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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

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 -



For stiff members the formula is appropriate

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


Hw10 appropriate

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


Hw11 appropriate

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



The maximum eccentricity to avoid tension appropriate

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.


918 appropriate

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.


N.A. appropriate


921 appropriate

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



9 5 stress at a point
9-5 Stress at A Point appropriate

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


symmetry

state of stress เมื่อแสดงด้วยระบบโคออร์ดิเนต (xyz)

symmetry

state of stressเมื่อแสดงด้วยระบบโคออร์ดิเนต (xyz)


  • Plane Stress represented by 6 components, - 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.


Plane Stress represented by 6 components,

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

  • Analytical approach

  • Using of Mohr’s circle



Find maximum or minimum represented by 6 components,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)


At zero shearing stress represented by 6 components,t = 0

Eq.(9-5)

Eq.(9-6)

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


มุม represented by 6 components,qและ 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 represented by 6 components,

Otto Mohr (1882)

Eq.(9-5)

Eq.(9-6)

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


x represented by 6 components,-axis

y-axis

Rule for Applying Mohr Circle to Combined Stresses


x represented by 6 components,-axis

C

y-axis


x represented by 6 components,-axis

n-axis

R

2q

q

C

y-axis


x represented by 6 components,-axis

n-axis

R

2q

q

C

y-axis


R represented by 6 components,

2q1

x-axis

C

y-axis

2q2


y represented by 6 components,-axis

R

C

2q1

x-axis


y represented by 6 components,-axis

R

60o

C

x-axis

45o


9 8 absolute maximum shearing stress

s represented by 6 components,2

s1

s2

s1

s1

s2

9-8 Absolute Maximum Shearing Stress

Mohr’s circle: Rotation around z-axis


s represented by 6 components,2

s1

Mohr’s circle: Rotation around x-axis

Mohr’s circle: Rotation around y-axis


s represented by 6 components,2

s1

s2

s1


s represented by 6 components,2

s1

Mohr’s circles for plane stress

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


s represented by 6 components,2

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


20 represented by 6 components,

50

Maximum in-plane shearing stress =

Absolute maximum shearing stress is the largest of


20 represented by 6 components,

50

Ex.

Maximum in-plane shearing stress =

Absolute maximum shearing stress is the largest of


the figure represented by 6 components,

( สำหรับข้อนี้ให้คำนวณ ค่าabsolute maximum shearing stress ด้วยโดยกำหนดให้ sz=0 )

Hw17

ค่าz1-z3ได้จากเลขประจำตัวนิสิต ดังต่อไปนี้46xxxz1z2z3


9 9 application of mohr s circle to combined loadings

Combined stresses represented by 6 components,

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 represented by 6 components,


Torsional failure modes

  • A ductile specimen breaks along a plane of maximum shear

  • A brittle specimen breaks along planes perpendicular to s1

Torsional Failure Modes

45o


Stress Trajectories for Torsion represented by 6 components,

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


y- represented by 6 components,axis

x-axis

Mohr’s Circle

Stress Trajectories for Beam


Mohr’s Circle represented by 6 components,


Mohr’s Circle represented by 6 components,


If represented by 6 components,


BM represented by 6 components,zD

TMD

BMyD


E represented by 6 components,

D

C

B

BMyD

A

BMzD

TMD

|M|

D

A

B

E

C

Cross section of solid shaft

and the resultant moment


At section C represented by 6 components,

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



Absolute maximum shearing stress represented by 6 components,


Mohr’s Circle represented by 6 components, at point A

x-axis

y-axis


Mohr’s Circle represented by 6 components, at point B

x-axis

y-axis


Hw1 represented by 6 components,8

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.


Hw1 represented by 6 components,9

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



Strain and deformation of line element represented by 6 components,


Eq.(9-5) represented by 6 components,

Eq.(9-6)



จงพิสูจน์ สมการ answer can be obtained(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)


ปริมาณทาง answer can be obtainedPhysics 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


ปริมาณทาง answer can be obtainedPhysics ไม่เปลี่ยนแปลงไปตามระบบโคออร์ดิเนตที่ใช้ในการวัด

แรงยังคงมีขนาดและทิศทางเท่าเดิม ไม่ว่าจะแสดง componentของเวคเตอร์ด้วยระบบโคออร์ดิเนตอื่น

สถานะของหน่วยแรง (state of stress)ยังคงมีคุณสมบัติเหมือนเดิม ไม่ว่าจะแสดงด้วยระบบโคออร์ดิเนตอื่น


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