Beam Theory There are two types of beam theory aailable to craft beam element formulations from. They are Bernoulli-Euler beam theory Timoshenko beam theory One learns the details of Bernoulli-Euler beam theory as undergraduates in a Strength of Materials course. Timoshenko beam theory is used when the effects on deformation from shear is significant. The conditions under which shear deformations can not be ignored are spelled out later in the section in the discussion on strain energy. The deriation for both theories are presented here. Bernoulli-Euler beam theory is discussed first. In the net figure a deformed beam is superimposed on a a beam element represented by a straight line. We represented aial force elements by a line element in a graph that maps deformations in a ery similar manner. Howeer, for a beam element we should be concerned with Aial translations Beam deflections Rotations (bending and torsional)
Howeer, in order to deelop the two beam theories torsional rotations and aial translations will be ignored for the moment and the focus will be on effects from beam bending. In the figure below the original position of the centroidal ais coincides with the -ais. The cross sectional area is constant along the length of the beam. A second moment of the cross sectional area with respect to z-ais is identified as I z. The beam is fabricated from a material with a Young s modulus E. Nodes are identified at either end of the beam element. y,
The key assumption for Bernoulli-Euler beam theory is that plane sections remain plane and also remain perpendicular to the deformed centroidal ais. The preious figure shows a beam element based on this theory of length,, with transerse end displacements, and, rotation of the end planes θ and θ and rotation of the neutral ais, μ and μ at nodes and. The requirement that cross-sections remain perpendicular to the neutral ais means that Note that d d d d where the sign conention that clockwise rotations are negatie is adopted.
Recall from elementary calculus the curature (lowercase Greek letter kappa, ) of a line at a point may be epressed as In the case of an elastic cure rotations are small, thus the deriatie dy/d is small and squaring a small quantity produces something smaller still, thus Gien a prismatic beam subjected to pure bending the curature is When the beam is subjected to transerse loads this equation is still alid proided St. Venant s principle is not iolated. Howeer the bending moment and thus the curature will 4 ary from section to section. d y d dy d d y d M EI
Thus And d d M EI M EI In addition d d tan EI M d C The assumption that plane sections remain perpendicular to the centroidal ais necessarily implies that shear strain, γ y, is zero. This in turn implies that shear stress and shear force are zero. The only load case that results in zero shear force is a constant (uniform) bending moment and thus Bernoulli-Euler theory strictly holds only for this case. 5
As we will see later Bernoulli-Euler beam theory is acceptable only for long slender beams. Errors incurred in displacements by ignoring shear effects are of the order of (d/), where d is the depth of a beam and is the length. In the case where a beam is relatiely short or deep, shear effects can, howeer, be significant (Timoshenko, 957). The critical difference in Timoshenko theory is that the assumption that plane sections remain plane and perpendicular to the neutral ais is relaed to allow plane sections to undergo a shear strain γ. The plane section still remains plane but rotates by an amount, θ, equal to the rotation of the neutral ais, μ, minus the shear strain γ as shown in the preious figure. The rotation of the cross section is thus Taking deriaties of both sides with respect to leads to d d d d d d d d 6
Thus for a Timoshenko beam d d d d M EI where for a Bernoulli-Euler beam d d The differences will come into play when strain energies are computed using one theory or the other. It turns out that a shear correction factor is introduced to the element stiffness matri for a Bernoulli-Euler beam element to obtain the element stiffness matri for a Timoshenko beam element. The issue of short deep beams can arise more often than one might think. For ciil engineers consider a girder supporting a short span in a bridge for a major transportation artery (rail or automotie). Another application would be the use of a short deep beam as a beam seat support. M EI 7
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In static analyses the differences between the two theories is not pronounced until the aspect ratio defined as AR Beam length Beam depth or l/h in the figures below, is less than 0. The differences become more pronounced in a ibrations problem. d 9
Preliminaries Beam Elements The stiffness matri for any structure is composed of the sum of the stiffnesses of the elements. Once again, we search for a conenient approach to sum the element stiffnesses in some systematic fashion when the structure is a beam. Consider the following prismatic beam element: This element can be from a three-dimensional frame so that all possible forces and moments can be depicted at each node. The member is fully restrained and it is conenient to adopt an orthogonal aes oriented to the member. 0
The m -ais is a centroidal ais, the m -y m plane and the m -z m plane are principle planes of bending. We assume that the shear center and the centroid of the member coincide so that twisting and bending are not coupled. The properties of the member are defined in a systematic fashion. The length of the member is, the cross sectional area is A. The principle moments of inertial relatie to the y m and z m aes are I y and I z, respectiely. The torsional constant J (= I y + I z only for a circular cross section, aka, the polar moment of inertia only for this special case) will be designated I in the figures that follow. Beams with non-circular cross sections (structural shapes) will warp when twisted with a torsional moment or when the plane of the applied loads does not pass through the shear center. Restraint at fied ends of a beam will induce both shear stresses and aial normal stresses that are not uniform within the cross section. In addition, these stresses will propagate out from the fied end for a distance that is dependent on the geometry of the cross section. Shape functions typically will not account for this propagation of stresses. To account for out of plane warping an additional degree of freedom must be introduced at each load that corresponds to the rate of twist (dq/d). Typical commercial codes do not account for the degree of freedom. The member stiffnesses for the twele possible types of end displacement are summarized in the net section.
