Chapter 9 combined stresses
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Chapter 9 Combined Stresses. 9-1 Introduction. Basic types of loading: axial, torsional and flexural Stress formulas: Axial loading - Torsional loading - Flexural loading -. 9-2 Combined Axial & Flexural Loads.

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Chapter 9 Combined Stresses

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Chapter 9 combined stresses

Chapter 9Combined Stresses


9 1 introduction

9-1 Introduction

  • Basic types of loading: axial, torsional and flexural

  • Stress formulas:

    Axial loading -

    Torsional loading -

    Flexural loading -


9 2 combined axial flexural loads

9-2 Combined Axial & Flexural Loads


Chapter 9 combined stresses

For stiff members the formula is appropriate

For long slender members or columns, the effect of P-d is significant


Chapter 9 combined stresses

Hw10

sallow

B

D2

D1

Fig. P-908

ค่า z1-z6 ได้จากเลขประจำตัวนิสิต ดังต่อไปนี้

46z1z2z3z4z5z6

D1=(1+z1) in. D2 = D1(1+z2) in.

I1-1=1000(1+z3) in4 Area=10(1+z4) in2

B =10(1+z5) in. sallow=10(1+z6) ksi.

หมายเหตุD2= D1(1+z2) in.

เพื่อให้หน้าตัดมีประสิทธิภาพดีในการรับหน่วยแรง


Chapter 9 combined stresses

Hw11

L4

L2

L3

b

h

L1

ค่า z1-z6 ได้จากเลขประจำตัวนิสิต ดังต่อไปนี้

46z1z2z3z4z5z6

L1= (1+z1) in. L2 = (1+z2) in.

L3= (1+z3) in. L4 = (1+z4) in.

b = 0.2(1+z5) in. h = b(1+z6) in.

P =(1+z5) kips. F =(1+z6) kips.

หมายเหตุh = b(1+z6) in.

เพื่อให้คานมีความลึกไม่น้อยกว่าความกว้างเสมอ


9 3 kern of section loads applied off axes of symmetry

9-3 Kern of Section: Loads Applied off Axes of Symmetry


Chapter 9 combined stresses

The maximum eccentricity to avoid tension

The general case:

The position of neutral axis (line of zero stress)

That is in designing of masonry or other structures weak in tension, the resultant load should fall in the middle third of the section.


Chapter 9 combined stresses

918

A compressive load P= 12 kips is applied, as in Fig. 9-8a, at a point 1 in. to the right and 2 in. above the centroid of a rectangular section for which h=10 in. and b=6 in. Compute the stress at each corner and the location of the neutral axis. Illustrate the answers with a sketch similar to Fig. 9-8b.


Chapter 9 combined stresses

N.A.


Chapter 9 combined stresses

921

Calcualte and sketch the kern of a W360 X 122 section.


9 4 variation of stress with inclination of element

9-4 Variation of Stress with Inclination of Element


9 5 stress at a point

9-5 Stress at A Point

Stress at a point really defines the uniform stress distributed over a differential area.


Chapter 9 combined stresses

  • The most general state of stress at a point may be represented by 6 components,

symmetry

state of stress เมื่อแสดงด้วยระบบโคออร์ดิเนต (xyz)

symmetry

state of stressเมื่อแสดงด้วยระบบโคออร์ดิเนต (xyz)


Chapter 9 combined stresses

  • Plane Stress - state of stress in which two faces of the cubic element are free of stress. For the illustrated example, the state of stress is defined by

  • State of plane stress occurs in a thin plate subjected to forces acting in the midplane of the plate.

  • State of plane stress also occurs on the free surface of a structural element or machine component, i.e., at any point of the surface not subjected to an external force.


Chapter 9 combined stresses

Plane Stress

Two methods to compute the maximum stresses i.e.,

  • Analytical approach

  • Using of Mohr’s circle


9 6 variation of stress at a point analytical derivation

9-6 Variation of Stress at A Point: Analytical Derivation


Chapter 9 combined stresses

Find maximum or minimum s differentiating Eq.(9-5) w.r.t. q and setting the derivative equal to zero

Find maximum or minimum t differentiating Eq.(9-6) w.r.t. q and setting the derivative equal to zero

Eq.(9-5)

Eq.(9-6)


Chapter 9 combined stresses

At zero shearing stress t = 0

Eq.(9-5)

Eq.(9-6)

ซึ่งเป็นมุมเดียวกับสมการ Eq.(9-7)ดังนั้น ค่า maximum or minimumsจะเกิดขึ้นเมื่อ t = 0


Chapter 9 combined stresses

มุมqและ qs ต่างกัน 45O

Maximum or minimum t

Maximum or minimum s (Principal stresses)


9 7 variation of stress at a point mohr s circle

9-7 Variation of Stress at A Point: Mohr’s Circle

Otto Mohr (1882)

Eq.(9-5)

Eq.(9-6)

Eq.(a)2 + Eq.(b)2


Chapter 9 combined stresses

x-axis

y-axis

Rule for Applying Mohr Circle to Combined Stresses


Chapter 9 combined stresses

x-axis

C

y-axis


Chapter 9 combined stresses

x-axis

n-axis

R

2q

q

C

y-axis


Chapter 9 combined stresses

x-axis

n-axis

R

2q

q

C

y-axis


Chapter 9 combined stresses

R

2q1

x-axis

C

y-axis

2q2


Chapter 9 combined stresses

y-axis

R

C

2q1

x-axis


Chapter 9 combined stresses

y-axis

R

60o

C

x-axis

45o


9 8 absolute maximum shearing stress

s2

s1

s2

s1

s1

s2

9-8 Absolute Maximum Shearing Stress

Mohr’s circle: Rotation around z-axis


Chapter 9 combined stresses

s2

s1

Mohr’s circle: Rotation around x-axis

Mohr’s circle: Rotation around y-axis


Chapter 9 combined stresses

s2

s1

s2

s1


Chapter 9 combined stresses

s2

s1

Mohr’s circles for plane stress

Absolute maximum shearing stress for plane stress is equal to the largest of the following three values


Chapter 9 combined stresses

s2

s1

s3

z

Mohr’s circles for general state of stress

Absolute maximum shearing stress for general state of stress is equal to the largest of the following three values


Chapter 9 combined stresses

20

50

Maximum in-plane shearing stress =

Absolute maximum shearing stress is the largest of


Chapter 9 combined stresses

20

50

Ex.

