Boundary Conditions

To obtain a unique solution for a particular beam, it is required to specify sufficient end conditions termed boundary conditions in addition to loading data and flexural rigidity. These boundary

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conditions play an important role in the determination of arbitrary constants resulting from integration of the differential equation in the solution procedure.

From fundamental theorems on differential equations, it is required to specify n boundary conditions to render the two-point, elliptic, ordinary differential equation of order n in terms of a variable v to be well-posed (the term two-point is used to emphasize that a domain has two end points). All such boundary conditions can only involve quantities associated with v and its derivatives of order lower than n, i.e. dv/dx, d²v/dx², …, d^n-1v/dx^n-1. For instance, a two-point, elliptic, ordinary differential equation of order 4 in terms of a variable v requires 4 boundary conditions in terms of v, dv/dx, d²v/dx² or d³v/dx³. For a special class of two-point, elliptic, ordinary differential equations of even order (i.e. n is even), boundary conditions can be divided into two categories: essential boundary conditions and natural boundary conditions. The first category involves quantities such as v and its derivative of order less than n/2 (i.e. v, dv/dx, d²v/dx²,

…, d^n/2-1v/dx^n/2-1) while the other involves the remaining derivatives (i.e. d^n/2v/dx^n/2, d^n/2+1v/dx^n/2+1,

…, d^n-1v/dx^n-1). In addition, exactly half of boundary conditions must be specified at each end point.

As is evident from the previous subsection, the full-order differential equation governing the beam deflection, i.e. equation (3.27), is elliptic and is of order 4. As a result, boundary conditions at both ends of the beam involve prescribed values of either the deflection v, the rotation dv/dx, the bending moment EId²v/dx², or the shear force d(EId²v/dx²)/dx. The first two are essential boundary conditions and the last two are natural boundary conditions. For each end of the beam, exactly two boundary conditions must be prescribed. To specify proper boundary conditions for each end of a particular beam, following two guidelines are useful:

 Both the deflection and the shear force cannot be prescribed independently at the end of the beam. One boundary condition can be deduced from following three cases: (i) the deflection is prescribed while the shear force is unknown a priori (e.g. roller support and pinned-end support); (ii) the shear force is prescribed while the deflection is unknown a priori (e.g. free end and guided support); and both the deflection and shear force are unknown a priori but there exists a relation (generally obtained from considering force equilibrium at the beam end) relating both quantities (e.g. beam end with a translational spring).

 Both the rotation and the bending moment cannot be prescribed independently at the end of the beam. One boundary condition can be deduced from following three cases: (i) the rotation is prescribed while the bending moment is unknown a priori (e.g. fixed-end support and guided support); (ii) the bending moment is prescribed while the rotation is unknown a priori (e.g. free end, roller support and pinned-end support); and both the rotation and bending moment are unknown a priori but there exists a relation (generally obtained from considering moment equilibrium at the beam end) relating both quantities (e.g. beam end with a rotational spring).

With the above guidelines, we obtain boundary conditions for certain types of beam ends that are mostly found in beam as demonstrated below. Comprehensive summary of Boundary conditions for various beam ends can be found in Table 3.1.

3.3.1 Fixed-end support

For a beam end with a fixed-end support, both the deflection and rotation at this point are fully prevented while the bending moment and the shear force are unknown a priori. Two boundary conditions are therefore given by

v(0) 0 and v (0) 0  if a support is at the left end (3.30a)

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v(L) 0 and v (L) 0  if a support is at the right end (3.30b) where v (x) denotes the first derivative of v(x), i.e. v (x) dv/dx  . If the vertical and rotational settlements occur at this support, the prescribed values of the deflection and the rotation are now non-zero and must be set equal to the settlements.

