Demonstrative Example

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    SM-01: Static Analysis of Euler-Bernoulli Beams

    Boundary Conditions

    Left Boundary

    Right Boundary

    Beam Parameters

    Young's Modulus (E):

    Area of Moment Inertia (I):

    Length (L):


    Load Type

    Load Location (x):

    Force (f):

    Plot Option

    Number of points: (≤1000)

    About the Tool

    The tool SM-01 computes and plots static response (transverse displacement, slope, bending moment and shear force) of a uniform Euler-Bernoulli beam with various end conditions and subject to transverse loads. The tool also gives analytical expressions of beam response, determines the reactions at the end supports, and finds the locations of maximum bending moment and shear force along the beam length.

    In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.


    Consider a simply-supported beam in the following figure, which has length L = 1 and bending stiffness EI = 25, and is subject to a pointwise force f0=1.2 at its midpoint.

    The input in the tool is

    After clicking "Run", the results will be shown in another page. The static responses of the uniform beam are

    The analytical results are also shown as follows

    Analytical Expression of Beam Response

    Beam Response for 0 <= x <=0.5

    Displacement w(x) = (-0.004)x3 + (-0)*x2 + (0.003)*x + (-0)

    Rotation dw(x)/dx = (-0.012)*x2 + (-0)*x + (0.003)

    Bending moment M(x) = (-0.6)*x + (-0)

    Shear force Q(x) = (-0.6)

    Beam Response for 0.5 <= x <=1.0

    Displacement w(x) = (0.004)*x3 + (-0.012)*x2 + (0.009)*x + (-0.001)

    Rotation dw(x)/dx = (0.012)*x2 + (-0.024)*x + (0.009)

    Bending moment M(x) = (0.6)*x + (-0.6)

    Shear force Q(x) = (0.6)

    Maximum Beam Response

    Maximum displacement = 0.001, location x = 0.5

    Maximum rotation (degree) = 0.17188734, location x = 0

    Maximum bending moment = -0.2996997, location x = 0.5

    Maximum shear force = -0.6, location x = 0

    Reactions at Two Ends of the Beam


    Sign Convention for support reactions:

    * Positive moment Mc: counterclockwise

    * Positive force Rc: upward


    At left boundary (x = 0): Mc = -0, Rc = -0.6

    At right boundary (x = L): Mc = -0, Rc = -0.6

    Subject Review

    Static of beam

    Fig. 1 A uniform Euler-Bernoulli beam under static loads

    Governing Equation

    The transverse displacement (deflection) w(x) of a uniform Euler-Bernoulli beam under static loads, as shown in Fig. 1, is governed by the differential equation

    Static of beam governing equation

    where E is Young’s modulus, I and L are the moment of inertia of beam cross section area and beam length, respectively. The EI is known as the bending stiffness of the beam. Here by “uniform beam”, the beam cross section area and EI are assumed to be constant along the beam length (x). For the loads shown in Fig. 1, f(x) can be written as

    Static of beam load

    where q(x) is a distributed external load, f0 is a pointwise force applied at xf, τ is a torque applied at xτ, and δ(*) is the Dirac delta function.

    The boundary conditions of the beam can be written as

    Static of beam boundary conditions

    where B01, B02, BL1, and BL2 are spatial differential operators. Four types of beam boundary conditions are given in Table 1.

    Table 1. Boundary conditions

    Static of beam typical boundary conditions

    The static response of a Euler-Bernoulli beam includes:

    • Transverse displacement or deflection w(x)
    • Rotation or slope
    • Static of beam rotation
    • Bending moment
    • Static of beam bending moment
    • Shear force
    • Static of beam shear force

    The positive direction of w(x) is upward; the positive direction of rotation is counterclockwise; and the sign convention of bending moment and shear force is given in Fig. 2.

    Static of beam sign of bending and moment

    Fig. 2 Sign convention of bending moment and shear force

    Except for free ends, the boundaries listed in Table 1 exert reaction moments and/or reaction forces to the beam. Because the reactions support the beam by balancing the external forces, the boundaries are also called supports. The reactions are represented by

    Static of beam boundary reaction

    where w'= dw/ dx, Mc and Rc are reaction moment and force, respectively. The sign convention of reactions is shown in Fig. 3.

    Static of beam sign of reaction

    Fig. 3 Sign convention of reactions

    The problem of static analysis of a uniform Euler-Bernoulli beam is to determine the beam response (displacement, slope, bending moment and shear force) by solving the governing differential equation subject to the boundary conditions.

    Solution Methods

    Several methods are available for static analysis of Euler-Bernoulli beams. For analytical solutions, the method of singularity functions, the boundary value approach, and the distributed transfer function method can be used. For numerical solutions, the finite element method is often applied. For detail of these methods, interested users can refer to the references at the end of this document. The on-line tool SM-01 delivers exact beam response via the distributed transfer function method in a systematic and efficient manner; see Pages 21-23 of Reference 1.

    Beam Deflection in Some Cases

    Static beam deflections in some cases are given in Table 2. The bending moment and shear force in each case can be determined by M=EIw'' and Q=EIw''' .

    Table 2. Static beam deflection

    Static beam deflection


    1. Yang, B., 2005, Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB toolboxes, Elsevier Science.

    2. Young, W.C., 1989, Roark's Formulas for Stress and Strain, 6th edition, McGraw-Hill, New York.

    3. Bedford, A., and Liechti, K. M., 2000, Mechanics of Materials, Prentice Hall, Upper Saddle River, New Jersey.

    4. Gere, J. M., and Stephen P. Timoshenko, S. P., 1997, Mechanics of Materials, 4th edition, PWS Publishing Co., Boston.

    5. Popov, E. P., 1998, Engineering Mechanics of Solids, 2nd ed., Prentice Hall, Upper Saddle River, New Jersey.

    6. Riley, W.F., Sturges, L. D., and Morris, D. H., 1999, Mechanics of Materials, 5th ed., John Wiley & Sons, Inc., New York.

    7. Shames, I. H., and Pitarresi, J. M., 2000, Introduction to Solid Mechanics, 3rd ed., Prentice Hall, Upper Saddle River, New Jersey.