucent plastic sheet. - - strain calculated through DIC of straight-curved specimen with an applied load of 114 N from TAM 456, UIUC. + + Average shear strain + -     + + Engineering (shear) strain: Compute angle from length changes and original (undeformed) total length. + True (shear) strain: Integrate infinitesimal angle changes. + + diff --git a/src/pages/sol/stress_transformation.astro b/src/pages/sol/superposition.astro similarity index 99% rename from src/pages/sol/stress_transformation.astro rename to src/pages/sol/superposition.astro index ceeb61d94..083928016 100644 --- a/src/pages/sol/stress_transformation.astro +++ b/src/pages/sol/superposition.astro @@ -22,7 +22,7 @@ import SubSubSubSection from "../../components/SubSubSubSection.astro" import CalloutContainer from "../../components/CalloutContainer.astro" import CalloutCard from "../../components/CalloutCard.astro" --- - +
      @@ -52,7 +52,7 @@ import CalloutCard from "../../components/CalloutCard.astro"
    -
    +
    diff --git a/src/pages/sol/thermal_loading.astro b/src/pages/sol/thermal_loading.astro new file mode 100644 index 000000000..63ff7df4b --- /dev/null +++ b/src/pages/sol/thermal_loading.astro @@ -0,0 +1,48 @@ +--- +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 Row from "../../components/Row.astro" +import Col from "../../components/Col.astro" +import CalloutCard from "../../components/CalloutCard.astro" +import CalloutContainer from "../../components/CalloutContainer.astro" +import Warning from "../../components/Warning.astro" +--- + + + + + + + +
    + +     From 9068c1d7435f3d00c3b4e00e31752f09aab30218 Mon Sep 17 00:00:00 2001 From: mihikad2 Date: Sun, 14 Sep 2025 20:32:23 -0500 Subject: [PATCH 2/3] Reorganizing reference pages --- src/pages/sol/appendix.astro | 43 +++++- src/pages/sol/material_properties.astro | 12 +- src/pages/sol/simple_loading.astro | 137 +++++------------- .../statically_indeterminate_problems.astro | 29 +++- src/pages/sol/stress.astro | 58 -------- src/pages/sol/superposition.astro | 10 ++ src/pages/sol/thermal_loading.astro | 42 ++++++ src/pages/sol/torsion.astro | 38 +---- 8 files changed, 165 insertions(+), 204 deletions(-) diff --git a/src/pages/sol/appendix.astro b/src/pages/sol/appendix.astro index 8c0020805..e7ee83a3a 100644 --- a/src/pages/sol/appendix.astro +++ b/src/pages/sol/appendix.astro @@ -52,6 +52,12 @@ Dimensionless [mm/mm, in/in, or %] + + +Force X distance [lb-in or N-m] + + + The units of stress can be found in the table below. @@ -100,13 +106,48 @@ The units of stress can be found in the table below. + + + + -Typical sign convention for axial (normal) stress: + Tension: Compression: +Notation + +
      +
    • Displacement of points in space: the movement of a point relative to its initial position in space (ex; )
      • +
    • Change in length of a segment: the relative displacement of a point with respect to another point (ex; ). This can be written multiple ways: +
      • +
    + +Sign Conventions for internal forces in a bar. +
      +
    • : tension
      • +
    • : compression
      • +
    • : elongation
      • +
    • : contraction
      • +
    + +     + + + + Sign convention for internal toruqe in a shaft + + + : counter clockwise + : clockwise + + + Right hand rule + + Torque and angle of twist follow the right hand rule sign convention. When positive, using the right hand, the thumb points outward from the shaft and the fingers will curl in the direction of the positive twist/torque. + diff --git a/src/pages/sol/material_properties.astro b/src/pages/sol/material_properties.astro index 7ac2fcd65..2b45a47ae 100644 --- a/src/pages/sol/material_properties.astro +++ b/src/pages/sol/material_properties.astro @@ -28,12 +28,12 @@ import Warning from "../../components/Warning.astro"
      -
    • Assumptions
      • -
    • Linear Elasticity
      • -
    • Plasticity
      • -
    • Strain Energy Density
      • -
    • Complex Material Behaviors
      • -
    • Other Derived Properties +
    • Assumptions
      • +
    • Linear Elasticity
      • +
    • Plasticity
      • +
    • Strain Energy Density
      • +
    • Complex Material Behaviors
      • +
    • Other Derived Properties
      • Directional Materials
      • Poisson's Ratio
        • diff --git a/src/pages/sol/simple_loading.astro b/src/pages/sol/simple_loading.astro index 59b1c8a00..667b7cb5e 100644 --- a/src/pages/sol/simple_loading.astro +++ b/src/pages/sol/simple_loading.astro @@ -11,12 +11,14 @@ import RedText from "../../components/RedText.astro" import InlineEquation from "../../components/InlineEquation.astro" import DisplayEquation from "../../components/DisplayEquation.astro" --- - + -
        - - - -Notation - -
          -
        • Displacement of points in space: the movement of a point relative to its initial position in space (ex; )
          • -
        • Change in length of a segment: the relative displacement of a point with respect to another point (ex; ). This can be written multiple ways: -
          • -
        - -Sign Conventions for internal forces in a bar. -
          -
        • : tension
          • -
        • : compression
          • -
        • : elongation
          • -
        • : contraction
          • -
        +
        + + +REWORD CONTENT + Saint-Venant's Principle: slender beam case Notation

