Calculus of the Elastic Properties of a Beam Cross-Section. Shear stress in fluids: Any real fluids (liquids and gases included) moving along solid boundary will incur a shear stress on that boundary. The no-slip condition dictates that the speed of the fluid at the boundary (relative to the boundary) is zero, but at some height from the boundary the flow speed must equal that of the fluid. The region, Similarly, if m(x ) is a partially distributed moment in the interval (x1, x2), it is equivalent to a distributed force in this interval and two concentrated forces at x x 1 and x x 2 : According to Timoshenko and Woinowsky-Krieger (1959), Kelvin and Tait transformed the twisting moments along an edge of a classical plate to a system of.

### What is Shear Stress? Definition Equation & Units

Stress (mechanics) Wikipedia. Application of Principle of Virtual Work to Find Displacement in Statically Indeterminate Structures Example: Find B, the vertical displacement at B. Consider flexural response only; assume EI is constant., Bending Moment 1. CH28 p355 How to find Bending Moment Calculate BM: M = Fr (Perpendicular to the force) Bending moment is a torque applied to each side of the beam if it was cut in two - anywhere along its length. The hinge applies a clockwise (+) moment (torque) to the RHS, and a counter-clockwise (-) moment to the LHS. Example Bending-Moment.

tion of the torsional and shear behavior re-quire the solution of a set of 2D Laplace dif-ferential equations on the section domain with Neumann boundary conditions. The aim of this work is the calculus of the elastic properties of beams with general cross-sections extending the approach proposed by Petrolo and Casciaro in [3] to the evaluation Shear force and bending moment diagram example #1: single point load. 5/7/2017 Comments are closed. Course Links. C++ Programming Calculus Chemistry Differential Equations

Shear stress in fluids: Any real fluids (liquids and gases included) moving along solid boundary will incur a shear stress on that boundary. The no-slip condition dictates that the speed of the fluid at the boundary (relative to the boundary) is zero, but at some height from the boundary the flow speed must equal that of the fluid. The region Application of Principle of Virtual Work to Find Displacement in Statically Indeterminate Structures Example: Find B, the vertical displacement at B. Consider flexural response only; assume EI is constant.

The Fundamental Theorem of Calculus IM&E Workshop, March 27{29, 2010 Wanda Bussey, Peter Collins, William McCallum, Scott Peterson, Marty Schnepp, Matt Thomas 1 Introduction The following interesting example prepares the way for an intuitive understanding of the Funda-mental Theorem of Calculus (FTC). The first free, easy to use customizable Bending Moment Diagram and Shear Force Diagram Calculator for simply supported Beams Main menu

Application of Principle of Virtual Work to Find Displacement in Statically Indeterminate Structures Example: Find B, the vertical displacement at B. Consider flexural response only; assume EI is constant. Example 4: Axial Force, Shear Force and Bending Momemt Diagram (1/2) by SpoonFeedMe в†ђ Video Lecture 55 of 67 в†’

What is an internal force? If you go through the quiz and worksheet, you can test your knowledge of this type of force and examples of it. The quiz... Shear stress in fluids: Any real fluids (liquids and gases included) moving along solid boundary will incur a shear stress on that boundary. The no-slip condition dictates that the speed of the fluid at the boundary (relative to the boundary) is zero, but at some height from the boundary the flow speed must equal that of the fluid. The region

Aug 12, 2017В В· For example, when a load of 5kN/m is applied on a 10m simply supported beam, the shear stress will be a straight line decreasing from 25kN/m^2 on one support till it reaches вЂ¦ Shear and Bending вЂ“ The presence of a shear force indicates a variable bending moment in the beam. вЂ“ The relationship between the shear force and the change in bending moment is given by dx dM V = (42) LECTURE 14. BEAMS: SHEARING STRESS (6.1 вЂ“ 6.4) Slide No. 5 Shearing Stress in Beams ENES 220 В©Assakkaf Shear and Bending

Example Problem For the beam and loading shown calculate. Similarly, if m(x ) is a partially distributed moment in the interval (x1, x2), it is equivalent to a distributed force in this interval and two concentrated forces at x x 1 and x x 2 : According to Timoshenko and Woinowsky-Krieger (1959), Kelvin and Tait transformed the twisting moments along an edge of a classical plate to a system of, The outline of the paper is as follows: In Section 2, a brief introduction to the tangential differential calculus (TDC) is given. In Section 3, the classical linear ReissnerвЂ“Mindlin shell equations are recast in terms of the TDC. Stress resultants such as membrane forces, bending moments and transverse shear forces are defined..

