Structural Analysis Problems for Practice

CE-207 STRUCTURAL ANALYSIS I TUTORIAL 1: Reactions, Shear Force and Bending Moment Diagrams, Free-body Diagrams

1.

Figure P1.1

Figure P1.1 shows a cantilever beam AB, subjected to a triangularly varying load as shown. Find the reactions at the fixed end B:

(a) By using the equations of static equilibrium. (b) By using the virtual displacement principle/Influence Line Method.

Also sketch the Shear Force and Bending Moment Diagrams for the beam, showing the values at intervals.

2.

Figure P1.2

Figure P1.2 shows a simply-supported beam AB with a roller at end A and a hinge at end B, subjected to a uniformly distributed load as shown. Find the reactions at the two ends A and B:


Also sketch the Shear Force and Bending Moment Diagrams for the beam, showing the values at intervals.

3.

Figure P1.3

Figure P1.3 shows a frame subjected to loads as shown. Find the reactions at roller support C and at hinge support E:

(a) By using the equations of static equilibrium. (b) By using the virtual displacement principle/Influence Line

Method. Also sketch the Shear Force and Bending Moment Diagrams for the frame. For each segment (AB, BC, CD and DE) sketch the free-body diagrams and verify that each segment is in equilibrium.

4.

Figure P1.4

A frame ABC has a roller support at A and hinge support at C. The frame has a rigid connection at B. The frame is subjected to a linearly varying load with intensity of at B to at C on segment BC as shown. For the purpose of finding the reactions this load can be substituted by point loads and , representing a uniform load of intensity on BC and a triangularly varying load of intensity at B and at C, respectively. This substitution is done because the centroids of a uniform load and of a triangularly varying load are known. Consequently, the point loads, and act at location of this centroids, as shown. Find the reactions at roller support A and at hinge support C:


Also sketch the Shear Force and Bending Moment Diagrams for the frame. For each segment (AB and BC) sketch the free-body diagrams and verify that each segment is in equilibrium.

5.

Figure P1.5

Find the reactions at all supports: (a) By using the equations of static equilibrium. (b) By using the virtual displacement principle.

Also sketch the Shear Force and Bending Moment Diagrams for the frame. For each segment (AB, BC and CD) sketch the free-body diagrams and verify that each segment is in equilibrium.

6.

Figure P1.6

Find the reactions at all supports: (a) By using the equations of static equilibrium. (b) By using the virtual displacement

principle/Influence Line Method. Also sketch the Shear Force and Bending Moment Diagrams for the frame. Also find the axial force in the truss element AB. For each segment (AB, BC, CD and BE) sketch the free-body diagrams and verify that each segment is in equilibrium.

CE-207 STRUCTURAL ANALYSIS I TUTORIAL 2: Rotations and Deflections in Beams by Kinematic, Conjugate Beam and Unit Load Methods

1.

Figure P2.1

Sketch the Curvature Diagram and the Deflected shape for the cantilever beam AB show in Figure P2.1. Compute the rotation,

, and vertical deflection, at B and also the rotation, , and vertical deflection, at C, the mid-span point of beam AB.

2.

Figure P2.2

Sketch the Curvature Diagram and the Deflected shape for the simply supported beam AB show in Figure P2.2. Compute the

rotation, , at A and also the rotation, , and vertical deflection, at C, the mid-span point of beam AB. EI is constant for the beam.

3. The beam of Figure P1.1 has constant EI. Compute the rotation, , and vertical deflection, at A and also

the rotation, , and vertical deflection, at C, the mid-span point of beam AB.

4.

The beam of Figure P1.2 has constant EI. Compute the rotation, , at A and the vertical deflection, at C, the mid-span point of beam AB.

5. The frame of Figure P1.3 has constant EI. Find the rotations at supports C and E. Also find the horizontal

deflection, , vertical deflection, , and rotation, , at A. Also find the vertical deflection, , at D.

6. The beam of Figure P1.5 has constant EI. Find the vertical deflection at internal hinge B, as well as the rotation just to the left and just to the right of internal hinge B.

7.

Figure P2.7

For the frame shown in Figure P2.7, compute the horizontal displacement at B and D. For all members and

Figure P2.8

8. The reinforced concrete beam shown in Figure P2.8 is pre-stressed by a steel cable that induces a compression force of with an eccentricity of . The external effect of the pre-stressing is to apply an axial force of and equal end moments as shown. The axial force causes the beam to shorten, but produces no bending deflections. The end-moments bend the beam upward. Determine the initial camber of the beam at mid span immediately after the cable is tensioned. Given:

, , , beam weight Note: Over time the initial deflection will increase due to creep by about 100 to 200 percent.

