strength of materials courseware by a.a jimoh

1. COURSE NAME & CREDIT LOAD:

Course : CVE 363-Strength of Materials (2 Credits-Compulsory)

Course Duration : Two hours per week for 15 weeks (30 hours), As taught in 2010/2011

2. LECTURER DETAILS:

Lecturer: Dr. A.A. Jimoh

B.Eng., M.Eng, Ph.D., Civil Engineering

E-mail address: [email protected]

Block 8 ground floor (8G25)

Consultation hours: 10 am -2 pm Monday-Friday every week

3.0 COURSE DETAILS:

3.1 Course Content

Generalized stress-strain relationship, Biaxial and triaxial state of stress. Stress transformation. Mohr’s circle, failure theories. Theories of bending of beams,. Unsymmetrical bending and shear centre. Strain Energy application. Torsion of non-circular and thin-walled hollow members.

3.2 Course Description

Generalized stress-strain relationship, Biaxial and triaxial state of stress. Stress transformation, Mohr’s circle, failure theories-Give stress-strain relationship of Young’s modulus = stress/strain, apply stress to beam section of rectangular and T- section. Maximum and minimum stress.

Theories of bending of beams- explain various equations and patterns.

Unsymmetrical bending and shear centre- apply this to T and rectangular sections.

Strain Energy application- Apply to beams and trusses

Torsion of non-circular and thin-walled hollow members- explain circular hollow members and torsion expressions and angle of twist.

3.3 Course Justification: Knowing how to analyse for stress imposed on materials before construction is important and will reduce failures

3.4 Course Objectives

Aims and objectives : At the end of the course the students are expected to understand the following

(i) Understand stress – Strain relationship , derivation of expression for young modulus (E). Stress at a point, Stress transformation and application

(ii) Be able to understand uniaxial, biaxial and triaxial state of stress using analytical method (iii) Understand above using graphical method (iv) Understand failure theories and theory of bending of beams (v) Computation in unsymmetrical bending for circular, hat and channel sections (vi) Computation for point of shear centre for channels, hat section (vii) Strain energy application to beam and trusses (viii) Torsion of non-circular and thin-walled hollow material.

3.5 Course Requirements: A student is expected to scored not less than 30 in GET 252

3.6 Methods of Grading

Courses shall be graded using continuous assessment (CA) -30% and exam (70 %). The CA shall include the following : take home assignment, test and Quiz

3.7 Course Delivery Strategies and Practical Schedules: The course shall be delivered by giving notes , extract from relevant text books code of practice and showing practical examples and so on, solving problems in the class giving out problem to solve by students themselves and giving reference books as reading and problem exercise.

4.0 LECTURE CONTENT:

Week 1 : Stress- strain curves of mild steel high yield steel, rubber, E derivation from curve. Application of Youngs Modulus E

Week 2 : Plane stress computation : Determination of stress at any arbitrary plane computation for principal Stress and Sketch.

Week 3 : Application of Mohr’s circles to the above.

Week 4: Failures theories types in beams, Columns Slabs and foundations

Week 5:Theories of bending of beams

Week6: Unsymmetrical bending

Week 7: Shear centre.

Week 8 : Strain energy application.

Week 9: Torsion of non-circular and thin walled hollowed members

Week 10-15: Revision test and examination.

5.0 GENERAL READING LIST

William A.N. & CEN STORGESS, 1977 Schaum,s outline of Theory and Problems of Strength of Materials Second Edition MCGrawhill, Newyork.

W.T. Marshall & H.M Nelson 1975 Structures Low-Priced Edition English Language Brok society and pitman London.

J.D. Todd 1982 Structural Theory and Analysis Macmillan, London.

6.0 LEGEND/ Tutorial questions / Exercise

Q1 (a) A bar of cross sectional area 850 mm2 is acted upon by an axial tensile force of 60 kN, applied at each end of the bar , determine ,

a) the normal and shearing stresses on a plane inclined at 30 to the direction of loading (15 mks)

b) the maximum shearing stress in the bar (5 mks)

Q2 At a point in a body, stresses are as shown in Fig Q2

Determine ,

(a) the normal and tangential stresses on the plane that makes an angle of 45o with the horizontal (10 mks)

(b) the principal stresses and the planes where they are acting (10 mks)

