Lecture 16

ENGR-1100 Introduction to Engineering Analysis

Today Lecture Outline

• Structures:•Trusses •Frames

• Trusses analysis- method of joints• Stability criteria.

Two important structures types

• Trusses: Structures composed entirely of two force members.

• Frames: Structures containing at least one member acted on by forces at three or more points.

• Plane trusses: lie in a single plane.

• Space trusses: not contained in a single plane and/or loaded out of the structure plane.

Plane Trusses

Assumptions

1) Truss members are connected together at their ends only.

2) Truss members are connected together by frictionless pins.

3) The truss structure is loaded only at the joints.

4) The weight of the member may be neglected.

The Four Assumptions

Truss members are two-force members

F

F

Truss member

F

F

Straight Members

Forces act along the axis of the member

F

F

Compressive forces tend to

shorten the member.

F

F

Tensile forces tend to elongate the

member.

“Rigid” trusses

“Rigid”- the truss will retain its shape when removed from its support

Simple truss- constructed by attaching several triangles together.

Allows a simple way to check rigidity.

What are we looking for?

The support reaction .

The force in each member.

How many equations are available? How many unknowns?

Each joint- 2 equations

Unknowns- number of members+ support reaction.

Stability Criteria

m=2j-32j- number of equations to be solved.

m- number of members.

3- number of support reaction

m<2j-3 Truss unstable

m>2j-3 Statically indeterminate

Example

m (Number of members) =

m=2j-3

13

j (Number of joints) =

Number of supports=

8

3

Method of Joints

Separate free-body diagrams for:

each member

each pin

Equilibrium equations for each pin:

SF=0

no moment equation

Example 7-3

Use the method of joints to determine the force in each member of the truss shown in Fig. P7-3. State whether each member is in tension or compression.

Solution

TAB TBD

Joint B---------

TBC

y

x

From a free-body diagram on joint B:

Fx = TBC sin 30º - TAB sin 60º = 0

Fy = - TBC cos 30º - TAB cos 60º - TBD = 0

TCDTAD

TBD

Joint D---------

y

x

From a free-body diagram on joint D:

Fy = TBD - 3000 = 0

TBD = 3000 lb = 3000 lb (T)

TAB = - 1500.0 lb = 1500 lb (c)

TBC = - 2598 lb 2600 lb (c)

known

TCD

Cy

Joint C---------

TBCy

xFrom a free-body diagram on joint C:

Fx = - TCD - TBCsin 30 = - TCD -(2598) sin 30 = 0

Fx = -TAD + TCD = - TAD + 1299 = 0

From a free-body diagram on joint D:

TCDTAD

TBD

Joint D---------

y

xTAD = 1299 lb = 1299 lb (T)

TCD = 1299 lb = 1299 lb (T)

Class Assignment: Exercise set 7-4please submit to TA at the end of the lecture

Answer:

TBC=14.1 kN (T)

TAC=5.13 kN (T)

TAB=28.2 kN (C)

Class Assignment: Exercise set 7-6please submit to TA at the end of the lecture

Answer:

TBD=3.23 kN (C)

TCD=4.62 kN (T)

TAD=0.567 kN (T)

TBC=4 kN (C)

TAB=2.38kN (C)

Zero Force Members

Member BC and DC are zero force members

Free body diagram on joint C

FBC

FCD

y

x

SFy=0 FCD=0

SFx=0FBC=0

SFy=0 FBD=0

Free body diagram on joint B

FABFCB

y

x

FBD

Free body diagram on joint D

FAD

FED

y

x

FDC

FAD=0

Example 7-19

The truss shown in figure P7-19 support one side of a bridge; an identical truss supports the other side. Floor beams carry vehicle loads to the truss joints. Calculate the forces in members BC, BG, and CG when a truck weighing 7500 lb is stopped in the middle of the bridge as shown. The center of gravity of the truck is midway between the front and rear wheels.

Solution

From symmetry: FH = FF

Free body diagram on floor beams

Free body diagram on floor beams FH

Fy = 2FG - 2(3750) + 4(1125) = 0

FG= 1500 lb

GFMG = 2FF (10) – 3750 (6) = 0FF = 1125 lb

f          = tan-1 8/5 = 57.99

= tan-1 5/10 =26.57

= tan-1 8/10 =38.66

Free body diagram on complete truss

ME= Ay (30) -1125 (25) - 1500(15) - 1125(5) = 0

Fx = Ax = 0 Ax = 0

Ay = 1875 lb = 1875 lb

Free body diagram on joint A

Fx = TAB cos 57.99 + TAH = 0

Fy = TAB sin 57.99 + 1875 = 0

TAB = - 2211 lb 2210 lb (c)

TBH = 1125 lb = 1125 lb (T)

Fy = TBH - 1125 = 0

Free body diagram on joint H

Free body diagram on joint B

Fx = TBC cos 26.57 + TBG cos 38.66

+ 2211cos 57.99 = 0Fy = TBC cos 26.57 - TBG sin 38.66

+ 2211 sin 57.99 - 1125 = 0

TBC = -1451.2 lb 1451 lb (C)

TBG = 161.31 lb 161.3 lb (T)

Free body diagram on joint C

Fx = TCD cos 26.57 + 1451.2 cos 26.57 = 0

Fy = TCD sin 26.57 - TCG + 1451.2 sin 26.57 = 0

TCD = -1451.2 lb 1451 lb (C)

TCG = 1298.2 lb 1298 lb (T)

Class Assignment: Exercise set 7-23please submit to TA at the end of the lecture