Title: Engineering Mechanics: STATICS
1Engineering MechanicsSTATICS
- Anthony Bedford and Wallace Fowler
- SI Edition
Teaching Slides Chapter 6 Structures in
Equilibrium
2Chapter Outline
- Trusses
- The Method of Joints
- The Method of Sections
- Space Trusses
- Frames Machines
- Computational Mechanics
36.1 Trusses
- Truss structures such as the beams supporting a
house
- Starting with very simple examples
- 3 bars pinned together at their ends to form a
triangle with supports as shown ? structure that
will support a load F
46.1 Trusses
- Construct more elaborate structures by adding
more triangles - The bars are the members of these structures
the places where the bars are pinned together are
called joints - Even though these examples are quite simple, they
begin to resemble the structures used to support
bridges roofs of houses
56.1 Trusses
- If these structures are supported loaded at
their joints we neglect the weight of the bars,
each bar is a 2-force member - We call such a structure a truss
66.1 Trusses
- Drawing the free-body diagram of a truss
- Force T axial force in the member
- When the forces are directed away from each
other, the member is in tension - When the forces are directed toward each other,
the member is in compression
- Because it is a 2-force member, the forces at the
ends, which are the sums of the forces exerted on
the member at its joints, must be equal in
magnitude, opposite in direction directed along
the line between the joints
76.1 Trusses
- If we cut the member by a plane draw the
free-body diagram of the part of the member on 1
side of the plane - We represent the system of internal forces
moments exerted by the part not included in the
free-body diagram by a force F acting at the
point P where the plane intersects the axis of
the member a couple M
86.1 Trusses
- The sum of the moments about P must equal zero,
so M 0 - Therefore, we have a 2-force member, which means
that F must equal in magnitude opposite in
direction to the force T acting at the joint - The internal force is a tension or compression
equal to the tension or compression exerted at
the joint - Notice the similarity to a rope or cable, in
which the internal force is a tension equal to
the tension applied at the ends
96.1 Trusses
- Although many actual structures, including roof
trusses bridge trusses, consist of bars
connected at the ends, very few have pinned
joints - E.g. a joint of a bridge truss
- The ends of the members are welded at the joint
are not free to rotate - Such a joint can exert couples on the members
106.1 Trusses
- However, these structures are designed to
function as trusses - They support loads primarily by subjecting their
members to axial forces - They can usually be modeled as trusses, treating
the joints as pinned connections under the
assumption that couples they exert on the members
are small in comparison to axial forces
116.2 The Method of Joints
- The method of joints involves
- Drawing free-body diagrams of the joints of a
truss 1 by 1 - Using equilibrium equations to determine the
axial forces in the members
126.2 The Method of Joints
- Before beginning, it is usually necessary to draw
a free-body diagram of the entire truss (i.e.
treat the truss as a single object) determine
the reactions at its supports - E.g. consider the Warren truss which has members
2 m in length support loads at B D
136.2 The Method of Joints
- From the equilibrium equations
- S Fx Ax 0
- S Fy Ay E ? 400 N ? 800 N 0
- S Mpoint A ?(1 m)(400 N) ? (3 m)(800 N) (4
m)E 0 - We obtain the reactions
- Ax 0, Ay 500 N E 700 N
- The next step is to choose a joint draw its
free-body diagram - Isolate joint A by cutting members AB AC
146.2 The Method of Joints
156.2 The Method of Joints
- The terms TAB TAC are the axial forces in
members AB AC, respectively - Although the directions of the arrows
representing the unknown axial forces can be
chosen arbitrarily, notice that we have chosen
them so that a member is in tension if we obtain
a positive value for the axial force - Consistently choosing the directions in this way
helps avoid errors
166.2 The Method of Joints
- The equilibrium equations for joint A are
- S Fx TAC TAB cos 60 0
- S Fy TAB sin 60 500 N 0
- Solving these equations, we obtain the axial
force TAB ?577 N TAC 289 N - Member AB is in compression member AC is in
tension
176.2 The Method of Joints
- Next, obtain a free-body diagram of joint B by
cutting members AB, BC BD - From the equilibrium equations for joint B
- S Fx TBD TBC cos 60 577 cos 60 0
- S Fy ?400 N 577 sin 60 ? TBC sin 60 0
- We obtain TBC 115 N TBD ?346 N
- Member BC is in tension member BD is in
compression
186.2 The Method of Joints
- By continuing to draw free-body diagrams of the
joints, we can determine the axial forces of all
the members - In 2 dimensions, you can obtain only 2
independent equilibrium equations from the
free-body diagram of a joint - Summing the moments about a point does not result
in an additional independent equation because the
forces are concurrent
196.2 The Method of Joints
- Therefore when applying the method of joints, you
should choose joints to analyze that are
subjected to no more than 2 unknown forces - In our example, we analyzed joint A first because
it was subjected to the known reaction exerted by
the pin support 2 unknown forces, the axial
forces TAB TAC - We could then analyze joint B because it was
subjected to 2 known forces 2 unknown forces,
TBC TBD - If we had attempted to analyze joint B first,
there would have been 3 unknown forces
206.2 The Method of Joints
- When determining the axial forces in the members
of a truss, it will be simpler if you are
familiar with 3 particular types of joints - 1.Truss joints with 2 collinear members no
load the sum of the forces must equal zero, T1
T2. The axial forces are equal.
