1. ENGR-1100 Introduction to Engineering Analysis
Lecture 13

2. Today Lecture Outline - Equilibrium equations in 2-D
- Solving problems of equilibrium of a rigid body in 2-D
Free body diagram
Application of physical laws
- Statically indeterminate reaction and partial constraints

3. Equilibrium Equations in 2-D
The necessary and sufficient conditions for a body to be in equilibrium in 2-D (3 independent equations) are:
‘A’ is ANY point on or off the rigid body.
The moment equilibrium equation means that the resultant must pass through A
Then apply the force equilibrium conditions along x and y to ensure that the resultant has zero magnitude (i.e. the body is in equilibrium) x y A

4. Equilibrium Equations in 2-D: Alternate Forms
‘A’ and ‘B’ are points on a line NOT PERPENDICULAR to the x-axis.
The condition MA=0 means that the resultant
must pass through A.
The condition Rx=0 means that the resultant (if
other than zero) is perpendicular to the x-axis.
Finally to MB=0 can only be satisfied if Ry=0.
In other words: x y B A
Lid
MB=0
Ry=0

5. Equilibrium equations in 2D: Alternate Forms
‘A’ ,‘B’ and ‘C’ three non-collinear points
The condition MA=0 means that the resultant
must pass through A.
The condition MB=0 means that the resultant
must pass through B. and thus must be along the line AB.
Finally to MC=0 can only be satisfied if R=0 since C does not lie on AB. B C A
Lid

6. Special Force Systems
Special case 1. A “two force member” is a body which is acted
on by only two forces. The forces are equal, opposite and
collinear (Newton’s Third Law!) F
Don’t let the shape deceive you F F F
A strut

7. Special Force Systems
Special case 2. A “three force member” is a body which is
acted on by only three forces. The forces MUST BE
CONCURRENT (otherwise there will be a resultant moment of
the third force about the point of concurrency of the first two). F3 F2 F1

8. Statically Determinate vs. Indeterminate Problems
If the equilibrium equations are sufficient to determine all the support reactions, then the body is said to be statically determinate with adequate constraints
Lid x y A B
Hinge at A
Roller at B x y Ax Ay By
Can determine Ax, Ay and By from the 3 equilibrium equations

9. Statically Determinate vs. Indeterminate Problems
If a body has more supports than are necessary for equilibrium then the equilibrium equations alone are not sufficient to determine all the support reactions, then the body is said to be statically indeterminate
Statically determinate
Statically indeterminate y y x C x A B A
Lid B
Lid
Roller at B
Roller at B
Hinge at A
Hinge at A

10. Problems with partial constraints
Three support reactions in a 2D problem do not necessarily mean that the body is adequately constrained. Sometimes the body may be partially constrained and the equilibrium equations will not be sufficient to compute the support reactions.
Statically determinate
Statically indeterminate with inadequate constraints y x y x A B B
Roller at B A
Cannot determine Ax, Ay and Bx from the 3 equilibrium equations
Hinge at A

11. Problems with partial constraints
A body with adequate number of reaction is improperly constrained when the constraints are arranged in such a way that the support forces are either concurrent or parallel.

12. Example P6-40
An angle bracket is loaded and supported as shown in Fig. P6-40.
Determine the reactions at supports A and B.

13. Solution First equation -350*0.22-500*0.1+Bx*0.2=0 x Known forces
unknown forces y Ay Ax
500 N
350 N Bx
Bx=635 N
Second equation
-Bx+500+ Ax=0
Ax=135 N
Third equation
Ay-350=0
A= 135 i j N
Ay=350 N

14. Class Assignment: Exercise set 6-38
please submit to TA at the end of the lecture
A beam is loaded and supported as shown in Fig. P The beam has a uniform cross section and a mass of 20 kg. Determine the reaction at support A and the tension T in the cable. Ay Ax T
Answer:
T= 512 N
A= -293 i j N.

15. Example P6-63
GThe wrecker truck of Fig. P6-63 has a weight of 15,000 lb and a center of gravity at G. The force exerted on the rear (drive) wheels by the ground consists of both a normal component By and a tangential component Bx, while the force exerted on the front wheel consists of a normal force Ay only. Determine the maximum pull P the wrecker can exert when 0 = 30 if Bx, cannot exceed 0.8 By (because of friction considerations) and the wrecker does not tip over backward (the front wheels remain in contact with the ground).

16. Solution First equation W*8 – 10*P*sin(300) – 5*P*cos(300) =0
P=12.86 kip
To prevent tipping: Ay>0 P Ay Bx By
W=15 Kip x y
Second equation
By-W- P*cos(300) =0
By=26.1 kip
Third equation
Bx+ P*sin(300) =0
Bx=-6.43 kip

17. Bx=6.43 kip < kip=0.8* By
Which satisfy the requirement for Bx <0.8 By

18. Class Assignment: Exercise set 6-64
please submit to TA at the end of the lecture
Bar AB of Fig. P6-64 has a uniform cross section, a mass of 25 kg, and a length of 1m. Determine the angle q for equilibrium.
Answer: q= 20.10