Many postmenopausal women treated with teriparatide for osteoporosis have previously received antiresorptive therapy. In women treated with alendronate (ALN) or raloxifene (RLX), adding versus switching to teriparatide produced different responses in areal bone mineral density (aBMD) and biochemistry; the effects of these approaches on volumetric BMD (vBMD) and bone strength are unknown. In this study, postmenopausal women with osteoporosis receiving ALN 70 mg/week (n = 91) or RLX 60 mg/day (n = 77) for ≥18 months were randomly assigned to add or switch to teriparatide 20 µg/day. Quantitative computed tomography scans were performed at baseline, 6 months, and 18 months to assess changes in vBMD; strength was estimated by nonlinear finite element analysis. A statistical plan specifying analyses was approved before assessments were completed. At the spine, median vBMD and strength increased from baseline in all groups (13.2% to 17.5%, p < 0.01); there were no significant differences between the Add and Switch groups. In the RLX stratum, hip vBMD and strength increased at 6 and 18 months in the Add group but only at 18 months in the Switch group (Strength, Month 18: 2.7% Add group, p < 0.01 and 3.4% Switch group, p < 0.05). In the ALN stratum, hip vBMD increased in the Add but not in the Switch group (0.9% versus –0.5% at 6 months and 2.2% versus 0.0% at 18 months, both p ≤ 0.004 group difference). At 18 months, hip strength increased in the Add group (2.7%, p < 0.01) but not in the Switch group (0%); however, the difference between groups was not significant (p = 0.076). Adding or switching to teriparatide conferred similar benefits on spine strength in postmenopausal women with osteoporosis pretreated with ALN or RLX. Increases in hip strength were more variable. In RLX‐treated women, strength increased more quickly in the Add group; in ALN‐treated women, a significant increase in strength compared with baseline was seen only in the Add group.