How to Calculate Tension & Deflection in a Rod

Calculating the tension and deflection of a rod is important when building structures such as bridges.
••• 'Kissing Bridge' Covered Bridge near Stowe in Vermont image by Rob Hill from Fotolia.com

When designing a structure such as a building or a bridge, it is important to understand the many forces that are applied to the structural elements such as beams and rods. Two especially important structural forces are deflection and tension. The tension is the magnitude of a force that is applied to a rod, while the deflection is the amount the rod is displaced under a load. Knowledge of these concepts will determine how stable the structure will be, and how feasible it is to use certain materials when building the structure.

Tension on the Rod

    Draw a diagram of the rod and set up a coordinate system (e.g. forces applied to the right are "positive," forces applied to the left are "negative").

    Label all forces that are applied to the object with an arrow that is pointing in the direction the force is applied. This is what's known as a "free-body diagram."

    Separate the forces into horizontal and vertical components. If the force is applied at an angle, draw a right triangle with the force acting as the hypotenuse. Use the rules of trigonometry to find the adjacent and opposite sides, which will be the horizontal and vertical components of the force.

    To find the resultant tension, add up the total forces on the rod in the horizontal and vertical directions.

Deflection of the Rod

    Find the bending moment of the rod. This is found by subtracting the length of the rod L by the position variable z, and then multiplying the result by the vertical force applied to the rod -- denoted by the variable F. The formula for this is M = F x (L - z).

    Multiply the modulus of elasticity of the beam by the moment of inertia of the beam about the non-symmetric axis.

    Divide the bending moment of the rod from Step 1 by the result from Step 2. The ensuing result will be a function of the position along the rod (given by the variable z).

    Integrate the function from Step 3 with respect to z, with the limits of integration being 0 and L, the length of the rod.

    Integrate the resulting function again with respect to z, with the limits of integration again ranging from 0 to L, the length of the rod.

    Things You'll Need

    • Calculator
    • Knowledge of integral calculus

    Tips

    • The modulus of elasticity is hard to estimate experimentally, so they must be given or you must assume the rod has an ideal shape, such as a cylinder, or it has some geometric symmetry. You generally look this up in a table.

    Warnings

    • The calculation for the deflection of the rod assumes a symmetrical rod.

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