Q3a

Fig Q3ai presents the stresses at a point in a structural element and Fig Q3aii is the Mohr’s circle of centre C describing the stresses at the point. Use the Mohr’s circle to determine the following:

(a) the stresses on the vertical and the horizontal planes at the point (V and H on the circle) (5 mks)

(b) the principal stresses at the point and the planes where they are acting (5 mks) Q3b

(a) What is unsymmetric bending and what causes it? (5 mks)

(b) Show the shear centre of the beam sections in Fig Q3b (no calculation is required) (5 mks)

Q4 Fig Q4 is a section of a simply supported beam of span 5 m. The beam section is acted upon by a bending moment of 100 kNm applied on a plane inclined at an angle of 30o to the vertical . The vertical and horizontal axis in the section are y-y and x-x respectively and z-z axis passes through the longitudinal centroidal axis of the beam . From the figure, determine the following

(i) the values of the moment about each of the three axis (i.e Mx-x, My-y and Mz-z) (10 mks)

(ii) the values of maximum tensile and compressive stresses (10 mks)

Q5 Compute the total strain energy stored in the truss in Fig Q5 in terms of E and A (E and A are constant) (20 mks)

60MPa

25 Mpa

450

60

50 Mpa

50 Mpa

Fig Q2

Fig Q3aii

300 C (40, 0)

+t

-t

-s +s O(0,0

)+t

A-A(60,20)

200

H

V

Fig Q3ai

A

A

Fig Q3b beam sections T- section channel section rectangular section Z- section

D E F

Q1. A bar of cross section 850 mm2 is acted upon by an axial tensile force of 60 kN applied at each end of the bar . Determine

c) the normal and shearing stresses on a plane inclined at 30 to the direction of loading , d) the maximum shearing stress in the bar , and, e) if the capacity of this bar is such that the maximum normal tensile stress is 150 N/mm2 and the

maximum shearing stress is 60 N/mm2, , what is the maximum axial tensile load the bar can take without exceeding these values and in what plane is the failure going to occur?

Fig Q4 The beam section with the applied moment

100 mm 40 mm

y

x

y

x

Z

100 kNm

30o

3 m 3 m

4 m

50 kN

50 kN

Fig Q5 A loaded truss

3 m 3 m

Q2. At a point in a body, stresses are as shown in Fig Q2 .

45 60 MPa 40 MPa

55 MPa

Fig Q2 Stress at a point diagrams

35o

75

80 MPa

(i) (ii) (iii)

120 MPa

60 MPa

60 MPa

120 MPa

Use the analytical method or the derived equations to determine on the inclined planes in the figures (Fig Q2) ,

(a) the normal and the shearing stresses , and,

(b) the principal stresses .

Q3. Use the Mohr’s circle to determine the normal and the shearing stresses on the inclined planes for the three figures in Fig Q2.

Q4. (a) What is unsymmetric bending and what causes it?

(b) Determine the shear centre of the channel section shown in Fig Q4. The thickness is constant.

Fig Q4 A Channel section

Q5. For each section in Fig Q5, plane A-A is the plane of load. Show with letters and lines, the positions of the peak tensile and the peak compressive stresses and the approximate neutral axis positions (no calculation is required). What are the components of moment about the X and Y axis for each figure.

50

100

20

20 20

v

Fig Q5 Structural section diagrams

1-27 Specifications for the timber block of Fig. P1-27 require that the stresses not exceed the following: shear parallel to the grain 0.75 MPa compression perpendicular to the grain 1.20 MPa. Determine the maximum value of the axial load p that can be applied without exceeding the given requirement.

1-28 Solve Prob 1-27 with the following data changes: dimensions of block 80mm deepx130 mm widex250 mm long; slope of grain 5 horizontal to 12 vertical; specified stresses 0.80 MPa shear and 1.40 MPa compression.

1-29 Because of internal pressure in a boiler, the stresses at a particular point in the boiler plate were found to be as shown in the stress picture of Fig.P1-29 dertermine the normer stress at this point on the inclind plan shown.

X

Y Y Y

X

X

(i) (ii) (iii)

60 kNm

25o

40 kNm

35 kNm 50o

75o

A

A

A A

A

A

1-30 At a point in a stressed body , there are normal stresses of 120 MPa C on a vertical plane and 60 MPa T on a horizontal plane. The

1. COURSE NAME & CREDIT LOAD:

Course : CVE 366-Structural Analysis II (2 Credits-Compulsory)








3.0 COURSE DETAILS:

3.1 Course Content

Theory and problems in indeterminate structures. Static and kinematic indeterminacy, classical methods of analysis , virtual work and energy methods, slope deflection method and moment distribution method. Influence line for statically indeterminate structures.


