Design Checks

Dr. Frame2D provides basic but useful design capabilities intended to help the designer rapidly select reasonable member sizes. Details are discussed and illustrated below with respect to axial and shear, and bending loading.

In general, additional design checks beyond those performed by Dr. Frame2D will be necessary, but in most cases members chosen using Dr. Frame2D's design calculations will turn out to be either suitable final choices or within a section or two.

To enable design checking, choose "Normalize Values" from the Modeling menu. As shown below, this will cause values and plots to be displayed in terms of percentages of each member's relevant capacity:

In the figure above, tensile axial loads and moments are depicted relative to the respective member capacities, while compressive load intensity is expressed in terms of the effective length factor, K_cr, that would cause the member to fail a design check. Shear diagrams and envelope plots behave in a similar fashion to moments and tensile loads. The value of K_cr is shown reported with an additional subscript indicating whether the member's weak or strong axis is used in the calculation. See below for a description of how this weak/strong axis determination is made.

By default, Dr. Frame2D uses standard AISC strength reduction factors. Turning off the "Resistance Factors On" toggle in the Modeling menu can be used to set all reduction factors to 1.0 to make it relatively simple to use alternative values. (Simply limit one's percentages to say, 80% of a member's tensile capacity for a strength reduction factor of 0.8, and so on.)

At this point, Dr. Frame2D is biased toward steel design involving wide-flange sections and the AISC LRFD code.

Dr. Frame2D's real-time analysis capabilities can be used in at least two useful ways to aid in member design:

Properties of members or groups of members can be adjusted while observing the capacity readouts. One can quickly and conveniently achieve suitable member sizes in this manner, while still being involved in the process (i.e., without turning selection control over to the computer).
For a given load/geometry/properties combination, one can determine the overall capacity of the structure and the manner in which failure could occur by adjusting loads until relevant capacities reach 100%. For any failure modes involving stability effects or load interactions, there is not necessarily a linear relation between load magnitude and a particular member's percentage capacity, so this can give a much better feel for a design's margins of safety than would otherwise be the case.

The utility of all this hinges on the ways in which capacities are computed, and this is explained in the following sections. For each potential failure mode, capacity or strength is calculated in terms of a nominal resistance multiplied by a strength reduction factor:

Limiting stresses (e.g., yield stress, F_y), member orientation, and bracing specifications are entered via Property Inspector (see Working with Members).

Axial Capacity

Tensile Case

Tensile capacity is based on gross yielding of the cross section:

No effective net section calculations are performed.

Compressive Case

Dr. Frame2D models column stability (not including torsional buckling modes) in a more general and accurate way than one can achieve with a typical member-oriented, effective-length factor approach. As discussed in Geometric Nonlinearities, one can monitor both member and overall frame stability via the onscreen "stability thermometer" displayed while doing geometrically nonlinear analysis.

The critical K-factor calculated for an individual member is thus an auxiliary check for buckling capacity, somewhat similar in spirit to typical local buckling checks. The compressive axial capacity is computed using a rearranged form of the standard AISC LRFD equations:

Since K_cr appears on both sides of the top relation, iteration is used. The radius of gyration, r, used in the calculation above depends on the orientation and bracing condition of the member:

When the member is oriented with the weak axis in-plane (i.e., the "Use weak axis properties" box is checked in the Member Info Inspector), then r_y is used.
When the member is oriented with the strong axis in-plane, r_x is used if out-plane-buckling is prevented (i.e., if the "Laterally braced" box is checked in the Member Info Inspector); otherwise r_y is used.

No local buckling checks are performed.

Shear Capacity

The relations used to compute shear capacity are given below:

The shear area, A_v, is taken as d * t_w for W-sections loadedabout their strong axis, and as 2*t_f*b_f for W-sections loaded about their weak axis. For custom sections, A_v is taken as the gross area.