### Lecture 52 Example 5 Axial Force Shear CosmoLearning

Real life application of Fundamental theorem of calculus. Apr 12, 2006В В· I am a graduate student working through my math education coursework and am currently in Calculus II. I can use all the help I can get. Initially I started searching Calculus applications for engineering in Hurricane Katrina and the levee system in New Orleans but have come up fruitless thus far. I'm not sure what information to look for in the 250 page documents I came across., Chapter 2 Review of Forces and Moments 2.1 Forces In this chapter we review the basic concepts of forces, and force laws. to material covered in EN030, and is provided here as a review. There are a few additional sections вЂ“ for example forces exerted by a damper or dashpot, an inerter, and interatomic forces The simplest application.

### Distributed loading on a beam example #3 parabolic loads

Morrow & Kokernak Statics and Strength of Materials 7th. Example 4: Axial Force, Shear Force and Bending Momemt Diagram (1/2) by SpoonFeedMe в†ђ Video Lecture 55 of 67 в†’ https://en.wikipedia.org/wiki/Talk:Lift_(force)/Archive_5 For all courses in statics and materials strength, and for courses on structural principles. This fully updated text presents logically organized, clear coverage of all major topics in statics and strength of materials, including the latest developments in materials technology and manufacturing.

May 10, 2019В В· Depending on your job as an EE, you will use calculus with a frequency ranging from every day, to never at all. But you canвЂ™t bypass calculus, by deciding that you will get one of those вЂњnever at allвЂќ jobs. You canвЂ™t even get a degree in EE, witho... For all courses in statics and materials strength, and for courses on structural principles. This fully updated text presents logically organized, clear coverage of all major topics in statics and strength of materials, including the latest developments in materials technology and manufacturing

On applications of generalized functions to beam bending problems. shear forces. W. a component of beam deflection. a i. real coefficients. m, m 0. force of a concentrated force and a concentrated moment are obtained using the discontinuities they produce in the bending moment and shearing forces of an EulerвЂ“Bernoulli beam. Jun 26, 2010В В· I need a problem with real life application that apply both portions of fundamental theorem of calculus. Any thoughts? Please describe how they would be applied (meaning if I can see the math so I can understand that would be great!) Thank you!!!

A bending moment is the reaction induced in a structural element when an external force or moment is applied to the element causing the element to bend. The most common or simplest structural element subjected to bending moments is the beam.The diagram shows a вЂ¦ semester) and integral (second semester) calculus courses were offered during the duration of the project. The application projects involved both teamwork and individual work, and we required use of both programmable calculators and Matlab for these projects. Some projects involved use of real data often collected by the involved faculty.

tion of the torsional and shear behavior re-quire the solution of a set of 2D Laplace dif-ferential equations on the section domain with Neumann boundary conditions. The aim of this work is the calculus of the elastic properties of beams with general cross-sections extending the approach proposed by Petrolo and Casciaro in [3] to the evaluation Mar 30, 2017В В· We rely heavily on the mechanics and calculus which Newton developed in the 17th century, together with linear algebra. Most problems boil down to solving the EulerвЂ“Bernoulli beam equation (usually the time-independent one, where inertial forces a...

Shear stress in fluids: Any real fluids (liquids and gases included) moving along solid boundary will incur a shear stress on that boundary. The no-slip condition dictates that the speed of the fluid at the boundary (relative to the boundary) is zero, but at some height from the boundary the flow speed must equal that of the fluid. The region and shear force resultant of the normal and shear stresses Figure 7.4.5: sign convention for moments and shear forces Note that the sign convention for the shear stress conventionally used the beam theory conflicts with the sign convention for shear stress used in the rest of mechanics, introduced in Chapter 3. This is shown in Fig. 7.4.6.