        Stress analysis very near to the point of application of load . Saint-Venant's principle: the stress and strain produced at points in a body sufficiently removed* from the region of external load application will be the same as the stress and strain produced by any other applied external loading that has the same statically equivalent resultant and is applied to the body within the same region.

        *farther than the widest dimension of the cross section +         -         + +ADD CONTENT HERE + - + @@ -77,9 +67,7 @@ import DisplayEquation from "../../components/DisplayEquation.astro" - - - + Axially Varying Properties For non-uniform load, material property and cross-section area: @@ -93,16 +81,6 @@ For non-uniform load, material property and cross-section area: Assume variations with are "mild" (on length scale longer than cross-sectional length scales) - - - - -Superposition: If the displacements are (1) small and (2) linearly related to the force components acting, the displacements caused by the components can be added up: - - - - -
        1. Draw a FBD @@ -114,70 +92,6 @@ Superposition: If the displacements are (1) small and (2) linearly related to th
        2. Compatibility equations: geometric constraints
        -         - - - - - - - Statically determinate - - -All internal forces can be obtained from equilibrium analysis only - - - - - - - Statically indeterminate - - - Equilibrium does not determine all internal forces. - - - - - -Notation -
          -
        • : Compression
          • -
        • : Tension
          • -
        - -

        - , present in addition to elastic , (from internal forces). Superposition (small strains): - - -