Stiffness Matrices for Beams Matri Methods Approach In the stiffness method the unknown quantities are nodal (joint) displacements. Hence, the number of unknowns is equal to the degree of kinematic indeterminacy for the stiffness method. Briefly, the unit load (force or moment) method to determine components of a stiffness matri proceeds as follows: Unknown nodal displacements are identified and structure is restrained Restrained structure is kinematically determinate, i.e., all displacements are zero Stiffness matri is formulated by successiely applying unit loads at the now restrained unknown nodal displacements To demonstrate the approach for beams we first consider a single span beam. This beam is depicted on the following page. Nodes (joints in Matri Methods) are positioned at either end of the beam.
y M Actual Beam M Neglecting aial deformations, as well as displacements and rotations in the z M -direction the beam in the first figure to the left is kinematically indeterminate to the first degree. The only unknown is a joint displacement at B, i.e., the rotation at this node. Restrained Beam # Restrained Beam # We alter the beam in the second figure such that it becomes kinematically determinate by making the rotation θ B equal to zero. This is accomplished by making the end B a fied end. This new beam is then called the restrained structure. The beam in the third figure is the restrained structure with the rotation θ B imposed from the original structure. Superposition of restrained beams # and # yields the original beam.
Due to the uniform load w the moment in the first restrained structure, i.e., M B M B w In restrained beam # an additional couple at B is deeloped due to the rotation θ B. The additional moment is equal in magnitude but opposite in direction to that on the loaded restrained beam. The alue of this moment is M Imposing equilibrium at the joint B in the superimposed restrained structures B 4EI B is yields M w 4EI B 0 B w 48EI 4
There is a way to analyze the preious simple structure through the use of a unit load. The superposition principle can be utilized. Both superposition and the application of a unit load will help deelop a systematic approach to analyzing beam elements as well as structures that hae much higher degrees of kinematic indeterminacy. The effect of a unit rotation on the preious beam is depicted below. The moment applied, m B, will produce a unit rotation at B. Restrained Beam with unit rotation 4EI m B We now somewhat informally define a stiffness coefficient as the action (force or moment) that produces a unit displacement (translation or rotation). 5
We now utilize superposition again. The moment in restrained beam # will be added to the moment m B multiplied by the rotation in the actual beam, B. The sum of these two terms must gie the moment in the actual beam, which is zero, i.e., M B m B B 0 or w 4EI B 0 In finite element analysis the first term aboe is an equialent nodal force generated by the distributed load. Soling for B yields once again B w 48EI The positie sign indicates the rotation is counterclockwise. Note that in finite element analysis, B is an unknown displacement (rotation) and m B is a stiffness coefficient for the beam element matri. The beam element is defined by nodes A and B. 6
Beam Tables Useful Cases The net seeral beam cases will proe useful in establishing components of the stiffness matri. Consult the AISC Steel Design Manual or Roark s tet for many others not found here. 7
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Note that eery eample cited hae fied-fied end conditions. All are kinematically determinate beam elements. 0
y M Multiple Degrees of Kinematic Indeterminacy M Consider the beam to the left with a constant fleural rigidity, EI. Since rotations can occur at nodes B as well as C, and we do not know what they are, the structure is kinematically indeterminate to the second degree. Aial displacements as well as displacements and rotations in the z M - direction are neglected. D = D = 0 D = 0 D = Designate the unknown rotations as D (and the associated bending moment from the load applied between nodes as A D ) and D (and bending moment A D from the load between nodes). Assume counterclockwise rotations as positie. The unknown displacements are determined by applying the principle of superposition to the bending moments at joints B and C. Note: D displacement (translation or rotation) A action (force or moment)
All loads ecept those corresponding to the unknown joint displacements are assumed to act on the restrained structure. Thus only actions P and P are shown acting on the restrained structure. The moments A D and A D are the actions at the restrained nodes associated with D (A D in Matri Methods, M B, the global moment at node B in Finite Element Analysis) and D (A D in Matri Methods, M C, the global moment at node C in Finite Element Analysis) respectiely. The notation in parenthesis will help with the matri notation momentarily. In addition, numerical subscripts are associated with the unknown nodal displacements, i.e., D and D. These displacements occur at nodes B and C. ater when this approach is generalized eery node will be numbered, not lettered. In addition, the actions in the figure aboe that occur between nodes are subscripted for conenience and these subscripts bear no relationship to the nodes.
In order to generate indiidual alues of the beam element stiffness matri, unit alues of the unknown displacements D and D are induced in separately restrained structures. D = In the restrained beam to the left a unit rotation is applied to joint B. The actions (moments) induced in the restrained structure corresponding to D and D are the stiffness coefficients S and S, respectiely D = In the restrained beam to the left a unit rotation is applied to joint B. The actions (moments) induced in this restrained structure corresponding to D and D are the stiffness coefficients S and S, respectiely All the stiffness coefficients in the figures hae two subscripts (S ij ). The first subscript identifies the location of the action induced by the unit displacement. The second subscript denotes where the unit displacement is being applied. Stiffness coefficients are taken as positie when the action represented by the coefficient is in the same direction as the i th displacement.