Bending Moment of a Beam . Simply supported beam at both ends with a single point load acting on it. #moment . Deformation of a Beam . Simply supported beam at both ends with a single point load acting on it (shear deformations are neglectable in this example). In this procedure the normal forces in the beam will be plotted. To show the Force Method for Analysis of Indeterminate Structures Number of unknown Reactions or Internal forces > Number of equilibrium equations Note: Most structures in the real world are statically indeterminate. Find the reactions and draw the Shear Force and Bending Moment Diagrams of the beam. ForceMethod Page 5 . Example: Frames

The existence of Complementary Shear Forces may be an important factor in the failure of anisotropic materials such as timber, which is weak in Shear along the grain. Near to a Free Boundary there are no externally applied Forces and it can be seen that The Shear Stress on any cross section must act in a direction parallel to the Boundary. Mathematics in Structural Engineering Dr Colin Caprani About Me вЂў Degree in Structural Engineering 1999 вЂў Full time consultancy until 2001 вЂў PhD in UCD from 2001 to 2006 вЂў Lecturing in DIT and UCD вЂў Consultant in buildings & bridges Guess my Leaving result! C1 in Honours Maths You donвЂ™t have to be a geniusвЂ¦

## Application of Principle of Virtual Work to Find

Lecture 52 Example 5 Axial Force Shear CosmoLearning. Shear Force Point of Application of Load and Sign Convention. we will learn that it is because of shear force bending is occurring and how we can determine their magnitude by doing the integration. Sanjeev Tiwari. UNACADEMY TOP EDUCATOR 2017/TEACHING EXPERIENCE OF 5 Years in GATE/Cleared GATE 8 times,COAL INDIA2017,BARC,NPCIL And Other, ii Leah Edelstein-Keshet List of Contributors Leah Edelstein-Keshet Department of Mathematics, UBC, Vancouver Author of course notes. Justin Martel Department of Mathematics, UBC, Vancouver Wrote and extended chapters on sequences, series and improper integrals вЂ“ January.

### What is Shear Stress? Definition Equation & Units

Stress (mechanics) Wikipedia. The outline of the paper is as follows: In Section 2, a brief introduction to the tangential differential calculus (TDC) is given. In Section 3, the classical linear ReissnerвЂ“Mindlin shell equations are recast in terms of the TDC. Stress resultants such as membrane forces, bending moments and transverse shear forces are defined., semester) and integral (second semester) calculus courses were offered during the duration of the project. The application projects involved both teamwork and individual work, and we required use of both programmable calculators and Matlab for these projects. Some projects involved use of real data often collected by the involved faculty..

goo.gl/Y0S3v4 for more FREE video tutorials covering Mechanics of Solids and Structural Mechanics Continuing the example 5 from previous video 2.10, this video very first shows how to take second cut for the stated problem and does the internal reactions and bending moment calculations similar to procedures previously done. The outline of the paper is as follows: In Section 2, a brief introduction to the tangential differential calculus (TDC) is given. In Section 3, the classical linear ReissnerвЂ“Mindlin shell equations are recast in terms of the TDC. Stress resultants such as membrane forces, bending moments and transverse shear forces are defined.

Apr 12, 2006В В· I am a graduate student working through my math education coursework and am currently in Calculus II. I can use all the help I can get. Initially I started searching Calculus applications for engineering in Hurricane Katrina and the levee system in New Orleans but have come up fruitless thus far. I'm not sure what information to look for in the 250 page documents I came across. Bending Moment 1. CH28 p355 How to find Bending Moment Calculate BM: M = Fr (Perpendicular to the force) Bending moment is a torque applied to each side of the beam if it was cut in two - anywhere along its length. The hinge applies a clockwise (+) moment (torque) to the RHS, and a counter-clockwise (-) moment to the LHS. Example Bending-Moment