Bending Capacity

No Axial Compression Present

When no axial compression is present, bending capacity is calculated as a function of unbraced length, L_b. In particular, a given member can be specified as being either braced or unbraced with respect to lateral displacement using the Member Info Table. For the fully braced case, the moment capacity is given by the section's plastic moment:

For the unbraced case, the moment capacity is computed accounting for lateral-torsional buckling, using the AISC LRFD relations for C_b, L_r, L_p, and so on. To model discrete lateral support, simply create a member with an internal joint as shown below:

To create a member of this sort, create one part of the member first, and then create the second part to fill the remaining space:

Note that such techniques can be used to approximately model tapered members by varying the section properties in each sub-segment of the member.

Example 1

The figure below shows a normalized plot of the moments in a simply-supported beam with a central lateral support:

This configuration and loading corresponds to Example 9.9.2 in Salmon and Johnson's Steel Structures--Design and Behavior, 4th Edition. Salmon and Johnson show that the maximum moment for this case is 506.5 k-ft, and the design capacity of the W27x84 is 534.6 k-ft. The ratio of 506.5 k-ft/534.6 k-ft = 0.95, which corresponds to the 95% depicted above. This illustrates how Dr. Frame2D's calculated capacities for beams match those predicted using the AISC LRFD code.

There are many cases for which Dr. Frame2D's simple approach for determining unbraced lengths is not adequate (e.g., internal member supports do not automatically register as brace points). In general one needs a discretization scheme for design distinct from that used for analysis. In a later release, we plan to incorporate lateral-torsional stability directly in the analysis, and this will remove the problems associated with unbraced length determination in the same way that effective length factors are not needed for column design because buckling is included directly in the analysis.

Axial Compression Present--Nonlinear Analysis

When axial compression is present and nonlinear analysis is used, member bending capacities are computed in one of the following ways:

When the "Resistance Factors On" option in the Modeling menu is selected, a rearranged version of AISC equation H1-1 is used:
The values used for M_n are calculated according to the procedures described above, , while P_n is calculated using the AISC column equations, the radius of gyration used in determining K_cr, and a fixed effective length factor of K = 1.0. Since P_n is based on a simple K value, this approach will be most effective for those cases in which a member's strong axis is oriented in the plane of analysis and buckling in the out-of-plane (weak axis) direction is possible.
When the "Resistance Factors On" option is turned off, the following simpler relation is used:
Here, P_y is the member's axial yield capacity and M_n is again calculated as described above. There are actually many different stub-column P-M interaction relations available, and in a subsequent release Dr. Frame2D is likely to provide a few options from which to choose.

Taken together, Dr. Frame2D's modified second order analysis and the simple P-M interaction relation can give quite accurate results, particularly for larger slenderness ratios. A typical comparison is shown below:

Note that in a case such as is shown above, Dr. Frame2D (and the "exact" analyses of Ketter) predicts somewhat higher capacities than the AISC LRFD equations for the case of relatively high axial load. The effect of this difference will be seen in the design example below.

Another comparison is shown below, and the relatively conservative nature of the P-M interaction relation can be seen in the portion of the curve corresponding to relatively low axial force.

Overall, for members with their weak axes in-plane or with their strong axes in-plane and buckling prevented out-of-plane, given its independence from estimated effective length factors, this latter approach is the better method to use. Although the strength reduction factor will be 1.0 in this case, one can, for example, limit the percentage to 90% to correspond to the usual 0.9 factor used for bending.

Note that the moments used along with these capacities to compute and display percentages will be amplified by axial effects, provided "2nd Order Analysis" is turned on.

Example 2:

The configuration shown below corresponds to Example 12.13.5 in Salmon and Johnson's Steel Structures--Design and Behavior, 4th Edition.

The columns have a lateral support at mid-height, which is modeled by means of an internal joint as described above (for lateral-torsional design checking), along with an explicit support (included directly by the built-in 2nd-order stability analysis). The trial column sections are W21x44's with their strong axes in-plane, as indicated, and the cross beam has a moment of inertia three times greater than that of the columns, as per the problem statement. This frame is braced against sway, so a plane roller has been inserted at the center of the cross beam. As can be seen, the axial load in the columns is 92 kips.

