Calculating the “actual” internal
force in truss bridge members
EGN1006
Learning Goals:
The student WILL be able to:
•
•
•
•
Calculate the components of a force vector.
Add two force vectors together.
Draw a free body diagram.
Calculate whether a truss is statically determinate or
indeterminate.
• Write and solve a force vector equilibrium equation.
• Use the Method of Joints to calculate the internal force in
every member in a truss.
• Evaluate a truss, to determine if it can carry a given load safely
by calculating factors of safety for individual members.
Static Determinacy and Stability
A structure that cannot be analyzed using the equations of equilibrium
alone is called statically indeterminate. A structure that can be analyzed
using the equations of equilibrium alone is called statically determinate.
Only statically determinate trusses can be analyzed with the Method of
Joints.
A statically determinate truss with two reactions must satisfy the
mathematical equation
Where j is the number of joints and m is the number
of members.
Static Determinacy and Stability
How many joints does this bridge have?
6
How many members does this bridge have?
10
Is this bridge statically determinate? In other words
does 2j = m+3?
No , 2(6) = 12 and 10+3 = 13
Thus 2J does NOT equal m+3
Static Determinacy and Stability
How many joints does this bridge have?
6
How many members does this bridge have?
9
Is this bridge statically determinate? In other words
does 2j = m+3?
Yes , 2(6) = 12 and 9+3 = 12
Thus 2J does equal m+3
Free Body Diagram
A free body diagram is a pictorial
representation of all of the forces which
act on an object. Suppose we have a box
being pushed by an applied force to the
right. What forces act on the box?
•Weight, aka, the Force due to gravity , - This force is ALWAYS drawn straight down.
•Normal Force – The force that a SURFACE applies on an object. Always drawn
PERPENDICULAR to the surface
•Applied Force - Either a PUSH or a PULL
•Friction force – the force that ALWAYS opposes the motion. Drawn at the surface.
Free Body Diagrams
Often times the free body diagram is drawn using what is
called a POINT MODEL. The object is drawn as a single point
with the forces labeled as “F”. A subscript is added according
to the type of force it is.
Free Body Diagrams
FT1
FT2
Fg
In the case of bridge members and ropes, we have a
special type of force called TENSION. Since you can’t
PUSH a rope, the tension is ALWAYS drawn as if your are
pulling the object. In other words, it is always drawn
AWAY from the object. If there are multiple ropes,
subscripts must be used to classify them separately.
Equations of Equilibrium
When an object is at rest, the SUM of all of the
FORCE vectors must be equal to zero.
F
F
x
0
y
0
So when you write your equations they MUST equal ZERO.
What is a vector?
A vector is any quantity which has both MAGNITUDE (#
and a unit) and DIRECTION. The vector is always
represented as an ARROW. Suppose the vector below
represents a displacement of 10m.
10 m, NORTH-EAST
MAGNITUDE
DIRECTION
A vector’s direction
Direction is best described by using a Cartesian Coordinate
system. Forces on the negative x or negative y axis must
have a negative sign. Using this idea allows us to write the
equation of equilibrium. Assume the object is at rest.
F
x
0
FA1  (  FA 2 )  0
FA1  FA 2  0
10  FA 2  0
FA 2  10N
Fa2=? N
Fa1=+10 N
Let’s look at JOINT B on our truss
FBI
FAB
Assume all forces are TENSION!
FBC
Joint B – Equations of Equilibrium
 Fy  0
FBI  0
F
x
FBI
FAB
FBC
0
FBC  ( FAB )  0
If you knew the FORCE in member AB, you would be able to solve for the
FORCE in member BC. Isolating just ONE JOINT to analyze the force is called
the METHOD OF JOINTS.
Some Basic Concepts from Trigonometry
A truss is a structure composed of members arranged in
interconnected triangles. For this reason, the geometry
of triangles is very important in structural analysis.