Earlier, we saw the methods to calculate the shear forces on a beam. Now we can analyze the stresses due to shear forces, like we did for stresses due to bending moments. We showed that the shear force V is given by V = dM/dx, where M is the moment acting at the point x. tion of the torsional and shear behavior re-quire the solution of a set of 2D Laplace dif-ferential equations on the section domain with Neumann boundary conditions. The aim of this work is the calculus of the elastic properties of beams with general cross-sections extending the approach proposed by Petrolo and Casciaro in [3] to the evaluation

Chapter 2 Review of Forces and Moments 2.1 Forces In this chapter we review the basic concepts of forces, and force laws. to material covered in EN030, and is provided here as a review. There are a few additional sections вЂ“ for example forces exerted by a damper or dashpot, an inerter, and interatomic forces The simplest application Bending Moment 1. CH28 p355 How to find Bending Moment Calculate BM: M = Fr (Perpendicular to the force) Bending moment is a torque applied to each side of the beam if it was cut in two - anywhere along its length. The hinge applies a clockwise (+) moment (torque) to the RHS, and a counter-clockwise (-) moment to the LHS. Example Bending-Moment

Bending Moment of a Beam . Simply supported beam at both ends with a single point load acting on it. #moment . Deformation of a Beam . Simply supported beam at both ends with a single point load acting on it (shear deformations are neglectable in this example). In this procedure the normal forces in the beam will be plotted. To show the ii Leah Edelstein-Keshet List of Contributors Leah Edelstein-Keshet Department of Mathematics, UBC, Vancouver Author of course notes. Justin Martel Department of Mathematics, UBC, Vancouver Wrote and extended chapters on sequences, series and improper integrals вЂ“ January

Force Method for Analysis of Indeterminate Structures Number of unknown Reactions or Internal forces > Number of equilibrium equations Note: Most structures in the real world are statically indeterminate. Find the reactions and draw the Shear Force and Bending Moment Diagrams of the beam. ForceMethod Page 5 . Example: Frames Force Method for Analysis of Indeterminate Structures Number of unknown Reactions or Internal forces > Number of equilibrium equations Note: Most structures in the real world are statically indeterminate. Find the reactions and draw the Shear Force and Bending Moment Diagrams of the beam. ForceMethod Page 5 . Example: Frames

tion of the torsional and shear behavior re-quire the solution of a set of 2D Laplace dif-ferential equations on the section domain with Neumann boundary conditions. The aim of this work is the calculus of the elastic properties of beams with general cross-sections extending the approach proposed by Petrolo and Casciaro in [3] to the evaluation May 10, 2019В В· Depending on your job as an EE, you will use calculus with a frequency ranging from every day, to never at all. But you canвЂ™t bypass calculus, by deciding that you will get one of those вЂњnever at allвЂќ jobs. You canвЂ™t even get a degree in EE, witho...

Aug 12, 2017В В· For example, when a load of 5kN/m is applied on a 10m simply supported beam, the shear stress will be a straight line decreasing from 25kN/m^2 on one support till it reaches вЂ¦ Jun 26, 2010В В· I need a problem with real life application that apply both portions of fundamental theorem of calculus. Any thoughts? Please describe how they would be applied (meaning if I can see the math so I can understand that would be great!) Thank you!!!

Jun 26, 2010В В· I need a problem with real life application that apply both portions of fundamental theorem of calculus. Any thoughts? Please describe how they would be applied (meaning if I can see the math so I can understand that would be great!) Thank you!!! tion of the torsional and shear behavior re-quire the solution of a set of 2D Laplace dif-ferential equations on the section domain with Neumann boundary conditions. The aim of this work is the calculus of the elastic properties of beams with general cross-sections extending the approach proposed by Petrolo and Casciaro in [3] to the evaluation

semester) and integral (second semester) calculus courses were offered during the duration of the project. The application projects involved both teamwork and individual work, and we required use of both programmable calculators and Matlab for these projects. Some projects involved use of real data often collected by the involved faculty. The outline of the paper is as follows: In Section 2, a brief introduction to the tangential differential calculus (TDC) is given. In Section 3, the classical linear ReissnerвЂ“Mindlin shell equations are recast in terms of the TDC. Stress resultants such as membrane forces, bending moments and transverse shear forces are defined.

