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 9Combined 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


For stiff members the formula is appropriate

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


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


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


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.


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.


N.A.


921

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


9-4 Variation of Stress with Inclination of Element


9-5 Stress at A Point

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


  • 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)


  • 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.


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


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)


At zero shearing stress t = 0

Eq.(9-5)

Eq.(9-6)

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


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

Otto Mohr (1882)

Eq.(9-5)

Eq.(9-6)

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


x-axis

y-axis

Rule for Applying Mohr Circle to Combined Stresses


x-axis

C

y-axis


x-axis

n-axis

R

2q

q

C

y-axis


x-axis

n-axis

R

2q

q

C

y-axis


R

2q1

x-axis

C

y-axis

2q2


y-axis

R

C

2q1

x-axis


y-axis

R

60o

C

x-axis

45o


s2

s1

s2

s1

s1

s2

9-8 Absolute Maximum Shearing Stress

Mohr’s circle: Rotation around z-axis


s2

s1

Mohr’s circle: Rotation around x-axis

Mohr’s circle: Rotation around y-axis


s2

s1

s2

s1


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


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


20

50

Maximum in-plane shearing stress =

Absolute maximum shearing stress is the largest of


20

50

Ex.

Maximum in-plane shearing stress =

Absolute maximum shearing stress is the largest of


the figure

( สำหรับข้อนี้ให้คำนวณ ค่าabsolute maximum shearing stress ด้วยโดยกำหนดให้ sz=0 )

Hw17

ค่าz1-z3ได้จากเลขประจำตัวนิสิต ดังต่อไปนี้46xxxz1z2z3


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


  • 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


Stress Trajectories for Torsion

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


y-axis

x-axis

Mohr’s Circle

Stress Trajectories for Beam


Mohr’s Circle


Mohr’s Circle


If


BMzD

TMD

BMyD


E

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

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


state of stress on the element on the surface of vessel


Absolute maximum shearing stress


Mohr’s Circle at point A

x-axis

y-axis


Mohr’s Circle at point B

x-axis

y-axis


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.


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


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


Strain and deformation of line element


Eq.(9-5)

Eq.(9-6)


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


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)


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


ปริมาณทางPhysics ไม่เปลี่ยนแปลงไปตามระบบโคออร์ดิเนตที่ใช้ในการวัด

แรงยังคงมีขนาดและทิศทางเท่าเดิม ไม่ว่าจะแสดง componentของเวคเตอร์ด้วยระบบโคออร์ดิเนตอื่น

สถานะของหน่วยแรง (state of stress)ยังคงมีคุณสมบัติเหมือนเดิม ไม่ว่าจะแสดงด้วยระบบโคออร์ดิเนตอื่น


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