This diagram shows a right triangle—a
triangle with one of its three angles
measuring exactly 90o. Sides a and b form the
90o angle. The other two angles, identified as
θ1 and θ2, are always less than 90o. Side c, the
side opposite the 90o angle, is always the
longest of the three sides. It is called the
hypotenuse of the right triangle. Thanks to an
ancient Greek mathematician named
Pythagoras, we can easily calculate the length
of the hypotenuse of a right triangle. The
Pythagorean Theorem tells us that:
Some Basic Concepts from Trigonometry
The Pythagorean Theorem shows how the lengths of the
sides of a right triangle are related to each other. But
how are the lengths of the sides related to the angles?
The sine of an angle (abbreviated “sin”) is defined
as the length of the opposite side divided by the
length of the hypotenuse. For example, the sine of
the angle θ1 would be calculated as:
The cosine of an angle (abbreviated “cos”) is
defined as the length of the adjacent side divided
by the length of the hypotenuse. Applying this
definition to our example, we have:
Breaking a Vector into its Components
Once the coordinate axis system is established, we can represent the direction of any
vector as an angle measured from either the x-axis or the y-axis. For example, the
force vector at right has a magnitude (F) of 20 Newtons and a direction (θ) of 50
degrees, measured counterclockwise from the x-axis.
This force can also be represented as two
equivalent forces, one in the x-direction
and one in the y-direction. Each of these forces
is called a component of the vector F.
What if a member is at an angle?
Calculating the Vector’s Components
Let’s get started
The required load our bridge must withstand is
49N or 5-kg.
Since there are TWO trusses held together by
lateral bracings, HOW much load does ONE
truss bridge hold?
24.5 N
Let’s get started
The load acts downward at joints J, K, and L. How much force acts at each one
of these locations?
8.17 N
Let’s get started
The two upward forces are both force normals. The “R” in
this case stands for REACTION as they are a reaction to the
load. How much force does each REACTION FORCE(force
normal) support?
12.25 N
Joint A
RA
FAI
FAB
Force AI must be broken into
components
RA
FAI
FAI
q
q
FAB
What is the VALUE of the angle THETA?
LengthAI
LengthBI
opp
1 opp
tanq 
 q  tan (
)
adj
adj
q
LengthAB
What other angle are also equal to theta?
Where is theta?
q
q
q
q
q
Force AI’s Components
RA
FAI
FAI
q
q
FAB
FAIcosq
Let’s now REDRAW the FBD!
FAIsinq
RA
FAIcosq
FAB
FAIsinq
Joint A’s Equations of Equilibrium
F  0
y
FAIsinq
RA
FAIcosq
FAB
RA  FAI sin q  0
RA
FAI  (
) A negative# ?
sin q
Whatdoes thismean?
This force is COMPRESSION and NOT tension, thus it
is a TUBE!
F
x
0
FAB  FAI cosq  0
FAB  ( FAI cosq )
Your task
Use the Method of Joints to solve for the rest of
the internal forces. Use the calculation guide
for reference and to keep organized.
Wait, there is ONE last thing…..
Factor of Safety
When an engineer designs a structure, he or she must consider many
different forms of uncertainty. There are three major types of uncertainty
that affect a structural design:
1. There is always substantial uncertainty in
predicting the loads a structure might experience
at some time in the future.
2. The strengths of the materials that are used to
build actual bridges are also uncertain.
3. The mathematical models we use for structural
analysis and design are never 100% accurate.
Factor of Safety
The engineer accounts for all forms of uncertainty by making the
structure somewhat stronger than it really needs to be—by
using a factor of safety in all analysis and design calculations.
In general, when it is used in the analysis of an existing
structure, the factor of safety is a defined as
In a truss, the actual force in a member is called the internal member
force, and the force at which failure occurs is called the strength. Thus
we can rewrite the definition of the factor of safety as
Factor of Safety
For example, if a structural member has an internal force of
5000 pounds and a strength of 7500 pounds, then its
factor of safety, FS, is
If the factor of safety is less than 1, then the member or structure is clearly
unsafe and will probably fail. If the factor of safety is 1 or only slightly greater
than 1, then the member or structure is nominally safe but has very little margin
for error—for variability in loads, unanticipated low member strengths, or
inaccurate analysis results.
Most structural design codes specify a factor of safety of 1.6 or larger
(sometimes considerably larger) for structural members and connections.
The next step
• Calculate the internal member force
• Use the previously found strengths to
calculate the factors of safety for each bridge
member