On applications of generalized functions to beam bending problems Article in International Journal of Solids and Structures 37(40):5675-5705 В· October 2000 with 190 Reads How we measure 'reads' On applications of generalized functions to beam bending problems. shear forces. W. a component of beam deflection. a i. real coefficients. m, m 0. force of a concentrated force and a concentrated moment are obtained using the discontinuities they produce in the bending moment and shearing forces of an EulerвЂ“Bernoulli beam.

exerts a bending moment of value -100 . on the portion of the beam to the right of x = 2. To find the bending moment at x = 6 consider the section of the beam to the left of x = 6 as shown in Fig. 5. There are now three forces, the reaction force at x = 0, the external force at x =5, and the shear force at x = 6. The first free, easy to use customizable Bending Moment Diagram and Shear Force Diagram Calculator for simply supported Beams Main menu

ii Leah Edelstein-Keshet List of Contributors Leah Edelstein-Keshet Department of Mathematics, UBC, Vancouver Author of course notes. Justin Martel Department of Mathematics, UBC, Vancouver Wrote and extended chapters on sequences, series and improper integrals вЂ“ January On applications of generalized functions to beam bending problems. shear forces. W. a component of beam deflection. a i. real coefficients. m, m 0. force of a concentrated force and a concentrated moment are obtained using the discontinuities they produce in the bending moment and shearing forces of an EulerвЂ“Bernoulli beam.

### Deп¬‚ections due to Bending MIT OpenCourseWare

Stress (mechanics) Wikipedia. Multivariable Calculus in this course. 2.2 Preliminary Analysis of a Simple SIR Model In many cases, analysis of two dimensional systems su ces in many applications. We illustrate this in the following example in which perform a preliminary anal-ysis of the SIR model developed in Example 2.1.1., Shear and Bending вЂ“ The presence of a shear force indicates a variable bending moment in the beam. вЂ“ The relationship between the shear force and the change in bending moment is given by dx dM V = (42) LECTURE 14. BEAMS: SHEARING STRESS (6.1 вЂ“ 6.4) Slide No. 5 Shearing Stress in Beams ENES 220 В©Assakkaf Shear and Bending.

### Determine Shear Force and Bending Moment in Unacademy

Shear force and bending moment diagram example #1 single. Aug 12, 2017В В· For example, when a load of 5kN/m is applied on a 10m simply supported beam, the shear stress will be a straight line decreasing from 25kN/m^2 on one support till it reaches вЂ¦ https://en.wikipedia.org/wiki/Shear_stress Shear Force Point of Application of Load and Sign Convention. we will learn that it is because of shear force bending is occurring and how we can determine their magnitude by doing the integration. Sanjeev Tiwari. UNACADEMY TOP EDUCATOR 2017/TEACHING EXPERIENCE OF 5 Years in GATE/Cleared GATE 8 times,COAL INDIA2017,BARC,NPCIL And Other.

Mar 30, 2017В В· We rely heavily on the mechanics and calculus which Newton developed in the 17th century, together with linear algebra. Most problems boil down to solving the EulerвЂ“Bernoulli beam equation (usually the time-independent one, where inertial forces a... Bending Moment 1. CH28 p355 How to find Bending Moment Calculate BM: M = Fr (Perpendicular to the force) Bending moment is a torque applied to each side of the beam if it was cut in two - anywhere along its length. The hinge applies a clockwise (+) moment (torque) to the RHS, and a counter-clockwise (-) moment to the LHS. Example Bending-Moment

A shear stress, often denoted by П„ (Greek: tau), is the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section of the material. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.. Shear stress arises from shear forces, which tion of the torsional and shear behavior re-quire the solution of a set of 2D Laplace dif-ferential equations on the section domain with Neumann boundary conditions. The aim of this work is the calculus of the elastic properties of beams with general cross-sections extending the approach proposed by Petrolo and Casciaro in [3] to the evaluation

The first free, easy to use customizable Bending Moment Diagram and Shear Force Diagram Calculator for simply supported Beams Main menu The first free, easy to use customizable Bending Moment Diagram and Shear Force Diagram Calculator for simply supported Beams Main menu

