Chapter 7 Columns.docx

[Department of Engineering ]

[Mechanical section] CHAPTER 7

STRAIN MEASUREMENT

Chapter Outline 7.1 Column and struts 7.2 Euler’s column theory:

Course outcomes covered 11. Set up the differential equation describing the elastic curve for a buckling of point's ended columns

7.3 Types of end condition

COLUMNS AND STRUTS A structural member, subjected to an axial compressive force, is called as Strut. A strut may be horizontal, inclined or vertical. But a vertical strut used in buildings or frames is called as Columns. Failure of a column or strut: When a column or strut is subjected to some compressive forces, then the compressive stress induced is σ= P/A, where P is the compressive force and A is the cross sectional area of the column. If the force or load is gradually increased, the column will reach a stage, when it will be subjected to the ultimate crushing stress. Beyond this stage, the column will fail by crushing. The load corresponding to the crushing stress is called as Crushing load. Long columns will have buckling load and short columns will have crushing load. Slenderness Ratio: (Le/k) It is defined as the ratio of equivalent length of column to the least radius of gyration. It has no units. For Long columns: Slenderness ratio is more than 80. For Short columns: Slenderness ratio is less than 80.

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MIME 3130 Mechanics of Materials

[Department of Engineering ]

[Mechanical section]

Euler’s column theory: Euler’s formula cannot be used in the case of short column, because the direct stress is considerable and cannot be neglected. Assumptions: 1. Initially the column is perfectly straight and the load applied is truly axial. 2. The cross section of the column is uniform throughout its length. 3. The column material is perfectly elastic, homogeneous and isotropic and obeys Hooke’s law. 4. The length of column is very large when compared to cross sectional dimensions. 5. The shortening of column, due to direct compression is neglected. 6. The failure of the column occurs due to buckling alone. Sign conventions: 1. A moment, which tends to bend the column with convexity towards its initial central line as shown in fig (a) is taken as positive. 2. A moment, which tends to bend the column with concavity towards its initial central line as shown in fig (b) is taken as negative.

Types of End conditions of columns: These four types of end conditions are important from subject point of view. 1. 2. 3. 4.

Both ends are hinged. Both ends are fixed. One end is fixed and other is hinged. One end is fixed and other is free.

Columns with both ends hinged: Consider a column AB of length ‘l’ hinged at both of its ends A and B and carrying a critical load at B. As a result of loading, let the column deflect in to a curved form AX1B as shown in the fig. P – Critical load on the column. Y – Deflection of the column at X.

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MIME 3130 Mechanics of Materials

[Department of Engineering ]

[Mechanical section]

Columns with one end fixed and the other free: Consider a column AB of length ‘l’ fixed at A and free at B and carrying a critical load at B. As a result of loading, let the column deflect in to a curved form AX1B1 as shown in the fig. P – Critical load on the column. Y – Deflection of the column at X.

Columns with both ends fixed: Consider a column AB of length ‘l’ fixed at both ends A and B and carrying a critical load at B. As a result of loading, the column deflects as shown in the fig. P – Critical load on the column. Y – Deflection of the column at X.

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MIME 3130 Mechanics of Materials

[Department of Engineering ]

[Mechanical section]

Columns with one end fixed and the other hinged: Consider a column AB of length ‘l’ fixed at A and hinged at B and carrying a critical load at B. As a result of loading, the column deflects as shown in the fig. P – Critical load on the column. Y – Deflection of the column at X.

Limitation of Euler’s formula: Euler’s formula holds good only for long columns. Euler’s formula cannot be used in the case of short column, because the direct stress is considerable and cannot be neglected. Problem 1:

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MIME 3130 Mechanics of Materials

[Department of Engineering ] Problem 2:

Problem 3: 5

MIME 3130 Mechanics of Materials

[Mechanical section]

[Department of Engineering ]

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MIME 3130 Mechanics of Materials

[Mechanical section]

[Department of Engineering ]

Problem 4:

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MIME 3130 Mechanics of Materials

[Mechanical section]

[Department of Engineering ] Problem 5:

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MIME 3130 Mechanics of Materials

[Mechanical section]

[Department of Engineering ]

[Mechanical section]

Sources: 1) Text Books E J Hearn. (1997).Mechanics of Material 1, Third Edition: Butterworth Heinemann, A division of Reed Educational and Professional Publishing Ltd., Oxford. 2) Reference Books 1. William F.Riley / Leroy D. Sturges / Don H.Morris. (2002).Statics and Mechanics of Materials: An Integrated Approach, Second Edition, John Wiley & Sons, INC. 2. R.C.Hibbeler, Mechanics of Materials, Ninth Edition:Pearson. 3. Ferdinand Beer , E Russell Johnston Jr, John T DeWolf: Mechanics of Material(2004),Tata McGraw Hill third Edition. 3) Websites 1. www.wikipedia.com 2. www. howstuffworks .com 3. www.nptel.ac.in.

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MIME 3130 Mechanics of Materials