        - -Temperature changes with no applied loads - - A rod rests freely on a smooth horizontal surface. Temperature of the rod is raised by . Rod elongates by an amount. - - - - Linear coefficient of thermal expansion , . This deformation is associated with an average thermal strain: - - - - - Temperature changes with statically indeterminate beam - - - Initially, rod of length is placed between two supports at a distance from each other. With no internal forces, there is no stress or strain. - - - - - After raising the temperature, total elongation of the rod is still zero. The total elongation is given by: - - - - The stress in the rod due to change in temperature is given by: - -         @@ -208,4 +122,23 @@ A misfit problem is one in which there is difference between a design distance a + +Torsion refers to the twisting of a specimen when it is loaded by couples (or moments) that produce rotation about the longitudinal axis. Applications include aircraft engines, car transmissions, and bicycles, etc. + + + + +ADD OTHER GENERAL ASSUMPTIONS +Gears: + + Gears are perfectly rigid. + The rotation axis is perfectly fixed in space. + Gear teeth are evenly spaced and perfectly shaped so there is no gap to create lost motion. + Gear tooth faces are perfectly smooth so there is no slip. + Mated gears twist through the same arc length. + + + + +         diff --git a/src/pages/sol/statically_indeterminate_problems.astro b/src/pages/sol/statically_indeterminate_problems.astro index a10f2101c..4e1f709a3 100644 --- a/src/pages/sol/statically_indeterminate_problems.astro +++ b/src/pages/sol/statically_indeterminate_problems.astro @@ -10,7 +10,7 @@ import BlueText from "../../components/BlueText.astro" import RedText from "../../components/RedText.astro" import Plotly from "../../components/Plotly.astro" --- - +
        -
        +
        +
        + + + + + Statically determinate + + +All internal forces can be obtained from equilibrium analysis only + + + + + + + Statically indeterminate + + + Equilibrium does not determine all internal forces. + + + +CONTENT BELOW MUST BE MOVED + +
        diff --git a/src/pages/sol/stress.astro b/src/pages/sol/stress.astro index 4690bdfc2..ea6827b15 100644 --- a/src/pages/sol/stress.astro +++ b/src/pages/sol/stress.astro @@ -48,64 +48,6 @@ The internal forces and moments generally vary from point to point. Obtaining th
        - -The units of stress can be found in the table below. - - - - - - - - - - - - - - - - - - - - - - - - - - -
        MeasureSI unitsImperial units
        Force[\(\rm N\)][\(\rm lb\)]
        Area[\(\rm m^2\)][\(\rm in^2\)]
        Stresspascal = [\(\rm Pa\)] = [\(\rm N/m^2\)]pound per square inch = [\(\rm psi\)] = [\(\rm lb/in^2\)]
        - - It is also important to recall some of the most common prefixes used to denote quantity. - - - - - - - - - - - - - - - - -
        kilopascal [\(\rm kPa\)]\(10^3\) [\(\rm Pa\)]
        megapascal [\(\rm MPa\)]\(10^6\) [\(\rm Pa\)]
        gigapascal [\(\rm GPa\)]\(10^9\) [\(\rm Pa\)]
        - -Typical sign convention for axial (normal) stress: - - Tension: - Compression: - - -         - - We consider a homogeneous distribution of the internal force over an infinitesimal area . The stress is defined by the infinitesimal force divided by the infinitesimal area. diff --git a/src/pages/sol/superposition.astro b/src/pages/sol/superposition.astro index 083928016..2e634b85e 100644 --- a/src/pages/sol/superposition.astro +++ b/src/pages/sol/superposition.astro @@ -55,6 +55,16 @@ import CalloutCard from "../../components/CalloutCard.astro"
        + + +Superposition: If the displacements are (1) small and (2) linearly related to the force components acting, the displacements caused by the components can be added up: + + + + + +CONTENT BELOW MUST BE MOVED + The general state of stress at a point is characterized by three independent normal stress components and three independent shear stress components, and is represented by the stress tensor. The combination of the state of stress for every point in the domain is called the stress field. diff --git a/src/pages/sol/thermal_loading.astro b/src/pages/sol/thermal_loading.astro index 63ff7df4b..8ca6ade2c 100644 --- a/src/pages/sol/thermal_loading.astro +++ b/src/pages/sol/thermal_loading.astro @@ -45,4 +45,46 @@ import Warning from "../../components/Warning.astro"
        + + +Notation +
          +
        • : Compression
          • +
        • : Tension
          • +
        + +

        + , present in addition to elastic , (from internal forces). Superposition (small strains): + + +

