Answered: Give an example of applying the method…

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Question

Give an example of applying the method of virtual force on beams?

Expert Solution

Step 1

Assume the value of EI is constant.

Consider the figure to determine the deflection of the beam at point C,

Step 2

The formula to calculate the deflection at point C is,

$∆_{C} = {[\int_{0}^{L} \frac{M_{x} m_{x}}{E I} d x]}_{A B} + {[\int_{0}^{L} \frac{M_{x} m_{x}}{E I} d x]}_{B C}$

Here, M_x is the moment due to external load, m_x is the moment due to virtual load, L is the length, I is the moment of inertia, and E is the modulus of elasticity.

Step 3

Calculate the reactions at the support.

Take summation of horizontal force.

$\begin{array}{rcl} \sum_{} F_{x} & = & 0 \\ A_{x} & = & 0 \end{array}$

Take summation of the moment at point A.

$\begin{array}{rcl} \sum_{} M_{A} & = & 0 \\ B_{y} (8) - (\frac{1}{2} \times 5 \times 15) (8 + \frac{5}{3}) & = & 0 \\ B_{y} & = & \frac{362.5}{8} \\ = & 45.3 kN \end{array}$

Take summation of vertical force.

$\begin{array}{rcl} \sum_{} F_{y} & = & 0 \\ A_{y} + B_{y} & = & (\frac{1}{2} \times 5 \times 15) \\ A_{y} & = & 37.5 - 45.3 \\ = & - 7.8 kN \end{array}$

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