Shear Forces And Bending Moments

1 Introduction

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Fig. 4-1 Examples of beams subjected to lateral loads.

Beam : planar structure

plane of bending : If all deflection occur in that plane.

4.2 Type of beams, loads, and reaction

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Fig. 4-2 Types of beams: (a) simple beam, (b) cantilever beam, and (c) beam with an

overhang

Simply supported beam (simple beam)

: a beam with a pin support at one end and a roller support at other.

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Fig. 4-3 Beam supported on a wall: (a) actual construction, and (b) representation as a roller support. Beam-to-column connection: (c) actual construction, and (d) representation as a pin support. Pole anchored to a concrete pier: (e) actual construction, and (f) representation as a fixed support.

■ Types of loads

concentrated load :

distribution load (uniformly distributed load) (uniform load) , linearly varying :

moment (couple) :

■ Reactions As an example, let us determine the reactions of the simple beam AB of Fig.4-2a.

?

As a second example , consider the cantilever beam of Fig.4-2b.

As a third example : The beam with an overhang(Fig.4-2c)

4.3 Shear forces and bending moments

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Fig. 4-4 Shear force V and bending moment M in a beam.

or

or

■ Sign Conventions

Fig. 4 - 5 Sign conventions for shear force V and bending moment M .

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Fig. 4-6Deformations (highly exaggerated) of a beam element caused by (a) shear forces, and (b) bending moments.

Sign convention are called deformation sign convention because they are based

upon how the material is deformed.

By contrast, when writing equations of equilibrium we use static sign convention, in

which forces are positive or negative according to their directions along the

coordinate axes.

■ EX 4-1 A simple beam AB supports two loads, a force P and a couple , acting as shown in Fig. 4-7a. Find the shear force V and bending moment M in the beam at cross sections located as follows: (a) a small distance to the left of the midpoint of the beam, and (b) a small distance to the right of the midpoint of the beam.

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Fig. 4-7 Example 4-1. Shear forces and bending moments in a simple beam.

Solution

Reaction.

1) total free-body diagram

2) left-hand half of beam as the free body (Figure 4-7(b)).

(a)

(b) ∴

(c)

3) (Figure 4-7(c)).

(d,e)

■ Example 4-2

A cantilever beam that is free at end A and fixed at end B is subjected to a distributed load of linearly varying intensity ｑ (Fig. 4-8a). The maximum intensity of the load occurs at the fixed support and is equal to . ｑ0

Find the shear force V and bending moment M at distance x from the free end of the beam.

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Fig. 4-8 Example 4-2. Shear force and bending moment in a cantilever beam.

Solution (FIg 4-8(b)).

The intensity of the distribution load at distance x from the end is

total load :

∴ (4-2a) :

(4-1) :

A( ) : V= 0,

B( ) : (4-2b)

∴ (4-3a)

( ) : M= 0,

( ) : (4-3b)

a

■ EX 4-3

A simple beam with an overhang is supported at points A and B(Fig. 4-9a). A uniform load of intensity acts throughout the length of the beam and a concentrated load acts at a point 9ft from the left-hand support. The span length is 24ft and the length of the overhang is 6ft. Calculate the shear force V and bending moment M at cross section D located 15ft from the left-hand support.

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Fig. 4-9 Example 4-3. Shear force and bending moment in a beam with an overhang.

Solution

1) Reaction at entire beam

2)(Fig 4-9(b)).

∴

☆¤(Fig 4-9(c))

∴

4.4 Relationships Between Loads, Shear Force, and Bending Moment

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Fig. 4-10 Element of a beam used in deriving the relationships between loads, shear forces and bending moments (all loads and stress resultants are shown in their positive directions.)

■ Distributed Load(Fig 4-10(a))

(4-4)

,

If q=0 , then

and shear force is constant in that part of the beam.

If q=constant , then = constant

and shear force changes linearly in that part of the beam.

● Example

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Fig. 4-8 Example 4-2. Shear force and bending moment in a cantilever beam.