Theory and problems in indeterminate structures-explain determinate and indeterminate structures and how to identify an indeterminate structure.

Static and kinematic indeterminacy, classical methods of analysis –discuss stability and meaning of Static and kinematic and discuss relevant equations with respect to beams frames and trusses.

virtual work and energy methods-explain this and apply to solving structural problems.

slope deflection method and moment distribution method- explain differences and apply on beams and frames.

Influence line for statically indeterminate structures-explain and apply to solve problems in beams and trusses.

3.3 Course Justification: Knowing how to analyse for forces in structures is important and prerequisite to design.


Aims and objectives : The students are expected to understand the following ;

B e able to compute for member reactions, forces, moments and strains in indeterminate structures using virtual work and energy methods. Solving indeterminate trusses using energy method and frames using slope deflection and moment distribution methods and application of influence lines to indeterminate structures

3.5 Course Requirements: A student is expected to have scored not less than 30 in CVE 365.



3.7 Course Delivery Strategies and Practical Schedules: The course shall be delivered by giving note , extract from text books code of practice and showing practical examples and so on, solving problems in the class and giving out problem to solve by students themselves , giving reference books as reading and problem exercise.


Week 1 Indeterminate simple beams analysis for state c and kinematic indeterminancy

Week 2 Solving indeterminate simple beams

Week 3 Solving continouse beams of n spams

Week 4 Solving indeterminate trusses

Week 5 Solving using slope – deflection method

Week 6 Solving using moment distribution method

Week 7 Influence line analysis and applications

Week 8-15 Revision, test and examination


Text books :




6.0 LEGEND

Exercise/tutorial Questions

Q1 For each of the structures shown in Fig Q1 ( i to v )

(a) State whether stable or not and give reasons for your answers (10 marks). (b) Determine the degree of static and kinematic indeterminacy (10 marks). (c) For any of the structures that is unstable, draw the stable form (5 marks).

Q2 (a) State the Castigliano’s second energy theorem and its usefulness (5 marks)

(b) Using the theorem or otherwise to determine the support reactions in the frame shown in Fig Q2 (20 marks)

Fig Q1 ( i to vi) Structural elements

i ii

iii iv

v

Q3 Fig Q3 is a pin jointed truss. Determine the degree of indeterminacy and member forces. Let the redundant be member BD (25 marks)

Q4 (a)Differentiate between moment distribution method and slope deflection equation method (5 marks).

(b) Use either of the two methods to determine the end moments in the Continuous beam shown in Fig Q4 (20 marks).

C B

A D

D

C

A

B

3m 4m

3.5m

80 kN

50 kN

4 m

3 m

Fig Q2 A Frame

Fig Q3 A truss

D A

50 kN 40 kN/m

Q5 (a) Define an influence line diagram (5 marks)

(b) A vehicle traverses a bridge from support A to B as shown in Fig Q5. The bridge is 20 m long and the front axle load of the car is 60 kN while the rear axle load is 45 kN. For the position of the front axle on the bridge and the rear wheels are just on the left support A, determine,

(i) the total reaction at each of the supports A and B (10 marks), and

(ii) the shear force and the bending moment at the centre of the bridge (10 marks).

1.0 COURSE NAME & CREDIT LOAD:

Course : CVE 562- Design of structures (3 Credits-Compulsory)

Course Duration : Three hours per week for 15 weeks (45 hours), As taught in 2010/2011




A

B

Moving axle loads

2m

60 kN

C B

3m 2m 2m

Fig Q4 A Continuous beam

45 kN

Fig Q5 Bridge span and moving loads

20 m

Front axle

Rear axle




3.0 COURSE DETAILS:

3.1 Course Content

Analysis and Design of Multi-storey buildings , pre-stressed concrete, hydraulic structures, water and earth –retaining structures, culverts, composite constructions, design considerations for bridges ,. Standards and codes of practice , methods of construction.


Analysis and Design of Multi-storey buildings – refer to timber, concrete and steel structures.

pre-stressed concrete- what is pretension and post tension and apply this to beams

hydraulic structures, water and earth –retaining structures-= On this discussion will be on retaining walls- gravity, cantilever counter-fort, concrete tanks and towers.

