UCD FUND OF STRUCTURAL ANALYSIS 5E
UCD FUND OF STRUCTURAL ANALYSIS 5E
5th Edition
ISBN: 9781264843923
Author: Leet
Publisher: MCG
Question
Book Icon
Chapter 6, Problem 27P

(a)

To determine

Calculate the reactions and member forces of three-hinged, trussed arch.

(a)

Expert Solution
Check Mark

Answer to Problem 27P

Case A:

The support reaction at A is Ay=45kN_ and Ax=67.5kN_.

The support reaction at G is Gy=45kN_ and Gx=67.5kN_.

Case B:

The support reaction at A is Ay=25kN_ and Ax=37.5kN_.

The support reaction at G is Gy=65kN_ and Gx=97.5kN_.

Explanation of Solution

Given information:

The structure of the arch is given in Figure 6P.27.

Case A: 90kN force only acting at joint D.

Assumption:

Apply the sign convention for calculating the equations of equilibrium as below.

  • For the horizontal forces equilibrium condition, take the force acting towards right side as positive (+) and the force acting towards left side as negative ().
  • For the vertical forces equilibrium condition, take the upward force as positive (+) and downward force as negative ().
  • For moment equilibrium condition, take the clockwise moment as positive and counter clockwise moment as negative.

Calculation:

Draw the free body diagram as shown in Figure 1.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 6, Problem 27P , additional homework tip  1

Calculate the support reactions Ay and Gy at support A and G.

Due to symmetry:

Ay=Gy=90kN2=45kN

Calculate the horizontal reaction at supports.

Take moment at D.

(45×27)(Ax×18)=0Ax=67.5kNGx=67.5kN

Therefore, the support reaction at A is Ay=45kN_ and Ax=67.5kN_.

Therefore, the support reaction at G is Gy=45kN_ and Gx=67.5kN_.

Find the member forces using method of joints as in Table 1

Free body diagram for jointsCalculation of Member forces
UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 6, Problem 27P , additional homework tip  2

Joint A.

Fx=0FABcos45°=67.5kNFAB=95.45kN(Compression)

Fy=0FAM=45kNFABsin45°FAM=45kN(95.45)sin45°=22.5kN(Tension)

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 6, Problem 27P , additional homework tip  3

Joint M.

Fy=0FMBsin45=FAMFMBsin45=22.45kNFMB=31.82kN(Compression)

Fx=0FML+FMBcos45=0FML+(31.82)cos45=0FML=22.5kN(Tension)

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 6, Problem 27P , additional homework tip  4

Joint B.

Fx=0FBCcos33.69FMBcos45FABcos45=0FBCcos33.69(31.82)cos45(95.45)cos45=0FBC=108.15kN(Compression)

Fy=0FBL+FBCsin33.69+FMBsin45FABsin45=0{FBL+(108.15)sin33.69+(31.82)sin45(95.45)sin45}=0FBL=15.3kNFBL15kN(tension)

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 6, Problem 27P , additional homework tip  5

Joint L.

Fy=0FBLFLCsin18.43=015FLCsin18.43=0FLC=47.4kN(Compression)

Fy=0FKLFML+FLCcos18.43=0FKL22.5+(47.4)cos18.43=0FKL=67.5kN(Tension)

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 6, Problem 27P , additional homework tip  6

Joint K.

Fy=0FKC=0Fx=0FKD=FKL=67.5kN(Tension)

 

Joint D.

Fy=0FKD12FCDcos18.43=067.512FCDcos18.43=0FCD=142.3kN(Compression)

Table 1

Show the calculated member forces as in Table 2

MemberMagnitude of Force
AB, GF95.45kN(Compression)
BC, FE108.15kN(Compression)
CD, ED142.3kN(Compression)
KD, DJ67.5kN(Tension)
KL, JI67.5kN(Tension)
LM, IH22.5kN(Tension)
AM, GH22.5kN(Tension)
BM, FH31.82kN(Compression)
BL, FI15kN(tension)
CL, EI47.4kN(Compression)
KC, JE0

Table 2

Consider case B.

Case B: Both 90kN and 60kN at joint D and M.

Draw the free body diagram as shown in Figure 7

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 6, Problem 27P , additional homework tip  7

Support reactions:

Take moment at G.

(Ay×54)(90×27)+(60×18)=0Ay=25kN

Apply Fy=0.

Ay+Gy=9025+Gy=90Gy=65kN

Calculate the horizontal reaction at supports.

Take moment at D.

(25×27)(Ax×18)=0Ax=37.5kN

Fx=0Ax+60=Gx37.5kN+60=GxGx=97.5kN

Gx=67.5kN

Therefore, the support reaction at A is Ay=25kN_ and Ax=37.5kN_.

Therefore, the support reaction at G is Gy=65kN_ and Gx=97.5kN_.

Find the member forces using method of joints similarly as case A.

Tabulate the member force values as shown in Table 3.

MemberMagnitude of Force
AB53kN(Compression)
BC60kN(Compression)
CD79kN(Compression)
KD22.5kN(compression)
KL22.5kN(compression)
LM47.5kN(compression)
AM12.5kN(Tension)
BM17.7kN(Compression)
BL8.33kN(tension)
CL26.4kN(Compression)
KC0
GF137.88kN(Compression)
FE156.2kN(Compression)
ED205.55kN(Compression)
DJ97.5kN(tension)
JI97.5kN(tension)
IH32.5kN(tension)
GH32.5kN(tension)
FH45.96kN(Compression)
FI21.7kN(tension)
EI68.5kN(Compression)
JE0

Table 3

Show the member forces as in Figure 8.

UCD FUND OF STRUCTURAL ANALYSIS 5E, Chapter 6, Problem 27P , additional homework tip  8

(b)

To determine

Find the maximum axial force in the arch.

(b)

Expert Solution
Check Mark

Answer to Problem 27P

The maximum axial force in the arch is FDE=205.55kN(compression)_, FEF=156.21kN(compression)_, and FFG=137.88kN(compression)_.

Explanation of Solution

Refer Table 3 and Figure 8.

The maximum axial force in the arch is,

FDE=205.55kN(compression)

FEF=156.21kN(compression)

FFG=137.88kN(compression)

Therefore, the maximum axial force in the arch is FDE=205.55kN(compression)_, FEF=156.21kN(compression)_, and FFG=137.88kN(compression)_.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Knowledge Booster
Background pattern image
Recommended textbooks for you
Text book image
Structural Analysis
Civil Engineering
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:Cengage,
Text book image
Structural Analysis (10th Edition)
Civil Engineering
ISBN:9780134610672
Author:Russell C. Hibbeler
Publisher:PEARSON
Text book image
Principles of Foundation Engineering (MindTap Cou...
Civil Engineering
ISBN:9781337705028
Author:Braja M. Das, Nagaratnam Sivakugan
Publisher:Cengage Learning
Text book image
Fundamentals of Structural Analysis
Civil Engineering
ISBN:9780073398006
Author:Kenneth M. Leet Emeritus, Chia-Ming Uang, Joel Lanning
Publisher:McGraw-Hill Education
Text book image
Sustainable Energy
Civil Engineering
ISBN:9781337551663
Author:DUNLAP, Richard A.
Publisher:Cengage,
Text book image
Traffic and Highway Engineering
Civil Engineering
ISBN:9781305156241
Author:Garber, Nicholas J.
Publisher:Cengage Learning