External Effects on Beams Internal Effects Sign...

Outline:

Review – External Effects on Beams

Beams – Internal Effects

Sign Convention

Shear Force and Bending Moment Diagrams

(text method)

Relationships between Loading, Shear Force

and Bending Moments (faster method for

SFD & BMD)

ENGR 1205 1 Chapter 10

Beams are bars of material that support

Beams are common structural members

Beams can support both concentrated and

distributed loads

We will only analyze statically determinate

beams in this class

ENGR 1205 2 Chapter 10

Typical Beam Configurations

ENGR 1205 3 Chapter 10

Beams can resist:

Tension / Compression

(happy face / frown face)

Shear force

Bending moment

Torsional moment

The amount of each kind of

internal load can change

throughout the length of

the beam depending on

external loads ENGR 1205 4 Chapter 10

We can ‘cut’ the beam at any point along its

length to analyze the internal forces at that

point

It is often difficult to tell the direction of the

shear and moment without calculations so

represent Vy and Mbz in their positive directions

and let the result tell you (+/-) whether you

drew it the right way

CONVENTION

ENGR 1205 5 Chapter 10

In North America, if the

moment tends to cause the

beam to curve

it is positive; if the moment

tends to cause it to curve

it is negative.

ENGR 1205 6 Chapter 10

The bending moment

is positive if

the upper side is in

it

it’s a

ENGR 1205 7 Chapter 10

A positive shear force at the

cut causes the beam to

rotate clockwise and

matches the “smiley face” shape

Positive Shear and Moment on a whole Beam

Positive Shear and Moment on a cut-out Beam

Note: Coming from the left or the right makes

a difference, so always start from left.

ENGR 1205 8 Chapter 10

The maximum bending moment (magnitude

– could be negative or positive) is often the

primary consideration in the design of a

beam

Variations in shear and moment are best

shown graphically

ENGR 1205 9 Chapter 10

Use equilibrium of whole beam to find

external reactions and supports

Then cut each section of the the beam and

replace any distributed loads with equivalent

concentrated loads and solve for unknown

axial, shear and moment at point x (your cut)

Plot Vy (shear force vs x) and Mbz (bending

moment vs x) along the beam

ENGR 1205 10 Chapter 10

Solve for R1 and R2

Isolate left section and solve for Vy

and Mbz between the left support

and 4kN load

Isolate right section and solve for

Vy and Mbz between the right

support and 4 kN load

Isolate a section between every

change in an external load, but do

not cut the section at the

concentrated load

ENGR 1205 11 Chapter 10

Draw the shear and bending moment diagrams

for the diving board, which supports the 80 kg

man. Determine the location and magnitude

of the maximum bending moment.

ENGR 1205 12 Chapter 10

ENGR 1205 Chapter 10 13

ENGR 1205 Chapter 10 14

Draw the shear and bending moment diagrams

for the beam and loading shown and

determine the location and magnitude of the

maximum bending moment.

ENGR 1205 15 Chapter 10

ENGR 1205 Chapter 10 16

Draw the shear and bending moment diagrams

for the beam and loading shown.

ENGR 1205 17 Chapter 10

Diagrams illustrate the value of the internal

forces (axial/shear/moment) that occur at

each point along a structure (beam).

Additionally,

ENGR 1205 18 Chapter 10

The shear diagram is the graphic

representation of the shear force at

successive points along the beam.

The shear force (Vy) at any point is equal to

the algebraic sum of the external loads and

reactions to the left of that point.

Since the entire beam must be in equilibrium

(the sum of Fy = 0), the shear diagram must

close at zero.

ENGR 1205 19 Chapter 10

The moment diagram is the graphical

representation of the magnitude of the

bending moment at successive points along the

beam.

The bending moment for the moment diagram

(Mbz) at any point equals the sum of moments

of the forces on the beam to the left about

that point.

Since the entire beam is in equilibrium (Sum of

M = 0), the bending moment diagram must

close to zero at right side.

ENGR 1205 20 Chapter 10

Cut the beam between each concentrated

load.

For each section solve for the unknown shear

force and bending moment

Equation for each in terms of x (the

distance along the beam

Sub in endpoint values of x to get numerical

values of Vy and Mbz at each cut

Plot Vy (shear force vs x) and Mbz (bending

moment vs x) along the beam

ENGR 1205 21 Chapter 10

The “text” method works every time, but it is

time consuming

You can draw the diagrams from the FBD if

you know the relationships between w (load

intensity), V (shear force) and M (bending

moment).

the negative slope of the slope of the

the shear diagram at moment diagram at the

a given point equals given point is shear

the load at the point at the point

ENGR 1205 22 Chapter 10

dx

dMV

dx

dVw

If there is no change in the load along the length

under consideration, the shear curve is a straight

horizontal line (or a curve of zero slope).

If a concentrated load exists, then there is a

vertical jump in the SFD ( force = drop down on

SFD, force = step up on SFD)

If a load exists, and is uniformly distributed, the

slope of the shear curve is constant and non-

horizontal.

If a load exists, and increases in magnitude over

successive increments, the slope of the shear

curve is positive (approaches the vertical); if the

magnitude decreases, the slope of the shear curve

is negative (approaches the horizontal). ENGR 1205 23 Chapter 10

If the slope of the SFD is zero, then the moment curve

has a constant slope that is equal to the value of the

shear for that increment.

If the slope of the SFD is positive, then the slope of

the moment curve is getting steeper.

If the slope of the SFD is negative, then slope of the

moment curve is getting flatter.

Changes in the shear diagram will produce changes in

the shape of the moment curve.

The area under the shear curve between two points is

equal to the change in bending moment between the

same two points.

ENGR 1205 24 Chapter 10

The area of the shear diagram to the left or to

the right of the section is equal to the moment

at that section.

The slope of the moment diagram at a given

point is the shear at that point.

The maximum moment occurs at the point of

zero shear. When the shear (also the slope of

the moment diagram) is zero, the tangent

drawn to the moment diagram is horizontal.

ENGR 1205 25 Chapter 10

ENGR 1205 26 Chapter 10

ENGR 1205 27 Chapter 10

Draw the shear and bending moment diagrams

for the beam and loading shown and

determine the location and magnitude of the

maximum bending moment.

ENGR 1205 28 Chapter 10

ENGR 1205 Chapter 10 29

ENGR 1205 Chapter 10 30

Draw the shear and bending moment diagrams for

the beam and loading shown and determine the

location and magnitude of the maximum bending

moment.

ENGR 1205 31 Chapter 10

ENGR 1205 Chapter 10 32

ENGR 1205 Chapter 10 33

ENGR 1205 Chapter 10 34

Draw the shear and bending moment diagrams

for the beam and loading shown.

ENGR 1205 35 Chapter 10