FramesME 202

1

Multi-force Member

A multi-force member has either

• three or more forces acting on it,

• or two or more forces AND one or more moments acting on it.

2

Frames & Machines

A frame/machine

• has at least one multi-force member,

• is stationary/non-stationary,

• and is designed to support/transmit loads.

3

Why?

• By definition, a truss is not a frame, because ....

• a truss has only two-force members and a frame must have at least one multi-force member.

4

Frame Example

5

Frame Example

6

Frame Example

7

Frame Example

8

(Online Homework 2-2-3)

Frame Example

9

(Test Two, Problem 2)

Analysis of Frames

• Because some forces are not directed along members,

• the method of joints is not sufficient;

• the method of sections is not sufficient;

• but Newton’s first and third laws are still sufficient.

10

11

x

y+

Statically indeterminate: 3 equations 4 unknowns

Let’s try to solve without recognizing that BD and ABC are a 2-force member and a 3-force member, respectively.

Ax

Ay

Dy

Dx

1000 lb

12

Statically determinate: 6 equations 6 unknowns

Now, let’s draw a separate FBD for each member and recount equations and unknowns.

x

y+

Ax

Ay

Dy

Dx

1000 lb

Bx

Bx

By

By

Ax = 3333 lbAy = −1500 lbBx = −3333 lbBy = 2500 lbDx = −3333 lbDy = 2500 lb

13

Observation

Sometimes the only way to find all the loads on a frame is to draw a separate FBD for each member.

14

ABC is now statically determinate, because it has only three unknowns.

Now, let’s use the fact that BD is a 2-force member.

x

y

ij

k^

^^

+

Ax

Ay

1000 lb FBDFBD

FBD

15

Now, let’s use the fact that ABC is a 3-force member.

MB∑ = 0 ⇒1000 6( )+ Ay 4( ) = 0Ay = −1500 lb

Ax = − 104.5

Ay =10000

3 lb

Fx∑ = 0 ⇒ Bx + Ax = 0 ⇒ Bx = −100003

lb

By = − 34Bx = 2500 lb

h6= 34⇒ h = 4.5 '

1000 lb

x

y

ij

k^

^^

+

FBD FA

h

Observation

Using what we know about two-force and three-force members reduces the number of force and moment summations required to find the unknown loads.

16

Frames

2 While it would sound reasonable to say that a two-force member is a multi-force member, that would contra-

dict the accepted usage of the term “multi-force member.” A two-force member is NOT a multi-force mem-

ber. (Although a two-force member has forces applied at two points, the member carries only one force, ei-

ther tension or compression, which is the same everywhere on the line connecting those two points.) Be-

cause a truss is a structure that has only two-force members, a truss is neither a frame nor a machine.

3 Unlike trusses, which have only two-force members, a frame or machine has at least one multi-force member.

Like a truss, a frame supports loads, which are forces and/or moments, and does not move. But a frame has at

least one multi-force member. That is also true for a machine, but a machine moves and is also called a

“mechanism.”

4 Given the previous two slides, this slide should be self-explanatory. Knowing whether a structure can be

treated as a truss is important. Trusses can be analyzed by the method of joints or the method of sections.

But these methods are not sufficient for analyzing a frame. If we were to try to use truss methods to analyze a

frame, we would be wasting time, which would not be efficient. It would also be ineffective, because we

would not be able to find the internal loads of the frame.

Page ��� of ���1 4

5 A FBD of member AB would have forces at only two points, namely A and B. But member AB also has an

applied moment, and so is a multi-force member.

Because a FBD of member BC would have forces at three points, member BC is a multi-force member.

The presence of even one multi-force member means that this structure is a frame, not a truss.

6 Although BD is a two-force member, ABC is a three-force member. It follows that the structure is a frame and

not a truss.

7 Although the link connecting B to the wall is a two-force member, ABD is a multi-force member. It follows

that the structure is a frame, and not a truss.

8 Because of the weight acting through G and the forces at both A and B, the bed of the truck is a multi-force

member and the structure is a frame, and not a truss.

9 Although the links connecting A and B to the ground are both two-force members, CAB is a multi-force mem-

ber. The structure is a frame, and not a truss.

Page ��� of ���2 4

10 The method of joints and the method of sections are each sufficient for analyzing trusses, which have only

two-force members. But, as mentioned in the commentary on slide 4, neither of these methods is sufficient

for analyzing a frame.

11 An overall FBD shows that this structure is statically indeterminate. But member BD is a two-force member,

so that we know the direction of the force at D. This means that we know the ratio of the components ��� and

��� , so that the only unknown at D is the magnitude of the force there. But as will be seen in the next slide,

even without using what we know about two-force members we can still find all of the loads.

12 We can write three independent equations of statics for each of the two FBDs. That gives us six equations,

which allows us to find all six unknowns. You should carry out the analysis to be sure that you can get the re-

sults shown here.

13 When a frame is statically indeterminate, we may or may not be able to find all of the reactions and internal

loads. But to find all that we can, we may need a separate FBD of each member of the frame.

Dx

Dy

Page ��� of ���3 4

14 Because ABC is now statically determinate, we can solve for all loads that act on it. And because one of

those is the force in BD, we can find that force without analyzing a FBD of member BD. The FBD of ABC is

all we need.

15 By using what we know about three-force members, we can find all of the unknowns with only one moment

summation and one force summation. The rest comes from trigonometry.

16 Using what we know about two-force and three-force members will often allow us to find unknowns by

trigonometry rather than by writing additional force or moment summations. For most of us, this is helpful.

Those who are weak in trigonometry might disagree.

Page ��� of ���4 4