Design Checks

Dr. Frame3D 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. Frame3D will be necessary, but in most cases members chosen using Dr. Frame3D'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, 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 based on buckling about the member's weak axis. Shear diagrams and envelope plots behave in a similar fashion to moments and tensile loads.

By default, Dr. Frame3D 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. Frame3D is biased toward steel design involving wide-flange sections and the AISC LRFD code.

Dr. Frame3D's real-time analysis capabilities can be used in at least two useful ways to aid in member design:

1. 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).
2. 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. Frame3D 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 out-plane-buckling is not prevented (i.e., if the "Laterally braced" box is not checked in the Member Info Table), r_y is used; otherwise r_x 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 507 k-ft, and the design capacity of the W27x84 is 534.6 k-ft. The ratio of 507 k-ft/534.6 k-ft = 0.95, which corresponds to the 95% depicted above. This illustrates how Dr. Frame3D's calculated capacities for beams match those predicted using the AISC LRFD code.

There are many cases for which Dr. Frame3D'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:

1. 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. P_n is not calculated using the AISC column equations, though. Instead, P_n is calculated simply as AF_y. This removes the dependence on effective length factors, the effects of which are included implicitly via direct accounting for nonlinear analysis. The ramifications of this are discussed further below.

2. 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. Frame3D is likely to provide a few options from which to choose.

Taken together, Dr' Frame3D'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. Frame3D (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 12.13.7 in Salmon and Johnson'sSteel Structures--Design and Behavior, 4th ed..

The member is pinned on both ends and there is a weak axis brace support at mid-height. The trial section shown is a W12x50.

The figure below shows the output corresponding to using second-order analysis with load-dependent EI's, normalized plots, and resistance factors enabled. (The moment diagrams may look a bit strange, but this is because they have been normalized using the interaction equation given above, which introduces coupling between the weak and strong axes.)

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

1. The moments are less than 100% capacity in the member.
2. The overall stability ratio is greater than 0.0. (Since in this case the overall frame stability is equivalent to the member's stability, this is consistent with the K_cry for the column needing to be less than 0.91, which it is.)

Salmon and Johnson determine that the W12x50 is adequate for the given load and support conditions according to the AISC LRFD criteria, but with somewhat less breathing room than indicated by the Dr. Frame3D results. In fact, Dr. Frame3D predicts that the next lighter section, a W12x45, would also be OK, but this would not pass the AISC LRFD check. This discrepancy is related to the additional capacity inherent in the Dr. Frame3D analysis as described above.

Example 3:

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 (adjusted to include axial effects). 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 figure below shows the output corresponding to using second-order analysis with load-dependent EI's, 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:

1. The moments are less than 100% capacity in all members.
2. The overall stability ratio is greater than 0.0.
3. 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 is somewhat higher than the 85% shown in the Dr. Frame3D analysis above. This is similar to the previous case and is related to Dr. Frame3D's higher member capacity predictions in the case of moderately high axial loading.

By selecting all the applied loads and dragging until the maximum moment reaches 100% in the columns, one can quickly determine the 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/27.7 k = 0.88, which can be compared directly to the 0.95 value Salmon and Johnson obtain using the AISC equation H1. Once again, the higher capacity prediction is evident.