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Stress Transformation Lectures #16-19 References: Beer, F.P. et al. (2011) Mechanics of materials. 6th Ed. SI. McGraw-Hill. Hibbeler, R.C. (2011) Mechanics of materials. 8th Ed. Prentice Hall.

Stress Transformation Problem 16.1 – Equilibrium Method Objectives & Review

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems

The state of plane stress at a point on the surface of the airplane fuselage is represented on the element oriented as shown. Represent the state of stress at the point on an element that is oriented 30° clockwise from the position shown.

Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act 2

𝜎𝑛 =

𝜎𝑥 + 𝜎𝑦 𝜎𝑥 − 𝜎𝑦 + cos(2𝜃) + 𝜏𝑥𝑦𝑠𝑖𝑛(2𝜃) 2 2

𝜏𝑛𝑡 = −

𝜎𝑥 − 𝜎𝑦 𝑠𝑖𝑛(2𝜃) + 𝜏𝑥𝑦𝑐𝑜𝑠(2𝜃) 2

Problem 16.2 Stress Transformation Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The state of plane stress at a point is represented by the element shown. Determine the state of stress at the point on another element oriented 30° counterclockwise from the position shown.

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

ANS: (Normal, Shear) V’ (59.46, -77.63) MPa H’ (10.54, 77.63) MPa

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

𝜎𝑛 =

𝜎𝑥 + 𝜎𝑦 𝜎𝑥 − 𝜎𝑦 + cos(2𝜃) + 𝜏𝑥𝑦𝑠𝑖𝑛(2𝜃) 2 2

𝜏𝑛𝑡 = −

𝜎𝑥 − 𝜎𝑦 𝑠𝑖𝑛(2𝜃) + 𝜏𝑥𝑦𝑐𝑜𝑠(2𝜃) 2

Problem 16.3 Stress Transformation Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The state of plane stress at a point on the lever is represented by the element shown. Determine the state of stress at the point on another element oriented 30° clockwise from the position shown.

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

Principal Stresses and Maximum In-Plane Shear Stress Objectives & Review

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

1. Calculate the stresses at the desired point 2. Perform stress transformation a.

Principal plane (eq.5) or the plane of maximum in-plane shear stress (eq.7) b. Calculate the corresponding principal stresses (eq.6), the average normal stress (eq.8) and the maximum in plane shear stress (eq.9) c. Draw the stress block (use eq.1 and eq. 2 as necessary) 𝜎𝑥+𝜎𝑦 𝜎𝑥−𝜎𝑦 + cos 2𝜃 + 𝜏𝑥𝑦 sin 2𝜃 2 2 𝜎 −𝜎 𝜏𝑛𝑡 = − 𝑥 𝑦 𝑠𝑖𝑛(2𝜃) + 𝜏𝑥𝑦𝑐𝑜𝑠(2𝜃) 2

Eq. 1

𝜎𝑛 =

Eq. 2 Eq. 5

tan 2𝜃𝑝 = 2𝜏𝑥𝑦 /(𝜎𝑥 − 𝜎𝑦 )

Eq. 6

𝜎1 , 𝜎2 , 𝜎3 =

Eq. 7

tan 2𝜃𝑠 = −

Eq. 8

𝜎𝑎𝑣𝑒 =

Eq. 9

𝜎𝑥+𝜎𝑦 2

𝑅 = 𝜏𝑚𝑎𝑥 = 𝑖𝑝

𝜎𝑥 +𝜎𝑦

2 𝜎𝑥−𝜎𝑦 2

± 𝑅, 0 /𝜏𝑥𝑦

𝜎𝑥−𝜎𝑦 2 2

+ 𝜏𝑥𝑦2

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

Problem 17.1 Stress Transformation Objectives & Review

Lecture #14 Combined Loadings Problems Homework

Determine the stresses at the principal plane and at the maximum in-plane shear stress plane

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

Problem 17.2 Stress Transformation Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The general state of plane stress at a point on the lever is shown. Determine the stresses at the principal plane and at the maximum in-plane shear stress plane

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

Recommended Analysis Objectives & Review

1.

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

2.

Calculate the stresses at the desired point [general state of stress] Perform stress transformation (using Mohr’s Cirlce) 1. 2. 3.

4.

Plot the given stresses (V and H) and locate the Center of the circle Calculate the Radius Locate the principal plane Determine: a. Principal Stresses b. Maximum in-plane shear stress c. Stress at any arbitrary point

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

Problem 18.1 – Mohr’s Circle Objectives & Review

Lecture #14 Combined Loadings Problems Homework

Determine the stresses at the principal plane and at the maximum in-plane shear stress plane

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

Problem 18.2 – Mohr’s Circle Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The general state of plane stress at a point on the lever is shown. Determine the stresses of the following planes: a. b.

Principal plane and at the maximum in-plane shear stress plane A plane oriented 30° clockwise from the position shown.

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

Problem 19.1 – Absolute Maximum Stress Objectives & Review

Determine the absolute maximum stress

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

Problem 19.2 – Absolute Maximum Stress Objectives & Review

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems

The state of stress at a point in a member is shown on the element. Use and draw the Mohr’s Circle to answer the following questions: a. b.

Calculate the maximum stresses (i.e. principal and absolute maximum shear stress) Calculate the stress components acting on the inclined plane AB and show the results on a sketch

Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

12

Problem 19.3 – Absolute Maximum Stress Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The general state of plane stress at a point on the lever is shown. Determine the absolute maximum stress

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

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Stress Transformation Problem 16.1 – Equilibrium Method Objectives & Review

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems

The state of plane stress at a point on the surface of the airplane fuselage is represented on the element oriented as shown. Represent the state of stress at the point on an element that is oriented 30° clockwise from the position shown.

Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act 2

𝜎𝑛 =

𝜎𝑥 + 𝜎𝑦 𝜎𝑥 − 𝜎𝑦 + cos(2𝜃) + 𝜏𝑥𝑦𝑠𝑖𝑛(2𝜃) 2 2

𝜏𝑛𝑡 = −

𝜎𝑥 − 𝜎𝑦 𝑠𝑖𝑛(2𝜃) + 𝜏𝑥𝑦𝑐𝑜𝑠(2𝜃) 2

Problem 16.2 Stress Transformation Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The state of plane stress at a point is represented by the element shown. Determine the state of stress at the point on another element oriented 30° counterclockwise from the position shown.

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

ANS: (Normal, Shear) V’ (59.46, -77.63) MPa H’ (10.54, 77.63) MPa

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

𝜎𝑛 =

𝜎𝑥 + 𝜎𝑦 𝜎𝑥 − 𝜎𝑦 + cos(2𝜃) + 𝜏𝑥𝑦𝑠𝑖𝑛(2𝜃) 2 2

𝜏𝑛𝑡 = −

𝜎𝑥 − 𝜎𝑦 𝑠𝑖𝑛(2𝜃) + 𝜏𝑥𝑦𝑐𝑜𝑠(2𝜃) 2

Problem 16.3 Stress Transformation Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The state of plane stress at a point on the lever is represented by the element shown. Determine the state of stress at the point on another element oriented 30° clockwise from the position shown.

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

Principal Stresses and Maximum In-Plane Shear Stress Objectives & Review

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

1. Calculate the stresses at the desired point 2. Perform stress transformation a.

Principal plane (eq.5) or the plane of maximum in-plane shear stress (eq.7) b. Calculate the corresponding principal stresses (eq.6), the average normal stress (eq.8) and the maximum in plane shear stress (eq.9) c. Draw the stress block (use eq.1 and eq. 2 as necessary) 𝜎𝑥+𝜎𝑦 𝜎𝑥−𝜎𝑦 + cos 2𝜃 + 𝜏𝑥𝑦 sin 2𝜃 2 2 𝜎 −𝜎 𝜏𝑛𝑡 = − 𝑥 𝑦 𝑠𝑖𝑛(2𝜃) + 𝜏𝑥𝑦𝑐𝑜𝑠(2𝜃) 2

Eq. 1

𝜎𝑛 =

Eq. 2 Eq. 5

tan 2𝜃𝑝 = 2𝜏𝑥𝑦 /(𝜎𝑥 − 𝜎𝑦 )

Eq. 6

𝜎1 , 𝜎2 , 𝜎3 =

Eq. 7

tan 2𝜃𝑠 = −

Eq. 8

𝜎𝑎𝑣𝑒 =

Eq. 9

𝜎𝑥+𝜎𝑦 2

𝑅 = 𝜏𝑚𝑎𝑥 = 𝑖𝑝

𝜎𝑥 +𝜎𝑦

2 𝜎𝑥−𝜎𝑦 2

± 𝑅, 0 /𝜏𝑥𝑦

𝜎𝑥−𝜎𝑦 2 2

+ 𝜏𝑥𝑦2

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

Problem 17.1 Stress Transformation Objectives & Review

Lecture #14 Combined Loadings Problems Homework

Determine the stresses at the principal plane and at the maximum in-plane shear stress plane

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

Problem 17.2 Stress Transformation Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The general state of plane stress at a point on the lever is shown. Determine the stresses at the principal plane and at the maximum in-plane shear stress plane

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Perform stress transformation to identify the orientation of elements upon which the maximum stresses act

Recommended Analysis Objectives & Review

1.

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

2.

Calculate the stresses at the desired point [general state of stress] Perform stress transformation (using Mohr’s Cirlce) 1. 2. 3.

4.

Plot the given stresses (V and H) and locate the Center of the circle Calculate the Radius Locate the principal plane Determine: a. Principal Stresses b. Maximum in-plane shear stress c. Stress at any arbitrary point

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

Problem 18.1 – Mohr’s Circle Objectives & Review

Lecture #14 Combined Loadings Problems Homework

Determine the stresses at the principal plane and at the maximum in-plane shear stress plane

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

Problem 18.2 – Mohr’s Circle Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The general state of plane stress at a point on the lever is shown. Determine the stresses of the following planes: a. b.

Principal plane and at the maximum in-plane shear stress plane A plane oriented 30° clockwise from the position shown.

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

Problem 19.1 – Absolute Maximum Stress Objectives & Review

Determine the absolute maximum stress

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

Problem 19.2 – Absolute Maximum Stress Objectives & Review

Lecture #14 Combined Loadings Problems Homework Lecture #15 Stress Transformation Problems

The state of stress at a point in a member is shown on the element. Use and draw the Mohr’s Circle to answer the following questions: a. b.

Calculate the maximum stresses (i.e. principal and absolute maximum shear stress) Calculate the stress components acting on the inclined plane AB and show the results on a sketch

Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

12

Problem 19.3 – Absolute Maximum Stress Objectives & Review

Lecture #14 Combined Loadings Problems Homework

The general state of plane stress at a point on the lever is shown. Determine the absolute maximum stress

Lecture #15 Stress Transformation Problems Lecture #16 Principal Stresses and Maximum In-Plane Shear Stresses Problems Lecture #17 Mohr's Circle Problems Lecture #18 Absolute Maximum Shear Stress Problems

Mohr’s Circle: Principal, maximum, in-plane and absolute maximum stresses

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