Maximum in-plane shearing stress =

Absolute maximum shearing stress is the largest of


Chapter 9 combined stresses

the figure

( สำหรับข้อนี้ให้คำนวณ ค่าabsolute maximum shearing stress ด้วยโดยกำหนดให้ sz=0 )

Hw17

ค่าz1-z3ได้จากเลขประจำตัวนิสิต ดังต่อไปนี้46xxxz1z2z3


9 9 application of mohr s circle to combined loadings

Combined stresses

Mohr’s Circle

Design Criteria,

y-axis

Principal stresses and, Maximum shearing stress

x-axis

9-9 Application of Mohr’s Circle to Combined Loadings

Combined Loadings (axial, torsional, flexural)


Stress trajectories

Stress Trajectories


Torsional failure modes

  • Ductile materials generally fail in shear.Brittle materials are weaker in tension than shear.

  • A ductile specimen breaks along a plane of maximum shear

  • A brittle specimen breaks along planes perpendicular to s1

Torsional Failure Modes

45o


Chapter 9 combined stresses

Stress Trajectories for Torsion

Stress Trajectories: lines of principal stress direction but of variable stress intensity


Chapter 9 combined stresses

y-axis

x-axis

Mohr’s Circle

Stress Trajectories for Beam


Chapter 9 combined stresses

Mohr’s Circle


Chapter 9 combined stresses

Mohr’s Circle


Chapter 9 combined stresses

If


Chapter 9 combined stresses

BMzD

TMD

BMyD


Chapter 9 combined stresses

E

D

C

B

BMyD

A

BMzD

TMD

|M|

D

A

B

E

C

Cross section of solid shaft

and the resultant moment


Chapter 9 combined stresses

At section C

BMyD

BMzD

From Prob. 951 and this problem.

Mohr’s Circle

y-axis

TMD

At section D

|M|

D

A

B

E

C

x-axis


Chapter 9 combined stresses

state of stress on the element on the surface of vessel


Chapter 9 combined stresses

Absolute maximum shearing stress


Chapter 9 combined stresses

Mohr’s Circle at point A

x-axis

y-axis


Chapter 9 combined stresses

Mohr’s Circle at point B

x-axis

y-axis


Chapter 9 combined stresses

Hw18

ค่า z1-z5 ได้จากเลขประจำตัวนิสิต ดังต่อไปนี้46xz1z2z3z4z5

L1= 4(1+z1) in. L2 = 4(1+z2) in.

L3= 4(1+z3) in. L4 = 4(1+z4) in.

D= 4(1+z5) in.


Chapter 9 combined stresses

Hw19

Also find the maximum shearing stress at point A. Show your results on a complete sketch of a differential element.

ค่า z1-z4 ได้จากเลขประจำตัวนิสิต ดังต่อไปนี้46xxz1z2z3z4

L= 0.4(1+z1)m. P= 4(1+z2)kN

H= 40(1+z3)mm. W= 40(1+z4)mm


Chapter 9 combined stresses

http://www.kyowa-ei.co.jp/english/products.htm


Chapter 9 combined stresses

Strain and deformation of line element


Chapter 9 combined stresses

Eq.(9-5)

Eq.(9-6)


Chapter 9 combined stresses

If we use the stress-strain relation directly the same answer can be obtained


Chapter 9 combined stresses

จงพิสูจน์ สมการ (9-19) (9-20) ด้วยภาษาของตัวเอง

Hw20a

Hw20b

Hw21

ค่า z1-z3 ได้จากเลขประจำตัวนิสิต ดังต่อไปนี้46xxxz1z2z3

ea= 100(1+z1)eb= -100(1+z2)

ec= 100(1+z3)


Chapter 9 combined stresses

ปริมาณทางPhysics สามารถแทนด้วยTensor

Order 0 = zero order Tensor (Scalar) – Magnitude (มวล, ความหนาแน่น)

Order 1 = first order Tensor (Vector) – Magnitude, Direction (ความเร็ว, แรง)

Order 2 = second order Tensor – Magnitudes, Directions(stress, strain)

… Higher order ….

ปริมาณทางPhysics ไม่เปลี่ยนแปลงไปตามระบบโคออร์ดิเนตที่ใช้ในการวัด


Chapter 9 combined stresses

ปริมาณทางPhysics ไม่เปลี่ยนแปลงไปตามระบบโคออร์ดิเนตที่ใช้ในการวัด

แรงยังคงมีขนาดและทิศทางเท่าเดิม ไม่ว่าจะแสดง componentของเวคเตอร์ด้วยระบบโคออร์ดิเนตอื่น

สถานะของหน่วยแรง (state of stress)ยังคงมีคุณสมบัติเหมือนเดิม ไม่ว่าจะแสดงด้วยระบบโคออร์ดิเนตอื่น


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