        + +Temperature changes with no applied loads + + A rod rests freely on a smooth horizontal surface. Temperature of the rod is raised by . Rod elongates by an amount. + + + + Linear coefficient of thermal expansion , . This deformation is associated with an average thermal strain: + + + + + Temperature changes with statically indeterminate beam + + + Initially, rod of length is placed between two supports at a distance from each other. With no internal forces, there is no stress or strain. + + + + + After raising the temperature, total elongation of the rod is still zero. The total elongation is given by: + + + + The stress in the rod due to change in temperature is given by: + + + + diff --git a/src/pages/sol/torsion.astro b/src/pages/sol/torsion.astro index 572fd7c62..da161ca76 100644 --- a/src/pages/sol/torsion.astro +++ b/src/pages/sol/torsion.astro @@ -23,7 +23,7 @@ import Enumerate from "../../components/Enumerate.astro"
      • Units
      • Notation and Convention
      • Equilibrium -
      • Assumptions +
      • Assumptions
      • Shear Stress and Strain
        • Shear Strain: Geometry of Deformation
          • @@ -44,30 +44,9 @@ import Enumerate from "../../components/Enumerate.astro"
          -Torsion refers to the twisting of a specimen when it is loaded by couples (or moments) that produce rotation about the longitudinal axis. Applications include aircraft engines, car transmissions, and bicycles, etc. -
          - - - -Force X distance [lb-in or N-m] - - - - - - Sign convention for internal toruqe in a shaft - - - : counter clockwise - : clockwise - - Right hand rule - - Torque and angle of twist follow the right hand rule sign convention. When positive, using the right hand, the thumb points outward from the shaft and the fingers will curl in the direction of the positive twist/torque. - - +
        • @@ -81,19 +60,8 @@ The stress distribution in the shaft is not known. - - - Torsion lines - - For circular shafts (hollow and solid): cross-sections remain plane and undistorted due to axisymmetric geometry - For non-circular shafts: cross-sections are distorted when subject to torsion - Linear and elastic deformation - - - - @@ -255,7 +223,7 @@ From observation: Gear system problems can be solved using the torsion equations. -

        Common assuptions:

        +

        Common assumptions:

        From 92206a2151866f990e158c3cb2636e841d547f86 Mon Sep 17 00:00:00 2001 From: mihikad2 Date: Sun, 14 Sep 2025 23:27:54 -0500 Subject: [PATCH 3/3] Reorganized pages --- src/pages/sol.astro | 2 +- src/pages/sol/appendix.astro | 343 +++++++++ src/pages/sol/bending.astro | 461 ------------ src/pages/sol/combined_loading.astro | 190 +++++ src/pages/sol/design_considerations.astro | 124 +--- .../sol/failure_engineering_materials.astro | 183 +++++ src/pages/sol/pressure_vessels.astro | 62 -- src/pages/sol/shear_moment_diagrams.astro | 1 + src/pages/sol/simple_loading.astro | 660 +++++++++++++++++- .../statically_indeterminate_problems.astro | 132 +--- src/pages/sol/stress.astro | 586 +++++++++++++++- src/pages/sol/superposition.astro | 578 +-------------- src/pages/sol/torsion.astro | 199 ------ src/pages/sol/transverse_shear.astro | 278 -------- 14 files changed, 1962 insertions(+), 1837 deletions(-) diff --git a/src/pages/sol.astro b/src/pages/sol.astro index b7549f39e..981f54e19 100644 --- a/src/pages/sol.astro +++ b/src/pages/sol.astro @@ -193,7 +193,7 @@ import "../../public/static/css/course_home_pages.css"
      -
    • Intro
      • +
    • Introduction
    • Strain
    • Stress
    • Material properties
      • diff --git a/src/pages/sol/appendix.astro b/src/pages/sol/appendix.astro index e7ee83a3a..d4152f2b1 100644 --- a/src/pages/sol/appendix.astro +++ b/src/pages/sol/appendix.astro @@ -13,6 +13,7 @@ import Col from "../../components/Col.astro" import CalloutCard from "../../components/CalloutCard.astro" import CalloutContainer from "../../components/CalloutContainer.astro" import Warning from "../../components/Warning.astro" +import DisplayTable from "../../components/DisplayTable.astro" ---