CE-207 STRUCTURAL ANALYSIS I TUTORIAL 3: Influence Lines by Muller-Breslau Principle

1. For the beam shown in Figure P1.1, draw the influence lines for shear force and bending moment for beam sections at every intervals. Using these influence lines compute the shear force and bending moment at these sections for the given loading. Verify with the values you had obtained while doing problem P1.1.

2. For the beam shown in Figure P1.2, draw the influence lines for shear force and bending moment for beam sections at every intervals. Using these influence lines compute the shear force and bending moment at these sections for the given loading. Verify with the values you had obtained while doing problem P1.2.

3. For the frame shown in Figure P1.3, draw the influence lines for shear force and bending moment at D. Using these influence lines compute the shear force and bending moment at D for the given loading. Verify with the values you had obtained while doing problem P1.3.

4. For the frame shown in Figure P1.4, draw the influence lines for bending moment at B. Using this influence line compute the bending moment at B for the given loading. Verify with the values you had obtained while doing problem P1.4.

5. For the frame shown in Figure P1.5, draw the influence lines for bending moment at support C, and for shear force just to the left and right of support C. Using these influence lines compute the bending moment at C and shear force just to the left and right of support C for the given loading. Verify with the values you had obtained while doing problem P1.5.

6. For the frame shown in Figure P1.6, draw the influence lines for bending moment at the location of the load at the mid-span of member BC. Using this influence lines compute the bending moment at the section for the given loading. Verify with the values you had obtained while doing problem P1.6.

7.

Figure P3.7

Beam AC is connected to a truss member DB at B. Draw the influence line for the axial force in DB, the vertical reaction at support A, and the bending moment at middle of AB.

8. Draw the influence line for bending moment and shear force at G.

Figure P3.8

9. Draw the influence line for vertical reactions at supports A and I.

Figure P3.9

10.

Figure P3.10

Assume that all the joints of the truss are pin-connected (they are shown riveted in the Figure P3.10). Draw the Influence Lines for reactions and axial forces in members AB, BC, GF, AG, GB and BF. Using these influence lines compute the maximum axial force that can arise in these members due to the movable loading shown.

11.

Figure P3.11

The beam shown is subjected to a moving concentrated load of . Construct the envelope of both maximum positive and negative bending moments for the beam by considering sections at intervals.

CE-207 STRUCTURAL ANALYSIS I TUTORIAL 4: Analysis of Trusses: Method of Joints and Method of Sections

1.

Figure P4.1

Analyze the truss by the method of Joints.

2.

Figure P4.2


3.

Figure P4.3


4.

Figure P4.4

Analyze the truss by the method of Sections.

5.

Figure P5.5

Analyze the truss.

6.

Figure P5.6

Analyze the truss.

7.

Figure P5.7

Analyze the truss.

8.

Figure P5.8

Analyze the truss.

CE-207 STRUCTURAL ANALYSIS I TUTORIAL 5: Analysis of Cable Structures

1.

Figure P5.1

Determine the reactions at the supports A and E. Also determine the cable sags at C and D. Also find the tension and slopes to the horizontal for all cable segments – AB, BC, CD and DE. In which segment is the cable tension and slope to the horizontal maximum.

2.

Figure P5.2

What value of is associated with the minimum volume of cable material required to support the load? The allowable stress in the cable is .

3.

Figure P5.3

The cables have been dimensioned so that a 3-kip tension force develops in each vertical strand when the main cables are tensioned. What value of jacking force must be applied at supports B and C to tension the system? Also find the sags of each main cable AB and CD at the location of the vertical strands. Then find the length of each vertical strand.

4.

Figure P5.4

Compute the support reactions and the maximum tension in the cable. Also find the equation for the cable sag measured from the chord AB. Is the equation for the cable sag a parabolic equation?

5.

Figure P5.5

Find the tensions in vertical hangers at B and C. Find the reactions at A and D. Find the sag in cable ABCD at B and C.

6.

Figure P5.6

The cable supported roof shown is composed of 24 equally spaced cables that span from a tension ring at the center to a compression ring on the perimeter. The tension ring lies below the compression ring. The roof weighs

based on the horizontal projection of roof area. If the sag at mid-span of each cable is , determine the tensile force each cable applies to the compression ring.

What is the required area of each cable if the allowable stress is ? Determine the weight of the tension ring required to balance the vertical components of the cable forces.

CE-207 STRUCTURAL ANALYSIS I TUTORIAL 6: Analysis of Arches

1.

Figure P6.1

Figure P6.1 is a three-hinged trussed arch. The geometry of the bottom chord of the truss is shaped to be funicular for uniform loads, such as for those shown. Member KJ, which is detailed so that it cannot transmit axial force, acts as a simple beam instead of a truss member. Find the reactions and forces in all truss members. Also solve the truss for a vertical load only at joint L.

2.

