Lecture Strength of Materials I: Chapter 2 - PhD. Tran Minh Tu

Chapter 2 - Axial force, shear force and bending moment. The following will be discussed in this chapter: Internal stress resultants; relationships between loads, shear forces, and bending moments; graphical method for constructing shear and moment diagrams; normal, shear force and bending moment diagram of frame.

STRENGTH OF MATERIALS

TRAN MINH TU -University of Civil Engineering, Giai Phong Str 55, Hai Ba Trung Dist Hanoi, Vietnam

CHAPTER

Axial Force, Shear Force and

Bending Moment

2.1 Introduction

2.2 Internal Stress Resultants

2.3 Example

2.4 Relationships between loads,

shear forces, and bending moments 2.5 Graphical Method for Constructing Shear

and Moment Diagrams 2.6 Normal, Shear force and

bending moment diagram of frame

Contents

2.1 Introduction

- Structural members are usually classified

according to the types of loads that they

support

- Planar structures: if they lie in a single

plane and all loads act in that same plane

2.1.1 Support connections.

- Structural members are joined together in various ways depending onthe intent of the designer The three types of joint most often specifiedare the pin connection, the roller support, and the fixed joint

H

- Pin support: prevents the translation at the

end of a beam but does not prevent the

rotation

- Roller support: prevents the translation in

the vertical direction but not in the horizontal

direction, and does not prevent the rotation

2.1 Introduction

A

A V

M

2.1 Introduction

1/10/2013 9

2.1.2 Types of beams

2.1 Introduction

2.2 Internal Stress Resultants

y

z

x Mx

My

NZ Qy

In general, internal stress resultants

(internal forces) consist of 6 components

• Nz – Normal force

• Qx, Qy – Shear forces

• Mx, My – Bending moments

• Mz – Torsional moment

 Planar structures: if they lie in a

single plane and all loads act in that

same plane => Only 3 internal stress

resultants exert on this plane (zoy)

z

x Mx

NZ Qy

• Nz – axial force (N);

• Qy – shear force (Q);

• Mx - bending moment (M)

1/10/2013 11

2.2 Internal force Resultants

 To determine the internal force resultants => Using the method of sections.

Q N

Q

 Sign convention:

2.2 Internal force Resultants

N

• Axial force: positive when outward

of an element, negative when

inward of an element

• Shear force: positive when acts

clockwise against an element,

negative when acts counterclockwise

against an element

• Bending moment: positive when

compresses the upper part of the

compresses the lower part of the

beam

1/10/2013 13

2.2 Axial, Shear and Moment diagram

• Because of the applied loadings, the beams develop an internalshear force and bending moment that, in general, vary from point topoint along the axis of the beam In order to properly design a beam ittherefore becomes necessary to determine the maximum shear andmoment in the beam

• One way to do this is expressing N, Q and M as the functions of theirarbitrary position z along the beam’s axis These axial, shear andmoment functions can then be plotted and represented by graph calledthe axial, shear and moment diagram

2.1.3 Axial, Shear and Moment diagram

2.2 Axial, Shear and Moment diagram

1/10/2013 15

Example 2.1: Draw the shear

and moment diagram for

the beam shown in the

V A z 1 Q

M N

V B

z 2 Q

M N

Segment AC

Segment BC

2.3 Example

a+b Fa

The section on which the

concentrated force acts, the

“jump”

Example 2.2: Draw the shear and

moment diagram for the beam

shown in the figure

A

q l V

B

q l V

The section on which the shear

force is equal to zero then the

bending moment is maximum

x B

Comment 3

The section on which the

concentrated moment acts,

the bending moment diagram

has “jump”

2.4 Relationships between transverse loads, shear forces,

and bending moments

- Consider the beam shown in the figure,

which is subjected to an arbitrary loading

segment dz of the beam:

- Positive distributed load q(z): acts

upward on the beam

q(z) > 0

1/10/2013 23

2.4 Relationships between loads, shear forces,

and bending moments

Application:

- Recognizing the type of Q and M diagrams when the distributedload’s function is known, i e if the distributed load’s function is n-degree, then the shear force’s function will be (n+1)-degree and thebending moment function will be (n+2)-degree

- The section, on which the shear force is equal to zero then thebending moment is maximum

- Determining Q, M on the arbitrary section, when knowing the value of

Q and M on the specific section

•Qright = Qleft + Sq ( Sq – Area of distributed load diagram q)

•Mright = Mleft + SQ ( SQ – Area of shear force diagram Q)

2.4 Relationships between loads, shear forces,

and bending moments

1/10/2013 25

2.4 Relationships between loads, shear forces,

and bending moments

2.5 Graphical Method for Constructing Shear

and Moment Diagrams

- Base on the relationships between loads, shear force, and bendingmoments

- Knows the distributed load q(z) => Predict the types of shear forceand bending moment diagram => Indentify the necessary number ofpoints to construct the diagram

Mmax=25qa2/18

4qa2/3

2.5 Graphical Method for Constructing Shear

and Moment Diagrams -Example

2.5 Graphical Method for Constructing Shear

and Moment Diagrams -Example

2.5 Graphical Method for Constructing Shear

and Moment Diagrams - Example

Example 2.5: Draw the shear force

and bending moment diagram for the

compound beam shown in the figure:

2.5 Graphical Method for Constructing Shear

and Moment Diagrams -Example

2.5 Graphical Method for Constructing Shear

and Moment Diagrams -Example

2.5 Graphical Method for Constructing Shear

and Moment Diagrams -Example

3.) The Shear force and bending

moment diagram of a system of

the beams

1/10/2013 33

2.6 Normal, Shear force and

bending moment diagram of frame

- The frame is composed of several connected members that are fixedconnection The design of these structures often requires drawing theshear and moment diagram for each of the members

- Using a method of section, we determine the axial force, the shearforce, and the bending moment acting on each members

- Always draw the moment diagram on the tensile side of the member

2.6 Normal, Shear force and

bending moment diagram of frame

VD

VA

HAa

0 A 5( )

X   H  F kN



1 1 1 .1 0

1/10/2013 35

2.6 Normal, Shear force and

bending moment diagram of frame

11

1

+

+

NkN

3 Shear force and bending moment diagram

kNm

2.6 Normal, Shear force and

bending moment diagram of frame

+

5

QkNM

1/10/2013 37

Draw the shear and moment diagram for the beam shown in thefigure

2.7 Home works

2.7 Homework

Draw the shear and moment diagram for the beam shown in thefigures

1/10/2013 39

Draw the shear and moment diagram for the compound beam shown

in the figures

2.7 Homework

THANK YOU FOR YOUR ATTENTION !