Taking the derivative gives

from

(a)

(4-5)

=-(Area of the loading diagram between A and B)

Let us now consider the moment equilibrium in Fig. 4-10a.

(4-6)

● Example

Again using the cantilever beam of Fig.4-8

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Fig. 4-8 Example 4-2. Shear force and bending moment in a cantilever beam.

from

(4-7)

=(area of the shear-force diagram between A and B)

a

■ Concentrated Loads(Fig. 4-10(b))

Now let us consider a concentrated load P acting on the beam element(Fig. 4-10(b))

From equilibrium of forces in the vertical direction, we get

(4-8)

From equilibrium of moments about the left-hand face of the element(Fig. 4-10(b)), we get

At the left-hand side

At the right-hand side

■ Loads in the form of couples(Fig.4-10(c))

From equilibrium of moments about the left-hand side of the element gives

(4-9)

4.5 Shear-force and bending-moment diagrams

■ Concentrated loads

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Fig. 4-11 Shear-force and bending moment diagrams for a simple beam with a concentrated load.

Diagrams showing the variation of N,V,M are very useful.

Because these diagrams quickly identify locations and values of maximum N, V, M needed for design.

with a concentrated load.

1) , (4-10 a, b)

( ) (4-11a,b)

( ) (4-12a)

(4-12b)

(4-13)

from

= 0 = V

A cross section where the shear force equals zero.

The maximum positive and negative bending moments in a beam may occur at the following places:

A cross section where a concentrated loads is applied and the shear force changes sign

A point of support where a vertical reaction is present A cross section where a couple is applied

■ Uniform load

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Fig. 4-12 Shear-force and bending moment diagrams for a simple beam with a uniform load.

= =

(4-14a)

(4-14b)

(4-15)

The maximum moment occurs where the shear force equals zero.

■ Several Concentrated loads

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Fig. 4-13 Shear-force and bending moment diagrams for a simple beam with several concentrated loads.

( ) (4-16a,b) ,

( ) (4-17a,b)

(4-18a)

( ) (4-18b)

( ) (4-18b)

(4-20a,b,c)

■ Ex 4-4

Draw the shear-force and bending-moment diagrams for a simple beam with a uniform load of intensity q acting over part of the span(Fig. 4-14a).

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Fig. 4-14 Example 4-4. Simple beam with a uniform load over part of the span.

Solution 1) Reaction.

(4-21a,b)

2) ( 0 < x < a ) (4-22a,b)

( ) (4-23a,b)

( ) (4-24a,b)

3) maximum bending moment

from V = 0 *( )

(4-25)

Now we substitute ----->

(4-26)

☆ Special case :

If ,

from (4-25) and (4-26)

(4-27a,b)

■ Example 4-5

Draw the shear-force and bending-moment diagrams for a cantilever beam with two concentrated loads(Fig. 4-15a)

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Fig. 4-15 Example 4-5. Cantilever beam with two concentrated loads

Solution

1)

(a,b)

2)

( 0 < x < a ) (c,d)

( a < x < L ) (e,f)

■ Example 4-6 A cantilever beam supporting a uniform load of constant intensity ｑ is shown in Fig. 4-16a. Draw the shear-force and bending-moment diagrams for this beam.

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Fig. 4-16 Example 4-6. Cantilever beam with a uniform load.

Solution

1)

(4-28a,b)

2)

(4-29a,b)

3) (4-30a,b)

☆¤

(g)

(h)

■ Example 4-7 A beam ABC with an overhang at the left-hand end is shown in Fig. 4-17a. The beam is subjected to a uniform load of intensity on the overhang AB and a counterclockwise couple acting midway between the supports at B and C. Draw the shear-force and bending-moment diagrams for this beam.

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Fig. 4-17 Example 4-7. Beam with an overhang.

Solution

1)

2)

The bending moment just to the left of the couple is

The bending moment just to the right of the couple is

The bending moment at the support C is as expected.