Culverts- rectangular and ring culverts . Load determination.

composite constructions and design considerations for bridges -Describe composite with respect to bridges etc

Standards and codes of practice – BS 5950, 8110 etc

Methods of construction- explain this with respect to various equipment on concrete , excavation, scaffolding etc.

3.3 Course Justification: Knowing how to analyse and design building before construction is important and will reduce failures


At the end of semester, students are expected to know how to design for buildings.

3.5 Course Requirements: A student is expected to scored not less than 30 in CVE 466 and CVE 362.



3.7 Course Delivery Strategies and Practical Schedules: Course shall be delivered by giving notes , extracts from the text books code of practice and showing practical examples and so on, solving problems in the class giving out problem to solve by students themselves , giving reference books as reading and problem exercise.


Week 1 : Analysis of multi-storey a reinforced concrete building

Week 2 : Analysis of steel structures

Week 3: Design of prestressed concrete structures

Week 4 Design of water retaining structures – water tank above and below ground surface Basement

Week 5: Design of earth retaining Structures – chanticleer, counter fort and gravity

Week 6: Design of culverts

Week 7 Design of bridges

Week8 : Composite construction

Week 9-15: Revision ,Test and Exam





J.F. Woodward (1975) Quantitative methods in Construction Management and Design Macmillan, London.

T.B BOFFEY Graph Theory in Operatives Research 1982, Macmillan London.

W.H. Mosley and J.H. Bungey 1990 Reinforcement Concrete Design Fourth Edition Macmillan London.

BS5950 Part 1 1990 Code for steel Design.

BS8110 Part 1,2, and 3 for structural use of concrete.

T.J. MacGinley & T.C Ang 1999 Structural Steel work Design to limit state Theory Second Edition Butter worth Heinemann.



Question 1

Figure 1 shows a frame of a residential building . The frames are at 3 m centres and braced against lateral forces . The floor loads and data are as follow: Slab thickness = 120 mm, finishes and partitions = 1 kN/m2

, live load = 2.5 kN/m2 ,

fcu = 25 N/mm2 ; fy = 460 N/mm2, concrete unit weight 24 kN/m3 , concrete cover =

25 mm , beam size 300 wide by 400 deep and column size 300 mm by 300 mm. Use 16 mm diameter bars for beams, 12 mm diameter bars for slabs and 8 mm diameter bars for stirrups .

Required : For the beam ABC in Fig 1, determine the following:

(i) the minimum and maximum load on the slab (5 marks); (ii) the bending moments in the slab (15 marks); (iii) the longitudinal reinforcement required (15 marks); and , (iv) the slab structural detail (5 marks) .

Question 2 (a) Write briefly on the following :- Gravity , Cantilever and Counterfort retaining walls. (9 marks)

(2b) Fig 2 is a cantilever retaining wall of height 3.5 m and base thickness 200 mm and width 2.6 m. Check for the stability of the retaining wall against sliding and overturning and design for only the wall reinforcement . Soil unit weight = 2000 kg/m3 and the concrete unit weight = 24 kN/m3 . The coefficient of active pressure, ka , is 0.33 , the coefficient of passive pressure kp , is 3 and the coefficient

of friction , , between the retaining wall base and the soil is 0.5. Fcu = 30 N/mm2 , fy = 460 N/mm2 and the soil bearing pressure = 100 N/mm2 (11 marks).

Question 3 Figure 3 is a water retaining structure . Using ultimate limit state determine the maximum moments at the wall and the base and determine their corresponding reinforcement detail. Fcu = 35 N/mm2 , fy = 460 N/mm2 , concrete cover = 40 mm, bar diameter = 12 mm and the water unit weight = 1000 kg/m3 (20 marks).

Question 4 (a)Write short notes on the following : (i) HA loading , (ii) HB loading , (iii) Types of failures in bridges. (9 marks)

(b) Sketch and write short notes on the following (i) truss bridge, (ii) cable stayed bridge, (iii) suspension bridge, (iv) arch bridge and (v) girder bridge.