3.3.2 Pinned-end support or Roller support

From the small rotation assumption and a proper constraint against the rigid translation, there is no component of the displacement parallel to the beam axis and this renders no difference between kinematical constraints provided by a pinned-end support and a roller support. These two supports provide a full constraint to beam end against the movement in the direction normal to the beam axis (i.e. no deflection) while the bending moment is fully prescribed. Two boundary conditions at these two supports are therefore given by

v(0) 0 and EI^{d v2}₂(0) M₀

dx   if a support is at the left end (3.31a) v(L) 0 and EI^{d v2}₂(L) M₀

dx  if a support is at the right end (3.31b) where M₀ is a moment acting at the supports; this applied moment is considered positive if it directs in the Z-direction or counterclockwise, otherwise it is negative. The negative sign appearing only in (3.31a) is due to that the counterclockwise applied moment acting at the left end produces a negative bending moment at that point (following the sign convention defined in subsection 2.5.2 in chapter 2) while the counterclockwise applied moment acting at the right end produces a positive bending moment at that point. Boundary conditions, for a special case when there is no applied moment M₀, can readily be obtained by substituting M₀ = 0 into (3.31a) and (3.31b). In addition, if the vertical settlement occurs at this support, the prescribed value of the deflection is now non-zero and must be set equal to the settlement.

3.3.3 Guided support

For a beam end with a guided support, the rotation is fully prevented at this point while the shear force is prescribed. Two boundary conditions are therefore given by

dv(0) 0

dx  and ^d EI^{d v2}₂ (0) P₀ dx dx

  

 

  if a support is at the left end (3.32a) dv(L) 0

dx  and ^d EI^{d v2}₂ (L) P₀ dx dx

 

   

  if a support is at the right end (3.32b) where P0 is a concentrated force acting at the support; this applied force is considered positive if it directs in the Y-direction or upward, otherwise it is negative. Again, the negative sign appearing only in (3.32b) is due to that the upward applied force acting at the left end produces a positive shear force at that point (following the sign convention defined in subsection 2.5.2 in chapter 2) while the upward applied force acting at the right end produces a negative shear force at that point.

Boundary conditions, for a special case when there is no applied force P0, can readily be obtained by substituting P0 = 0 into (3.32a) and (3.32b). In addition, if the rotational settlement occurs at this support, the prescribed value of the rotation is now non-zero and must be set equal to the settlement.

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Table 3.1: Boundary conditions for various types of beam end

Left end (x = 0) Boundary conditions Right end (x = L) Boundary conditions v(0) 0

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3.3.4 Free end

For a free end of a beam, there is no constraint on both the deflection and the rotation. As a result, both the bending moment and the shear force are fully prescribed and this leads to following two boundary conditions M₀ are the same as stated in the two previous cases. Boundary conditions, for a special case when there is no applied force and applied moment, can readily be obtained by substituting P0 = 0 and M0

= 0 into (3.33a) and (3.33b).

3.3.5 Beam end with a translational spring

Consider a beam end connected to a linear translational spring with a spring constant k_s. For this particular case, both the shear force and the deflection at this end are unknown a priori; therefore, neither of these two quantities can be treated as a boundary condition. Presence of a translational spring generally induces a force proportional to the deflection of the beam end (in fact it is equal to the product of the deflection and the spring constant) and in the direction opposite to the deflection.

To construct a proper boundary condition, we consider force equilibrium of an infinitesimal element containing the end point as shown schematically in Figure 3.4 and this leads to one boundary condition: where P₀ is an applied force at the beam end. The other boundary condition can be deduced from the rotational constraint.

Figure 3.4: FBD of beam end containing translational spring and subjected to force P₀ Right end (x = L)

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3.3.6 Beam end with a rotational spring

Consider a beam end connected to a linear rotational spring with a spring constant k. For this particular case, both the bending moment and the rotation at this end are unknown a priori;

therefore, neither of these two quantities can be treated as a boundary condition. Presence of a rotational spring generally induces a moment proportional to the rotation of the beam end (in fact it is equal to the product of the rotation and the spring constant) and in the direction opposite to the rotation. To construct a proper boundary condition, we consider moment equilibrium of an infinitesimal element containing the end point as shown schematically in Figure 3.5 and this leads to one boundary condition: where M0 is an applied moment at the beam end. The other boundary condition can be deduced from the translational constraint.

Figure 3.5: FBD of beam end containing rotational spring and subjected to moment M0

If the reduced-order differential equation (i.e. second-order and third-order differential equations) is chosen as a key governing equation, consideration of boundary conditions as described above still applies. However, it is important to emphasize that not all four boundary conditions at both ends of the beam apply to the reduced-order differential equations since some of them are used in the construction of either the shear force or the bending moment. If the second-order differential equation is employed, only two boundary conditions are needed for determining constants from the integration and they involve only the deflection and the rotation. If the third-order differential equation is employed, only three boundary conditions are needed to determine constants from the integration and they involve only the deflection, the rotation, and the bending moment.