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PTC_ME_FEB_2.2_us_mp.mcdx
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Finite Element Beginnings David A. Pintur © 2011 Parametric Technology Corp.
2 The discrete approach
Section 2.2: A Simple Elastic Spring
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diagrams and other information (collectively “Solution”) presented in this Mathcad worksheet are
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Array origin:
ORIGIN ≔ 1
A simple example of a discrete element is an elastic spring, as
shown in the figure below:
Figure 1: One Elastic Spring
The left end of the spring is rigidly connected to a wall, while the other end is free to move only in a
horizontal direction. A force, F, is applied at the free end. Using Hooke's law, a direct relationship
between the axial load, F, in the spring and the deflection,  at its free end is developed:
((1))
F ≔ 4.5 kip
k ≔ 103 ⋅ kip ⋅ ft −1
1
Δ ≔ ―⋅ F
k
Δ = 0.044 ft
where k is the spring stiffness or spring constant.
Isolating F of equation (1) results in the force-deflection relationship for the spring:
((2))
F≔k⋅Δ
F = 4.5 kip
The stiffness, k, can be interpreted as the force required to produce a unit deflection (= 1).
Observe that for this very simple example, the free end displacement of the spring, , can be
calculated directly using equation (2). No discretization is required.
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PTC_ME_FEB_2.2_us_mp.mcdx
User Notices
 Equations and numeric solutions presented in this Mathcad worksheet are applicable to the
specific example, boundary condition or case presented in the book. Although a reasonable effort
was made to generalize these equations, changing variables such as loads, geometries and
spans, materials and other input parameters beyond the intended range may make some
equations no longer applicable. Modify the equations as appropriate if your parameters fall
outside of the intended range.
 For this Mathcad worksheet, the global variable defining the beginning index identifier for vectors
and arrays, ORIGIN, is set as specified in the beginning of the worksheet, to either 1 or 0. If
ORIGIN is set to 1 and you copy any of the formulae from this worksheet into your own, you need
to ensure that your worksheet is using the same ORIGIN.
 Engineering and construction code values shown in US Customary units are converted from
original values in Metric units. They are NOT obtained from US codes unless specified.
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