Two superposition equations describing the moment conditions on the original structure may now be epressed at joints B and C. The superposition equations are y M M A A D AD SD SD D AD SD SD The two superposition equations epress the fact that the actions in the original structure (first figure defining the structure and loads) must be equal to the corresponding actions in the restrained structure due to the loads (the second figure) plus the corresponding actions in the restrained structure under the unit displacements (third and fourth figure) multiplied by the unknown displacements. D = D = 0 D = 0 D = 4
These equations can be epressed in matri format as where using the loads applied to the structure and we are looking for the unknown displacements. S S S S S A D A D S D D S A A D D 8 8 8 0 P P P A A A M M M A A A D D D C B D D D D D D 5
If then P P M P P P P P A D P 4 A D P 8 A D P 8 P P P P 4 8 8 D 0 P 8 A A D The net step is the formulation of the stiffness matri. Consider a unit rotation at B 6
thus 4EI S S 4EI S S S 8EI S EI With a unit rotation at C S 4EI S EI and the stiffness matri is S EI 8 4 7
Element Stiffness Matri for Bernoulli-Euler Beams The General Case Unit Displacement Unit Displacement Node j Node k Unit Displacement Unit Displacement Unit Displacement Unit Displacement 8
Unit Displacement Unit Displacement Unit Displacement Unit Displacement Unit Displacement Unit Displacement 9
The most general stiffness matri for the beam element depicted in the last two oerheads is a matri we will denote [ k ], the subscript me implies member. Each column represents the actions caused by a separate unit displacement. Each row corresponds to a figure in the preious two oerheads, i.e., a figure depicting where the unit displacement is applied. [ k ] = 0
The stiffness matri for beam elements where aial displacements as well as displacements and rotations in the z M -direction are neglected is smaller than the preious stiffness matri. The preious stiffness matri represents the general case for a Bernoulli-Euler beam element. What about stiffness coefficients for a Timoshenko beam? For a beam element where aial displacements as well as displacements and rotations in the z M -direction are neglected there are four types of displacements at a node. The actions associated with these displacements (one translation and one rotation at each node) are shown at the left and are identified as ectors through 4 in the figure. The four significant displacements correspond to a shear force and bending moment at each node.
The corresponding 44 member stiffness matri is gien below. The elements of this matri are obtained from cases (), (6), (8) and () in preious oerheads. [ k ] i = No torsion No aial deformation No biaial bending
Displacement Function Beam Element We can derie the 44 stiffness matri by taking a different approach. Recall that when displacement functions were first introduced for springs, the number of unknown coefficients corresponded to the number of degrees of freedom in the element. For a simple Bernoulli beam element there are four degrees of freedom two at each node of the element. At the beginning and end nodes the beam element can deflect () and rotate (). Consider the complete cubic displacement function With a a a a4 d d 0 0 d d a a a a 4 a a a a
Soling the preious four equations for the unknowns a, a, a, and a 4 leads to Thus Using shape functions where 4 a a a a d N 4
and then the shape functions become 4 N N N N d 5
From Strength of Materials the moment-displacement and shear displacement relationships are d d M EI V EI With the following sign conention for nodal forces, shear forces, nodal bending moments and nodal rotations within a beam element d d then at node # d d 0 EI f y V EI 6 6 6
d 0 EI m M EI 6 4 6 d Minus signs result from the following traditional sign conention used in beam theory At node # d d EI f y V EI 6 6 d EI m M EI 6 6 4 d These four equations relate actions (forces and moments) at nodes to nodal displacements (translations and rotations). 7
The four equations can be place into a matri format as follows where the stiffness matri for element i is designated 4 6 6 6 6 6 4 6 6 6 EI m f m f y y 4 6 6 6 6 6 4 6 6 6 EI k i 8
Shape Functions Beam Element As we will see when we deelop shape functions for two dimensional planar elements and three dimensional solid elements the shape functions for the spring element, the aial element and the truss element are agrange polynomials. Shape functions based on agrange polynomials sum to unity eerywhere in the element. The shape functions deried in the preious section for beam elements belong to a class of polynomials referred to as Hermite, or Hermitian polynomials. In some tets and publications they are designated H, H, H and H 4 in order to proide a clear distinction. As indicated in the figure below only N (H ) and N (H ) sum to unity. 9
With substitution leads to In addition, from the strain displacement relationships d N T d y du y,, 40
From undergraduate strength of materials thus and d d y y u,, d d y y 6 6 4 6 6, y y 4
Continuity requirements for beam elements should demand that displacements and rotations, i.e., and, must be continuous from element to element. This is referred to as C continuity. The simplest shape functions that meet the nodal degrees of freedom associated with a Bernoulli beam element as well as the continuity conditions are Hermitian shape functions. These functions can be cast in a dimensionless format, known as natural coordinates, if the following change of ariable is used With this change of ariable the shape functions take the following form 8 4 8 4 4 N N N N 4