(11 marks)

Question 5 (a) Discuss the following : (i) Prestress concrete (ii) Pretensioned concrete (iii) Post tensioned concrete (8 marks)

(c)

Roof

1st floor

5 m 1.5 m 1.5 m

300

2.2 m

Wall 150mm thick

Fig 3. A water retaining Beam

Continuous at this edge

Continuous at this edge

3

3

P

Q

G. F.

Fig 1a Cross-section

3m

3m

A C

B

Span of floor

Fig 1b Floor plan

Fig 1 Frame and floor plan of a residential

6 m 4 m

6 4

Question 1

Figure 1 shows a frame of a residential building . The frames are at 3 m centres and braced

against lateral forces . The floor loads and data are as follow: Slab thickness = 120 mm :

Finishes and partitions = 1 kN/m2 :Live load = 2.5 kN/m2 : Fcu = 25 N/mm2 ; fy = 460 N/mm2

: Concrete unit weight 24 kN/m3 ; Beam size 300 wide by 400 deep , column size 300 mm

by 300 mm , Use 16 mm diameter high yield steel bars for beams and 8 mm diameter bars

for the stirrups. Concrete cover = 25 mm

Required : (a) Analyse, design and prepare structural detail for beam ABC in Fig 1 (25) : (b)

Prepare a typical reinforcement detail for a foundation and a typical detail for a column .

Show plans and sections for each of them.(25)

Question 2

Q (2a) Write briefly on the following retaining walls :- Gravity (3 marks) , Cantilever (3 marks) and

Counterfort . (3 marks)

Q(2b) Fig 2 is a cantilever retaining wall of wall height 3.5 m and base width 2.6 m. Soil unit

weight = 2000 kg/m3 , concrete unit weight = 24 kN/m3 , The coefficient of active pressure,

ka , is 0.33 , the coefficient of passive pressure, kp , is 3 , The coefficient of friction , ,

between the retaining wall base and the soil is 0.5, fcu = 30 N/mm2 , fy = 460 N/mm2 , soil

bearing pressure = 100 N/mm2 .

0.8 1.60.2

Soil

0.2 m

Fig 2. A cantilever retaining wall

(i) Calculate the maximum horizontal force on the wall (6 marks) (ii) the maximum moment at

the base of the wall (4 marks) (iii) Prepare a typical reinforcement detail for an earth retaining

wall structure.(6 marks)

Question 3 Figure 3 is a water retaining structure . with fcu = 35 N/mm2 , fy = 460 N/mm2 ,

concrete cover = 40 mm, bar diameter = Y12 mm , water unit weight = 1000 kg/m3

(a) What is the maximum moment at the wall base when the water tank is full and when

empty (12 marks) :

(b) What number of bars per meter length are required in the wall (6 marks)

(c) Prepare a typical detail for a water retaining structure (7 marks)

Question 4

Q (4a) Write short notes on the following : (i) HA loading , (ii) HB loading , (iii) Types of failures in

bridges. (12 marks)

Q(4b) Sketch with labels the following types of bridges, stating the appropriate materials and

sections required for each of their structural elements. (i) An Arch bridge (ii) A suspension

bridge (iii) A girder bridge (iv) deck of bridges (8 marks),

Q(4c) Prepare a typical reinforcement detail for a box culvert (5 marks)

6 m 4 m

3

3

P

Q

Roof

1st floor

G. F.

Fig 1a Cross-section

3m

3m

A C

Span of floor

6 m 4 m

Roof Roof Roof

1.6

3.5 m

0.8 0.2

0.2 m

Fig 2 A cantilever retaining wall


Course : CVE 565- Structural Analysis IV (2 Credits-Compulsory)

Course Duration : Three hours per week for 15 weeks (30 hours), As taught in 2010/2011







3.0 COURSE DETAILS:

3.1 Course Content

Approximate method of analysis for frame structures, yield line analysis of slabs. Structural Forms.

Plastic analysis of multi-bay and multi-storey frame buildings.


Approximate method of analysis for frame structures, to be applied on portal , multibay frames

yield line analysis of slabs- apply on concrete slabs circular, rectangular and triangular yield patterns,

collapse load

Structural Forms-consider various structural forms for timber steel and reinforced concrete

structures

Plastic analysis of multi-bay and multi-storey frame buildings. Consider various collapse mechanisms,

collapse loads hinge formations etc.

3.3 Course Justification: Knowing how to analyse structures before construction is important and will reduce failures


At the end of semester, students are expected to know how to design for buildings.

