At any point angle made by tangent drawn to the deflected shape of a beam, with horizontal is called _________

Question

At any point angle made by tangent drawn to the deflected shape of a beam, with horizontal is called ____________ at that point.

Examoogle · Accepted Answer

The correct answer is Option B: Slope. In structural mechanics, the slope of a beam at any point is defined as the angle (in radians) that the tangent to the elastic curve makes with the original horizontal axis of the beam. It represents the first derivative of the deflection function with respect to the beam's longitudinal coordinate.

Step-by-Step Solution & Concepts:
In structural mechanics, the slope of a beam at any point is defined as the angle (in radians) that the tangent to the elastic curve makes with the original horizontal axis of the beam. It represents the first derivative of the deflection function with respect to the beam's longitudinal coordinate.

Working Principle: When a beam is subjected to transverse loads, it undergoes bending, causing it to deform into a curve known as the elastic curve or deflected shape. The slope at any cross-section is the inclination of the tangent to this curve, represented mathematically by $	heta = \frac{dy}{dx}$, where $y$ is the deflection and $x$ is the horizontal position.

Key Formulas: $	heta = \frac{dy}{dx}$ — The slope as the first derivative of deflection; $EI \frac{d^2y}{dx^2} = M$ — The governing differential equation of the elastic curve

Key Points: Slope is a dimensionless quantity measured in radians.; The slope is zero at points of maximum or minimum deflection.; For small deflections, $	an(	heta) \approx 	heta$ (in radians).; Slope and deflection are related through integration of the bending moment equation.

Advantages: Allows determination of boundary conditions for indeterminate structures.; Essential for calculating beam stiffness and deformation under load.

Disadvantages: Assumes linear elastic material behavior (Hooke's Law).; Standard slope-deflection equations are limited to small-angle approximations.

Applications: Structural analysis of continuous beams and frames.; Design of mechanical shafts and machine components to limit angular deformation.

Additional Info: Deflection is the vertical displacement of a point on the beam, whereas slope is the angular change at that point.. Option A is incorrect because deflection is the vertical distance measured from the original horizontal axis, not the angle.

Answer: B is correct — The angle made by the tangent to the deflected shape with the horizontal is defined as the slope of the beam.

Exam Tip: Always remember the relation: curvature is the second derivative of deflection ($d^2y/dx^2$), while slope is the first derivative ($dy/dx$).

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Option A: Deflection

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Option C: Both Deflection and Slope

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Option D: None of these

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