Calculation of support reactions of a beam

Problem 4-1

Find the reactions at the support of the following beam shown in figure 4-1(a).

Figure 4-1(a)

Solution:

The given beam is considered as 2-dimensional structure. The free-body diagram of the beam is given in figure 4-1 (b) which shows all the reaction components (Ax, Ay, By) and the applied loads. This beam is statically determinate as it can have only 3 reaction components; 2 at the hinged support at A and 1 at the roller support at B, and there are 3 possible equations of static equilibrium ΣFx = 0,  ΣFy = 0, ΣMz = 0; (The x-axis and y-axis are as shown in the figure and the z-axis is perpendicular to the x-y plane.)

Figure 4-1(b) free-body diagram

Applying equations of static equilibrium:

ΣFx = 0;  Ax = 0;                                              (eq. 1)

ΣFy = 0; Ay + By – 10 – 20 × 4 = 0;

Ay + By  = 90 kN;                             (eq. 2)

Considering z-axis passing through A, and taking moment of all the forces about z-axis (taking clockwise –ve and anticlockwise +ve);

ΣMz = 0;  By × 10 -10 × 8 – 20 × 4 × 2 = 0     (eq. 3)

Solving eq. 3 yields  By = 24 kN;

Substituting the value of By in eq. 4-2 gives Ay = 66 kN.

Excellent Calculators

Stress Transformation Calculator
Calculate Principal Stress, Maximum shear stress and the their planes

To determine Absolute Max. B.M. due to moving loads.

Bending Moment Calculator
Calculate bending moment & shear force for simply supported beam

Moment of Inertia Calculator
Calculate moment of inertia of plane sections e.g. channel, angle, tee etc.

Reinforced Concrete Calculator
Calculate the strength of Reinforced concrete beam

Moment Distribution Calculator
Solving indeterminate beams

Deflection & Slope Calculator
Calculate deflection and slope of simply supported beam for many load cases

Fixed Beam Calculator
Calculation tool for beanding moment and shear force for Fixed Beam for many load cases

BM & SF Calculator for Cantilever
Calculate SF & BM for Cantilever

Deflection & Slope Calculator for Cantilever
For many load cases of Cantilever

Overhanging beam calculator
For SF & BM of many load cases of overhanging beam

Civil Engineering Quiz
Test your knowledge on different topics of Civil Engineering

Research Papers
Research Papers, Thesis and Dissertation

List of skyscrapers of the world
Containing Tall buildings worldwide

Forthcoming conferences
Containing List of civil engineering conferences, seminar and workshops

Profile of Civil Engineers
Get to know about distinguished Civil Engineers

Professional Societies
Worldwide Civil Engineers Professional Societies