3.5 Course Requirements: A student is expected to scored not less than 30 in CVE 466 and CVE 362.



3.7 Course Delivery Strategies and Practical Schedules: Course shall be delivered by giving notes , extracts from the text books code of practice and showing practical examples and so on, solving problems in the class giving out problem to solve by students themselves , giving reference books as reading and problem exercise.


At the end of this course ,the students are expected to know how to carryout the following analysis:

Duration: 15 weeks

Week i: (i-v) Differentiate between approximate method and exact method of Analysis.

Week (vi-viii): Analyse indeterminate trusses, evaluate forces in members assuming that the diagonals are to be designed such that they are equally capable of carrying compressive and tensile forces.

Week (ix-x): Analyse buildings using Cantilever and Portal method of analysis.

Week (xi-xii) : Analyse using yield line theories , yield line pattern for a triangular, circular reinforced and a rectangular concrete slabs.

Week (xiii-xv): Analyse using plastic method of analysis and compare with the elastic method and derive the expression for collapse loads in term of plastic moment Mp . Revision tests and examinations.





J.F. Woodward (1975) Quantitative methods in Construction Management and Design Macmillan, London.

T.B BOFFEY Graph Theory in Operatives Research 1982, Macmillan London.

W.H. Mosley and J.H. Bungey 1990 Reinforcement Concrete Design Fourth Edition Macmillan London.

BS5950 Part 1 1990 Code for steel Design.

BS8110 Part 1,2, and 3 for structural use of concrete.

T.J. MacGinley & T.C Ang 1999 Structural Steel work Design to limit state Theory Second Edition Butter worth Heinemann.





Course - CVE 565 : Structural Analysis IV; Time 3 hrs : Answer all questions (Each 20 marks)

F E D

C B A

80 kN

D

Fig 1 A loaded Truss

(Q1) For the truss shown in Fig 1,

(a) determine the degree of indeterminacy (2 marks) and (b) evaluate approximately forces in members assuming that the diagonals are to be designed such that they are equally capable of carrying compressive and tensile forces (18 marks).

Q2 (a) Differentiate between Cantilever and Portal methods of analysis (5 marks).

(b) For the Portal Frame shown in Fig 2, determine using approximate method of analysis, the reactions at the supports assuming that hinges are formed at the middle point of the girder and columns (15 marks).

Q3 (a) List the yield line theories (5 marks)and sketch the yield line pattern for a circular reinforced concrete slab and a rectangular slab simply supported at their edges (4 marks).

(b) The slab shown in Fig 3 is isotropically reinforced and is required to carry an ultimate design load of 12 kN/m2. If the ultimate moment of resistance of the reinforcement is m per

3 m

4 m

3 m

20 kN

Fig 2 A Portal frame

unit width of slab in the directions shown, calculate the value of m for the given yield line pattern (11 marks).

(4) What are the differences between plastic method of analysis and elastic method (5 marks) and derive the expression for collapse load W in term of plastic moment Mp and length L for the simply supported beam shown in Fig 4 (15 marks).

Q5 (a) What are the assumptions made in the approximate method of analysis of a frame subjected to vertical loads only (4 marks). (b) Analyse the building frame shown in Fig 5 for vertical loads using the approximate methods to compute the vertical loads and bending moments in columns. What is the maximum bending moment in the beams (16 marks).

W

4m

4m

Fig 3 Isotropicaly reinforced concrete slab with yield line pattern shown

m m

L/2 L/2

Fig 4 A simply supported beam


Course : CVE 464 – Structural Analysis III (2 Credits-Compulsory)



A

B

C F I

H

G

E

D

8 kN/m

4 kN/m

6m 6m

5m

5m

Fig 5 A building frame






3.0 COURSE DETAILS:

3.1 Course Content

Matrix methods of structural analysis. Flexibility and stiffness methods. Elastic instability. Introduction to plastic theory of bending. Collapse loads.


3.3 Course Justification: Knowing how to analyse for stress imposed on materials before construction is important and will reduce failures


Aims and objectives : At the end of the course the students are expected to understand the following

Matrix methods of structural analysis. Introduction to matrix

Flexibility and stiffness methods. Discuss on simple beam and continuous beams, trusses

Elastic instability. On columns / beams

Introduction to plastic theory of bending. Collapse loads. Discuss on beams , frames and yield line analysis of slabs.


At the end of semester, Students are expected to understand the following:-

Matrix methods of structural analysis. Flexibility and stiffness methods on simple beam and continuous beams, trusses , Elastic instability analysis on columns / beams ,plastic analysis on beams , frames and yield line analysis of slabs.