216.2 The Method of Joints
- 2.Truss joints with 2 noncollinear members no
load because the sum of the forces in the x
direction must equal zero, T2 0. therefore T1
must also equal zero. The axial forces are zero.
226.2 The Method of Joints
- 3.Truss joints with 3 members, 2 of which are
collinear no load because the sum of the
forces in the x direction must equal zero, T3
0. The sum of the forces in the y direction must
equal zero, so T1 T2. The axial forces in the
collinear members are equal the axial force in
the 3rd member is zero.
23Example 6.1 Applying the Method of Joints
- Determine the axial forces in the members of
the truss in Fig. 6.12. - Strategy
- 1st, draw a free-body diagram of the entire
truss, treating it as a single object determine
the reactions at the supports. Then apply the
method of joints, simplifying the task by
identifying any special joints
24Example 6.1 Applying the Method of Joints
- Solution
- Determine the Reactions at the Supports
- Draw the free-body diagram of the entire truss
25Example 6.1 Applying the Method of Joints
- Solution
- From the equilibrium equations
- S Fx Ax B 0
- S Fy Ay ? 2 kN 0
- S Mpoint B ?(6 m) Ax ? (10 m)(2 kN) 0
- We obtain the reactions Ax ?3.33 kN, Ay 2 kN
- B 3.33 kN.
26Example 6.1 Applying the Method of Joints
- Solution
- Identify Special Joints
- Because joint C has 3 members, 2 of which are
- collinear no load, the axial force in member BC
is - zero, TBC 0 the axial forces in the collinear
- members AC CD are equal, TAC TCD.
- Draw Free-Body Diagrams of the Joints
- We know the reaction exerted on joint A by the
- support joint A is subjected to only 2 unknown
- forces, the axial forces in members AB AC.
27Example 6.1 Applying the Method of Joints
28Example 6.1 Applying the Method of Joints
- Solution
- The angle ? arctan (5/3) 59.0
- The equilibrium equations for joint A are
- S Fx TAC sin ? ? 3.33 kN 0
- S Fy 2 kN ? TAB ? TAC cos ? 0
- Solving these equations, we obtain TAB 0
- TAC 3.89 kN.
-
29Example 6.1 Applying the Method of Joints
- Solution
- Now draw the free-body diagram of joint B
30Example 6.1 Applying the Method of Joints
- Solution
- From the equilibrium equation
- S Fx TBD 3.33 kN 0
- We obtain TBD ?3.33 kN. The negative sign
- indicates that member BD is in compression.
- The axial forces in the members are
- AB 0
- AC 3.89 kN in tension (T)
- BC 0
- BD 3.33 kN in compression (C)
- CD 3.89 kN in tension (T)
31Example 6.1 Applying the Method of Joints
- Critical Thinking
- Observe how our solution was simplified by
recognizing that joint C is the type of special
joint with 3 members, 2 of which are collinear
no load - This allowed us to determine the axial forces in
all members of the truss by analyzing only 2
joints
32Example 6.2 Determining the Largest Force a Truss
Will Support
- Each member of the truss in Fig. 6.13 will
safely support a tensile force of 10 kN a
compressive force of 2 kN. What is the largest
downward load F that the truss will safely
support?
33Example 6.2 Determining the Largest Force a Truss
Will Support
- Strategy
- This truss is identical to the one we analyzed
in Example 6.1. By applying the method of joints
in the same way, the axial forces in the members
can be determined in terms of the load F. The
smallest value of F that will cause a tensile
force of 10 kN or a compressive force of 2 kN in
any of the members is the largest value of F that
the truss will support.
34Example 6.2 Determining the Largest Force a Truss
Will Support
- Solution
- By using the method of joints in the same way as
- in Example 6.1, we obtain the axial forces
- AB 0
- AC 1.94F (T)
- BC 0
- BD 1.67F (C)
- CD 1.94F (T)
35Example 6.2 Determining the Largest Force a Truss
Will Support
- Solution
- For a given load F, the largest tensile force
is 1.94F (in members AC CD) the largest
compressive force is 1.67F (in member BD). - The largest safe tensile force would occur
when 1.94F 10 kN or when F 5.14 kN. - The largest safe compressive force would occur
when 1.67F 2 kN or when F 1.20 kN. - Therefore, the largest load F that the truss
will safely support is 1.20 kN.
36Example 6.2 Determining the Largest Force a Truss
Will Support
- Critical Thinking
- This example demonstrates why engineers analyze
structures - By doing so, they can determine the loads that an
existing structure will support or design a
structure to support given loads - In this example, the tensile compressive loads
the members of the truss will support are given
37Example 6.2 Determining the Largest Force a Truss
Will Support
- Critical Thinking
- Information of that kind must be obtained by
applying the methods of mechanics of materials to
the individual members - Then statics can be used, as we have done in this
examples, to determine the axial loads in the
members in terms of the external loads on the
structure