Ahsanullah University of Science And Technology

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Prepared By Course no: CE 416

Objectives : Introduction of Couple Understanding the Properties of a Couple. Analyzing the Characteristics of a Couple. What a couple does Internal resisting couple

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Definition of a couple :

When two parallel forces are equal but opposite in sense and are not collinear they form a couple. The moment of a couple is the product of one of the forces and the perpendicular distance between the lines of action of the forces.

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The above figure indicates a counterclockwise couple where the moment of the couple

M = F * d here , F = Force acting on the body. d = Moment arm of the couple.

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F

F

F

F

d

d

Properties of a couple :

o Equal magnitude and opposite in direction . o Act along parallel lines of action .

o Separated by a perpendicular distance .

o The orientation of the plane of the couple in relation to the body upon which it acts .

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Characteristics of couples :

• A couple has the same moment about all points on a body.

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F

F

F

F

A couple may be transformed into another couple that produces the same external effect.

A force may be resolved into a force and a couple where a single force and a couple cannot balance one another.

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The above shows how a force can be replaced by a force-couple equivalent.

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B

F

A d

A B

F F

F

d

A B

F

MA = F x d

Force Couple Equivalent

The resultant of several couples in the same plane or parallel planes has a magnitude equal to the sum of several couple moments

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A

C D

B

100 lb

200 lb

250 lb

150 lb

What a couple does? A couple causes a body to rotate

only without translational motion since the two forces ‘cancels’ out each other giving zero resultant.

A couple acting in a system of forces will only contribute to the resulting moment but not to the resulting force.

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Internal resisting couple in a pre stressed concrete beam :

Fig: Internal resisting couple with arm a

In the figure a pre stressed concrete beam is shown where Steel is supplying the tensile force Tand the concrete is supplying the compressive force C. Whre C and T together form a couple,with arm a .

The couple C-T is formed in order to resist the external moment.

For a pre stressed concrete beam

Resistance to external moment is gained by lengthening of the lever arm between the couple C-T.

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In a prestressed concreted beam section under working load, as the external bending moment increases ,the magnitude of C and T reamins practically constant while the lever arm ‘a’ lengthens almost proportionately.

Fig: Variable a in a prestressed –concrete beam

When a section of a prestressed beam is subjected to a crack under working load the stress in steel increases and reaching to its ultimate strength the lever arm of the internal couple cannot be increased anymore which results in the point of failure.

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Nominal & Ultimate :resisting moment

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Maximum value of the internal resisting couple is termed as nominal moment strength.which must resists the bending moment applied by the external loads.

From the internal resisting couple the ultimate resisting moment can also be computed which represents the very last point of the beam failure. From the above figure for C-T force couple, the ultimate resisting moment according to ACI code, Mu = φ [Apsfps (d - a/2)]

The End ..........

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Special Thanks To: Mr.Galib sirLecturer atAUST