3.5 Course Requirements: A student is expected to score not less than 30 in CVE 366



3.7 Course Delivery Strategies and Practical Schedules: Course shall be delivered by giving note , extract from text books code of practice and showing practical examples and so on, solving problems in the class giving out problem to solve by students themselves , giving reference books as reading and problem exercise.


Week i Matrix methods of structural analysis. Introduction to matrix

Week ii-v Flexibility and stiffness methods. Discuss on simple beam and continuous beams, trusses

Week vi-ix Elastic instability. On columns / beams

Week x-xiii Introduction to plastic theory of bending. Collapse loads. Discuss on beams , frames and yield line analysis of slabs.

Week iv-v Revision Test and Examination


(1) P-= 1 23 4 , Q-= 1 2

5 6 , determine PQ and QP

(2) identity matrix I= 1 0 00 1 00 0 1

(3) Determine from (1) above P-1 and Q-1.

(4) Figure below shows the load cases required to produce unit lateral displacements alternately at joints 1 and 2 in a vertical cantilever beam. If the positive direction for the displacements is as shown , construct the ;lateral stiffnesss matrix for the cantilever and calculate THE FORCE REQUIRED at the joints to produce displacements of 0.5 units at joints 1 and 2.0 units at joint 2.

Answer

퐿1퐿2)=( 10 −3

−3 2 ) (0.52.0)

L1= -1.0, L2= 2.5

EXAMPLE 17.1 Determine the horizontal and vertical components of the deflection of node 2 and the forces in the members of the truss shown in Fig. 17.4. The product AE is constant for all members. We see from Fig. 17.4 that the nodes 1 and 3 are pinned to the foundation and are therefore not displaced. Hence, referring to the global coordinate system shown, w1 = v1 = w3 = v3 = 0

FIGURE 17.4 Truss of Ex. 17.1

3

10

2

1

2

3

2

1 2

1

3

L

W

X

450

Y

The external forces are applied at node 2 such that Fx,2 =0, Fy,2=−W; the nodal forces at 1 and 3 are then unknown reactions. The first step in the solution is to assemble the stiffness matrix for the complete framework by writing down the member stiffness matrices referred to the global axes using Eq. (17.23). The direction cosines λ and µ take different values for each of the three members; therefore, remembering that the angle θ is measured anticlockwise from the positive direction of the x axis we have the following:

See the reference book (2) below , for the rest computation

THEOREMS OF PLASTIC ANALYSIS Plastic analysis is governed by three fundamental theorems which are valid for elastoplastic structures in which the displacements are small such that the geometry of the displaced structure does not affect the applied loading system. THE UNIQUENESS THEOREM The following conditions must be satisfied simultaneously by a structure in its collapsed state: The equilibrium condition states that the bending moments must be in equilibrium with the applied loads. The yield condition states that the bending moment at any point in the structure must not exceed the plastic moment at that point. 592 18.2 Plastic Analysis of Beams • 593 The mechanism condition states that sufficient plastic hinges must have formed so that all, or part of, the structure is a mechanism. THE LOWER BOUND, OR SAFE, THEOREM If a distribution of moments can be found which satisfies the above equilibrium and yield conditions the structure is either safe or just on the point of collapse. THE UPPER BOUND, OR UNSAFE, THEOREM If a loading is found which causes a collapse mechanism to form then the loading must be equal to or greater than the actual collapse load. Generally, in plastic analysis, the upper bound theorem is used. Possible collapse mechanisms are formulated and the corresponding collapse loads calculated. From the upper bound theorem we know that all mechanisms must give a value of collapse load which is greater than or equal to the true collapse load so that the critical mechanism is the one giving the lowest load. It is possible that a mechanism, which would give a lower value of collapse load, has been missed. A check must therefore be carried out by applying the lower bound theorem.