The Salmon and Johnson solution is based on a simplified structural analysis that does not account for redistribution of moment due to inelastic axial effects. The figure below shows the output corresponding to using second-order analysis with load-dependent EI's turned off (to mimic the simplified analysis used in the example), normalized plots, and resistance factors enabled.

It can be inferred from this diagram that the design is adequate for the given loads for the following reasons:

The moments are less than 100% capacity in all members.
The overall stability ratio is greater than 0.0.
The out-of-plane effective length factor for the columns need only be less than 1.83, which it is.

In Salmon and Johnson, it is shown that a straight AISC LRFD design check gives the following:

From this it is simple to calculate the available moment capacity in the columns as follows:

Comparing the applied moments to this capacity gives 168 k-ft/182 k-ft = 0.92, which matches the value shown in the Dr. Frame2D analysis above.

By selecting all the applied loads and dragging until the maximum moment reaches 100% in the columns, one can quickly determine the factored design capacity of the structure for the given load pattern:

The ratio of the originally specified concentrated load magnitude to the capacity load magnitude here is 24.5 k/25.8 k = 0.95, which can be compared directly to the 0.95 value Salmon and Johnson obtain using the AISC equation H1.

Turning on the load-dependent EI (which is analogous to using inelastic correction factors when computing effective length factors using alignment charts), the redistribution of moment between the beam and columns is evident (compare the 147 k' moment to the original 168 k'):

The corresponding design values change, as well, since the columns have shed moment to the beam:

By turning off the resistance factors, selecting all the applied loads and dragging until the maximum moment reaches 100% in the columns, one can quickly determine the "actual" capacity of the structure for the given load pattern:

The ratio of the originally specified concentrated load magnitude to the capacity load magnitude here is 24.5 k/35.5 k = 0.69, which indicates that ignoring moment redistribution is a fairly conservative assumption in this case.

Example 3: Sway Frame

This example considers example 12.13.6 from Salmon and Johnson, which uses the same configuration as the previous example, but which allows the frame to sway and uses loads with both vertical and horizontal orientation. The loads shown below were chosen to give moments matching those assumed by Salmon and Johnson for their analysis--this required the addition of some gravity loads to the sway case, but since Dr. Frame determines P-delta moments directly using the total applied load, this causes no problems for the analysis:

The combination of these loads is shown below, including the effects due to geometric nonlinearity, but without using a load-dependent EI, to be consistent with the simplified approach used in the example:

In this case the (true) amplified moment of 170 k-ft is lower than the 173 k-ft determined by Salmon and Johnson using the M_lt + M_nt approach. In the figure above, the story drift is illustrated and labeled (0.6677in) for reference.

Normalizing plots with resistance factors enabled as in the previous example leads to the following figure:

As in the previous example, the W21x44 is again found to be adequate for similar reasons: the column moments are less than 100%; the frame is stable overall; and the out-of-plane critical effective length factor is greater than the actual factor.

Amplifying the applied loads to determine the AISC-based capacity gives an AISC-relative resistance factor of safety of 1.04, which matches closely the results in Salmon and Johnson for this case (1.0/0.97 = 1.031).

Turning on the load-dependent EI setting leads to the result shown below:

As in the previous example, the inclusion of load-dependent EI reduces the effective bending stiffness in the columns, and this leads to a reduction in the end moments and an increase in the sidesway. The stability ratio can be seen to have decreased substantially, but the moment demand for the columns is reduced as indicated in the capacity display below:

Turning off the resistance factors, leaving on the load-dependent EI, and using the load combination manager to increase the loads until the column bending capacity is reached gives the following result:

The factor of 1.39 gives a reasonably accurate estimate of the actual effective resistance factor (1.031/1.39 = 0.75) resulting from applying the standard AISC procedure.

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