Figure P6.2

For the parabolic arch shown, plot the variation of the thrust at the support A for values of If the arch section is shaped so that its sectional area is proportional to the thrust at the section, find the vertical displacement of the hinge B for different values of . Take to be constant.

Figure P6.3

3.

Determine the reactions at supports A and C for the three hinged circular arch. If the arch has a constant EI, find the horizontal and vertical displacements of the hinge B. Consider only bending deformations.

CE-207 STRUCTURAL ANALYSIS I TUTORIAL 7: Analysis of Indeterminate Structures by the Flexibility Method

Note: In all problems EI is constant.

1.

Figure P7.1

(a) Find all the reactions by taking the vertical reaction at C as the redundant.

(b) Find all the reactions by taking the bending moment at A as the redundant.

Which choice of redundant involved less effort?

2.

Figure P7.2

(a) Find all the reactions by taking the vertical reaction at B as the redundant.

(b) Find all the reactions by taking the bending moment at A as the redundant.


3.

Figure P7.3

(a) Find all the reactions by taking the vertical reaction at C as the redundant.

(b) Find all the reactions by taking the vertical reaction at B as the redundant.

(c) Find all the reactions by taking the bending moment at B as the redundant.


4.

Figure P7.4

Determine the reactions for the beam in Figure P7.4. When the uniform load is applied, the fixed support at A rotates clockwise 0.003 radians and support B settles by . Given: and .


Note: In all problems EI is constant, unless stated otherwise.

1.

Figure P8.1

Find the reactions and draw the shear force and bending moment diagrams by taking the following redundants: (a) vertical reaction at B and bending moment at B. (b) bending moment at A and bending moment at B. (c) shear and bending moment at the mid-span of the beam. Which choice of redundants involved less effort?

2.

Figure P8.2

Find the reactions and draw the shear force and bending moment diagrams by taking the following redundants: (a) vertical reaction at D and bending moment at D. (b) bending moment at A and bending moment at D. (c) shear and bending moment at the mid-span of the beam. Which choice of redundants involved less effort?

3. The beam and the loading are symmetric in both Figures P8.1 and P8.2. For the choice (c) of the redundants, you must have obtained the value of the redundant shear at the mid-span of the beam as zero. Now, solve both problems by using the knowledge that the bending moments at the two-fixed ends must be equal to each other because of symmetry. This would also reduce the number of redundants to just one.

4. Find the reactions and draw the shear force and bending moment diagrams by taking the following redundants:

(a) Horizontal reaction at D. (b) Vertical reaction at D. Which choice of redundants involved less effort?

Figure P8.4

5.

Figure P8.5

Solve and Draw the Bending Moment Diagram

6. Solve.

Figure P8.6

7. Solve. All members have same EA.

Figure P8.7

8. Solve. All members have same EI.

Figure P8.8

9. Solve. All members have same EA.

Figure P8.9

Is the force in members DB and AC nearly the same? Do they take nearly equal horizontal shear equal to ?

11 Solve. All members have same EA.

Figure P8.10


Note: In all problems EI or EA is constant, unless stated otherwise.

1. Solve.

Figure P9.1

2. Compute the reactions and draw the shear force and bending

moment diagrams for the beam. ,

Figure P9.2

3. Solve.

Figure P9.3

4. Determine the reactions and bar forces, when the top chord ABCD of the truss are subjected to temperature increase.

Given: , ,

Figure P9.4

5. Figure P9.5a shows the Influence Line for Bending Moment at support B for a continuous beam with two equal spans and constant EI.

Figure P9.5a

Figure P9.5b

Where, the bending moment at support B due to a concentrated Load shall be:

The ordinates of the I.L. are given at every and the sign is given inside the curve considering hogging moment to be negative and the load acting downwards. The value, , for example, gives the average height of the influence line in span AB. This means that if span AB is loaded by a uniform load , then the bending moment at B shall be:

Use the Muller-Breslau Principle to obtain the values of the influence coefficients on your own. Figure P9.5b shows the shape of the Influence line for shear at mid-span point of beam AB. By using the influence line coefficients for Bending Moment at Support B, obtain the coefficients for the influence line for shear at mid-span of beam AB at every .

CE-207 STRUCTURAL ANALYSIS I QUIZ with 2 Questions

Enrolment No. & Name of Student:

1.

Figure 1

Figure 1 shows a hanging weightless cable ABCDE with loads at B, C and D, as shown. The elevation difference between the ends A and E of the cable is as shown. The sag of the cable at B, as measured from the chord connecting ends A and E of the cable, is , as shown. Find the vertical sag at D as measured from the chord connecting ends A and E.

(5 Marks)

Please write your solution below this line:

2.

Figure 2

Figure 2 shows a three-hinged parabolic arch with hinges at A, C and E. Find the reactions.

(5 Marks)

Please write your solution below this line:

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Structural Analysis Problems for Practice