YIELD LINE THEORY There are two approaches to the calculation of the ultimate load-carrying capacity of a reinforced concrete slab involving yield line theory. One is an energy method which uses the principle of virtual work and the other, an equilibrium method, studies the equilibrium of the various parts of the slab formed by the yield lines; we shall restrict the analysis to the use of the principle of virtual work since this was applied in Chapter 18 to the calculation of collapse loads of beams and frames. YIELD LINES Aslab is assumed to collapse at its ultimate load through a system of nearly straight lines

which are called yield lines. These yield lines divide the slab into a number of panels and this pattern of yield lines and panels is termed the collapse mechanism; a typical collapse mechanism for a simply supported rectangular slab carrying a uniformly distributed load is shown in Fig. 19.1(a). The panels formed by the supports and yield lines are assumed to be plane (at fracture elastic deformations are small compared with plastic deformations and are ignored) 625 626 • Chapter 19 / Yield Line Analysis of Slabs and therefore must possess a geometric compatibility; the section AA in Fig. 19.1(b) shows a cross section of the collapsed slab. It is further assumed that the bending moment along all yield lines is constant and equal to the value corresponding to the yielding of the steel reinforcement. Also, the panels rotate about axes along the supported edges and, in a slab supported on columns, the axes of rotation pass through the columns, see Fig. 19.2(b). Finally, the yield lines on the sides of two adjacent panels pass through the point of intersection of their axes of rotation. Examples of yield line patterns are shown in Fig. 19.2. Note the conventions for the representation of different support conditions. In the collapse mechanisms of Figs 19.1(a) and 19.2(b) the supports are simple supports so that the slab is free to rotate along its supported edges. In Fig. 19.2(a) the left-hand edge of the slab is built in and not free to rotate. At collapse, therefore, a yield line will develop along this edge as shown. Along this yield line the bending moment will be hogging, i.e. negative, and the reinforcing steel will be positioned in the upper region of the slab; where the bending moment is sagging the reinforcing steel will be positioned in the lower region.

Structural Instability

So far, in considering the behaviour of structural members under load, we have been concerned with their ability to withstand different forms of stress. Their strength, therefore, has depended upon the strength properties of the material from which they are fabricated. However, structural members subjected to axial compressive loads may fail in a manner that depends upon their geometrical properties rather than their material properties. It is common experience, for example, that a long slender structural member will suddenly bow with large lateral displacements when subjected to an axial compressive load . This phenomenon is known as instability and the member is said to buckle. If the member is exceptionally long and slender it may regain its initial straight shape when the load is removed. Structural members subjected to axial compressive loads are known as columns or struts, although the former term is usually applied to the relatively heavy vertical members that are used to support beams and slabs; struts are compression members in frames and trusses. It is clear from the above discussion that the design of compression members must take into account not only the material strength of the member but also its stability against buckling. Obviously the shorter a member is in relation to its cross-sectional dimensions, the more likely it is that failure will be a failure in compression of the material rather than one due to instability. It follows that in some intermediate range a failure will be a combination of both. We shall investigate the buckling of long slender columns and derive expressions for the buckling or critical load; 21.1 EULER THEORY FOR SLENDER COLUMNS The first significant contribution to the theory of the buckling of columns was made in the 18th century by Euler. His classical approach is still valid for long slender columns possessing a variety of end restraints. Before presenting the theory, however, we shall investigate the nature of buckling and the difference between theory and practice. We have seen that if an increasing axial compressive load is applied to a long slender column there is a value of load at which the column will suddenly bow or buckle in some unpredetermined direction. This load is patently the buckling load of the column

or something very close to the buckling load. The fact that the column buckles in a particular direction implies a degree of asymmetry in the plane of the buckle caused by geometrical and/or material imperfections of the column and its load. Theoretically, however, in our analysis we stipulate a perfectly straight, homogeneous column in which the load is applied precisely along the perfectly straight centroidal axis. Theoretically, therefore, there can be no sudden bowing or buckling, only axial compression. Thus we require a precise definition of buckling load which may be used in the analysis of the perfect column. If the perfect column is subjected to a compressive load P, only shortening of the column occurs no matter what the value of P. Clearly if P were to produce a stress greater than the yield stress of the material of the column, then material failure would occur. However, if the column is displaced a small amount by a lateral load, F, then, at values of P below the critical or buckling load, PCR, removal of F results in a return of the column to its undisturbed position, indicating a state of stable equilibrium. When P =PCR the displacement does not disappear and the column will, in fact, remain in any displaced position so long as the displacement is small.

See reference (2) for the rest.

Reference book

(1) K.H.M. Bray, P.C.L. Croxton and L.H. Martin Matrix Analysis of Structures, Pitman Press, 1976

(2) T.H.G. Megson , Structural and Stress Analysis, Second Edition, Elsevier,2005, London.

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