Deп¬‚ections due to Bending 10.1 The Moment/Curvature Relation Just as we took the pure bending construction to be accurate enough to produce useful estimates of the normal stress due to bending for loadings that included shear, so too we will use the same moment/curvature relationship to produce a dif- A shear stress, often denoted by П„ (Greek: tau), is the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section of the material. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.. Shear stress arises from shear forces, which

Bending Moment 1. CH28 p355 How to find Bending Moment Calculate BM: M = Fr (Perpendicular to the force) Bending moment is a torque applied to each side of the beam if it was cut in two - anywhere along its length. The hinge applies a clockwise (+) moment (torque) to the RHS, and a counter-clockwise (-) moment to the LHS. Example Bending-Moment Shear force and bending moment diagram example #1: single point load. 5/7/2017 Comments are closed. Course Links. C++ Programming Calculus Chemistry Differential Equations

tion of the torsional and shear behavior re-quire the solution of a set of 2D Laplace dif-ferential equations on the section domain with Neumann boundary conditions. The aim of this work is the calculus of the elastic properties of beams with general cross-sections extending the approach proposed by Petrolo and Casciaro in [3] to the evaluation In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material which is not a physical quantity . For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately

Application of Principle of Virtual Work to Find Displacement in Statically Indeterminate Structures Example: Find B, the vertical displacement at B. Consider flexural response only; assume EI is constant. Similarly, if m(x ) is a partially distributed moment in the interval (x1, x2), it is equivalent to a distributed force in this interval and two concentrated forces at x x 1 and x x 2 : According to Timoshenko and Woinowsky-Krieger (1959), Kelvin and Tait transformed the twisting moments along an edge of a classical plate to a system of

The Fundamental Theorem of Calculus IM&E Workshop, March 27{29, 2010 Wanda Bussey, Peter Collins, William McCallum, Scott Peterson, Marty Schnepp, Matt Thomas 1 Introduction The following interesting example prepares the way for an intuitive understanding of the Funda-mental Theorem of Calculus (FTC). Force Method for Analysis of Indeterminate Structures Number of unknown Reactions or Internal forces > Number of equilibrium equations Note: Most structures in the real world are statically indeterminate. Find the reactions and draw the Shear Force and Bending Moment Diagrams of the beam. ForceMethod Page 5 . Example: Frames

Oct 15, 2014В В· The mass of an object of known density, the moment of inertia of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus. An example of the use of calculus in mechanics is Newton's second law of motion: 17. and shear force resultant of the normal and shear stresses Figure 7.4.5: sign convention for moments and shear forces Note that the sign convention for the shear stress conventionally used the beam theory conflicts with the sign convention for shear stress used in the rest of mechanics, introduced in Chapter 3. This is shown in Fig. 7.4.6.

In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material which is not a physical quantity . For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material which is not a physical quantity . For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately

ii Leah Edelstein-Keshet List of Contributors Leah Edelstein-Keshet Department of Mathematics, UBC, Vancouver Author of course notes. Justin Martel Department of Mathematics, UBC, Vancouver Wrote and extended chapters on sequences, series and improper integrals вЂ“ January Example Problem For the beam and loading shown calculate the shear and bending from MAE 314 at North Carolina State University. Example problem for the beam and loading shown shear and bending moment. вЂў Sum forces in the vertical direction.

Application of Principle of Virtual Work to Find Displacement in Statically Indeterminate Structures Example: Find B, the vertical displacement at B. Consider flexural response only; assume EI is constant. Shear stress in fluids: Any real fluids (liquids and gases included) moving along solid boundary will incur a shear stress on that boundary. The no-slip condition dictates that the speed of the fluid at the boundary (relative to the boundary) is zero, but at some height from the boundary the flow speed must equal that of the fluid. The region

In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material which is not a physical quantity . For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushes on the particles immediately Deп¬‚ections due to Bending 10.1 The Moment/Curvature Relation Just as we took the pure bending construction to be accurate enough to produce useful estimates of the normal stress due to bending for loadings that included shear, so too we will use the same moment/curvature relationship to produce a dif-